Collected Papers of Charles Sanders Peirce

The Collected Papers of Charles Sanders Peirce 1. CONTENTS. To view the complete table of contents of The Collected Papers of Charles Sanders Peirce, click on the Contents icon on the left border of your screen. Click on plus (+) symbols in the Contents window to expand the list of entries. A minus symbol (-) indicates that a list may not be further expanded. While viewing the contents, click on Levels under the Search menu to view the Contents at the Volume, Book, or Chapter levels. While in the Contents window, you may jump to any location by double-clicking on the line of interest or click on the Contents icon to go back to full database view. 2. SEARCH TEXT. To execute a search, click on the Query icon on the left border of your screen, type in your search terms in the Query For: window, then click on OK to execute your search. To reveal reference information (e.g. page numbers) for each paragraph, click on Hidden under the View menu.

Editorial Introduction by John Deely Past Masters Introduction Chronological Listing of Texts Groups of the database Key to Symbols Volume 1 Volume 3 Volume 5 Volume 7 Contents Contents Contents Contents Text Text Text Text Footnotes Footnotes Footnotes Footnotes Volume 2 Volume 4 Volume 6 Volume 8 Contents Contents Contents Contents Text Text Text Text Footnotes Footnotes Footnotes Footnotes All text only All footnotes only All text authored by Peirce

Peirce: CP Editorial Introduction to Electronic Edition Membra Ficte Disjecta (A Disordered Array of Severed Limbs) Editorial Introduction by John Deely to the electronic edition of The Collected Papers of Charles Sanders Peirce

reproducing Vols. I-VI ed. Charles Hartshorne and Paul Weiss (Cambridge, MA: Harvard University Press, 1931-1935), Vols. VII-VIII ed. Arthur W. Burks (same publisher, 1958)

1 June 1994

Peirce: CP Editorial Introduction to Electronic Edition Charles S. Peirce (the "S" stands for "Sanders" by Baptism and later for "Santiago" as Charles' way of honoring William James) has so far best been known in academia at large as some kind of a background figure to the rise of Pragmatism, as mentor to that movement's truly well-known protagonists, William James and John Dewey. That misleading identification is in the process of changing, and the literature supporting the understanding of Peirce in the established framework of modern philosophy, particularly with its opposition of "realism" to "idealism" such as the works of Buchler, Goudge, Manley Thompson already belong to the genre of depassé interpretation. Peirce: CP Editorial Introduction to Electronic Edition It is not merely a question of the curiously underassessed fact (excepting Apel's pioneering 1970 study, Der Denkweg von Charles S. Peirce: Eine Einführung in den amerikanischen Pragmatismus [Frankfurt am Main: Suhrkamp Verlag], presciently retitled From Pragmatism to Pragmaticism for its 1981 English translation by J. M. Krois [Amherst, MA: University of Massachusetts Press]) that, despite the willingness on all sides to attribute the original coining of the term "pragmatism" as a philosophical name to him, Peirce eschewed the classical pragmatist development to the point of giving to his own position a new name, "Pragmaticism". It is a question at bottom of the principal optic through which Peirce early and ever-after came to view the problems of philosophy, the optic of "semiotic", as he called it after Locke, or the doctrina signorum, as both Locke and Peirce called it, both unaware of the earlier Latin Iberian development of this optic through the successive work of Domingo de Soto (with his Summulae or Introductory Logic of 1529), Pedro da Fonseca (1564) and the Conimbricenses (1607) he started, Francisco Araujo (1617), and the culminating synthesis of John Poinsot's Tractatus de Signis (Treatise on Signs) of 1632 (also a full-text data-base in this Past Masters series). Peirce: CP Editorial Introduction to Electronic Edition I first came to take Peirce seriously as a result of Thomas A. Sebeok's 1978 NEH Summer Seminar on semiotics as a new foundation for the sciences. In that group of seminarians there were three expert Peirceans, Jarrett E. Brock, H. William Davenport, and George A. Benedict. It soon became clear that anyone studying Peirce today on the basis of the Harvard Collected Papers of Charles Sanders Peirce (henceforward CP) was essentially in the position of an animal wading into a pool of piranha fish. A whole generation of young Peirce scholars had come of age under the tutelage or indirect influence of Max Fisch, the most knowledgeable of all the senior Peirce scholars, who had almost alone come to grasp the semiotic trajectory animating the entire Peircean corpus. First through Kenneth Ketner's Institute for the Study of Pragmaticism at Texas Tech University, and later through the Peirce Edition Project at IUPUI, Fisch had shown the new generation not only the importance of the unpublished Peirce manuscripts, but, equally importantly, how to read them with

semiotic eyes. Oddly enough, as an index of how much remains to be done in achieving a balanced and integral presentation of the Peircean corpus, the recent An Introduction to C. S. Peirce by Robert Corrington (Lanham, MD: Rowman & Littlefield, 1993) stands out as the first introduction to give semiotic a co-ordinate billing with such traditional aspects of Peircean thought as his metaphysics (yet even in this ground-breaking over-all introduction, arguably the best so far, Corrington told me that "piety toward the elders" inhibited him in annotating his bibliography). Peirce: CP Editorial Introduction to Electronic Edition The story of the Harvard edition titled CP, which we here re-present in electronic form, is a story fairly well known, and a sad one. Hartshorne and Weiss, along with Burks later, deserve our thanks for getting the volumes out, but we must at the same time regret the manner of their editing, which was to construct a topical scheme of their own devising under which to sort and dissect the papers left whole to Harvard through the good intentions of Josiah Royce. How Harvard abused that trust! The story, at least, is now out with the bursting upon the scene of the newly-worked (after more than thirty years of repression) biographical dissertation of Joseph Brent in the form of the book, Charles Sanders Peirce. A Life (Bloomington, IN: Indiana University Press, 1993). This publication is a tribute in equal parts to the writing skill and historical tenacity of its author, to the editorial genius (to say nothing of the detective skills) of Thomas A. Sebeok, and to the publishing genius of John Gallman, the Director of the Indiana University Press. Peirce: CP Editorial Introduction to Electronic Edition But why re-publish the CP now, just when the chronological edition of the Writings (henceforward W) may be getting up steam? There are several answers to this question. The first reason is that the CP is not in competition with W. The chronological edition, when completed, will become the irreplaceable standard and, if brought to completion at its current level of scholarly excellence, will remain practically unsurpassable as a hardcopy critical source. But W is, simply put, taking too long, partly in the nature of the task which, after all, however much more quickly it might have been shepherded, cannot be rushed: it needs to be done rightly, and critical editing takes time. Still, those of us alive today and interested in Peirce would like to have access to as much of his work as possible as soon as possible. At present, as far as published writings go, that still means the CP. Peirce: CP Editorial Introduction to Electronic Edition A second reason is that CP contains some material which, at least according to current plans, will not be included in W. That means that, for the foreseeable future, the CP will remain an independent, and at least minor, source for Peircean scholarship. Peirce: CP Editorial Introduction to Electronic Edition The third reason, however, is the main reason for this edition. By bringing out the CP in electronic form, we not only keep available the so-far primary published source of Peirce material, but we present it in a form that enables the user in principle to overcome the primary defect of the original publication, namely, its artificial dismemberment of the Peircean corpus. Using the invaluable tool of the Burks bibliography from the last of the eight CP volumes, which gave scholars the necessary key to reconstruct the order of the Peirce manuscripts before the CP editors dissected them and shuffled the pieces (it is amazing, between the Burks bibliography and the Robin catalogue, not to mention many lesser essays, how much Peirce scholarship has been devoted to undoing that dismemberment), we have created hypertext links which

will enable the users of the electronic edition to reconstruct and print out for themselves Peirce's manuscripts in something like their original integrity. Peirce: CP Editorial Introduction to Electronic Edition An illustration of this advantage of the electronic CP may be given using Peirce's c.1895 essay "That Categorical and Hypothetical Propositions Are One in Essence". According to Burks (p. 286), paragraphs CP 2.332-339, 2.278-28, 1.564-567 (c.1899), and 2.340 "are from it in this order". Using the electronic CP, a reader can reconstruct this whole and print it out as such for scholarly or classroom use. Thus the "bodily parts" of the Peircean corpus, so far as they are included in the CP, may be easily rearrayed in proper order so as to appear in something closer to the light under which Peirce left them. Peirce: CP Editorial Introduction to Electronic Edition This illustration brings out the fourth reason for this electronic edition, namely, to stimulate self-appointed scholarly caretakers of the manuscript materials to hasten the making available of the whole of the Peirce documents in electronic form even while the critical published edition (for which there is no substitute) goes forward at its own pace. Joseph Ransdell has been tirelessly promoting the desirability of an on-line forum through the proposal of the Peirce electronic consortium and through the two Peirce bulletin boards in which he is closely involved (contact Professor Ransdell at for full details of the possibilities). By presenting this edition to the scholarly world, we have done the best that was possible at this actual historical moment in bringing Peirce as so far published "on line".

Peirce: Collected Papers - PAST MASTERS Introduction Past Masters Introduction

Below find the text of The Collected Papers of Charles Sanders Peirce. All footnotes have been placed at the ends of their respective volumes. We have numbered the footnotes of volumes 1-6 relative to the page (instead of using the symbols of volumes 1-6). Footnotes authored by Peirce in volumes 1-6 are identified by the letter "P" after the carat symbol (†) and before the numeral. Thus footnote †P1 is a footnote authored by Peirce (i.e. the numbered footnotes of the printed editions (CP 1-6). In volumes 7 and 8 we have followed Burks scheme. Peirce: Collected Papers - PAST MASTERS Introduction A number of substitutions were made for symbols. Please see the "Key to Symbols" for a complete list. Peirce: Collected Papers - PAST MASTERS Introduction A number of groups have been created to facilitate searches. Please see the "Groups of the database" for more information. Peirce: Collected Papers - PAST MASTERS Introduction A link token is found on every reference line which leads to the "Table of Cross-References." The "Table of Cross-References" correlates the bibliography with every paragraph of text of the CP. A link appears next to every bibliographic entry of

this table, which leads to the complete bibliographic record. Thus to see the complete bibliographic record which identifies the source of any particular paragraph: a) Go to the View menu, and execute the Hidden menu item (either by moving to the item with your Arrow keys or mouse then pressing Return/Enter or by clicking on the item with your mouse). Successful execution will result in a check mark to the left of the Hidden menu item. All reference lines in the text will be unhidden. The reference lines are located at the beginning of the paragraph, and appear purplish-red on color screens. b) Note your current paragraph number (e.g. CP 3.183); click with your mouse on top of the link to the right of "Cross-Ref:". You will be moved to a table of Cross-references for the volume in question. c) Move to the paragraph range in the table (using your arrow keys or mouse) in which your current paragraph falls, then click on the range with your mouse. You will be moved to the record which identifies the bibliographic source of the paragraph in question.

Peirce: Collected Papers Groups Groups of the Peirce database

A number of groups have been created to make searching the database easier. First, groups have been created from the divisions of Hartshorne, Weiss, and Burks. [Group: CP1], [Group: CP2], ... [Group: CP8] etc. identify volumes 1 through 8. Thus the search [Group: CP3] abnumeral would find all paragraphs in volume 3 containing the word abnumeral. Peirce: Collected Papers Groups A group exists for every book, chapter and section division as well. Thus the search [Group: cp4.i.ii] good would find all paragraphs from Volume 4, Book I, Lecture II containing the word "good." Peirce: Collected Papers Groups Secondly, every paragraph of the Collected Papers has been placed in a group which identifies the year in which the paragraph was authored. Thus the [Group: Peirce.1888] group contains all paragraphs identified in the bibliography as having been written in 1888. The search [Group: Peirce.1888] would find all paragraphs written by Peirce in 1888, which are in the CP. The search [Group: Peirce.1888] abnumeral

would find all paragraphs written by Peirce in 1888 (in the CP) which contain the word "abnumeral" (if any). Peirce: Collected Papers Groups This chronological grouping also exists at the 5-year and 10-year level. The [Group: Peirce5.1875] and [Group: Peirce10.1880] groups contain every paragraph authored by Peirce (in the CP, identified in the bibliography) in the years 1871-1875, and 1871-1880 respectively. Thus the search

[Group: Peirce10.1890] [Group: Peirce5.1895] abnumeral

would find every paragraph containing abnumeral authored by Peirce between the years 1881-1895 (in the CP). Peirce: Collected Papers Groups A group has been created from the text only and footnotes only of each volume. These groups are accessible from the opening screen of the database. Thus [Group: cp1.text] contains all paragraphs of the text of volume 1, and [Group: cp3.footnotes] contains all footnotes of volume 3. A [Group: cp.text] group excludes all footnotes, introductory and explanatory material, as well as table of contents entries. (Thus the [Group: cp.text] group = [Group: cp1.text] or [Group: cp2.text] ... [Group: cp8.text].) A [Group: cp.footnotes] group contains the footnotes from all 8 volumes. Finally, a [Group: peirce] group contains all and only material authored by Peirce, from both footnotes and text.

Peirce: CP Key to Electronic Symbols: Introduction Key to Symbols: Introduction

Many symbols which do not appear in the extended ANSI or ASCII character sets (or symbol font sets) appear in the text of the Collected Papers. In the Windows and Macintosh version of this database, we are creating a Peirce font set that will accurately display onscreen all symbols found in the Collected Papers. This new Peirce font will appear in an updated version of the database. In the meantime, below find all substitutions made, with (if necessary) an image which displays the symbol as it appears in the print edition. Peirce: CP Key to Electronic Symbols: Key to Symbols

All subscripts are enclosed between brackets. Thus A[1] is A followed by the subscript 1. Occasionally a bracket in the text is double-bracketed in the electronic edition, to avoid ambiguity. Thus A[[1]] would indicate that an unsubscripted 1 enclosed by brackets appears in the printed edition.

Occasionally parentheses have been introduced to disambiguate expressions made ambiguous by substituting notation. Parentheses were particularly necessary to disambiguate numerators and denominators in division from surrounding expressions. All Greek has been transliterated and is enclosed between braces {}. This transliterated Greek will be replaced with true Greek in an updated version of this database. Standard rules for transliteration were followed with the following exceptions:

{é} = lowercase eta {É} = uppercase eta {ö} = lowercase omega {Ö} = uppercase omega

The $ sign is used to represent "some". If the the curved line appears over an expression, the $ sign precedes the parenthesized expression.

= $A

A vertical bar above a symbol or expression has been replaced with a tilde preceding the expression. Thus:

= A ~-< B

= ~A

A vertical bar underneath a symbol is represented by following the symbol with _. If the vertical bar is underneath more than one symbol, the _ sign is placed after the parenthetical expression:

= (x

x)_

A dot over a symbol is represented by preceding the symbol with a dot. Thus:

=·

=·

The remainder of the symbol-equivalents are self-explanatory:

= =,

= +,

= `+

= - m[1]. Either or both the quantities t[1] and t[2] may be negative. Next with each of these quantities enter the table below, and take out 1/2Θt[1] and 1/2Θt[2] and give each the same sign as the t from which it is derived. Then

Σq = 1/2 Θt[2] - 1/2 Θt[1].

[Click here to view]

Peirce: CP 2.701 Cross-Ref:†† In rough calculations we may take Θt equal to t for t less than 0.7, and as equal to unity for any value above t = 1.4.

Peirce: CP 2.702 Cross-Ref:†† §3. INDUCTION †1

702. The principle of statistical deduction is that these †2 two proportions--namely, that of the P's among the M's, and that of the P's among the S's--are probably and approximately equal. If, then, this principle justifies our inferring the value of the second proportion from the known value of the first, it equally justifies our inferring the value of the first from that of the second, if the first is unknown but the second has been observed. We thus obtain the following form of inference:

FORM V

Induction.

S', S'', S''', etc. form a numerous set taken at random from among the M's, S', S'', S''', etc. are found to be--the proportion {r} of them--P's; Hence, probably and approximately the same proportion, {r}, of the M's are P's.

Peirce: CP 2.702 Cross-Ref:†† The following are examples. From a bag of coffee a handful is taken out, and found to have nine-tenths of the beans perfect; whence it is inferred that about nine-tenths of all the beans in the bag are probably perfect. The United States Census of 1870 shows that of native white children under one year old, there were 478,774 males to 463,320 females; while of colored children of the same age there were 75,985 males to 76,637 females. We infer that generally there is a larger proportion of female births among negroes than among whites. Peirce: CP 2.703 Cross-Ref:†† 703. When the ratio {r} is unity or zero, the inference is an ordinary induction; and I ask leave to extend the term "induction" to all such inference, whatever be the value of {r}. It is, in fact, inferring from a sample to the whole lot sampled. These two forms of inference, statistical deduction and induction, plainly depend upon the same principle of equality of ratios, so that their validity is the same. Yet the nature of the probability in the two cases is very different. In the statistical deduction, we know that among the whole body of M's the proportion of P's is {r}; we say, then, that the S's being random drawings of M's are probably P's in about the same proportion--and though this may happen not to be so, yet at any rate, on continuing the drawing sufficiently, our prediction of the ratio will be vindicated at last. On the other hand, in induction we say that the proportion {r} of the sample being P's, probably there is about the same proportion in the whole lot; or at least, if this happens not to be so, then on continuing the drawings the inference will be, not vindicated as in the other case, but modified so as to become true. The deduction, then, is probable in this sense, that though its conclusion may in a particular case be falsified, yet similar conclusions (with the same ratio {r}) would generally prove approximately true; while the induction is probable in this sense, that though it may happen to give a false conclusion, yet in most cases in which the same precept of inference was followed, a different and approximately true inference (with the right value of {r}) would be drawn.

Peirce: CP 2.704 Cross-Ref:†† §4. HYPOTHETIC INFERENCE

704. Before going any further with the study of Form V, I wish to join to it another extremely analogous form. Peirce: CP 2.704 Cross-Ref:†† We often speak of one thing being very much like another, and thus apply a vague quantity to resemblance. Even if qualities are not subject to exact numeration, we may conceive them to be approximately measurable. We may then measure

resemblance by a scale of numbers from zero up to unity. To say that S has a 1-likeness to a P will mean that it has every character of a P, and consequently is a P. To say that it has a 0-likeness will imply total dissimilarity. We shall then be able to reason as follows:

FORM II (bis).

Simple probable deduction in depth.

Every M has the simple mark P, The S's have an r-likeness to the M's; Hence, the probability is r that every S is P.

Peirce: CP 2.704 Cross-Ref:†† It would be difficult, perhaps impossible, to adduce an example of such kind of inference, for the reason that simple marks are not known to us. We may, however, illustrate the complex probable deduction in depth (the general form of which it is not worth while to set down) as follows: I forget whether, in the ritualistic churches, a bell is tinkled at the elevation of the Host or not. Knowing, however, that the services resemble somewhat decidedly those of the Roman Mass, I think that it is not unlikely that the bell is used in the ritualistic, as in the Roman, churches. Peirce: CP 2.705 Cross-Ref:†† 705. We shall also have the following:

FORM IV (bis).

Statistical deduction in depth.

Every M has, for example, the numerous marks P', P'', P''', etc., S has an r-likeness to the M's; Hence, probably and approximately, S has the proportion r of the marks P', P'', P''', etc.

Peirce: CP 2.705 Cross-Ref:†† For example, we know that the French and Italians are a good deal alike in their ideas, characters, temperaments, genius, customs, institutions, etc., while they also differ very markedly in all these respects. Suppose, then, that I know a boy who

is going to make a short trip through France and Italy; I can safely predict that among the really numerous though relatively few respects in which he will be able to compare the two people, about the same degree of resemblance will be found. Peirce: CP 2.705 Cross-Ref:†† Both these modes of inference are clearly deductive. When r = 1, they reduce to Barbara.†P1 Peirce: CP 2.706 Cross-Ref:†† 706. Corresponding to induction, we have the following mode of inference:

FORM V (bis).

Hypothesis.

M has, for example, the numerous marks P', P'', P''', etc., S has the proportion r of the marks P', P'', P''', etc.; Hence, probably and approximately, S has an r-likeness to M.

Peirce: CP 2.706 Cross-Ref:†† Thus, we know, that the ancient Mound-builders of North America present, in all those respects in which we have been able to make the comparison, a limited degree of resemblance with the Pueblo Indians. The inference is, then, that in all respects there is about the same degree of resemblance between these races. Peirce: CP 2.706 Cross-Ref:†† If I am permitted the extended sense which I have given to the word "induction," this argument is simply an induction respecting qualities instead of respecting things. In point of fact P', P'', P''', etc., constitute a random sample of the characters of M, and the ratio r of them being found to belong to S, the same ratio of all the characters of M are concluded to belong to S. This kind of argument, however, as it actually occurs, differs very much from induction, owing to the impossibility of simply counting qualities as individual things are counted. Characters have to be weighed rather than counted. Thus, antimony is bluish-gray: that is a character. Bismuth is a sort of rose-gray; it is decidedly different from antimony in color, and yet not so very different as gold, silver, copper, and tin are. Peirce: CP 2.707 Cross-Ref:†† 707. I call this induction of characters hypothetic inference,†1 or, briefly, hypothesis. This is perhaps not a very happy designation, yet it is difficult to find a better. The term "hypothesis" has many well established and distinct meanings. Among these is that of a proposition believed in because its consequences agree with experience. This is the sense in which Newton used the word when he said, Hypotheses non fingo. He meant that he was merely giving a general formula for the motions of the heavenly bodies, but was not undertaking to mount to the causes of the acceleration they exhibit. The inferences of Kepler, on the other hand, were

hypotheses in this sense; for he traced out the miscellaneous consequences of the supposition that Mars moved in an ellipse, with the sun at the focus, and showed that both the longitudes and the latitudes resulting from this theory were such as agreed with observation. These two components of the motion were observed; the third, that of approach to or regression from the earth, was supposed. Now, if in Form V (bis) we put r = 1, the inference is the drawing of a hypothesis in this sense. I take the liberty of extending the use of the word by permitting r to have any value from zero to unity. The term is certainly not all that could be desired; for the word hypothesis, as ordinarily used, carries with it a suggestion of uncertainty, and of something to be superseded, which does not belong at all to my use of it. But we must use existing language as best we may, balancing the reasons for and against any mode of expression, for none is perfect; at least the term is not so utterly misleading as "analogy" would be, and with proper explanation it will, I hope, be understood.

Peirce: CP 2.708 Cross-Ref:†† §5. GENERAL CHARACTERS OF DEDUCTION, INDUCTION, AND HYPOTHESIS

708. The following examples will illustrate the distinction between statistical deduction, induction, and hypothesis. If I wished to order a font of type expressly for the printing of this book, knowing, as I do, that in all English writing the letter e occurs oftener than any other letter, I should want more e's in my font than other letters. For what is true of all other English writing is no doubt true of these papers. This is a statistical deduction. But then the words used in logical writings are rather peculiar, and a good deal of use is made of single letters. I might, then, count the number of occurrences of the different letters upon a dozen or so pages of the manuscript, and thence conclude the relative amounts of the different kinds of type required in the font. That would be inductive inference. If now I were to order the font, and if, after some days, I were to receive a box containing a large number of little paper parcels of very different sizes, I should naturally infer that this was the font of types I had ordered; and this would be hypothetic inference. Again, if a dispatch in cipher is captured, and it is found to be written with twenty-six characters, one of which occurs much more frequently than any of the others, we are at once led to suppose that each character represents a letter, and that the one occurring so frequently stands for e. This is also hypothetic inference. Peirce: CP 2.709 Cross-Ref:†† 709. We are thus led to divide all probable reasoning into deductive and ampliative, and further to divide ampliative reasoning into induction and hypothesis. In deductive reasoning, though the predicted ratio may be wrong in a limited number of drawings, yet it will be approximately verified in a larger number. In ampliative reasoning the ratio may be wrong, because the inference is based on but a limited number of instances; but on enlarging the sample the ratio will be changed till it becomes approximately correct. In induction, the instances drawn at random are numerable things; in hypothesis they are characters, which are not capable of strict enumeration, but have to be otherwise estimated. Peirce: CP 2.710 Cross-Ref:†† 710. This classification of probable inference is connected with a preference

for the copula of inclusion over those used by Miss Ladd [Mrs. Christine Ladd-Franklin] and by Mr. Mitchell.†P1 De Morgan established eight forms of simple propositions; and from a purely formal point of view no one of these has a right to be considered as more fundamental than any other. But formal logic must not be too purely formal; it must represent a fact of psychology, or else it is in danger of degenerating into a mathematical recreation. The categorical proposition, "every man is mortal," is but a modification of the hypothetical proposition, "if humanity, then mortality"; and since the very first conception from which logic springs is that one proposition follows from another, I hold that "if A, then B" should be taken as the typical form of judgment. Time flows; and, in time, from one state of belief (represented by the premisses of an argument) another (represented by its conclusion) is developed. Logic arises from this circumstance, without which we could not learn anything nor correct any opinion. To say that an inference is correct is to say that if the premisses are true the conclusion is also true; or that every possible state of things in which the premisses should be true would be included among the possible states of things in which the conclusion would be true. We are thus led to the copula of inclusion. But the main characteristic of the relation of inclusion is that it is transitive--that is, that what is included in something included in anything is itself included in that thing; or, that if A is B and B is C, then A is C. We thus get Barbara as the primitive type of inference. Now in Barbara we have a Rule, a Case under the Rule, and the inference of the Result of that rule in that case. For example:

Rule. All men are mortal, Case. Enoch was a man; Result. .·. Enoch was mortal.

Peirce: CP 2.711 Cross-Ref:†† 711. The cognition of a rule is not necessarily conscious, but is of the nature of a habit, acquired or congenital. The cognition of a case is of the general nature of a sensation; that is to say, it is something which comes up into present consciousness. The cognition of a result is of the nature of a decision to act in a particular way on a given occasion.†P1 In point of fact, a syllogism in Barbara virtually takes place when we irritate the foot of a decapitated frog. The connection between the afferent and efferent nerve, whatever it may be, constitutes a nervous habit, a rule of action, which is the physiological analogue of the major premiss. The disturbance of the ganglionic equilibrium, owing to the irritation, is the physiological form of that which, psychologically considered, is a sensation; and, logically considered, is the occurrence of a case. The explosion through the efferent nerve is the physiological form of that which psychologically is a volition, and logically the inference of a result. When we pass from the lowest to the highest forms of inervation, the physiological equivalents escape our observation; but, psychologically, we still have, first, habit--which in its highest form is understanding, and which corresponds to the major premiss of Barbara; we have, second, feeling, or present consciousness, corresponding to the minor premiss of Barbara; and we have, third, volition, corresponding to the conclusion of the same mode of syllogism. Although these analogies, like all very broad generalizations, may seem very fanciful at first sight, yet the more the reader reflects upon them the more profoundly true I am confident they

will appear. They give a significance to the ancient system of formal logic which no other can at all share. Peirce: CP 2.712 Cross-Ref:†† 712. Deduction proceeds from Rule and Case to Result; it is the formula of Volition. Induction proceeds from Case and Result to Rule; it is the formula of the formation of a habit or general conception--a process which, psychologically as well as logically, depends on the repetition of instances or sensations. Hypothesis proceeds from Rule and Result to Case; it is the formula of the acquirement of secondary sensation--a process by which a confused concatenation of predicates is brought into order under a synthetizing predicate.†1 Peirce: CP 2.713 Cross-Ref:†† 713. We usually conceive Nature to be perpetually making deductions in Barbara. This is our natural and anthropomorphic metaphysics. We conceive that there are Laws of Nature, which are her Rules or major premisses. We conceive that Cases arise under these laws; these cases consist in the predication, or occurrence, of causes, which are the middle terms of the syllogisms. And, finally, we conceive that the occurrence of these causes, by virtue of the laws of Nature, results in effects which are the conclusions of the syllogisms. Conceiving of nature in this way, we naturally conceive of science as having three tasks--(1) the discovery of Laws, which is accomplished by induction; (2) the discovery of Causes, which is accomplished by hypothetic inference; and (3) the predictio of Effects, which is accomplished by deduction. It appears to me to be highly useful to select a system of logic which shall preserve all these natural conceptions. Peirce: CP 2.714 Cross-Ref:†† 714. It may be added that, generally speaking, the conclusions of Hypothetic Inference cannot be arrived at inductively, because their truth is not susceptible of direct observation in single cases. Nor can the conclusions of Inductions, on account of their generality, be reached by hypothetic inference. For instance, any historical fact, as that Napoleon Bonaparte once lived, is a hypothesis; we believe the fact, because its effects--I mean current tradition, the histories, the monuments, etc.--are observed. But no mere generalization of observed facts could ever teach us that Napoleon lived. So we inductively infer that every particle of matter gravitates toward every other. Hypothesis might lead to this result for any given pair of particles, but it never could show that the law was universal.

Peirce: CP 2.715 Cross-Ref:†† §6. INDUCTION AND HYPOTHESIS

INDIRECT STATISTICAL INFERENCES; GENERAL RULE FOR THEIR VALIDITY

715. We now come to the consideration of the Rules which have to be followed in order to make valid and strong Inductions and Hypotheses. These rules can all be reduced to a single one; namely, that the statistical deduction of which the Induction or Hypothesis is the inversion, must be valid and strong.

Peirce: CP 2.716 Cross-Ref:†† 716. We have seen that Inductions and Hypotheses are inferences from the conclusion and one premiss of a statistical syllogism to the other premiss. In the case of hypothesis, this syllogism is called the explanation. Thus in one of the examples used above, we suppose the cryptograph to be an English cipher, because, as we say, this explains the observed phenomena that there are about two dozen characters, that one occurs more frequently than the rest, especially at the end of words, etc. The explanation is--

Simple English ciphers have certain peculiarities, This is a simple English cipher; Hence, this necessarily has these peculiarities.

Peirce: CP 2.717 Cross-Ref:†† 717. This explanation is present to the mind of the reasoner, too; so much so, that we commonly say that the hypothesis is adopted for the sake of the explanation. Of induction we do not, in ordinary language, say that it explains phenomena; still, the statistical deduction, of which it is the inversion, plays, in a general way, the same part as the explanation in hypothesis. From a barrel of apples, that I am thinking of buying, I draw out three or four as a sample. If I find the sample somewhat decayed, I ask myself, in ordinary language, not "Why is this?" but "How is this?" And I answer that it probably comes from nearly all the apples in the barrel being in bad condition. The distinction between the "Why" of hypothesis and the "How" of induction is not very great; both ask for a statistical syllogism, of which the observed fact shall be the conclusion, the known conditions of the observation one premiss, and the inductive or hypothetic inference the other. This statistical syllogism may be conveniently termed the explanatory syllogism. Peirce: CP 2.718 Cross-Ref:†† 718. In order that an induction or hypothesis should have any validity at all, it is requisite that the explanatory syllogism should be a valid statistical deduction. Its conclusion must not merely follow from the premisses, but follow from them upon the principle of probability. The inversion of ordinary syllogism does not give rise to an induction or hypothesis. The statistical syllogism of Form IV is invertible, because it proceeds upon the principle of an approximate equality between the ratio of P's in the whole class and the ratio in a well-drawn sample, and because equality is a convertible relation. But ordinary syllogism is based upon the property of the relation of containing and contained, and that is not a convertible relation. There is, however, a way in which ordinary syllogism may be inverted; namely, the conclusion and either of the premisses may be interchanged by negativing each of them. This is the way in which the indirect, or apagogical,†P1 figures of syllogism are derived from the first, and in which the modus tollens is derived from the modus ponens. The following schemes show this:

First Figure.

Rule.

All M is P,

Case.

S is M;

Result. S is P.

Second Figure.

|

Third Figure.

| Rule.

All M is P, | Denial of Result. S is not P,

Denial of Result. S is not P; | Case.

S is M;

Denial of Case. .·.S is not M. | Denial of Rule. .·. Some M is not P.

Modus Ponens.

Rule. If A is true, C is true, Case. In a certain case A is true; Result. .·. In that case C is true.

Modus Tollens.

|

Modus Innominatus.

| Rule. If A is true, C is true,|Case. In a certain case A is true, Denial of Result. In a certain| Denial of Result. In that case, case C is not true;

| C is not true;

Denial of Case. .·. In that | Denial of Rule. .·. If A is true, C is not necessarily true.

case A is not true.

Peirce: CP 2.719 Cross-Ref:†† 719. Now suppose we ask ourselves what would be the result of thus apagogically inverting a statistical deduction. Let us take, for example, Form IV:

The S's are a numerous random sample of the M's, The proportion r of the M's are P's;

|

Hence, probably about the proportion r of the S's are P's.

Peirce: CP 2.720 Cross-Ref:†† 720. The ratio r, as we have already noticed, is not necessarily perfectly definite; it may be only known to have a certain maximum or minimum; in fact, it may have any kind of indeterminacy. Of all possible values between 0 and 1, it admits of some and excludes others. The logical negative of the ratio r is, therefore, itself a ratio, which we may name {r}; it admits of every value which r excludes, and excludes every value of which r admits. Transposing, then, the major premiss and conclusion of our statistical deduction, and at the same time denying both, we obtain the following inverted form:

The S's are a numerous random sample of the M's, The proportion {r} of the S's are P's; Hence, probably about the proportion {r} of the M's are P's.†P1

Peirce: CP 2.721 Cross-Ref:†† 721. But this coincides with the formula of Induction. Again, let us apagogically invert the statistical deduction of Form IV (bis). This form is--

Every M has, for example, the numerous marks P', P'', P''', etc., S has an {r}-likeness to the M's; Hence, probably and approximately, S has the proportion r of the marks P', P'', P''', etc.

Peirce: CP 2.721 Cross-Ref:†† Transposing the minor premiss and conclusion, at the same time denying both, we get the inverted form--

Every M has, for example, the numerous marks P', P'', P''',etc., S has the proportion {r} of the marks P', P'', P''', etc.; Hence, probably and approximately, S has a {r}-likeness to the class of M's.

Peirce: CP 2.722 Cross-Ref:†† 722. This coincides with the formula of Hypothesis. Thus we see that Induction and Hypothesis are nothing but the apagogical inversions of statistical deductions. Accordingly, when r is taken as 1, so that {r} is "less than 1," or when r is

taken as 0, so that {r} is "more than 0," the induction degenerates into a syllogism of the third figure and the hypothesis into a syllogism of the second figure. In these special cases, there is no very essential difference between the mode of reasoning in the direct and in the apagogical form. But, in general, while the probability of the two forms is precisely the same--in this sense, that for any fixed proportion of P's among the M's (or of marks of S's among the marks of the M's) the probability of any given error in the concluded value is precisely the same in the indirect as it is in the direct form--yet there is this striking difference, that a multiplication of instances will in the one case confirm, and in the other modify, the concluded value of the ratio. Peirce: CP 2.723 Cross-Ref:†† 723. We are thus led to another form for our rule of validity of ampliative inference; namely, instead of saying that the explanatory syllogism must be a good probable deduction, we may say that the syllogism of which the induction or hypothesis is the apagogical modification (in the traditional language of logic, the reduction) must be valid. Peirce: CP 2.724 Cross-Ref:†† 724. Probable inferences, though valid, may still differ in their strength. A probable deduction has a greater or less probable error in the concluded ratio. When r is a definite number the probable error is also definite; but as a general rule we can only assign maximum and minimum values of the probable error. The probable error is, in fact--

_____________ 0.477√(2r(1 - r))/n

where n is the number of independent instances. The same formula gives the probable error of an induction or hypothesis; only that in these cases, r being wholly indeterminate, the minimum value is zero, and the maximum is obtained by putting r = 1/2.

Peirce: CP 2.725 Cross-Ref:†† §7. FIRST SPECIAL RULE FOR SYNTHETIC INFERENCE.

SAMPLING MUST BE FAIR. ANALOGY

725. Although the rule given above really contains all the conditions to which Inductions and Hypotheses need to conform, yet inasmuch as there are many delicate questions in regard to the application of it, and particularly since it is of that nature that a violation of it, if not too gross, may not absolutely destroy the virtue of the

reasoning, a somewhat detailed study of its requirements in regard to each of the premisses of the argument is still needed. Peirce: CP 2.726 Cross-Ref:†† 726. The first premiss of a scientific inference is that certain things (in the case of induction) or certain characters (in the case of hypothesis) constitute a fairly chosen sample of the class of things or the run of characters from which they have been drawn. Peirce: CP 2.726 Cross-Ref:†† The rule requires that the sample should be drawn at random and independently from the whole lot sampled. That is to say, the sample must be taken according to a precept or method which, being applied over and over again indefinitely, would in the long run result in the drawing of any one set of instances as often as any other set of the same number. Peirce: CP 2.727 Cross-Ref:†† 727. The needfulness of this rule is obvious; the difficulty is to know how we are to carry it out. The usual method is mentally to run over the lot of objects or characters to be sampled, abstracting our attention from their peculiarities, and arresting ourselves at this one or that one from motives wholly unconnected with those peculiarities. But this abstention from a further determination of our choice often demands an effort of the will that is beyond our strength; and in that case a mechanical contrivance may be called to our aid. We may, for example, number all the objects of the lot, and then draw numbers by means of a roulette, or other such instrument. We may even go so far as to say that this method is the type of all random drawing; for when we abstract our attention from the peculiarities of objects, the psychologists tell us that what we do is to substitute for the images of sense certain mental signs, and when we proceed to a random and arbitrary choice among these abstract objects we are governed by fortuitous determinations of the nervous system, which in this case serves the purpose of a roulette. Peirce: CP 2.727 Cross-Ref:†† The drawing of objects at random is an act in which honesty is called for; and it is often hard enough to be sure that we have dealt honestly with ourselves in the matter, and still more hard to be satisfied of the honesty of another. Accordingly, one method of sampling has come to be preferred in argumentation; namely, to take of the class to be sampled all the objects of which we have a sufficient knowledge. Sampling is, however, a real art, well deserving an extended study by itself: to enlarge upon it here would lead us aside from our main purpose. Peirce: CP 2.728 Cross-Ref:†† 728. Let us rather ask what will be the effect upon inductive inference of an imperfection in the strictly random character of the sampling. Suppose that, instead of using such a precept of selection that any one M would in the long run be chosen as often as any other, we used a precept which would give a preference to a certain half of the M's, so that they would be drawn twice as often as the rest. If we were to draw a numerous sample by such a precept, and if we were to find that the proportion {r} of the sample consisted of P's, the inference that we should be regularly entitled to make would be, that among all the M's, counting the preferred half for two each, the proportion {r} would be P's. But this regular inductive inference being granted, from it we could deduce by arithmetic the further conclusion that, counting the M's for one each, the proportion of P's among them must ({r} being over 2/3) lie between 3/4{r}

+ 1/4, and 3/2{r} - 1/2. Hence, if more than two thirds of the instances drawn by the use of the false precept were found to be P's, we should be entitled to conclude that more than half of all the M's were P's. Thus, without allowing ourselves to be led away into a mathematical discussion, we can easily see that, in general, an imperfection of that kind in the random character of the sampling will only weaken the inductive conclusion, and render the concluded ratio less determinate, but will not necessarily destroy the force of the argument completely. In particular, when p approximates towards 1 or 0, the effect of the imperfect sampling will be but slight. Peirce: CP 2.729 Cross-Ref:†† 729. Nor must we lose sight of the constant tendency of the inductive process to correct itself. This is of its essence. This is the marvel of it. The probability of its conclusion only consists in the fact that if the true value of the ratio sought has not been reached, an extension of the inductive process will lead to a closer approximation. Thus, even though doubts may be entertained whether one selection of instances is a random one, yet a different selection, made by a different method, will be likely to vary from the normal in a different way, and if the ratios derived from such different selections are nearly equal, they may be presumed to be near the truth. This consideration makes it extremely advantageous in all ampliative reasoning to fortify one method of investigation by another.†P1 Still we must not allow ourselves to trust so much to this virtue of induction as to relax our efforts towards making our drawings of instances as random and independent as we can. For if we infer a ratio from a number of different inductions, the magnitude of its probable error will depend very much more on the worst than on the best inductions used. Peirce: CP 2.730 Cross-Ref:†† 730. We have, thus far, supposed that although the selection of instances is not exactly regular, yet the precept followed is such that every unit of the lot would eventually get drawn. But very often it is impracticable so to draw our instances, for the reason that a part of the lot to be sampled is absolutely inaccessible to our powers of observation. If we want to know whether it will be profitable to open a mine, we sample the ore; but in advance of our mining operations, we can obtain only what ore lies near the surface. Then, simple induction becomes worthless, and another method must be resorted to. Suppose we wish to make an induction regarding a series of events extending from the distant past to the distant future; only those events of the series which occur within the period of time over which available history extends can be taken as instances. Within this period we may find that the events of the class in question present some uniform character; yet how do we know but this uniformity was suddenly established a little while before the history commenced, or will suddenly break up a little while after it terminates? Now, whether the uniformity observed consists (1) in a mere resemblance between all the phenomena, or (2) in their consisting of a disorderly mixture of two kinds in a certain constant proportion, or (3) in the character of the events being a mathematical function of the time of occurrence--in any of these cases we can make use of an apagoge from the following probable deduction:

Within the period of time M, a certain event P occurs, S is a period of time taken at random from M, and more than half as long; Hence, probably the event P will occur within the time S.

Peirce: CP 2.730 Cross-Ref:†† Inverting this deduction, we have the following ampliative inference:

S is a period of time taken at random from M, and more than half as long, The event P does not happen in the time S; Hence, probably the event P does not happen in the period M.

Peirce: CP 2.730 Cross-Ref:†† The probability of the conclusion consists in this, that we here follow a precept of inference, which, if it is very often applied will more than half the time lead us right. Analogous reasoning would obviously apply to any portion of an unidimensional continuum, which might be similar to periods of time. This is a sort of logic which is often applied by physicists in what is called extrapolation of an empirical law. As compared with a typical induction, it is obviously an excessively weak kind of inference. Although indispensable in almost every branch of science, it can lead to no solid conclusions in regard to what is remote from the field of direct perception, unless it be bolstered up in certain ways to which we shall have occasion to refer further on. Peirce: CP 2.731 Cross-Ref:†† 731. Let us now consider another class of difficulties in regard to the rule that the samples must be drawn at random and independently. In the first place, what if the lot to be sampled be infinite in number? In what sense could a random sample be taken from a lot like that? A random sample is one taken according to a method that would, in the long run, draw any one object as often as any other. In what sense can such drawing be made from an infinite class? The answer is not far to seek. Conceive a cardboard disk revolving in its own plane about its centre, and pretty accurately balanced, so that when put into rotation it shall be about †P1 as likely to come to rest in any one position as in any other; and let a fixed pointer indicate a position on the disk: the number of points on the circumference is infinite, and on rotating the disk repeatedly the pointer enables us to make a selection from this infinite number. This means merely that although the points are innumerable, yet there is a certain order among them that enables us to run them through and pick from them as from a very numerous collection. In such a case, and in no other, can an infinite lot be sampled. But it would be equally true to say that a finite lot can be sampled only on condition that it can be regarded as equivalent to an infinite lot. For the random sampling of a finite class supposes the possibility of drawing out an object, throwing it back, and continuing this process indefinitely; so that what is really sampled is not the finite collection of things, but the unlimited number of possible drawings. Peirce: CP 2.732 Cross-Ref:†† 732. But though there is thus no insuperable difficulty in sampling an infinite lot, yet it must be remembered that the conclusion of inductive reasoning only consists in the approximate evaluation of a ratio, so that it never can authorize us to conclude that in an infinite lot sampled there exists no single exception to a rule. Although all the planets are found to gravitate toward one another, this affords not the

slightest direct reason for denying that among the innumerable orbs of heaven there may be some which exert no such force. Although at no point of space where we have yet been have we found any possibility of motion in a fourth dimension, yet this does not tend to show (by simple induction, at least) that space has absolutely but three dimensions. Although all the bodies we have had the opportunity of examining appear to obey the law of inertia, this does not prove that atoms and atomicules are subject to the same law. Such conclusions must be reached, if at all, in some other way than by simple induction. This latter may show that it is unlikely that, in my lifetime or yours, things so extraordinary should be found, but [does] not warrant extending the prediction into the indefinite future. And experience shows it is not safe to predict that such and such a fact will never be met with. Peirce: CP 2.733 Cross-Ref:†† 733. If the different instances of the lot sampled are to be drawn independently, as the rule requires, then the fact that an instance has been drawn once must not prevent its being drawn again. It is true that if the objects remaining unchosen are very much more numerous than those selected, it makes practically no difference whether they have a chance of being drawn again or not, since that chance is in any case very small. Probability is wholly an affair of approximate, not at all of exact, measurement; so that when the class sampled is very large, there is no need of considering whether objects can be drawn more than once or not. But in what is known as "reasoning from analogy," the class sampled is small, and no instance is taken twice. For example: we know that of the major planets the Earth, Mars, Jupiter, and Saturn revolve on their axes, and we conclude that the remaining four, Mercury, Venus, Uranus, and Neptune, probably do the like. This is essentially different from an inference from what has been found in drawings made hitherto, to what will be found in indefinitely numerous drawings to be made hereafter. Our premisses here are that the Earth, Mars, Jupiter, and Saturn are a random sample of a natural class of major planets--a class which, though (so far as we know) it is very small, yet may be very extensive, comprising whatever there may be that revolves in a circular orbit around a great sun, is nearly spherical, shines with reflected light, is very large, etc. Now the examples of major planets that we can examine all rotate on their axes; whence we suppose that Mercury, Venus, Uranus, and Neptune, since they possess, so far as we know, all the properties common to the natural class to which the Earth, Mars, Jupiter, and Saturn belong, possess this property likewise. The points to be observed are, first, that any small class of things may be regarded as a mere sample of an actual or possible large class having the same properties and subject to the same conditions; second, that while we do not know what all these properties and conditions are, we do know some of them, which some may be considered as a random sample of all; third, that a random selection without replacement from a small class may be regarded as a true random selection from that infinite class of which the finite class is a random selection. The formula of the analogical inference presents, therefore, three premisses, thus:

S', S'', S''', are a random sample of some undefined class X, of whose characters P', P'', P''', are samples, Q is P', P'', P'''; S', S'', S''', are R's;

Hence, Q is an R.

Peirce: CP 2.733 Cross-Ref:†† We have evidently here an induction and an hypothesis followed by a deduction; thus:

Every X is, for example, P', | S', S'', S''', etc., are samples P'', P''', etc.,

| of the X's,

Q is found to be P',P'',P''', | S', S'', S''', etc., are found etc.;

| to be R's;

Hence, hypothetically, Q is | Hence, inductively, every X an X.

| is an R.

Hence, deductively, Q is an R.†P1

Peirce: CP 2.734 Cross-Ref:†† 734. An argument from analogy may be strengthened by the addition of instance after instance to the premisses, until it loses its ampliative character by the exhaustion of the class and becomes a mere deduction of that kind called complete induction, in which, however, some shadow of the inductive character remains, as this name implies.

Peirce: CP 2.735 Cross-Ref:†† §8. SECOND SPECIAL RULE FOR SYNTHETIC INFERENCE, THAT OF PREDESIGNATION

735. Take any human being, at random--say Queen Elizabeth. Now a little more than half of all the human beings who have ever existed have been males; but it does not follow that it is a little more likely than not that Queen Elizabeth was a male, since we know she was a woman. Nor, if we had selected Julius Cæsar, would it be only a little more likely than not that he was a male. It is true that if we were to go on drawing at random an indefinite number of instances of human beings, a slight excess over one-half would be males. But that which constitutes the probability of an inference is the proportion of true conclusions among all those which could be derived from the same precept. Now a precept of inference, being a rule which the mind is to follow, changes its character and becomes different when the case presented to the mind is essentially different. When, knowing that the proportion r of all M's are P's, I draw an instance, S, of an M, without any other knowledge of whether it is a P or not, and infer with probability, r, that it is P, the case presented to

my mind is very different from what it is if I have such other knowledge. In short, I cannot make a valid probable inference without taking into account whatever knowledge I have (or, at least, whatever occurs to my mind) that bears upon the question. Peirce: CP 2.736 Cross-Ref:†† 736. The same principle may be applied to the statistical deduction of Form IV. If the major premiss, that the proportion r of the M's are P's be laid down first, before the instances of M's are drawn, we really draw our inference concerning those instances (that the proportion r of them will be P's) in advance of the drawing, and therefore before we know whether they are P's or not. But if we draw the instances of the M's first, and after the examination of them decide what we will select for the predicate of our major premiss, the inference will generally be completely fallacious. In short, we have the rule that the major term P must be decided upon in advance of the examination of the sample; and in like manner in Form IV (bis) the minor term S must be decided upon in advance of the drawing. Peirce: CP 2.737 Cross-Ref:†† 737. The same rule follows us into the logic of induction and hypothesis. If in sampling any class, say the M's, we first decide what the character P is for which we propose to sample that class, and also how many instances we propose to draw, our inference is really made before these latter are drawn, that the proportion of P's in the whole class is probably about the same as among the instances that are to be drawn, and the only thing we have to do is to draw them and observe the ratio. But suppose we were to draw our inferences without the predesignation of the character P; then we might in every case find some recondite character in which those instances would all agree. That, by the exercise of sufficient ingenuity, we should be sure to be able to do this, even if not a single other object of the class M possessed that character, is a matter of demonstration. For in geometry a curve may be drawn through any given series of points, without passing through any one of another given series of points, and this irrespective of the number of dimensions. Now, all the qualities of objects may be conceived to result from variations of a number of continuous variables; hence any lot of objects possesses some character in common, not possessed by any other. It is true that if the universe of quality is limited, this is not altogether true; but it remains true that unless we have some special premiss from which to infer the contrary, it always may be possible to assign some common character of the instances S', S'', S''', etc., drawn at random from among the M's, which does not belong to the M's generally. So that if the character P were not predesignate, the deduction of which our induction is the apagogical inversion would not be valid; that is to say, we could not reason that if the M's did not generally possess the character P, it would not be likely that the S's should all possess this character. Peirce: CP 2.738 Cross-Ref:†† 738. I take from a biographical dictionary †1 the first five names of poets, with their ages at death. They are,

Aagard, died at 48. Abeille, died at 76. Abulola, died at 84.

Abunowas, died at 48. Accords, died at 45.

These five ages have the following characters in common: 1. The difference of the two digits composing the number, divided by three, leaves a remainder of one. 2. The first digit raised to the power indicated by the second, and then divided by three, leaves a remainder of one. 3. The sum of the prime factors of each age, including one as a prime factor, is divisible by three. Peirce: CP 2.738 Cross-Ref:†† Yet there is not the smallest reason to believe that the next poet's age would possess these characters. Peirce: CP 2.738 Cross-Ref:†† Here we have a conditio sine qua non of valid induction which has been singularly overlooked by those who have treated of the logic of the subject, and is very frequently violated by those who draw inductions. So accomplished a reasoner as Dr. Lyon Playfair, for instance, has written a paper of which the following is an abstract. He first takes the specific gravities of the three allotropic forms of carbon, as follows:

Diamond, 3.48. Graphite, 2.29. Charcoal, 1.88.

He now seeks to find a uniformity connecting these three instances; and he discovers that the atomic weight of carbon, being 12,

[Click here to view]

Sp. gr. diamond nearly = 3.46 = 2√12

Sp. gr. graphite nearly = 2.29 = 3√12 Sp. gr. charcoal nearly = 1.86 = 4√12

This, he thinks, renders it probable that the specific gravities of the allotropic forms of other elements would, if we knew them, be found to equal the different roots of their atomic weight. But so far, the character in which the instances agree not having been predesignated, the induction can serve only to suggest a question, and ought not to create any belief. To test the proposed law, he selects the instance of silicon, which like carbon exists in a diamond and in a graphitoidal condition. He finds for the specific gravities--

Diamond silicon, 2.47 Graphite silicon, 2.33.†P1

Peirce: CP 2.738 Cross-Ref:†† Now, the atomic weight of silicon, that of carbon being 12, can only be taken as 28. But 2.47 does not approximate to any root of 28. It is, however, nearly the cube

root of 14, [Click here to 3 view]( √1/2 x 28 = 2.41), while 2.33 is nearly the fourth root of 28

[Click here to view](4√28 = 2.30). Dr. Playfair claims that silicon is an instance satisfying his formula. But in fact this instance requires the formula to be modified; and the modification not being predesignate, the instance cannot count. Boron also exists in a diamond and a graphitoidal form; and accordingly Dr. Playfair takes this as his next example. Its atomic weight is 10.9, and its specific gravity is 2.68; which is the square root of 2/3 x 10.9. There seems to be here a further modification of the formula not predesignated, and therefore this instance can hardly be reckoned as confirmatory. The next instances which would occur to the mind of any chemist would be phosphorus and sulphur, which exist in familiarly known allotropic forms. Dr. Playfair admits that the specific gravities of phosphorus have no relations to its atomic weight at all analogous to those of carbon. The different forms of sulphur have nearly the same specific gravity, being approximately the fifth root of the atomic weight 32. Selenium also has two allotropic forms, whose specific gravities are 4.8 and 4.3; one of these follows the law, while the other does not. For tellurium the law fails altogether; but for bromine and iodine it holds. Thus the number of specific gravities for which the law was predesignate are 8; namely, 2 for phosphorus, 1 for sulphur, 2 for selenium, 1 for tellurium, 1 for bromine, and 1 for iodine. The law holds for 4 of these, and the proper inference is that about half the specific gravities of metalloids are roots of some

simple ratio of their atomic weights. Peirce: CP 2.738 Cross-Ref:†† Having thus determined this ratio, we proceed to inquire whether an agreement half the time with the formula constitutes any special connection between the specific gravity and the atomic weight of a metalloid. As a test of this, let us arrange the elements in the order of their atomic weights, and compare the specific gravity of the first with the atomic weight of the last, that of the second with the atomic weight of the last but one, and so on. The atomic weights are--

Boron,

10.9

Tellurium, 128.1

Carbon,

12.0

Iodine,

Silicon,

28.0

126.9

Bromine,

Phosphorus, 31.0

80.0

Selenium, 79.1

Sulphur, 32.

There are three specific gravities given for carbon, and two each for silicon, phosphorus, and selenium. The question, therefore, is, whether of the fourteen specific gravities as many as seven are in Playfair's relation with the atomic weights, not of the same element, but of the one paired with it. Now, taking the original formula of Playfair we find

[Click here to view] Sp. gr. boron

= 2.68 5√Te = 2.64

3d Sp. gr. carbon = 1.88 5√I = 1.84 2d Sp. gr. carbon = 2.29 6√I = 2.24 1st Sp. gr. phosphorus = 1.83 7√Se = 1.87 2d Sp. gr. phosphorus = 2.10 6√Se = 2.07

or five such relations without counting that of sulphur to itself. Next, with the modification introduced by Playfair, we have

[Click here to view] Sp. gr. silicon = 2.47 4√1/2 x Br = 2.51 2d Sp. gr. silicon = 2.33 6√2 x Br = 2.33 Sp. gr. iodine = 4.95 3√2 x C = 4.90 1st Sp. gr. carbon = 3.48 3√1/3 x I = 3.48

It thus appears that there is no more frequent agreement with Playfair's proposed law than what is due to chance.†P1 Peirce: CP 2.739 Cross-Ref:†† 739. Another example of this fallacy was "Bode's law" of the relative distances of the planets, which was shattered by the first discovery of a true planet after its enunciation. In fact, this false kind of induction is extremely common in science and in medicine.†P1 In the case of hypothesis, the correct rule has often been laid down; namely, that a hypothesis can only be received upon the ground of its having been verified by successful prediction. The term predesignation used in this paper appears to be more exact, inasmuch as it is not at all requisite that the ratio {r} should be given in advance of the examination of the samples. Still, since {r} is equal to 1 in all ordinary hypotheses, there can be no doubt that the rule of prediction, so far as it goes, coincides with that here laid down. Peirce: CP 2.740 Cross-Ref:†† 740. We have now to consider an important modification of the rule. Suppose that, before sampling a class of objects, we have predesignated not a single character but n characters, for which we propose to examine the samples. This is equivalent to making n different inductions from the same instances. The probable error in this case is that error whose probability for a simple induction is only (1/2)n, and the theory of probabilities shows that it increases but slowly with n; in fact, for n = 1000 it is only about five times as great as for n = 1, so that with only 25 times as many instances the inference would be as secure for the former value of n as with the latter; with 100 times as many instances an induction in which n = 10,000,000,000 would be equally secure. Now the whole universe of characters will never contain such a number as the last; and the same may be said of the universe of objects in the case of hypothesis. So that, without any voluntary predesignation, the limitation of our imagination and experience amounts to a predesignation far within those limits; and we thus see that if the number of instances be very great indeed, the failure to predesignate is not an

important fault. Of characters at all striking, or of objects at all familiar, the number will seldom reach 1,000; and of very striking characters or very familiar objects the number is still less. So that if a large number of samples of a class are found to have some very striking character in common, or if a large number of characters of one object are found to be possessed by a very familiar object, we need not hesitate to infer, in the first case, that the same characters belong to the whole class, or, in the second case, that the two objects are practically identical; remembering only that the inference is less to be relied upon than it would be had a deliberate predesignation been made. This is no doubt the precise significance of the rule sometimes laid down, that a hypothesis ought to be simple--simple here being taken in the sense of familiar. Peirce: CP 2.740 Cross-Ref:†† This modification of the rule shows that, even in the absence of voluntary predesignation, some slight weight is to be attached to an induction or hypothesis. And perhaps when the number of instances is not very small, it is enough to make it worth while to subject the inference to a regular test. But our natural tendency will be to attach too much importance to such suggestions, and we shall avoid waste of time in passing them by without notice until some stronger plausibility presents itself.

Peirce: CP 2.741 Cross-Ref:†† §9. UNIFORMITIES

741. In almost every case in which we make an induction or a hypothesis, we have some knowledge which renders our conclusion antecedently likely or unlikely. The effect of such knowledge is very obvious, and needs no remark. But what also very often happens is that we have some knowledge, which, though not of itself bearing upon the conclusion of the scientific argument, yet serves to render our inference more or less probable, or even to alter the terms of it. Suppose, for example, that we antecedently know that all the M's strongly resemble one another in regard to characters of a certain order. Then, if we find that a moderate number of M's taken at random have a certain character, P, of that order, we shall attach a greater weight to the induction than we should do if we had not that antecedent knowledge. Thus, if we find that a certain sample of gold has a certain chemical character--since we have very strong reason for thinking that all gold is alike in its chemical characters--we shall have no hesitation in extending the proposition from the one sample to gold in general. Or if we know that among a certain people--say the Icelanders--an extreme uniformity prevails in regard to all their ideas, then, if we find that two or three individuals taken at random from among them have all any particular superstition, we shall be the more ready to infer that it belongs to the whole people from what we know of their uniformity. The influence of this sort of uniformity upon inductive conclusions was strongly insisted upon by Philodemus,†1 and some very exact conceptions in regard to it may be gathered from the writings of Mr. Galton. Again, suppose we know of a certain character, P, that in whatever classes of a certain description it is found at all, to those it usually belongs as a universal character; then any induction which goes toward showing that all the M's are P will be greatly strengthened. Thus it is enough to find that two or three individuals taken at random from a genus of animals have three toes on each foot, to prove that the same is true of the whole genus; for we know that this is a generic character. On the other hand, we shall be slow to infer that all the animals of a genus have the same color, because

color varies in almost every genus. This kind of uniformity seemed to J. S. Mill to have so controlling an influence upon inductions, that he has taken it as the centre of his whole theory of the subject. Peirce: CP 2.742 Cross-Ref:†† 742. Analogous considerations modify our hypothetic inferences. The sight of two or three words will be sufficient to convince me that a certain manuscript was written by myself, because I know a certain look is peculiar to it. So an analytical chemist, who wishes to know whether a solution contains gold, will be completely satisfied if it gives a precipitate of the purple of cassius with chloride of tin; because this proves that either gold or some hitherto unknown substance is present. These are examples of characteristic tests. Again, we may know of a certain person, that whatever opinions he holds he carries out with uncompromising rigor to their utmost logical consequences; then, if we find his views bear some of the marks of any ultra school of thought, we shall readily conclude that he fully adheres to that school. Peirce: CP 2.743 Cross-Ref:†† 743. There are thus four different kinds of uniformity and non-uniformity which may influence our ampliative inferences: Peirce: CP 2.743 Cross-Ref:†† (1) The members of a class may present a greater or less general resemblance as regards a certain line of characters. Peirce: CP 2.743 Cross-Ref:†† (2) A character may have a greater or less tendency to be present or absent throughout the whole of whatever classes of certain kinds. Peirce: CP 2.743 Cross-Ref:†† (3) A certain set of characters may be more or less intimately connected, so as to be probably either present or absent together in certain kinds of objects. Peirce: CP 2.743 Cross-Ref:†† (4) An object may have more or less tendency to possess the whole of certain sets of characters when it possesses any of them. Peirce: CP 2.743 Cross-Ref:†† A consideration of this sort may be so strong as to amount to demonstration of the conclusion. In this case, the inference is mere deduction--that is, the application of a general rule already established. In other cases, the consideration of uniformities will not wholly destroy the inductive or hypothetic character of the inference, but will only strengthen or weaken it by the addition of a new argument of a deductive kind.

Peirce: CP 2.744 Cross-Ref:†† §10. CONSTITUTION OF THE UNIVERSE

744. We have thus seen how, in a general way, the processes of inductive and hypothetic inference are able to afford answers to our questions, though these may relate to matters beyond our immediate ken. In short, a theory of the logic of verification has been sketched out. This theory will have to meet the objections of two opposing schools of logic.

Peirce: CP 2.744 Cross-Ref:†† The first of these explains induction by what is called the doctrine of Inverse Probabilities, of which the following is an example: Suppose an ancient denizen of the Mediterranean coast, who had never heard of the tides, had wandered to the shore of the Atlantic Ocean, and there, on a certain number m of successive days had witnessed the rise of the sea. Then, says Quetelet, he would have been entitled to conclude that there was a probability equal to (m + 1)/(m + 2) that the sea would rise on the next following day.†P1 Putting m = 0, it is seen that this view assumes that the probability of a totally unknown event is 1/2; or that of all theories proposed for examination one half are true. In point of fact, we know that although theories are not proposed unless they present some decided plausibility, nothing like one half turn out to be true. But to apply correctly the doctrine of inverse probabilities, it is necessary to know the antecedent probability of the event whose probability is in question. Now, in pure hypothesis or induction, we know nothing of the conclusion antecedently to the inference in hand. Mere ignorance, however, cannot advance us toward any knowledge; therefore it is impossible that the theory of inverse probabilities should rightly give a value for the probability of a pure inductive or hypothetic conclusion. For it cannot do this without assigning an antecedent probability to this conclusion; so that if this antecedent probability represents mere ignorance (which never aids us), it cannot do it at all. Peirce: CP 2.745 Cross-Ref:†† 745. The principle which is usually assumed by those who seek to reduce inductive reasoning to a problem in inverse probabilities is, that if nothing whatever is known about the frequency of occurrence of an event, then any one frequency is as probable as any other. But Boole has shown that there is no reason whatever to prefer this assumption, to saying that any one "constitution of the universe" is as probable as any other. Suppose, for instance, there were four possible occasions upon which an event might occur. Then there would be 16 "constitutions of the universe," or possible distributions of occurrences and non-occurrences. They are shown in the following table, where Y stands for an occurrence and N for a non-occurrence.

[Click here to view]

Peirce: CP 2.745 Cross-Ref:†† It will be seen that different frequencies result some from more and some from fewer different "constitutions of the universe," so that it is a very different thing to assume that all frequencies are equally probable from what it is to assume that all constitutions of the universe are equally probable.

Peirce: CP 2.746 Cross-Ref:†† 746. Boole says that one assumption is as good as the other. But I will go further, and say that the assumption that all constitutions of the universe are equally probable is far better than the assumption that all frequencies are equally probable. For the latter proposition, though it may be applied to any one unknown event, cannot be applied to all unknown events without inconsistency. Thus, suppose all frequencies of the event whose occurrence is represented by Y in the above table are equally probable. Then consider the event which consists in a Y following a Y or an N following an N. The possible ways in which this event may occur or not are shown in the following table:

[Click here to view]

Peirce: CP 2.746 Cross-Ref:†† It will be found that assuming the different frequencies of the first event to be equally probable, those of this new event are not so--the probability of three occurrences being half as large again as that of two, or one. On the other hand, if all constitutions of the universe are equally probable in the one case, they are so in the other; and this latter assumption, in regard to perfectly unknown events, never gives rise to any inconsistency. Peirce: CP 2.746 Cross-Ref:†† Suppose, then, that we adopt the assumption that any one constitution of the universe is as probable as any other; how will the inductive inference then appear, considered as a problem in probabilities? The answer is extremely easy;†P1 namely, the occurrences or non-occurrences of an event in the past in no way affect the probability of its occurrence in the future. Peirce: CP 2.747 Cross-Ref:†† 747. Boole frequently finds a problem in probabilities to be indeterminate. There are those to whom the idea of an unknown probability seems an absurdity. Probability, they say, measures the state of our knowledge, and ignorance is denoted by the probability 1/2. But I apprehend that the expression "the probability of an event" is an incomplete one. A probability is a fraction whose numerator is the frequency of a specific kind of event, while its denominator is the frequency of a genus embracing that species. Now the expression in question names the numerator of the fraction, but omits to name the denominator. There is a sense in which it is true that the probability of a perfectly unknown event is one half; namely, the assertion of its occurrence is the answer to a possible question answerable by "yes" or "no," and of

all such questions just half the possible answers are true. But if attention be paid to the denominators of the fractions, it will be found that this value of 1/2 is one of which no possible use can be made in the calculation of probabilities. Peirce: CP 2.748 Cross-Ref:†† 748. The theory here proposed does not assign any probability to the inductive or hypothetic conclusion, in the sense of undertaking to say how frequently that conclusion would be found true. It does not propose to look through all the possible universes, and say in what proportion of them a certain uniformity occurs; such a proceeding, were it possible, would be quite idle. The theory here presented only says how frequently, in this universe, the special form of induction or hypothesis would lead us right. The probability given by this theory is in every way different--in meaning, numerical value, and form--from that of those who would apply to ampliative inference the doctrine of inverse chances. Peirce: CP 2.749 Cross-Ref:†† 749. Other logicians hold that if inductive and hypothetic premisses lead to true oftener than to false conclusions, it is only because the universe happens to have a certain constitution. Mill and his followers maintain that there is a general tendency toward uniformity in the universe, as well as special uniformities such as those which we have considered. The Abbe Gratry believes that the tendency toward the truth in induction is due to a miraculous intervention of Almighty God, whereby we are led to make such inductions as happen to be true, and are prevented from making those which are false.†1 Others have supposed that there is a special adaptation of the mind to the universe, so that we are more apt to make true theories than we otherwise should be. Now, to say that a theory such as these is necessary to explaining the validity of induction and hypothesis is to say that these modes of inference are not in themselves valid, but that their conclusions are rendered probable by being probable deductive inferences from a suppressed (and originally unknown) premiss. But I maintain that it has been shown that the modes of inference in question are necessarily valid, whatever the constitution of the universe, so long as it admits of the premisses being true. Yet I am willing to concede, in order to concede as much as possible, that when a man draws instances at random, all that he knows is that he tries to follow a certain precept; so that the sampling process might be rendered generally fallacious by the existence of a mysterious and malign connection between the mind and the universe, such that the possession by an object of an unperceived character might influence the will toward choosing it or rejecting it. Such a circumstance would, however, be as fatal to deductive as to ampliative inference. Suppose, for example, that I were to enter a great hall where people were playing rouge et noir at many tables; and suppose that I knew that the red and black were turned up with equal frequency. Then, if I were to make a large number of mental bets with myself, at this table and at that, I might, by statistical deduction, expect to win about half of them--precisely as I might expect, from the results of these samples, to infer by induction the probable ratio of frequency of the turnings of red and black in the long run, if I did not know it. But could some devil look at each card before it was turned, and then influence me mentally to bet upon it or to refrain therefrom, the observed ratio in the cases upon which I had bet might be quite different from the observed ratio in those cases upon which I had not bet. I grant, then, that even upon my theory some fact has to be supposed to make induction and hypothesis valid processes; namely, it is supposed that the supernal powers withhold their hands and let me alone, and that no mysterious uniformity or adaptation interferes with the action of chance. But then this negative fact supposed by my theory plays a totally different part from

the facts supposed to be requisite by the logicians of whom I have been speaking. So far as facts like those they suppose can have any bearing, they serve as major premisses from which the fact inferred by induction or hypothesis might be deduced; while the negative fact supposed by me is merely the denial of any major premiss from which the falsity of the inductive or hypothetic conclusion could in general be deduced. Nor is it necessary to deny altogether the existence of mysterious influences adverse to the validity of the inductive and hypothetic processes. So long as their influence were not too overwhelming, the wonderful self-correcting nature of the ampliative inference would enable us, even if they did exist, to detect and make allowance for them. Peirce: CP 2.750 Cross-Ref:†† 750. Although the universe need have no peculiar constitution to render ampliative inference valid, yet it is worth while to inquire whether or not it has such a constitution; for if it has, that circumstance must have its effect upon all our inferences. It cannot any longer be denied that the human intellect is peculiarly adapted to the comprehension of the laws and facts of nature, or at least of some of them; and the effect of this adaptation upon our reasoning will be briefly considered in the next section. Of any miraculous interference by the higher powers, we know absolutely nothing; and it seems in the present state of science altogether improbable. The effect of a knowledge of special uniformities upon ampliative inferences has already been touched upon. That there is a general tendency toward uniformity in nature is not merely an unfounded, it is an absolutely absurd, idea in any other sense than that man is adapted to his surroundings. For the universe of marks is only limited by the limitation of human interests and powers of observation. Except for that limitation, every lot of objects in the universe would have (as I have elsewhere shown)†1 some character in common and peculiar to it. Consequently, there is but one possible arrangement of characters among objects as they exist, and there is no room for a greater or less degree of uniformity in nature. If nature seems highly uniform to us, it is only because our powers are adapted to our desires.

Peirce: CP 2.751 Cross-Ref:†† §11. FURTHER PROBLEMS

751. The questions discussed in this essay relate to but a small part of the Logic of Scientific Investigation. Let us just glance at a few of the others. Peirce: CP 2.752 Cross-Ref:†† 752. Suppose a being from some remote part of the universe, where the conditions of existence are inconceivably different from ours, to be presented with a United States Census Report--which is for us a mine of valuable inductions, so vast as almost to give that epithet a new signification. He begins, perhaps, by comparing the ratio of indebtedness to deaths by consumption in counties whose names begin with the different letters of the alphabet. It is safe to say that he would find the ratio everywhere the same, and thus his inquiry would lead to nothing. For an induction is wholly unimportant unless the proportions of P's among the M's and among the non-M's differ; and a hypothetic inference is unimportant unless it be found that S has either a greater or a less proportion of the characters of M than it has of other characters. The stranger to this planet might go on for some time asking inductive

questions that the Census would faithfully answer, without learning anything except that certain conditions were independent of others. At length, it might occur to him to compare the January rainfall with the illiteracy. What he would find is given in the following table †P1:

REGION

| January Rainfall | Illiteracy

---------------------------------|------------------|-----------|

Inches

|

Atlantic seacoast, Portland to } | Washington

Per Cent 0.92

}| |

Vermont, Northern and Western } |

0.78

}|

7

|

Upper Mississippi River

|

|

0.52

|

3

|

Ohio River Valley

| |

0.74

|

8

|

Lower Mississippi, Red River, } | and Kentucky

1.08

}|

|

50

|

57

|

|

|

Mississippi Delta and Northern } | |

1.09 |

| Southeastern Coast

|

|

|

Gulf Coast

11

|

|

New York

|

| |

0.68

|

40

-----------------------------------------------------------------

He would infer that in places that are drier in January there is, not always but generally, less illiteracy than in wetter places. A detailed comparison between Mr. Schott's map of the winter rainfall with the map of illiteracy in the general census, would confirm the result that these two conditions have a partial connection. This is a very good example of an induction in which the proportion of P's among the M's is different, but not very different, from the proportion among the non-M's. It is unsatisfactory; it provokes further inquiry; we desire to replace the M by some different class, so that the two proportions may be more widely separated. Now we,

knowing as much as we do of the effects of winter rainfall upon agriculture, upon wealth, etc., and of the causes of illiteracy, should come to such an inquiry furnished with a large number of appropriate conceptions; so that we should be able to ask intelligent questions not unlikely to furnish the desired key to the problem. But the strange being we have imagined could only make his inquiries haphazard, and could hardly hope ever to find the induction of which he was in search. Peirce: CP 2.753 Cross-Ref:†† 753. Nature is a far vaster and less clearly arranged repertory of facts than a census report; and if men had not come to it with special aptitudes for guessing right, it may well be doubted whether in the ten or twenty thousand years that they may have existed their greatest mind would have attained the amount of knowledge which is actually possessed by the lowest idiot. But, in point of fact, not man merely, but all animals derive by inheritance (presumably by natural selection) two classes of ideas which adapt them to their environment. In the first place, they all have from birth some notions, however crude and concrete, of force, matter, space, and time; and, in the next place, they have some notion of what sort of objects their fellow-beings are, and of how they will act on given occasions. Our innate mechanical ideas were so nearly correct that they needed but slight correction. The fundamental principles of statics were made out by Archimedes. Centuries later Galileo began to understand the laws of dynamics, which in our times have been at length, perhaps, completely mastered. The other physical sciences are the results of inquiry based on guesses suggested by the ideas of mechanics. The moral sciences, so far as they can be called sciences, are equally developed out of our instinctive ideas about human nature. Man has thus far not attained to any knowledge that is not in a wide sense either mechanical or anthropological in its nature, and it may be reasonably presumed that he never will.†1 Peirce: CP 2.754 Cross-Ref:†† 754. Side by side, then, with the well established proposition that all knowledge is based on experience, and that science is only advanced by the experimental verifications of theories, we have to place this other equally important truth, that all human knowledge, up to the highest flights of science, is but the development of our inborn animal instincts.

Peirce: CP 2.755 Cross-Ref:†† CHAPTER 9

THE VARIETIES AND VALIDITY OF INDUCTION†1

§1. CRUDE, QUANTITATIVE, AND QUALITATIVE INDUCTION

755. Retroduction and Induction face opposite ways. The function of retroduction is not unlike those fortuitous variations in reproduction which played so important a rôle in Darwin's original theory. In point of fact, according to him every

step in the long history of the development of the moner into the man was first taken in that arbitrary and lawless mode. Whatever truth or error there may be in that, it is quite indubitable, as it appears to me, that every step in the development of primitive notions into modern science was in the first instance mere guess-work, or at least mere conjecture. But the stimulus to guessing, the hint of the conjecture, was derived from experience. The order of the march of suggestion in retroduction is from experience to hypothesis. A great many people who may be admirably trained in divinity, or in the humanities, or in law and equity, but who are certainly not well trained in scientific reasoning, imagine that Induction should follow the same course. My Lord Chancellor Bacon was one of them. On the contrary, the only sound procedure for induction, whose business consists in testing a hypothesis already recommended by the retroductive procedure, is to receive its suggestions from the hypothesis first, to take up the predictions of experience which it conditionally makes, and then try the experiment and see whether it turns out as it was virtually predicted in the hypothesis that it would. Throughout an investigation it is well to bear prominently in mind just what it is that we are trying to accomplish in the particular stage of the work at which we have arrived. Now when we get to the inductive stage what we are about is finding out how much like the truth our hypothesis is, that is, what proportion of its anticipations will be verified. Peirce: CP 2.756 Cross-Ref:†† 756. It is well to distinguish three different varieties of induction. The first and weakest kind of inductive reasoning is that which goes on the presumption that future experience as to the matter in hand will not be utterly at variance with all past experience.†P1 Example: "No instance of a genuine power of clairvoyance has ever been established: So I presume there is no such thing." I promise to call such reasoning crude induction.†1 Bacon seems to refer to this when he speaks of "inductio quae procedit per enumerationem simplicem." But I hardly think he meant to say that that phrase exactly describes it. It certainly does not; since in most cases no enumeration is attempted; and the enumeration, even if given, would not be the reasoner's chief reliance, which is rather the absence of instances to the contrary. Peirce: CP 2.757 Cross-Ref:†† 757. Crude induction is the only kind of induction that is capable of inferring the truth of what, in logic, is termed a universal proposition. For what is called "complete induction" is not inductive reasoning, but is logistic deduction. We might further say, if we chose, that every crude induction concludes a universal proposition; but this would be merely the expression of a way of regarding matters. For any proposition concerning the general run of future experience may be regarded as universal, even if it be "A pair of dice will, every now and then, turn up doublets." The undipped heel of crude induction is that if its conclusion be understood as indefinite, it will be of little use, while if it be taken definitely, it is liable at any moment to be utterly shattered by a single experience; for a series of experiences, if the whole constitutes but a single one of the instances to which an inductive conclusion refers, is to be regarded as a single experience.†P1 Peirce: CP 2.758 Cross-Ref:†† 758. From the weakest kind of induction let us pass at once to the strongest. This investigates the interrogative suggestion of retroduction, "What is the 'real probability' that an individual member of a certain experiential class, say the S's, will have a certain character, say that of being P?" This it does by first collecting, on scientific principles, a "fair sample" of the S's, taking due account, in doing so, of the

intention of using its proportion of members that possess the predesignate character of being P. This sample will contain none of those S's on which the retroduction was founded. The induction then presumes that the value of the proportion, among the S's of the sample, of those that are P, probably approximates, within a certain limit of approximation, to the value of the real probability in question. I propose to term such reasoning Quantitative Induction. Now, if I were writing a treatise on logic, I should here be obliged, not only to teach the art of sampling, including all that Dr. Karl Pearson †1 and others have taught us about distributions of specific instances among general ones, and the consequent proper inferences in such cases, but I should have to state and expound the exact definitions of "real probability," "independent," "fair sample," "predesignate," etc. As it is, I will limit myself to a single needful explanation that, so far as I know, the reader could not find definitely stated in any of the books. It is that when we say that a certain ratio will have a certain value in "the long run," we refer to the probability-limit of an endless succession of fractional values; that is, to the only possible value from 0 to ∞, inclusive, about which the values of the endless succession will never cease to oscillate; so that, no matter what place in the succession you may choose, there will follow both values above the probability-limit and values below it; while if V be any other possible value from 0 to ∞, but not the probability-limit there will be some place in the succession beyond which all the values of the succession will agree, either in all being greater than V, or else in all being less. Peirce: CP 2.759 Cross-Ref:†† 759. The remaining kind of induction, which I shall call Qualitative Induction, is of more general utility than either of the others, while it is intermediate between them, alike in respect to security and to the scientific value of its conclusions. In both these respects it is well separated from each of the other kinds. It consists of those inductions which are neither founded upon experience in one mass, as Crude Induction is, nor upon a collection of numerable instances of equal evidential values, but upon a stream of experience in which the relative evidential values of different parts of it have to be estimated according to our sense of the impressions they make upon us. Peirce: CP 2.759 Cross-Ref:†† Qualitative Induction consists in the investigator's first deducing from the retroductive hypothesis as great an evidential weight of genuine conditional predictions as he can conveniently undertake to make and to bring to the test, the condition under which he asserts them being that of the retroductive hypothesis having such degree and kind of truth as to assure their truth. In calling them "predictions," I do not mean that they need relate to future events but that they must antecede the investigator's knowledge of their truth, or at least that they must virtually antecede it. I will give an illustration of such "virtual antecedence." Suppose that to avoid wasting a great deal of time upon a hypothesis which the first comparisons with the facts may show to be utterly worthless, an investigator of a certain conjecture draws up and resolves to follow a well-considered initial program for work upon the question, and that this consists mainly in working out and testing as many consequences of the hypothesis as he can work out by a certain mathematical method and can ascertain the truth or falsity of at a cost of not more than $100 for each. But suppose that among the half dozen predictions to which that method will carry him, there, quite unexpectedly, turns up one whose truth has long been known to him, though it is a surprise to him to find that it is deducible from the hypothesis under examination. What course does sound logic impose upon him under these

circumstances? The answer is that he must reexamine the process of retroduction that suggested the hypothesis; and if the fact that is now repredicted in any degree influenced that hypothesis, it has had its due effect, and must not be used again. But if not, will he then be free to use the prediction if he likes? Not at all: the validity of his Qualitative Induction will be found to depend upon his following a rational and decisive method; he has no more right, but rather less, to favor the inductive rejection of the retroductive suggestion, than to favor its inductive adoption; and he is bound, as a man who means to reason as honestly as the imperfections of his nature and training will permit, to admit the true prediction into his counsels. The predictions must eventually be so varied as to test every feature of the hypothesis; yet the interests of science command constant attention to economy, especially in the earlier inductive stages of research. Peirce: CP 2.759 Cross-Ref:†† Having made his initial predictions the investigator proceeds to ascertain their truth or falsity; and then, having taken account of such subsidiary arguments as there may be, goes on to judge of the combined value of the evidence, and to decide whether the hypothesis should be regarded as proved, or as well on the way toward being proved, or as unworthy of further attention, or whether it ought to receive a definite modification in the light of the new experiments and be inductively reexamined ab ovo, or whether finally, that while not true it probably presents some analogy to the truth, and that the results of the induction may help to suggest a better hypothesis. Peirce: CP 2.760 Cross-Ref:†† 760. I will now state, with slight hints of argument, the conclusions which I have reached as to the warrant, or basis of validity, of the inferential processes in the three stages of inquiry. I have been actively studying this subject, for the sake of completely satisfying my own mind about it, for 50 or 51 years. To be sure, I have, some half dozen times during the half-century, let my mind lie fallow, as to this subject, during one or two dozens of months, hoping so to rid myself of any inveterate bad habits of thinking that I may insensibly have fallen into. I have six times published my views, in 1867,†1 1868,†2 1878,†3 1882 [1883],†4 1892,†5 and 1902 [1901].†6 The last of these publications, compared with my present brief abstract, shows that my last week of years has by no means been an idle one, and encourages me to hope that I may yet be able to detect errors and omissions in my views, even if others do not confer upon me the benefit of such amendments.

Peirce: CP 2.761 Cross-Ref:†† §2. MILL ON INDUCTION

761. In regard to the theory of the validity of Induction the great majority still follow the System of Logic set forth in 1843 by John Stuart Mill, who was certainly a clear thinker, and apparently a remarkably candid thinker, in spite of his long training in writing for one of the old "quarterlies," and his consequent unfortunate taste for and skill in controversy, which, combined with his having imbibed his father's sterilizing nominalism with his mother's milk, rendered him, for example, incapable of appreciating Whewell, whose acquaintance with the processes of thought of science was incomparably greater than his own. J. S. Mill's beautiful style, of truly

French perfection, together with the bulk of the two volumes, prevent all but the keenest readers from perceiving that he unconsciously wavers between three (not to say four) incompatible theories of the validity of induction. The first (stated in Bk. III, Chap. 3, Sec. 1) is [that] the whole force of Induction is the same as that of a syllogism of which the major premiss is the same for all inductions, being a certain "Axiom of the uniformity of the course of nature" (so described in the table of "Contents"). This was substantially Whately's theory of 1826. The second theory (which seems to be usually uppermost in Mill's mind; especially in Bk. II, Chap. 3, Sec. 7 and in Bk. III, Chap. 4, Sec. 2), is that induction proceeds as if upon the principle that a predicate which throughout a more or less extensive experience has been uniformly found to be true of all the members of a given class that have been examined in this respect may, with little risk, be presumed to be true of every member of that class, without exception; and that while it is not necessary that the inductive reasoner should have this principle clearly in mind, the logician, whose business it partly is to explain why inductions turn out to be true, must recognize the fact that nature is sufficiently uniform to render that quasi principle true, and must recognize that [nothing] else renders induction a safe and justifiable procedure. This theory is little more than the old maxim that "we must judge of the future by the past," which Mill--into such unfairness can an inclination toward controversy betray even an eminently fair mind!--attacks as if it merely meant that future history will repeat past history, instead of what it has meant, that future experience must be presumed to resemble past experience under sufficiently similar conditions. The third theory (see Bk. III, Chap. 3, Sec. 3), is that nature as a whole is not absolutely uniform, variety being a far more prominent characteristic of it; and that such uniformity as there is, is "a mere tissue of partial regularities," each consisting in the fact that some classes of objects show a greater, and some a less, tendency to a resemblance of all their members in respect to certain lines of characters; and that whoever knows this "has solved the problem of Induction." This theory was original with Mill; and though it is not the sole, nor the main, support of induction, it certainly does bring a powerful additional support to many inductions. But it is curious that Mill should have chanced to say, whoever might be acquainted with this theory "knows more of the philosophy of logic than the wisest of the ancients." For a quarter of a century later Gomperz †1 published so much as remained of the contents of a papyrus from Herculaneum, which was a defence of induction and a theory of its validity by the Epicurean Philodemus, under whose instruction Cicero studied; and the theory of Philodemus, like that of Mill, is that this kind of reasoning (the only valid reasoning in his opinion) derives its validity from the existence in nature of special uniformities. Only, the uniformities that attracted the attention of Philodemus, instead of characterizing certain classes, characterize certain characters, and consist in their having a special tendency to be present (or to be absent) throughout all the members of certain kinds of classes. In fact, still other types of uniformities may affect the strength of inductions. Peirce: CP 2.761 Cross-Ref:†† A yet fourth theory of induction, that of Laplace, received, by implication, the assent of Mill; and since this theory is taught as correct in all the textbooks of the Doctrine of Chances, it behooves me, in adopting another, to state, with the utmost brevity wherein Laplace's theory is false and harmful. I shall also give my explanation of Mill's assenting to it. Peirce: CP 2.762 Cross-Ref:†† 762. If, upon any occasion, we were to devise a method of forming a

numerous sample of any class, say the S's, which should be suitable for use in determining to a given degree of approximation what proportion of future experiences of S's would, in the long run, be found to have the character of being P, in case existing general conditions should undergo no alteration, then in case there were any definite reason to expect that, among S's coming to our attentive experience from any particular sub-class, say among the S's that belong to the sub-class of T's, a markedly different proportion would turn out to be P from the proportion among the S's that should not be T's, then that method of forming the sample, since we have supposed it to be "suitable" for showing the proportion of P's among all future experiences of S's, must needs insure that the proportion of S's that are T's should be nearly the same in the sample as it was destined to be among all the S's of our future experience; though, as I need not repeat again, this would be so only under the supposition of unchanged general circumstances, and need not be more precisely true than would suffice to keep the errors of the concluded proportion of subsequently experienced S's that should be P within the intended limit of approximation. Moreover, should there be any serious reason to suspect that any identifiable S presenting itself for admission to the sample was so connected with any S already admitted as to have a special liability whether to being like or to being unlike that already admitted instance in respect to being or not being P (under the same limitation that is not to be repeated), then our "suitable" method would have to exclude that instance from the sample. And once again, should there happen to be any reason to suspect that an instance had attracted our attention owing to causes connected, whether directly or indirectly, with its being P, or to such causes as should be connected with its not being P, then our suitable method must exclude that instance. Furthermore, our suitable method must so operate that the sample shall contain a sufficient number of instances to give the intended degree of approximation. For instance, if it will suffice that the figure next following the decimal point in the decimal expression of the proportion among all the S's of such as are P should be exact, 9 instances may first be taken, and if these make the ratio less than 0.05 or greater than 0.95, they will suffice. If not, 14 more instances may be collected; and if the whole 23 make the ratio less than 0.15 or greater than 0.85, they will suffice. If not, then if 11 more instances being taken, the whole 34 make the ratio less than 0.25 or greater than 0.75, they will suffice. If not, add 7 more, and if the ratio appears as less than 0.35 or more than 0.65, the 41 will suffice. If not, take 4 more and if the ratio then appears as less than 0.45 or greater than 0.55, the 45 will suffice. If not, one more instance will in any case be enough. If the first two figures of the decimal fraction must be correct, a hundred times as many instances will be requisite. Peirce: CP 2.763 Cross-Ref:†† 763. Every person of common-sense must, upon reflexion, acknowledge, what is familiar to everybody habituated to inductive reasoning, that all the above precautions are requisite, except that the concluding rule need not have been so detailed, and that, if the instances were sufficiently multiplied, it would suffice that the other rules should not be too frequently and grossly violated and that they should not prevailingly be violated in the same direction. Let all such diminutions from them be made, and it still remains true for sound reason, that such an induction does not follow merely from the fact that P is true of such and such of the S's of a collection of S's, but that it is necessary to take account of the manner in which these S's were brought to the inquirer's attention. This fixes a great gulf between Induction and Deduction. It is quite true that we may describe the general conditions of a valid quantitative induction, and may convince ourselves that if the sample be drawn strictly at random from among the S's and be made sufficiently numerous, then, the

general conditions remaining unchanged, it necessarily follows that future experience, under the same general conditions, will on the average of an indefinite multitude of such inductions, bear out the Inductive conclusion. Still, this in no wise suffices to reduce the quantitative induction to any kind of induction [deduction?]. For even if I were to grant that the truth of the inductive conclusion would necessarily follow if the conditions of a fair sample were to be ideally fulfilled, which, for a reason that I will presently state, I find myself unable to do, still the person who really draws the inductive inference cannot possibly have any demonstrative evidence that those conditions are fulfilled even to the imperfect degree that is needful for an approximation to the true ratio. He knows, if you will, that [he] has made strenuous efforts to make his sample a fair one; but he cannot be quite sure that deep down in the caverns of his heart there may not lurk, unsuspected by him, a determination to force himself to believe in a certain value for the ratio, nor that this has not frustrated all his efforts to make the sample a fair one; and if he cannot be absolutely certain even of his own honesty, how can he so much as approach certainty as to the correctness of his concluded approximation to the ratio not having been destroyed by external circumstances? A theoretician--or rather, a papyrobite, a man whose vitality is that of sentences written down or imagined--may reply that that contingency is covered by proviso that general conditions remain sufficiently unchanged. But that is to overlook the principal end of inquiry, as regards human life. What is the chief end of man? Answer: To actualize ideas of the immortal, ceaselessly prolific kind. To that end it is needful to get beliefs that the believer will take satisfaction in acting upon, not mere rules set down on paper, with lethal provisos attached to them. The inductive reasoner cannot possibly find any strictly demonstrative reasoning that could take the place of his induction, since every demonstrative is strictly limited to the field of that part of its copulate premiss that corresponds to the minor premiss of a syllogism; while to serve his purpose, that of forming a basis for conduct, it must transcend that limit in concluding future from past experience. Now every valid mathematical reasoning is demonstrative and is limited to an ideal state of things. The reasoning of the calculus of probabilities consists simply of demonstrations concerning "probabilities," which, in all useful applications of the calculus, are real probabilities, or ratios of frequency in the "long run" of experiences of designated species among experiences designated, or obviously designable, genera over those species; which real probabilities are ascertained by quantitative inductions from statistics laboriously collected and critically tabulated. But the phrase "the probability of an event," which is perpetually recurring in the treatises, and which is not free from objection, even when the real probability is meant (because it seems to refer to a singular experience considered by itself, and because it does not mention that two classes of experiences are essentially concerned), is used in various different senses, owing to the ambiguity of the word "probability"; and the writers of the mathematical treatises on the subject have not had sufficient power of logical analysis to found any useful theory upon it. . . . Peirce: CP 2.764 Cross-Ref:†† 764. Laplace maintains that it is possible to draw a necessary conclusion regarding the probability of a particular determination of an event based on not knowing anything at all about [it]; that is, based on nothing. When a man thinks himself to know nothing at all as to which of a number of alternatives is the truth, his mind can no more incline toward or against any one of them or any combination of them than a mathematical point can have an inclination toward any point of the compass. Suppose the question concerns the color of an object which we know has a high color, but are otherwise in a state of blank ignorance [about it]. Then, according

to Laplace, if one were to draw two lines across a map of the spectrum, it would be probable that the color did not match any part of the spectrum included between those lines; no matter how nearly they might include the whole spectrum. Laplace holds that for every man there is one law (and necessarily but one) of dissection of each continuum of alternatives so that all the parts shall seem to that man to be "également possibles" in a quantitative sense, antecedently to all information. But he presents not the slightest reason for thinking this to be so, and seems to admit that to different men different modes of dissection will seem to give alternatives that are également possibles. It is only by basing the theory of probability upon this doctrine, and thus rendering probability without interest except to a student of human eccentricities, that it is possible to assign any mathematical probability to an inductive conclusion. Much might be added in refutation of Laplace's position. Peirce: CP 2.765 Cross-Ref:†† 765. In the first edition of his Logic,†1 Mill presents arguments against Laplace's view; but in his third, without answering his former arguments, as far as I see, he abandons them, and thus assents to all that is necessary for calculating a necessary probability for the inductive conclusion, without any regard to the manner in which the instances have been collected. Peirce: CP 2.766 Cross-Ref:†† 766. I will now sketch one or more ways of refuting each of Mill's three professed theories of Induction. Peirce: CP 2.766 Cross-Ref:†† To the first theory, that an Induction is equivalent to a syllogism whose major premiss is the axiom of the uniformity of nature, while its minor premiss states the observed facts about the instances, the conclusion being identical with that of the induction, each of the following objections is conclusive: first, that an induction, unlike a demonstration, does not rest solely upon the facts observed, but upon the manner in which those facts have been collected; secondly, that a syllogism infers its conclusion apodictically, while an induction does not; thirdly, that a syllogism enriches our knowledge of ideas, but not our information, which is what Kant meant in saying that it only explicates but does not amplify knowledge, while an induction does amplify our knowledge; fourthly, that the proposed syllogism would be fallacious, because its major premiss is vague, so that it could be fairly thrown into the form of a fallacy of undistributed middle, since all we really know of the general uniformity of nature is that some pairs of phenomena (an apparently infinitesimal proportion of all pairs) are connected as logical antecedent and consequent; fifthly, because a sound syllogism must not conclude beyond the breadth, or logical extension of its minor premiss (when this is suitably stated), while to represent a true induction it must do so. There are other objections, fully as strong as these five; but it seems needless to mention them. Peirce: CP 2.767 Cross-Ref:†† 767. The second theory correctly describes the procedure of the mind in crude inductions, but in no others; and Mill's celebrated four methods (chiefly based on the Novum Organum), though they may be of some help to minds that need such aids, yet furnish nothing but crude inductions, after all. The principle of this theory also sufficiently explains how it is that we meet such frequent opportunities to draw crude inductions as we do. But the moment the attempt is made to apply this theory to justifying, or explaining the validity even of crude inductions (and it is still worse

with other kinds of induction), it lays itself open to all the objections to the first method, including the five that were specified above. For this second theory, which is the point where Mill's vain attempt to make reasoning able to get along without generalization becomes the most futile, and verges closely upon overt absurdity, differs from the first merely in not allowing, as essential to induction, that it should have any of such force as it might derive from employing the uniformity of experience as a premiss. Now this point of difference cannot confer upon induction as explained by the second theory any validity that it would not have if it were explicable by the first theory. Peirce: CP 2.768 Cross-Ref:†† 768. The third theory presents two decided advantages. For it may remove entirely the vagueness of the general principle of uniformity; and in some cases makes the special uniformity predicate a probability, so as to render the refutation of the theory, on the ground that induction does not conclude apodictically, considerably more difficult in those cases. Moreover, this theory does correctly state a part of the argument for very many inductive conclusions. But this part of the argument is not inductive but deductive. For these special uniformities (such, for example, as that every chemical element has the same combining weight, no matter from what mineral or from what part of the globe it has come), have only become known by induction, often only by elaborate investigations, and are not logical principles; so that they need to be stated as premisses when the argument is to be set forth in full. The special uniformities, when they become known, enable us to dispense with certain inductive inquiries that would otherwise be requisite. But they leave other inductions (such as that which led Mendeléeff to enunciate his periodic law), quite untouched, not explaining them in any sense. Peirce: CP 2.769 Cross-Ref:†† 769. The true guarantee of the validity of induction is that it is a method of reaching conclusions which, if it be persisted in long enough, will assuredly correct any error concerning future experience into which it may temporarily lead us. This it will do not by virtue of any deductive necessity (since it never uses all the facts of experience, even of the past), but because it is manifestly adequate, with the aid of retroduction and of deductions from retroductive suggestions, to discovering any regularity there may be among experiences, while utter irregularity is not surpassed in regularity by any other relation of parts to whole, and is thus readily discovered by induction to exist where it does exist, and the amount of departure therefrom to be mathematically determinable from observation where it is imperfect. The doctrine of chances, in all that part of it that is sound, is nothing but the science of the laws of irregularities. I do not deny that God's beneficence is in nothing more apparent than in how in the early days of science Man's attention was particularly drawn to phenomena easy to investigate and how Man has ever since been led on, as through a series of graduated exercises, to more and more difficult problems; but what I do say is that there is no possibility of a series of experiences so wanting in uniformity as to be beyond the reach of induction, provided there be sufficiently numerous instances of them, and provided the march of scientific intelligence be unchecked. Peirce: CP 2.770 Cross-Ref:†† 770. Quantitative induction approximates gradually, though in an irregular manner to the experiential truth for the long run. The antecedent probable error of it at any stage is calculable as well as the probable error of that probable error. Besides that, the probable error can be calculated from the results, by a mixture of induction

and theory. Any striking and important discrepancy between the antecedent and a posteriori probable errors may require investigation, since it suggests some error in the theoretical assumptions. But the fact which is here important is that Quantitative Induction always makes a gradual approach to the truth, though not a uniform approach. Peirce: CP 2.771 Cross-Ref:†† 771. Qualitative Induction is not so elastic. Usually either this kind of induction confirms the hypothesis or else the facts show that some alteration must be made in the hypothesis. But this modification may be a small detail. Peirce: CP 2.772 Cross-Ref:†† 772. Experiments †1 which I have conducted in great numbers and great elaboration have convinced me of the extremely important advantages of making use, in Qualitative Induction, of numbers in place of such adverbs of comparison of the intensity of feelings as, "slightly," "a little," "somewhat," "tolerably," "moderately," "considerably," "much," "greatly," "excessively," etc. It is not necessary to use the adverbs; but in some cases I have found it convenient to employ a few of them. What is necessary is to get certain feelings so fixed in one's mind that they can be exactly and severally reproduced in the imagination at any time, these feelings forming such a series of ten or so, beginning with the zero of intensity and running up to high intensities; and further being such that any one of them being contemplated by the investigator and compared with the next intenser in the series, the interval of intensity between them shall appear, to the contemplator's feeling, to be equal to the interval between any other one of the series and the member next intenser than it. It is certain that this can be done, since all sidereal astronomers since Ptolemy have practised this; and many psychologists beside me have done something similar for other feelings than that of luminosity. It has been demonstrated that a series of positive numbers, integer and fractional, expresses in itself nothing more than an order of succession. But this scale is made to express, besides, a feeling of a difference of feeling in one respect; and the experiments of many persons prove conclusively that people generally can form such a scale; and further that the scales of different persons are concordant to a pretty high degree. The next step has been executed by but few persons so far as I know; but the experiments of these few render it all but certain that all normal persons can do so with good accord. This consists in comparing a difference of feeling in one respect with a difference in a single other respect; such as luminosity and pressure-feelings, or the relative bitterness of two solutions of quassia and that of self-blame for two former actions. Such comparisons as these last are, to be sure, of no direct applicability so far as I am aware; but they are good exercises in that prescissive abstraction of intensity from its subject which is required for estimating the equality of two differences of intensity. Such estimations enable us to add and take the arithmetical mean of intensities referred to the same standard; and not only the practice of all photometricians, both astronomers and gas-examiners, but also very many thousands of experiments by me upon a wide variety of qualities of sensation, establishes, to my full satisfaction, the great utility of such applications of number in giving a control over qualitative inductions. I have not found multiplications of such numbers useful, for example, in establishing the laws of such comparisons as the relative photometric value of two lights of different colors, where I need not say that it is one thing to ask what intensity of a light A, of fixed hue and chroma but variable luminosity best matches a light B, that is altogether fixed, and quite another and independent question what photometric intensity of B, if this be made to vary, best matches an A of given fixed intensity. The meaning of the product

of two differences of intensity which refer in general to different qualities is obvious enough: it is the number to be attached as a measure to a phenomenon which involves two feelings of the intensities indicated by the multiplicand and multiplier, these two feelings being [in] a certain fixed relation to one another in which they are as independent as possible. But there is no advantage in attaching any single measure to such a complex phenomenon, unless there are different ways of analyzing it, more or less similar to the different systems of coördinates in geometry. I mean that, for example, different horizontal areas are not only measured by the sum of the parallelogram into which [they] may be cut up, each parallelogram having its sides in the directions of ENE and N by W. Were that the sole method of measurement, nothing would be gained by combining the linear measures in the two directions, but rather the reverse. But in fact we may measure the area by parallelograms in any other two dimensions; and the ratio between any two areas will be the same by any two such methods of measurement. Moreover, we may employ polar, in place of Cartesian, coördinates, and cut the area up into a circle and broken, concentric, and very thin rings. The area will be the sum of the areas, each of which will be X x Y, where X is the difference of the two radii, while Y is the proportion of their sum, the proportion being that of the entire ring which forms a part of the area.

Peirce: CP 2.773 Cross-Ref:†† CHAPTER 10

NOTES ON AMPLIATIVE REASONING

§1. REASONING †1

773. Reasoning is a process in which the reasoner is conscious that a judgment, the conclusion, is determined by other judgment or judgments, the premisses, according to a general habit of thought, which he may not be able precisely to formulate, but which he approves as conducive to true knowledge. By true knowledge he means, though he is not usually able to analyse his meaning, the ultimate knowledge in which he hopes that belief may ultimately rest, undisturbed by doubt, in regard to the particular subject to which his conclusion relates. Without this logical approval, the process, although it may be closely analogous to reasoning in other respects, lacks the essence of reasoning. Every reasoner, therefore, since he approves certain habits, and consequently methods, of reasoning, accepts a logical doctrine, called his logica utens. Reasoning does not begin until a judgment has been formed; for the antecedent cognitive operations are not subject to logical approval or disapproval, being subconscious, or not sufficiently near the surface of consciousness, and therefore uncontrollable. Reasoning, therefore, begins with premisses which are adopted as representing percepts, or generalizations of such percepts. All the reasoner's conclusions ought to refer solely to the percepts, or rather to propositions expressing facts of perception. But this is not to say that the general conceptions to which he attains have no value in themselves.

Peirce: CP 2.774 Cross-Ref:†† 774. Reasoning is of three elementary kinds; but mixed reasonings are more common. These three kinds are induction, deduction, and presumption (for which the present writer proposes the name abduction). Peirce: CP 2.775 Cross-Ref:†† 775. Induction takes place when the reasoner already holds a theory more or less problematically (ranging from a pure interrogative apprehension to a strong leaning mixed with ever so little doubt); and having reflected that if that theory be true, then under certain conditions certain phenomena ought to appear (the stranger and less antecedently credible the better), proceeds to experiment, that is, to realize those conditions and watch for the predicted phenomena. Upon their appearance he accepts the theory with a modality which recognizes it provisionally as approximately true. The logical warrant for this is that this method persistently applied to the problem must in the long run produce a convergence (though irregular) to the truth; for the truth of a theory consists very largely in this, that every perceptual deduction from it is verified. It is of the essence of induction that the consequence of the theory should be drawn first in regard to the unknown, or virtually unknown, result of experiment; and that this should virtually be only ascertained afterward. For if we look over the phenomena to find agreements with the theory, it is a mere question of ingenuity and industry how many we shall find. Induction (at least, in its typical forms) contributes nothing to our knowledge except to tell us approximately how often, in the course of such experience as our experiments go towards constituting, a given sort of event occurs. It thus simply evaluates an objective probability. Its validity does not depend upon the uniformity of nature, or anything of that kind. The uniformity of nature may tend to give the probability evaluated an extremely great or small value; but even if nature were not uniform, induction would be sure to find it out, so long as inductive reasoning could be performed at all. Of course, a certain degree of special uniformity is requisite for that. Peirce: CP 2.775 Cross-Ref:†† But all the above is at variance with the doctrines of almost all logicians; and, in particular, they commonly teach that the inductive conclusion approximates to the truth because of the uniformity of nature. They only contemplate as inductive reasoning cases in which, from finding that certain individuals of a class have certain characters, the reasoner concludes that every single individual of the class has the same character. According to the definition here given, that inference is not inductive, but is a mixture of deduction and presumption. Cf. Probable Inference [§4.] See also Scientific Method [vol. 7.] Peirce: CP 2.776 Cross-Ref:†† 776. Presumption, or, more precisely, abduction (which the present writer believes to have been what Aristotle's twenty-fifth chapter of the second Prior Analytics imperfectly described under the name of {apagögé}, until Apellicon substituted a single wrong word and thus disturbed the sense of the whole), furnishes the reasoner with the problematic theory which induction verifies. Upon finding himself confronted with a phenomenon unlike what he would have expected under the circumstances, he looks over its features and notices some remarkable character or relation among them, which he at once recognizes as being characteristic of some conception with which his mind is already stored, so that a theory is suggested which would explain (that is, render necessary) that which is surprising in the phenomena.

Peirce: CP 2.776 Cross-Ref:†† He therefore accepts that theory so far as to give it a high place in the list of theories of those phenomena which call for further examination. If this is all his conclusion amounts to, it may be asked: What need of reasoning was there? Is he not free to examine what theories he likes? The answer is that it is a question of economy. If he examines all the foolish theories he might imagine, he never will (short of a miracle) light upon the true one. Indeed, even with the most rational procedure, he never would do so, were there not an affinity between his ideas and nature's ways. However, if there be any attainable truth, as he hopes, it is plain that the only way in which it is to be attained is by trying the hypotheses which seem reasonable and which lead to such consequences as are observed. Peirce: CP 2.777 Cross-Ref:†† 777. Presumption is the only kind of reasoning which supplies new ideas, the only kind which is, in this sense, synthetic. Induction is justified as a method which must in the long run lead up to the truth, and that, by gradual modification of the actual conclusion. There is no such warrant for presumption. The hypothesis which it problematically concludes is frequently utterly wrong itself, and even the method need not ever lead to the truth; for it may be that the features of the phenomena which it aims to explain have no rational explanation at all. Its only justification is that its method is the only way in which there can be any hope of attaining a rational explanation. This doctrine agrees substantially with that of some logicians; but it is radically at variance with a common theory and with a common practice. This prescribes that the reasoner should be guided by balancing probabilities, according to the doctrine of inverse probability. This depends upon knowing antecedent probabilities. If these antecedent probabilities were solid statistical facts, like those upon which the insurance business rests, the ordinary precepts and practice would be sound. But they are not and cannot, in the nature of things, be statistical facts. What is the antecedent probability that matter should be composed of atoms? Can we take statistics of a multitude of different universes? An objective probability is the ratio of frequency of a specific to a generic event in the ordinary course of experience. Of a fact per se it is absurd to speak of objective probability. All that is attainable are subjective probabilities, or likelihoods, which express nothing but the conformity of a new suggestion to our prepossessions; and these are the source of most of the errors into which man falls, and of all the worst of them. An instance of what the method of balancing likelihoods leads to is the "higher criticism" of ancient history, upon which the archaeologist's spade has inflicted so many wounds. Peirce: CP 2.778 Cross-Ref:†† 778. The third elementary way of reasoning is deduction, of which the warrant is that the facts presented in the premisses could not under any imaginable circumstances be true without involving the truth of the conclusion, which is therefore accepted with necessary modality. But though it be necessary in its modality, it does not by any means follow that the conclusion is certainly true. When we are reasoning about purely hypothetical states of things, as in mathematics, and can make it one of our hypotheses that what is true shall depend only on a certain kind of condition--so that, for example, what is true of equations written in black ink would certainly be equally true if they were written in red--we can be certain of our conclusions, provided no blunders have been committed. This is "demonstrative reasoning." Fallacies in pure mathematics have gone undetected for many centuries. It is to ideal states of things alone--or to real states of things as ideally conceived, always more or

less departing from the reality--that deduction applies. The process is as follows, at least in many cases: Peirce: CP 2.778 Cross-Ref:†† We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular; but this is an opinion not founded on any systematic examination. If visual, it will either be geometrical, that is, such that familiar spatial relations stand for the relations asserted in the premisses, or it will be algebraical, where the relations are expressed by objects which are imagined to be subject to certain rules, whether conventional or experiential. This diagram, which has been constructed to represent intuitively or semi-intuitively the same relations which are abstractly expressed in the premisses, is then observed, and a hypothesis suggests itself that there is a certain relation between some of its parts--or perhaps this hypothesis had already been suggested. In order to test this, various experiments are made upon the diagram, which is changed in various ways. This is a proceeding extremely similar to induction, from which, however, it differs widely, in that it does not deal with a course of experience, but with whether or not a certain state of things can be imagined. Now, since it is part of the hypothesis that only a very limited kind of condition can affect the result, the necessary experimentation can be very quickly completed; and it is seen that the conclusion is compelled to be true by the conditions of the construction of the diagram. This is called "diagrammatic, or schematic, reasoning."

Peirce: CP 2.779 Cross-Ref:†† §2. VALIDITY †1

779. The possession by an argumentation or inference of that sort of efficiency in leading to the truth, which it professes to have; it is also said to be "valid." Peirce: CP 2.780 Cross-Ref:†† 780. Every argument or inference professes to conform to a general method or type of reasoning, which method, it is held, has one kind of virtue or another in producing truth. In order to be valid the argument or inference must really pursue the method it professes to pursue, and furthermore, that method must have the kind of truth-producing virtue which it is supposed to have. For example, an induction may conform to the formula of induction; but it may be conceived, and often is conceived, that induction lends a probability to its conclusion. Now that is not the way in which induction leads to the truth. It lends no definite probability to its conclusion. It is nonsense to talk of the probability of a law, as if we could pick universes out of a grab-bag and find in what proportion of them the law held good. Therefore, such an induction is not valid; for it does not do what it professes to do, namely, to make its conclusion probable. But yet if it had only professed to do what induction does (namely, to commence a proceeding which must in the long run approximate to the truth), which is infinitely more to the purpose than what it professes, it would have been valid. Validity must not be confounded with strength. For an argument may be perfectly valid and yet excessively weak. I wish to know whether a given coin is so accurately made that it will turn up heads and tails in approximately equal

proportions. I therefore pitch it five times and note the results, say three heads and two tails; and from this I conclude that the coin is approximately correct in its form. Now this is a valid induction; but it is contemptibly weak. All simple arguments about matters of fact are weak. The strength of an argument might be theoretically defined as the number of independent equal standard unit arguments upon the other side which would balance it. But since it is next to impossible to imagine independent arguments upon any question, or to compare them with accuracy, and since moreover the "other side" is a vague expression, this definition only serves to convey a rough idea of what is meant by the strength of an argument. It is doubtful whether the idea of strength can be made less vague. But we may say that an induction from more instances is, other things being equal, stronger than an induction from fewer instances. Of probable deductions the more probable conclusion is the stronger. In the case of hypotheses adopted presumptively on probation, one of the very elements of their strength lies in the absence of any other hypothesis; so that the above definition of strength cannot be applied, even in imagination, without imagining the strength of the presumption to be considerably reduced. Perhaps we might conceive the strength, or urgency, of a hypothesis as measured by the amount of wealth, in time, thought, money, etc., that we ought to have at our disposal before it would be worth while to take up that hypothesis for examination. In that case it would be a quantity dependent upon many factors. Thus a strong instinctive inclination towards it must be allowed to be a favouring circumstance, and a disinclination an unfavourable one. Yet the fact that it would throw a great light upon many things, if it were established, would be in its favour; and the more surprising and unexpected it would be to find it true, the more light it would generally throw. The expense which the examination of it would involve must be one of the main factors of its urgency. Peirce: CP 2.781 Cross-Ref:†† 781. Returning to the matter of validity, an argument professing to be necessary is valid in case the premisses could not under any hypothesis, not involving contradiction, be true, without the conclusion being also true. If this is so in fact, while the argument fails to make it evident, it is a bad argument rhetorically, and yet is valid; for it absolutely leads to the truth if the premisses are true. It is thus possible for an argument to be valid and yet bad. Yet an argument ought not to be called bad because it does not elucidate steps with which readers may be assumed to be familiar. A probable deductive argument is valid, if the conclusions of precisely such arguments (from true premisses) would be true, in the long run, in a proportion of times equal to the probability which this argument assigns to its conclusion; for that is all that is pretended. Thus, an argument that out of a certain set of sixty throws of a pair of dice about to be thrown, about ten will probably be doublets, is rendered valid by the fact that if a great number of just such arguments were made, the immense majority of the conclusions would be true, and indeed ten would be indefinitely near the actual average number in the long run. The validity of induction is entirely different; for it is by no means certain that the conclusion actually drawn in any given case would turn out true in the majority of cases where precisely such a method was followed; but what is certain is that, in the majority of cases, the method would lead to some conclusion that was true, and that in the individual case in hand, if there is any error in the conclusion, that error will get corrected by simply persisting in the employment of the same method. The validity of an inductive argument consists, then, in the fact that it pursues a method which, if duly persisted in, must, in the very nature of things, lead to a result indefinitely approximating to the truth in the long run. The validity of a presumptive adoption of a hypothesis for examination consists in this, that the hypothesis being such that its consequences are capable of being

tested by experimentation, and being such that the observed facts would follow from it as necessary conclusions, that hypothesis is selected according to a method which must ultimately lead to the discovery of the truth, so far as the truth is capable of being discovered, with an indefinite approximation to accuracy.

Peirce: CP 2.782 Cross-Ref:†† §3. PROOF †1

782. An argument which suffices to remove all real doubt from a mind that apprehends it. Peirce: CP 2.782 Cross-Ref:†† It is either mathematical demonstration; a probable deduction of so high probability that no real doubt remains; or an inductive, i.e., experimental, proof. No presumption can amount to proof. Upon the nature of proof see Lange, Logische Studien, who maintains that deductive proof must be mathematical; that is, must depend upon observation of diagrammatic images or schemata. Mathematical proof is probably accomplished by appeal to experiment upon images or other signs, just as inductive proof appeals to outward experiment.

Peirce: CP 2.783 Cross-Ref:†† §4. PROBABLE INFERENCE †2

783. Any inference which does not regard its own conclusion as being necessarily true (though the facts be as the premisses assert). Peirce: CP 2.783 Cross-Ref:†† In such an inference the facts asserted in the premisses are regarded as constituting a sign of the fact stated in the conclusion in one or other of three senses, as follows: i.e., that relation of the premissed facts to the concluded fact which is regarded as making the former a sign of the latter (1) may be such as could not exist until the conclusion was problematically recognized; this is inductive or experimental inference. Such a relation (2) may be altogether irrespective of whether the conclusion is recognized or not, yet such that it could not subsist if the concluded fact were not probable; this is probable deduction. Such a relation (3) may consist merely in the premissed facts having some character which may agree with, or be in some other relation to, a character which the concluded fact would possess if it existed; this is presumptive inference. Peirce: CP 2.784 Cross-Ref:†† 784. (1) The first case is that in which we begin by asking how often certain described conditions will, in the long run of experience, be followed by a result of a predesignate description; then proceeding to note the results as events of that kind present themselves in experience; and finally, when a considerable number of instances have been collected, inferring that the general character of the whole endless succession of similar events in the course of experience will be approximately of the character observed. For that endless series must have some character; and it

would be absurd to say that experience has a character which is never manifested. But there is no other way in which the character of that series can manifest itself than while the endless series is still incomplete. Therefore, if the character manifested by the series up to a certain point is not that character which the entire series possesses, still, as the series goes on, it must eventually tend, however irregularly, towards becoming so; and all the rest of the reasoner's life will be a continuation of this inferential process. This inference does not depend upon any assumption that the series will be endless, or that the future will be like the past, or that nature is uniform, nor upon any material assumption whatever. Peirce: CP 2.784 Cross-Ref:†† Logic imposes upon us two rules in performing this inference. The first is this: so far as in us lies, the conditions of the experience should remain the same. For we are reasoning exclusively from experience, that is, from the cognitions which the history of our lives forces upon us. So far as our will is allowed to interfere, it is not experience; so we must take pains that we do not, in taking the instances from which we are to reason, restrict the conditions or relax them from those to which the question referred. The second prescription of logic is that the conclusion be confined strictly to the question. If the instances examined are found to be remarkable in any other respect than that for which they were selected, we can draw no inference of the present kind from that. It would be merely an infinitely weaker inference of the third kind (below). The present kind of inference derives its great force from the circumstance that the result is virtually predicted. Peirce: CP 2.785 Cross-Ref:†† 785. (2) The second kind of probable inference is, by the definition of it, necessary inference. But necessary inference may be applied to probability as its subject-matter; and it then becomes, under another aspect, probable inference. If of an endless series of possible experiences a definite proportion will present a certain character (which is the sort of fact called an objective probability), then it necessarily follows that, foreseen or not, approximately the same proportion of any finite portion of that series will present the same character, either as it is, or when it has been sufficiently extended. This is governed by precisely the same principle as the inductive inference, but applied in the reverse way. The same prescriptions of logic apply as before; but, owing to that being now inferred which was in the other case a premiss, and conversely, it is not here true that the relation of the facts laid down in the premisses to the fact stated in the conclusion, which makes the former significant of the latter, requires the recognition of the conclusion. This is probable deduction. It covers all the ordinary and legitimate applications of the mathematical doctrine of probability. Peirce: CP 2.785 Cross-Ref:†† The legitimate results of the calculus of probability are of enormous importance, but others are unfortunately vitiated by confusing mere likelihood, or subjective probability, with the objective probability to which the theory ought to be restricted. An objective probability is the ratio in the long run of experience of the number of events which present the character of which the probability is predicated to the total number of events which fulfill certain conditions often not explicitly stated, which all the events considered fulfill. But the majority of mathematical treatises on probability follow Laplace in results to which a very unclear conception of probability led him. Laplace and other mathematicians, though they regard a probability as a ratio of two numbers, yet, instead of holding that it is the limiting

ratio of occurrences of different kinds in the course of experience, hold that it is the ratio between numbers of "cases," or special suppositions, whose "possibilities" (a word not clearly distinguished, if at all, from "probabilities") are equal in the sense that we are aware of no reason for inclining to one rather than to another. This is an error often appearing in the books under the head of "inverse probabilities." Peirce: CP 2.786 Cross-Ref:†† 786. (3) Probable inference of the third kind includes those cases in which the facts asserted in the premisses do not compel the truth of the fact concluded, and where the significant observations have not been suggested by the consideration of what the consequences of the conclusion would be, but have either suggested the conclusion or have been remarked during a search in the facts for features agreeable or conflicting with the conclusion. The whole argument then reduces itself to this, that the observed facts show that the truth is similar to the fact asserted in the conclusion. This may, of course, be reinforced by arguments of some other kind; but we should begin by considering the case in which it stands alone. As an example to fix ideas, suppose that I am reading a long anonymous poem. As I proceed, I meet with trait after trait which seems as if the poem were written by a woman. In what way do the premisses justify the acceptance of that conclusion, and in what sense? It does not necessarily, nor with any necessitated objective probability, follow from the premisses; nor must the method eventually lead to the truth. The only possible justifications which it might have would be that the acceptance of the conclusion or of the method might necessarily conduce, in the long run, to such attainment of truth as might be possible by any means, or else to the attainment of some other purpose. All these alternatives ought to be carefully examined by the logician in order that he may be assured that no mode of probable inference has been overlooked. Peirce: CP 2.786 Cross-Ref:†† It appears that there is a mode of inference in which the conclusion is accepted as having some chance of being true, and as being at any rate put in such a form as to suggest experimentation by which the degree of its truth can be ascertained. The only method by which it can be proved that a method, without necessarily leading to the truth, has some tolerable chance of doing so, is evidently the empirical, or inductive, method. Hence, as induction is proved to be valid by necessary deduction, so this presumptive inference must be proved valid by induction from experience. Peirce: CP 2.786 Cross-Ref:†† The presumptive conclusion is accepted only problematically, that is to say, as meriting an inductive examination. The principal rule of presumption is that its conclusion should be such that definite consequences can be plentifully deduced from it of a kind which can be checked by observation. Among the wealth of methods to which this kind of inference (perhaps by virtue of its experiential origin) gives birth, the best deserving of mention is that which always prefers the hypothesis which suggests an experiment whose different possible results appear to be, as nearly as possible, equally likely. Peirce: CP 2.787 Cross-Ref:†† 787. Among probable inferences of mixed character, there are many forms of great importance. The most interesting, perhaps, is the argument from Analogy, in which, from a few instances of objects agreeing in a few well-defined respects, inference is made that another object, known to agree with the others in all but one of those respects, agrees in that respect also.

Peirce: CP 2.788 Cross-Ref:†† §5. PREDESIGNATE †1

788. (A word formed by Sir W. Hamilton by composition from Lat. prae, in front of, and designatus, marked out): (1) A term applied by Hamilton to verbal propositions whose quantity, as universal or particular, is expressed (Lectures on Logic, xiii). Peirce: CP 2.789 Cross-Ref:†† 789. (2) By C. S. Peirce applied to relations, characters, and objects which, in compliance with the principles of the theory of probability, are in probable reasonings specified in advance of, or, at least, quite independently of, any examination of the facts. See Probable Inference [785]. Peirce: CP 2.790 Cross-Ref:†† 790. For example, the laws of England will, in the long run, cause the majority of English sovereigns to be males. In that sense it was unlikely that the successor of William IV would be a queen. But it would be absurd to say this after knowing that there was no heir to the crown so near as the Princess Victoria; and, in like manner, to say that it was not very unlikely that Queen Victoria's successor would be a queen was true enough as long as the character of her progeny was not known, or, if known was not taken account of, but false considering the number of her sons and grandsons. In such cases of deductive probable inference the necessity of the predesignation is too obvious to be overlooked. But in indirect statistical inferences, which are mere transformations of similar deductive consequences, and the validity of which, therefore, depends upon precisely the same conditions, the necessity of the predesignation is more often overlooked than remarked. Thus Macaulay, in his essay on the inductive philosophy, collects a number of instances of Irish whigs--which we may suppose constitute a random sample, as they ought, since they are to be used as the basis of an induction. By the exercise of ingenuity and patience, the writer succeeds in finding a character which they all possess, that of carrying middle names; whereupon he seems to think that an unobjectionable induction would be that all Irish whigs have middle names. But he has violated the rule, based on the theory of probabilities, that the character for which the samples are to be used as inductive instance must be specified independently of the result of that examination. Upon the same principle only those consequents of a hypothesis support the truth of the hypothesis which were predicted, or, at least, in no way influenced the character of the hypothesis. But this rule does not forbid the problematic acceptance of a hypothesis which has nothing to do with the theory of probability.

Peirce: CP 2.791 Cross-Ref:†† §6. PRESUMPTION †1

791. In logic: a more or less reasonable hypothesis, supported, it may be, by circumstances amounting all but to proof, or, it may be, all but baseless.

Peirce: CP 2.791 Cross-Ref:†† Logical or philosophical presumption is non-deductive probable inference which involves a hypothesis. It might very advantageously replace hypothesis in the sense of something supposed to be true because of certain facts which it would account for. See Probable Inference [786].

Peirce: CP 2.792 Cross-Ref:†† APPENDIX

MEMORANDA CONCERNING THE ARISTOTELIAN SYLLOGISM †1

792. The Quantity of Propositions is the respect in which Universal and Particular Propositions differ. The Quality of Propositions is the respect in which Affirmative and Negative Propositions differ.

NAMES AND SIGNS FOR PROPOSITIONS.

Universal Affirmative: A: Any S is

P.

Particular Affirmative: I: Some S is

P.

Universal Negative:

E: Any S is not P.

Particular Negative:

O: Some S is not P.

Terms occupying the places of S and P in the above, are called the logical Subject and Predicate.

Peirce: CP 2.793 Cross-Ref:†† RELATIONS OF PROPOSITIONS

793. In the following diagram, the different propositions are supposed to have the same logical Subject and Predicate. The lines connecting A with O, and E with I, are meant to indicate that these connected propositions contradict one another. The

sign [Click here to view] has its broad end towards a proposition which implies another, and its point toward the proposition implied.

[Click here to view]

Peirce: CP 2.794 Cross-Ref:†† RULE, CASE, AND RESULT.

794. A syllogism in the first figure argues from a Rule, and the subsumption of a Case, to the Result of that rule in that case.

Rule:

Any man is mortal,

Case:

Napoleon III is a man;

Result: .·. Napoleon III is mortal.

Peirce: CP 2.794 Cross-Ref:†† The Rule must be universal; and the Case affirmative. And the subject of the Rule must be the predicate of the Case. The Result has the quality of the Rule and the quantity of the Case; and has for its subject the subject of the Case, and for its predicate the predicate of the Rule.

THE THREE FIGURES.

Figure 1.

Assertion of Rule,

AE

Assertion of Case;

AI

Assertion of Result. E A O I

Figure 2.

Assertion of Rule,

A E

Denial of Result;

OIEA

Denial of Case.

O

E

Figure 3.

Denial of Result,

IOAE

Assertion of Case;

AI

Denial of Rule.

OI

Peirce: CP 2.794 Cross-Ref:†† The letters A, E, I, O, in the above diagram are so arranged that inferences can be made along the straight lines. Peirce: CP 2.795 Cross-Ref:†† 795. It is important to observe that the second and third figures are apagogical, that is, infer a thing to be false in order to avoid a false result which would follow from it. That which is thus reduced to an absurdity is a Case in the second figure, and a Rule in the third. Peirce: CP 2.795 Cross-Ref:†† To contrapose two terms or propositions is to transpose them, and at the same time substitute for each its contradictory. The second figure is derived from the first by the contraposition of the Case and Result, the third by the contraposition of the Rule and Result. The Rule and Case of the first figure cannot be contraposed, because they already occupy the same logical position, namely, that of a premiss; their contraposition in either of the other figures converts these figures into one another. Peirce: CP 2.795 Cross-Ref:†† Let F, S, T denote syllogisms of the first, second, and third figures, respectively. And let s, t, f denote the processes of contraposition of the Case and Result, Rule and Result, and Rule and Case, respectively. Then

sF = S sS = F tF = T tT = F

fS = T fT = S s2 = t2 = f2 = 1 f = st = ts s = ft = tf t = fs= sf

Peirce: CP 2.796 Cross-Ref:†† 796. The following table exhibits all the moods of Aristotelian syllogism (varieties resulting from variations of the Quantity and Quality of the propositions). Enter at the top, the proposition asserting or denying the rule; enter at the side, the proposition asserting or denying the case; find in the body of the table the proposition asserting or denying the result. In the body of the table, propositions indicated by italics belong to the first figure, those by black letter to the second figure, and those by script to the third figure.

[Click here to view]

Peirce: CP 2.796 Cross-Ref:†† Two moods of the third figure, namely, A A I and E A O, are omitted, for two reasons. The first is that they correspond by contraposition to two moods in the first figure, A A I and E A O, never given by logicians, who, therefore, act inconsistently in admitting these. The second reason is, that, like those moods in the first figure, they are virtually enumerated already, if the change of a proposition from universal to particular be not an inference; but if it be, then, again like those moods of the first figure, the argument they embody may be analyzed into a syllogism and an inference from universal to particular. Peirce: CP 2.797 Cross-Ref:†† 797. The celebrated lines of William Shyreswood (?) are here given. The vowels of the first three syllables of each word indicate the three propositions of the syllogisms. He enumerates, along with the moods of the first figure, the Theophrastean moods (two of which we omit for the same reason that we do those

two in the third figure):

Barbara: Celarent: Darii: Ferio: Baralipton: Celantes: Dabitis: Fapesmo: Frisesomorum: Cesare: Camestres: Festino: Baroco: Darapti: Felapton: Disamis: Datisi: Bocardo: Ferison.

Peirce: CP 2.798 Cross-Ref:†† 798. The diagram below shows the relations in which the second and third figures stand to the first. In order to understand the seven syllogistic formulas there set down, it is necessary to notice that propositions may be divided into four parts: first the Any or Some, second the Subject, third the is or is not, and fourth the Predicate. When a proposition admits of varieties in either of these parts, they are shown in the diagram by two words or letters, one above the other, as is/is not in the rule of the first figure. Two independent variations may occur in one formula, and the variations of different parts are independent, but in the same part either the upper or lower line must always be read, in any one syllogism. Peirce: CP 2.798 Cross-Ref:†† For example, the result in the first figure has four forms: any or some S is or is not P; but if Some has been read in the Case, Some must also be read in the Result. So, in the second figure, where a variation is possible in the quality of either premiss; but the same line of the third part of both propositions must be taken.

Figure 1

Figure 2

Figure 3

[Click here to view]

[Click here to view] [Click here to view]

Peirce: CP 2.798 Cross-Ref:†† At the top of the diagram are given the formulæ of the first figure, and of the second and third, as derived from that of the first by contraposition of the propositions. Under the second and third figures, respectively, are given forms expressing the same arguments in the first figure. It is necessary to study carefully the manner in which this reduction to the first figure is effected. Peirce: CP 2.799 Cross-Ref:†† 799. It will be perceived that the arrangements of the terms in the three figures, as determined by the rules given in 794, are as follows: where the first letter of each pair indicates the subject of a proposition of the syllogism and the second its predicate:

Figure 1. Figure 2. First.

{B} {A}

Second. {G} {B} Third.

{G} {A}

Figure 3.

{N} {M} {X} {M} {X} {N}

{S} {P} {S} {R} {R} {P}

Peirce: CP 2.800 Cross-Ref:†† 800. It is plain that there are two ways of transposing the arrangements of the terms of the second and third figures without removing a term from the conclusion, so as to give the term the same arrangement as that of the first figure. This is shown in

the following table, where the columns headed s show the propositions whose terms are to be transposed, while those headed m show the propositions to be transposed.†P1

------------------------------|

Figure 2. Figure 3. |

|-----------------------------| s | m | s | m | |

|

|

|

|Short Reduction| 1st |

| | 2d |

|

|Long Reduction | 2d 3d |2d 1st|1st 3d|1st 2d| -----------------------------------------------

Peirce: CP 2.800 Cross-Ref:†† The effect of these transpositions is here shown.

SECOND FIGURE

Short Reduction Long Reduction {N} {M}

{M} {N}

{M} {X}

{X} {M}

{X} {M}

{N} {M}

{X} {N}

{X} {N}

{N} {X}

THIRD FIGURE

Short Reduction Long Reduction {S} {P}

{S} {P}

{S} {R}

{S} {R}

{R} {S}

{P} {S}

{R} {P}

{R} {P}

{P} {R}

Peirce: CP 2.800 Cross-Ref:†† It must next be shown how these transpositions may be made, in syllogisms

themselves. Peirce: CP 2.801 Cross-Ref:†† 801. The short reduction of the second figure is shown in the second syllogism of that column of the large diagram headed Figure 2. The term not-P is introduced. This we define as that class to which some or any S belongs, when it is not P. Accordingly, for "some or any S is not P," we can substitute "some or any S is not-P," and this substitution is made in the reduction. But we cannot, on that account, substitute "any M is not-P" for "any M is not P." For "any M is not P," is substituted, in the reduction, "any P is not M;" and for "any M is P" is substituted "any not-P is not M." The only syllogisms by which these substitutions can be justified are these:

Any M is not P, Any P is .·.Any P is not M.

P;

Any

M is

P,

Any not-P is not P;

.·. Any not-P is not M.

Both these are syllogisms in the second figure.

Peirce: CP 2.802 Cross-Ref:†† 802. The short reduction of the third figure is shown in the second syllogism of the column headed Figure 3. The term some-S is introduced. The definition of this term is that it is that part of S which is or is not P when some S is or is not P. Hence, we can and do substitute "Any some-S is or is not P" for "Some S is or is not P," though we could not substitute "Any some-S is M" for "Some S is M." For "Some S is M" we substitute "Some M is S"; and for "Any S is M" we substitute "Some M is some-S"; and these substitutions are justified by inferences which can be expressed syllogistically only thus:

Any S is S, Some S is M; .·.Some M is S.

Some S is some-S, Any S is

M;

.·. Some M is some-S.

These are both syllogisms in the third figure.

Peirce: CP 2.803 Cross-Ref:†† 803. The long reduction of the second syllogism is shown in the third syllogism of the column headed Figure 2. Here not-P is defined as that class to which any M belongs which is not P. Hence we can substitute "Any M is not-P" for "Any M is not P." Some-S is defined as in the short reduction of the third figure. Hence, for "Some S is or is not P," we can say "Any Some-S is or is not P." Then, we use the inferences which are expressed syllogistically, thus:

Some

Some

Any S is not P,

Any P

is P;

Any S is P,

Any not P is not P;

Some S .·.Any P is not S

Some S

.·. Any not P is not S.

Peirce: CP 2.803 Cross-Ref:†† These are both syllogisms of the second figure. Substituting their conclusions for the second premiss of the second figure and transposing the premisses we obtain the premisses of the reduction. The conclusion of the reduction justifies that of the second figure, by inferences which are expressed syllogistically as follows:

Any M is not some-S, Some S is .·.Some S is not

M.

some-S;

Any M is not S, Any S is

S;

.·. Any S is not M.

Both these are syllogisms of the second figure.

Peirce: CP 2.804 Cross-Ref:†† 804. The long reduction of the third figure is shown in the third syllogism of the column headed Figure 3. Some S is here defined as that part of S which is M when some S is M. Hence, for "Some S is M," we can substitute "Any Some-S is M." Not-P is defined as in the short reduction of the second figure. Hence, in place of "Some or any S is not P," we can put "Some or any S is not-P." In place of "Some S is P or not-P" we again substitute "Some P or not-P is S," and in place of "Any S is P or not-P" we substitute "Some P or not-P is some-S," in virtue of inferences which are expressed syllogistically thus:

Any

S is

S,

Not P Some S is P;

Some S is some-S,

not P Any S is

P;

not P

not P

.·.Some P is S .·. Some P is some S.

These are syllogisms of the third figure.

Peirce: CP 2.804 Cross-Ref:†† Then, the premisses being transposed, we have the premisses of the reduction. The conclusion of the reduction justifies that of the third figure by inferences which are expressed syllogistically thus:

.·.Some

Any not-P is

P,

Some not-P is

M;

M is not-P.

Any P is P, Some P is M;

.·. Some M is P.

These are syllogisms of the third figure.

Peirce: CP 2.805 Cross-Ref:†† 805. The reduction called reductio per impossibile is nothing more than the repetition or inverse repetition of that contraposition by which the second and third figures have been obtained. It is not ostensive (that is, does not yield an argument with essentially the same premisses and conclusion as that of the argument thus to be reduced), but apagogical, that is, shows by the first figure that the contradiction of the conclusion of the second or third leads to the contradiction of one of the premisses. Contradiction arises from a difference in both quantity and quality. But it is to be observed that in the contraposition which gives the second figure, a change of the quality alone, and in that which gives the third figure, a change of the quantity alone of the contraposed propositions is sufficient. This shows that the two contrapositions are of essentially different kinds. The reductions per impossibile of the second and third figures respectively involve, therefore, these inferences:

Figure 2.

The Result follows from the Case; .·.The negative of the Case follows from the negative of the Result.

Figure 3.

The Result follows from the Rule; .·.The Rule changed in Quantity follows from the Result changed in Quantity.

These inferences may also be expressed thus:

Figure 2.

P; Whatever (S) is M is

not P;

not P .·.Whatever (S) is P is not M.

Figure 3.

Any some S is whatever (P or not-P) M is;

some S .·.Some M is whatever (P or not P) S is.

Peirce: CP 2.805 Cross-Ref:†† And if we omit the limitations in parentheses, which do not alter the essential nature of the inferences, we have:

Figure 2.

P; Any M is

not P

not P;

.·.Any P is not M.

Figure 3.

S Any some S is M;

Some S. .·.Some M

is S.

Peirce: CP 2.805 Cross-Ref:†† We have seen above that the former of these can only be reduced to a syllogism in the second figure, and the latter only to one in the third figure. Peirce: CP 2.806 Cross-Ref:†† 806. The ostensive reductions of each figure are also apagogical reductions of the other. There are also the following:

not S,

some S,

Any not-M is not some S,

not P

some P

Any P is not-M;

not P

Any some-M is S,

not S.

Any some not P is some-M;

some P

Any P is not some S.

some S.

Any some not P is

S.

Peirce: CP 2.806 Cross-Ref:†† But all these reductions involve the peculiar inferences we have found in those which have been examined, inasmuch as they are but complications of the latter. Peirce: CP 2.807 Cross-Ref:†† 807. Hence, it appears that no syllogism of the second or third figure can be reduced to the first, without taking for granted an inference which can only be expressed syllogistically in that figure from which it has been reduced. These inferences are not strictly syllogistic, because one of the propositions taken as a

premiss in the syllogistic expression is a logical fact. But the fact that each can only be expressed in the second or third figure of syllogism, as the case may be, shows that those figures alone involve the respective principles of those inferences. Hence, it is proved that every figure involves the principle of the first figure, but the second and third figures contain other principles, besides.

Peirce: CP 2.1 Fn 1 p 3 †1 "Intended Characters of this Treatise," ch. 1 of the "Minute Logic" (1902). Peirce: CP 2.1 Fn 2 p 3 †2 See 200. Peirce: CP 2.3 Fn 1 p 4 †1 See 186ff. Peirce: CP 2.4 Fn 2 p 4 †2 Cf. vol. 1, bk. IV, ch. 5. Peirce: CP 2.4 Fn 3 p 4 †3 See vol. 6, bk. II, ch. 7. Peirce: CP 2.4 Fn 4 p 4 †4 See 4.114ff. Peirce: CP 2.4 Fn 5 p 4 †5 See 6.304ff. Peirce: CP 2.4 Fn 6 p 4 †6 See 101ff. and bk. III passim. Peirce: CP 2.7 Fn 1 p 5 †1 For an extended discussion of this "Pragmatic" doctrine see vol. 5. Peirce: CP 2.7 Fn P1 p 5 Cross-Ref:†† †P1 The latter word is not, at least to one individual whom I wot of, particularly pleasing. The verb, normo, to square, is in the dictionary, but what ordinary reader of Latin can remember having met with it? Yet if the presumable motive for the substitution of the new adjective, namely, its avoidance of an apparent implication in "directive" that logic is a mere art, or practical science, approves itself to us, the twentieth century would laugh at us if we were too squeamish about the word's legitimacy of birth. [The Vocabulaire de la Philosophie, ed. by A. Lalande, vol. 2, p. 521, attributes the creation of this word, or at least its introduction into common speech, to Wundt. The Century Dictionary gets normative from normo, but not all the other standard works agree on this.] Peirce: CP 2.9 Fn 1 p 6 †1 Interesting papers on Benjamin Peirce are to be found in Benjamin Peirce, Biographical Sketch and Bibliography, Mathematical Association of America, (1925), Oberlin, Ohio. Peirce: CP 2.9 Fn 1 p 7

†1 Royce was originally specified, his name being deleted, apparently, at the suggestion of William James. A number of similar changes seem to have been made throughout this chapter. Peirce: CP 2.9 Fn 1 p 7 †1 See Preface to vol. 6 for Peirce's views regarding "scientific" metaphysics. Peirce: CP 2.11 Fn P1 p 8 Cross-Ref:†† †P1 "Mem. Before this goes to press I have to go over three books: 1. Barthélemy St. Hilaire's Ed. of Aristotle's Historia Animal; 2. Littré's Hippocrates; 3. The best German history of medicine"--marginal note. Peirce: CP 2.12 Fn 1 p 8 †1 See the Opus Majus, Parts I and VI. Peirce: CP 2.19 Fn 1 p 10 †1 Logic (1895), translated by H. Dendy. Peirce: CP 2.19 Fn 2 p 10 †2 Algebra der Logik. There is no fourth volume. Volume 2.2, which perhaps was what was meant, however, was published posthumously in 1905. Peirce: CP 2.19 Fn 3 p 10 †3 Cf. Logik, §3, 1. Peirce: CP 2.20 Fn 1 p 11 †1 See also 5.85ff. Peirce: CP 2.21 Fn 2 p 11 †2 Cf., e.g., La Logique, Paris (1855), vol. 2, pp. 196-7. Peirce: CP 2.22 Fn 1 p 12 †1 See, e.g., 769, 1.118 and 6.487ff. Peirce: CP 2.24 Fn P1 p 13 Cross-Ref:†† †P1 Lao-Tze's Tao-Teh-King, Paul Carus, Chicago (1898), ch. 52, sec. 3. Peirce: CP 2.26 Fn 1 p 13 †1 In his Aristotle, 2d ed. (1880), p. 259, and p. 562. Peirce: CP 2.26 Fn P1 p 14 Cross-Ref:†† †P1 Indeed this is precisely the position he takes in the Metaphysics Γ iii, 1005b, 19. Peirce: CP 2.27 Fn 1 p 14 †1 In Posterior Analytics I, ii. Peirce: CP 2.27 Fn 1 p 15 †1 Cf. Lewis Carroll's "What the Tortoise said to Achilles," Mind, N. S. vol. 4, p. 278. Peirce: CP 2.28 Fn 1 p 17 †1 Discourse on Method, pts. 2 and 4; Principles of Philosophy, I, 45, 46. Peirce: CP 2.28 Fn 2 p 17 †2 "Meditationes de Cognitione, Veritate et Ideis" ed. Gerhardt, vol. 4, p. 422;

transl. by G. M. Duncan in Philosophical Works of Leibniz, article III. Cf. also Discours de la Métaphysique, sec. 24 and Nouveaux Essais, II, ch. 29. Peirce: CP 2.29 Fn 1 p 18 †1 See, e.g., Herbert Spencer, First Principles, Appleton & Co., N.Y., 4th ed. (1882), pp. 34-36. Peirce: CP 2.29 Fn 2 p 18 †2 Cf. Logic, bk. II, ch. 5, sec. 6. Peirce: CP 2.29 Fn 3 p 18 †3 Though the text is unmistakable, this should be "true" and not "false." See below and 48. Peirce: CP 2.30 Fn 1 p 19 †1 The fact that part of all the integers can be put in one-to-one correspondence with all the integers was known long before Cantor, though not the same use was made of the information. See, e.g., Galileo's Mathematical Discourses, Weston's translation, p. 46; Renouvier, Année Philosophique (1868), p. 37. Peirce: CP 2.32 Fn 1 p 22 †1 1852-1919. Editor of the Open Court and the Monist. Peirce: CP 2.34 Fn 2 p 22 †2 And the highest of all possible aims is to further concrete reasonableness. See 1.602, 1.615, 5.121, 5.433. Peirce: CP 2.36 Fn 1 p 23 †1 See 1.487, 1.624, 1.625. Peirce: CP 2.37 Fn 2 p 23 †2 The 11th Scholarch of the Peripatetics, circa 70 B.C. Peirce: CP 2.37 Fn P1 p 23 Cross-Ref:†† †P1 "Organon" was first the name of the science, given to it by the early Peripatetics, because logic did not satisfy Aristotle's definitions either of Science or of Art. I fully accept the usual story about Aristotle's writings lying some centuries perdus until they were rescued and put in order first by Apellicon [circa 90 B.C., a famous book collector of Athens] and later by Andronicus. I consider the rejection of this story and consequent partial refusal to admit the authenticity of Aristotle's works as we have them, one of the extravagancies of "higher criticism." Peirce: CP 2.38 Fn 1 p 24 †1 Syntagma philosophicum, pt. I, (1658). Peirce: CP 2.38 Fn 2 p 24 †2 See T. Gomperz, Philodemi de ira Libra, (1864). Peirce: CP 2.38 Fn 3 p 24 †3 La Logique, (1712). Peirce: CP 2.38 Fn 4 p 24 †4 Logick, (1724). Peirce: CP 2.38 Fn 5 p 24 †5 La Logique, (1805) and Langue des Calculs, (1798).

Peirce: CP 2.38 Fn 6 p 24 †6 Logik oder Denklehre, (1806). Peirce: CP 2.38 Fn 7 p 24 †7 System der Logik (1823). Peirce: CP 2.38 Fn 8 p 24 †8 E.g., F. Hoffmann was a follower of Baader; Reinhold, Forberg, Schad were followers of Fichte; while Klein and Troxler were followers of Schelling. Peirce: CP 2.39 Fn 1 p 25 †1 E.g. G. B. Bülffinger, L. P. Thümming. Peirce: CP 2.39 Fn 2 p 25 †2 Critique of Pure Reason. B, viii. Peirce: CP 2.39 Fn 3 p 25 †3 Empirical Logic, (1889). Peirce: CP 2.42 Fn 1 p 26 †1 See 5.244ff. Peirce: CP 2.44 Fn 2 p 26 †2 See 227, 364, 422 and 428. Peirce: CP 2.45 Fn 1 p 27 †1 Association is treated in detail in vol. 8. Peirce: CP 2.46 Fn 2 p 27 †2 Vol. 1, bk. II, ch. 2. Peirce: CP 2.47 Fn 3 p 27 †3 In his Examination of Hamilton, ch. XXI. Peirce: CP 2.54 Fn 1 p 30 †1 In 27. Peirce: CP 2.56 Fn 1 p 31 †1 Peirce seems to have considered the construction of a logical machine. He cut out and arranged a number of overlapping papers to represent specific arguments; but it does not appear that he ever completed it or devised a key. In vol. 8 Peirce gives a detailed criticism of logical machines. Peirce: CP 2.61 Fn 1 p 34 †1 Logik, (1892), Bd. I, S. 9. Peirce: CP 2.62 Fn 2 p 34 †2 Cf. 205 Peirce: CP 2.64 Fn 1 p 35 †1 Ibid., S.10. Peirce: CP 2.66 Fn 1 p 36 †1 See 1.333. Peirce: CP 2.69 Fn 1 p 37

†1 A. H. Sayce, Introduction to the Science of Language (1880), vol. 2, p. 329. Peirce: CP 2.71 Fn 1 p 38 †1 Karl Pearson, in his Grammar of Science (1893), pp. 67-74. Peirce: CP 2.77 Fn 1 p 40 †1 Ch. 3 of the "Minute Logic" published in vol. 4, bk. I, as No. 7. Peirce: CP 2.77 Fn 2 p 40 †2 See vol. 4, bk. II. Peirce: CP 2.77 Fn 3 p 40 †3 Prior Analytics, 1, 2a. Peirce: CP 2.79 Fn 1 p 42 †1 The remainder of ch. 1 of the "Minute Logic." Peirce: CP 2.79 Fn 2 p 42 †2 Not only was this book never completed, but many of the proposed discussions here outlined were never begun. Peirce: CP 2.81 Fn 3 p 42 †3 E.g. Dedekind and Whitehead. Peirce: CP 2.82 Fn 1 p 43 †1 See ch. 4 of the "Minute Logic" published in vol. 1, bk. IV as ch. 2. Peirce: CP 2.84 Fn 1 p 44 †1 Vol. 1, bk. III contains a detailed study of the categories. Peirce: CP 2.84 Fn 1 p 45 †1 Cf. 3.93n, 3.611ff, 6.6. Peirce: CP 2.85 Fn 1 p 46 †1 Cf. 1.322. Peirce: CP 2.86 Fn 1 p 48 †1 See, for example, Die Mechanik, ch. II, vi, 6 and 9. Peirce: CP 2.87 Fn 1 p 49 †1 See, e.g., 1.298, 1.347. Peirce: CP 2.91 Fn 1 p 51 †1 See, e.g., Nomenclature and Divisions of Dyadic Relations, Paper XVIII, vol. 3 for an extended treatment of Dyads. Peirce: CP 2.95 Fn 1 p 52 †1 Icons can be only terms; indices can be only terms or propositions (dicisigns), while symbols can be all three. Peirce: CP 2.95 Fn 1 p 53 †1 Today the rhema, or rheme, is conventionally symbolized as φx and is called a propositional function. Peirce: CP 2.96 Fn P1 p 53 Cross-Ref:†† †P1 The reader should refer to the definitions of Index, Icon, and Symbol [in

92]. Peirce: CP 2.99 Fn 1 p 56 †1 "The Fixation of Belief" (1877) and "How to Make Our Ideas Clear" (1878), chs. 4 and 5 of bk. II, vol. 5. The maxim is first stated in the latter paper. Peirce: CP 2.99 Fn 2 p 56 †2 i.e., in 1902-3. Peirce: CP 2.100 Fn 3 p 56 †3 In bk. III. Peirce: CP 2.101 Fn 1 p 57 †1 Bk. III, B. Peirce: CP 2.101 Fn 2 p 57 †2 E.g. Laplace and Quetelet. Peirce: CP 2.101 Fn 3 p 57 †3 See his Logic of Chance, (1866) and Empirical Logic (1889). Peirce: CP 2.101 Fn 4 p 57 †4 See Keynes' Treatise on Probability for a bibliography of Edgeworth's writings. Peirce: CP 2.102 Fn 1 p 58 †1 See, e.g., bk. III, ch. 9. Peirce: CP 2.102 Fn 2 p 58 †2 Bk. III, ch. 8. Peirce: CP 2.102 Fn 1 p 59 †1 That part of the "Minute Logic" was not written. But see vol. 5, bk. I, ch. 7. Peirce: CP 2.105 Fn 1 p 60 †1 There is no systematic treatment of this subject. Remarks on the conditions of research and the principles of discovery are scattered through the volumes. Peirce: CP 2.108 Fn 1 p 61 †1 Cf. 3.364. Peirce: CP 2.108 Fn 2 p 61 †2 See vol. 1, bk. II, ch. 2, § 1-2. Peirce: CP 2.111 Fn 1 p 62 †1 The closest approach to this subject is to be found in vol. 6, bk. I, ch. 7. But see 4.80. Peirce: CP 2.113 Fn 1 p 64 †1 See Molière's Le Malade imaginaire. Peirce: CP 2.118 Fn 1 p 66 †1 Cf. vol. 1, bk. IV. Peirce: CP 2.119 Fn 1 p 67 †1 Second section of the second chapter, "Pre-logical Notions" of the "Minute Logic." For the first section, "The Classification of the Sciences," see vol. 1, bk. II,

ch. 2. Peirce: CP 2.135 Fn 1 p 71 †1 See 1.172. Peirce: CP 2.141 Fn P1 p 74 Cross-Ref:†† †P1 I speak here too much of myself personally. Many men and women have imaginations resembling percepts up to the point of being mistaken for them. In others imagination is less vivid, down to my point, who see no resemblance between an imagination and a percept except that somehow they can be compared. Peirce: CP 2.141 Fn P2 p 74 Cross-Ref:†† †P2 For some people, however, this stenographic report seems to be illustrated with photographs. I can only adequately describe my own experience. Peirce: CP 2.141 Fn P1 p 75 Cross-Ref:†† †P1 This I suppose would be true even for the most vivid imaginations. Peirce: CP 2.142 Fn P2 p 75 Cross-Ref:†† †P2 Personally, until old age began to mark its effects, I hardly had any pictorial dreams; and even now they are mere scraps of images to which incongruous abstract ideas are attached. Peirce: CP 2.142 Fn P1 p 76 Cross-Ref:†† †P1 Personally, I never had anything like a hallucination except in the delirium of fever. Peirce: CP 2.144 Fn P2 p 76 Cross-Ref:†† †P2 I here use the word "criticize" in the philosophical sense. Criticism proper, literary criticism, does not necessarily approve or disapprove. Peirce: CP 2.148 Fn 1 p 79 †1 442. Peirce: CP 2.151 Fn 1 p 82 †1 Cf. 5.85ff. Peirce: CP 2.151 Fn 2 p 82 †2 Cf. 5.3. Peirce: CP 2.152 Fn 1 p 83 †1 E.g., Boole, De Morgan, Whewell, J. S. Mill, Jevons, Venn, Pearson, MacColl. Peirce: CP 2.152 Fn 2 p 83 †2 E.g., Sigwart, Wundt, Schuppe, Erdmann, Bergmann, Glogau, Husserl. Peirce: CP 2.163 Fn 1 p 95 †1 No such list has been found. Peirce: CP 2.166 Fn 1 p 97 †1 See, e.g., Schröder's Algebra der Logik, 3.1; and the Principia Mathematica, *23ff. Peirce: CP 2.168 Fn 1 p 99 †1 In his Logica, e.g., in I, xiv.

Peirce: CP 2.176 Fn 1 p 104 †1 Cf. 5.234. Peirce: CP 2.177 Fn 1 p 105 †1 Cf. 1.672. Peirce: CP 2.191 Fn 1 p 111 †1 See 614. Peirce: CP 2.196 Fn 1 p 114 †1 Mathematics is dealt with in the next chapter of the "Minute Logic," published in vol. 4, bk. I, as no. VII. Peirce: CP 2.197 Fn 2 p 114 †2 See vol. 1, bk. III. Peirce: CP 2.197 Fn 1 p 115 †1 A classmate--"a noble-hearted, sterling-charactered young gentleman . . . almost the only real companion I have ever had." Peirce: CP 2.197 Fn 2 p 115 †2 Only three and a half chapters of the "Minute Logic" were written, of which none is devoted to esthetics. Peirce: CP 2.198 Fn 3 p 115 †3 See vol. 1, bk. IV and vol. 5, bk. I. Peirce: CP 2.203 Fn 1 p 119 †1 Dictionary of Philosophy and Psychology (1901-2, 1911), edited by J. M. Baldwin, Macmillan, New York, vol. 2, pp. 20-23, by Peirce and Mrs. Ladd-Franklin. Peirce: CP 2.204 Fn 1 p 120 †1 773ff. Peirce: CP 2.205 Fn 2 p 120 †2 Life of Aristotle, bk. V, ch. 13. Peirce: CP 2.205 Fn 1 p 121 †1 Cf. Politicus, 260. Peirce: CP 2.205 Fn 2 p 121 †2 Cf. Critik der Reinen Vernunft, 2te Auflage, Einleitung VII. Peirce: CP 2.219 Fn 1 p 129 †1 Syllabus of Certain Topics of Logic (1903), pp. 10-14, Alfred Mudge & Son, Boston, Continuing 1.202. Cf. 5.413, 5.502, and vol. 5, Appendix §4. Peirce: CP 2.227 Fn 1 p 134 †1 From an unidentified fragment, c. 1897. Peirce: CP 2.228 Fn 1 p 135 †1 E.g., in 1.551. Peirce: CP 2.230 Fn 1 p 136 †1 From "Meaning," 1910.

Peirce: CP 2.233 Fn 1 p 138 †1 §§ 3-10 are from "Nomenclature and Divisions of Triadic Relations, as far as they are determined,"--a manuscript continuation of the "Syllabus," c. 1903. See note to vol. 1, Bk. II, ch. 1, and 3.608n. Peirce: CP 2.235 Fn 1 Para 1/2 p 139 †1 On Peirce's principle that possibilities determine only possibilities and laws are determined only by laws, the terms "First Correlate" and "Third Correlate" should be interchanged in 235-38. In this way one secures, in harmony with other writings, the ten classes mentioned in 238. They are as follows: If the Third Correlate is a possibility, then

First (I)1. Possibility

Second

Third

Possibility

Possibility

(II)2. Existent (III)3. Existent

Possibility

Possibility

Existent

Possibility

(V)4. Law

Possibility

Possibility

(VI)5. Law

Existent

Possibility

Law

Possibility

(VIII)6. Law

If the Second is an existent, then also (IV)7. Existent

Existent

Existent

(VII)8. Law

Existent

Existent

If the First is a law, then also (IX)9. Law

Law

Existent

(X)10. Law

Law

Law

By 242 and 274 the Representamen, Object, and Interpretant are the first, second, and third correlate respectively, while by 243ff. the representamen in itself, in relation to its object, and as interpreted, is the first, second, and third correlate respectively. The former division yields ten trichotomies and sixty-six classes of signs, the latter three trichotomies and ten classes of signs. Peirce: CP 2.235 Fn 1 Para 2/2 p 139 The bracketed roman numerals in the above table give the order of discussion in §7 and the designations in the table in 264. See also 243n. Peirce: CP 2.237 Fn 1 p 140 †1 The truth of this last clause is to be seen from case numbered 5 above. The truth of the rest of the proposition is to be seen from cases numbered 1, 7, and 10. Peirce: CP 2.238 Fn 2 p 140

†2 The three ways are given in the notes to 243. Peirce: CP 2.239 Fn 3 Para 1/2 p 140 †3 Though Peirce has laid down the condition that a dyadic relation to be an existent requires both its correlates to be existents (cf. 283), he does not seem ever to have given the conditions involved in determining a dyadic relation to be of the nature of a law. In fact, his usual view is that there are no such dyadic relations. However, what seems to be meant here is that a dyadic relation is of the nature of law if both its correlates are laws. If, in addition, we accept the unstated propositions that a dyadic relation is a possibility if one correlate is a possibility, while a dyadic relation is an existent if one correlate is an existent and the other a law, we should get the following table: At least one dyadic relation of the nature of possibility:

First

Second

Third

1. Possibility-------------Possibility---------------Possibility |_____________________________________________| 2. Existent----------------Possibility---------------Possibility |_______________________________________________|

3. Existent......2.........Existent------------------Possibility |_______________________________________________|

4. Law--------------------Possibility----------------Possibility |___________________________________________________|

5. Law..........2.........Existent-------------------Possibility

|___________________________________________________|

6. Law..........3.........Law------------------------Possibility |___________________________________________________|

Those which have at least two existent dyadic relations:

7. Existent---------------Existent------------------Existent

|_____________________________________________|

8. Law--------------------Existent------------------Existent |____________________________________________ ___|

9. Law..........3.........Law-----------------------Existent |_________________________________________________|

All dyadic relations are laws:

10. Law-------------------Law-----------------------Law |_______________________________________________|

Peirce: CP 2.239 Fn 3 Para 2/2 p 140 The black lines between the correlates are marks of the presence of the specified relation; the ".....2....." and ".....3....." stand for existential and rational dyadic relations respectively. Peirce: CP 2.240 Fn 1 p 141 †1 I.e., 5 has its correlates all of different natures; 1, 7, 10 have their correlates all of the same nature; and the rest have two and only two correlates of the same nature; while 1, 2, 4, 7, 10, have all the dyadic relations of the same nature and 3, 5, 6, 8, 9, have only two of the same nature. Peirce: CP 2.241 Fn 2 p 141 †2 In 1-6, the third correlate is determined by the first to have a quality; in 7-9, it is determined to have an existential relation to the second, and in 10 it is determined to have a relation of thought to the second for another correlate. Peirce: CP 2.243 Fn 1 Para 1/2 p 142 †1 Peirce later (c. 1906, see e.g., 1.291, 4.530) discovered that there are ten trichotomies and sixty-six classes of signs. The analysis of the additional divisions was never satisfactorily completed; the best statement of them is to be found in the letters to Lady Welby, vol. 9. The present book, it is believed, contains most of Peirce's most thorough and authoritative work on signs. Peirce: CP 2.243 Fn 1 Para 2/2 p 142 The ten classes of signs derived from the three trichotomies here given are diagrammatically presented by Peirce in 264. If "Representamen," "Representamen as related to object," and "Interpreted Representamen" be substituted for first, second, and third correlate respectively, the tables of 235n and 239n should prove helpful schemata in §4-§7. The present section treats of the firstness, secondness, and thirdness of the Representamen.

Peirce: CP 2.243 Fn 2 p 142 †2 If we make the suggested substitutions we get the three groups consisting of: I; II, III, IV; and V-X. Peirce: CP 2.243 Fn 3 p 142 †3 I.e., I, II, V; III, IV, VI, VII; VIII, IX, X. Peirce: CP 2.243 Fn 4 p 142 †4 I.e., The three groups of 241n--1-6, 7-9; 10 i.e., I, II, III, V, VI, VIII; IV, VII, IX; X. Peirce: CP 2.250 Fn 1 p 144 †1 See the second note to 95. Peirce: CP 2.252 Fn 1 p 145 †1 See 315. Peirce: CP 2.253 Fn 1 p 146 †1 See 582-3. Peirce: CP 2.264 Fn 1 p 150 †1 See 235n and 243n for explanation of the roman numerals. Peirce: CP 2.271 Fn 1 p 154 †1 See ch. 1. Peirce: CP 2.273 Fn 1 p 155 †1 Dictionary of Philosophy and Psychology, vol. 2, p. 464. Peirce: CP 2.273 Fn 2 p 155 †2 303-4. Peirce: CP 2.274 Fn 1 p 156 †1 274-7, 283-4, 292-4 are from "Syllabus," c. 1902, no part of which was ever published (cf. note to ch. 1). 278-80 are from "That Categorical and Hypothetical Propositions are one in essence, with some connected matters," c. 1895, following 339; 281, 285, 297-302 are from chapter 2 of "The Art of Reasoning," c. 1895, while 282, 286-91 and 295-6 are from "The Short Logic," c. 1893. Peirce: CP 2.283 Fn 1 p 160 †1 "Seme" is usually reserved for indexical dicisigns which are only a subclass of the indices. Peirce: CP 2.287 Fn 1 p 162 †1 New Latin Grammar, p. 131n (ed. 1884). Peirce: CP 2.287 Fn P1 p 163 Cross-Ref:†† †P1 Modern grammars define a pronoun as a word used in place of a noun. That is an ancient doctrine which, exploded early in the thirteenth century, disappeared from the grammars for several hundred years. But the substitute employed was not very clear; and when a barbarous rage against medieval thought broke out, it was swept away. Some recent grammars, as Allen and Greenough's, set the matter right again. There is no reason for saying that I, thou, that, this, stand in place of nouns; they indicate things in the directest possible way. It is impossible to express what an assertion refers to except by means of an index. A pronoun is an

index. A noun, on the other hand, does not indicate the object it denotes; and when a noun is used to show what one is talking about, the experience of the hearer is relied upon to make up for the incapacity of the noun for doing what the pronoun does at once. Thus, a noun is an imperfect substitute for a pronoun. Nouns also serve to help out verbs. A pronoun ought to be defined as a word which may indicate anything to which the first and second persons have suitable real connections, by calling the attention of the second person to it. Allen and Greenough say, "pronouns indicate some person or thing without either naming or describing" [p. 128, edition of 1884]. This is correct--refreshingly correct; only it seems better to say what they do, and not merely what they don't. Peirce: CP 2.290 Fn P1 p 164 Cross-Ref:†† †P1 If a logician had to construct a language de novo--which he actually has almost to do--he would naturally say, I shall need prepositions to express the temporal relations of before, after, and at the same time with, I shall need prepositions to express the spatial relations of adjoining, containing, touching, of in range with, of near to, far from, of to the right of, to the left of, above, below, before, behind, and I shall need prepositions to express motions into and out of these situations. For the rest, I can manage with metaphors. Only if my language is intended for use by people having some great geographical feature related the same way to all of them, as a mountain range, the sea, a great river, it will be desirable to have prepositions signifying situations relatively to that, as across, seaward, etc. But when we examine actual languages, it would seem as though they had supplied the place of many of these distinctions by gestures. The Egyptians had no preposition nor demonstrative having any apparent reference to the Nile. Only the Esquimos are so wrapped up in their bearskins that they have demonstratives distinguishing landward, seaward, north, south, east, and west. But examining the cases or prepositions of any actual language we find them a haphazard lot. Peirce: CP 2.291 Fn P1 p 165 Cross-Ref:†† †P1 The nomenclature of grammar, like that of logic, is derived chiefly from a late Latin, the words being transferred from the Greek, the Latin prefix translating the Greek prefix and the Latin stem the Greek stem. But while the logical words were chosen with fastidious care, the grammarians were excessively careless, and none more so than Priscian. The word indicative is one of Priscian's creations. It was evidently intended to translate Aristotle's {apophantiké}. But this is precisely equivalent to declarative both in signification and according to the rules of transferrence, de, taking the place of {apo} as is usual in these artificial formations (demonstration for {apodeixis}, etc.), and clarare representing {phainein} to make clear. Perhaps the reason Priscian did not choose the word declaratiuus was that Apuleius [see Prantl's Geschichte der Logik, I, 581], a great authority on words, had used this in a somewhat different sense. Peirce: CP 2.293 Fn P1 p 166 Cross-Ref:†† †P1 There are two ways in which a Symbol may have a real Existential Thing as its real Object. First, the thing may conform to it, whether accidentally or by virtue of the Symbol having the virtue of a growing habit, and secondly, by the Symbol having an Index as a part of itself. But the immediate object of a symbol can only be a symbol and if it has in its own nature another kind of object, this must be by an endless series. Peirce: CP 2.297 Fn 1 p 168

†1 De Interpretatione, II, 16a, 12. Peirce: CP 2.301 Fn 1 p 169 †1 Cf. Tractatus Logicæ, I, xiv. Peirce: CP 2.303 Fn 2 p 169 †2 Dictionary of Philosophy & Psychology, vol. 2, p. 527. Peirce: CP 2.305 Fn 1 p 170 †1 Ibid., vol. 1, pp. 531-2. Peirce: CP 2.307 Fn 1 p 172 †1 Ibid., vol. 2, p. 640. Peirce: CP 2.308 Fn 2 p 172 †2 Ibid., vol. 2, pp. 691-2. Peirce: CP 2.309 Fn 1 p 174 †1 §§1-4 are from "Syllabus," c. 1902, continuing 294. Peirce: CP 2.309 Fn P1 p 174 Cross-Ref:†† †P1 To explain the judgment in terms of the "proposition" is to explain it by that which is essentially intelligible. To explain the proposition in terms of the "judgment" is to explain the self-intelligible in terms of a psychical act, which is the most obscure of phenomena or facts. Peirce: CP 2.315 Fn P1 p 178 Cross-Ref:†† †P1 But if anybody prefers a form of analysis which gives more prominence to the unquestionable fact that a proposition is something capable of being assented to and asserted, it is not my intention to make any objection to that. I do not think my analysis does put quite the emphasis on that that it justly might. Peirce: CP 2.315 Fn 1 p 179 †1 If "some" be taken to involve the existence of what it quantifies, then I and O propositions of non-existents must both be false; by the square of opposition both E and A would then be true, so that all universals, whether affirmative or negative, are true of the non-existent. See also 324, 327, 369. Peirce: CP 2.315 Fn 1 p 180 †1 See vol. 4, bk. II. Peirce: CP 2.316 Fn P1 p 181 Cross-Ref:†† †P1 Conditional is the right appellation, and not hypothetical, if the rules of the author's Ethics of Philosophical Terminology [bk. II, ch. 1] are to be followed. The meaning of {hypothetikos} was quite unsettled with the Greeks; but the word seems ultimately to have come to be applied to any compound proposition; and so Apuleius, under Nero, uses the translation conditionalis; saying, "Propositionum igitur, perinde ut ipsarum conclusionum, duae species sunt: altera praedicativa, quae etiam simplex est; ut si dicamus, qui regnat, beatus est: altera substitutiva, vel conditionalis, quae etiam composita est; ut si aias: qui regnat, si sapit, beatus est. Substituis enim conditionem, qua, nisi sapiens est, non sit beatus." [See Prantl's Geschichte der Logik, I, 580-581.] But as early as Boëthius and Cassiodorus, that is, about A.D. 500, it was settled that hypothetica applies to any compound proposition, and conditionalis to a proposition asserting one thing only in case a condition set forth in a separate clause be fulfilled. This was the universally accepted use of the

terms throughout the middle ages. Therefore, hypotheticals should have been divided into disjunctives and copulatives. They were usually divided into conditionals, disjunctives, and copulatives. But conditionals are really only a special kind of disjunctives. To say, "If it freezes tonight, your roses will be killed" is the same as to say, "It either will not freeze, or your roses will tonight be killed." A disjunctive does not exclude the truth of both alternatives, at once [cf. 345-7]. Peirce: CP 2.317 Fn P1 p 182 Cross-Ref:†† †P1 Mill's term connote is not very accurate. Connote properly means to denote along with in a secondary way. Thus "killer" connotes a living thing killed. When the scholastics said that an adjective connoted, they meant it connoted the abstraction named by the corresponding abstract noun. But the ordinary use of an adjective involves no reference to any abstraction. The word signify has been the regular technical term since the twelfth century, when John of Salisbury (Metalogicus, II, xx) spoke of "quod fere in omnium ore celebre est, aliud scilicet esse quod appellativa (i.e., adjectives) significant, et aliud esse quod nominant. Nominantur singularia (i.e., existent individual things and facts), sed universalia (i.e., Firstnesses) significantur." See my paper of Nov. 13, 1867 [next chapter], to which I might now [1902] add a multitude of instances in support of what is here said concerning connote and signify. Peirce: CP 2.323 Fn P1 p 185 Cross-Ref:†† †P1 The Summulae Logicales of Petrus Hispanus, which Prantl [Geschichte der Logik, II, 266ff, a writer of little judgment and over-rated learning, whose useful history of Logic is full of blunders, misappreciations, and insensate theories, and whose own Billingsgate justifies almost any tone toward him, absurdly maintains that this book was substantially translated from a Greek book, which is manifestly from the Latin. The Summulae of Petrus Hispanus are nearly identical with some other contemporary works and evidently show a doctrine which had been taught in the schools from about A.D. 1200. After Boëthius, it is the highest authority for logical terminology, according to the present writer's ethical views. Peirce: CP 2.323 Fn 1 p 186 †1 I.e., it can be stated in terms of a material or Philonian implication. See 348n. Peirce: CP 2.324 Fn 2 p 186 †2 Prantl, op. cit., I, 581. Peirce: CP 2.328 Fn 1 p 187 †1 Prantl, op. cit., I, 696. Peirce: CP 2.328 Fn 1 p 188 †1 {Horon de kalö eis hon dialyetai é protasis oion to te katégoroumenon kai to kath ou katégoreitai}, says Aristotle 24b.16. Peirce: CP 2.328 Fn 2 p 188 †2 Prantl, op. cit., II, 197. Peirce: CP 2.328 Fn 3 p 188 †3 See also 3.459. Peirce: CP 2.331 Fn 1 p 190 †1 See Prantl, op. cit., II, 272.

Peirce: CP 2.332 Fn 2 p 190 †2 §5 and §6 are from "That Categorical and Hypothetical Propositions are one in essence, with some connected matters," c. 1895. Peirce: CP 2.332 Fn 3 p 190 †3 "On a New List of Categories," vol. 1, bk. III, ch. 6. Peirce: CP 2.339 Fn 1 p 193 †1 See 1.567 for a definition of this term. Peirce: CP 2.340 Fn 1 p 194 †1 1.559. Peirce: CP 2.345 Fn 1 p 196 †1 Cf. Schröder, Logik, §28. Peirce: CP 2.345 Fn 2 p 196 †2 E.g., in his Formal Logic, ch. 4, and his Syllabus, §21ff. See 366. Peirce: CP 2.345 Fn 1 p 197 †1 See Studies in Logic, edited by C. S. Peirce, Little, Brown and Co., Boston, 1883. "On the Algebra of Logic," by Christine Ladd, p. 61ff. Peirce: CP 2.345 Fn 2 p 197 †2 Fabian Franklin, "A Point of Logical Notation," Johns Hopkins University Circular, p. 131, April 1881. Peirce: CP 2.348 Fn 1 p 199 †1 A Philonian is one who defines implication "materially" i.e., one who takes "P implies Q" to mean the same as "Not P or Q." There is a reference to the controversy between Philo the Megarian, Diodorus Cronus, and Chrysippus on this point in Cicero's Acad. Quaest. II, 143; the issues between Philo and Diodorus are mentioned also in Sextus Empiricus, Adv. Math. VIII, 113-17. See also 3.441ff. Peirce: CP 2.349 Fn 2 p 199 †2 On the Algebra of Logic," vol. 3, no. VI. Peirce: CP 2.349 Fn 1 p 200 †1 Vol. 3, nos. XII and XIII, §3. Peirce: CP 2.349 Fn 2 p 200 †2 In Algebra der Logik. Peirce: CP 2.352 Fn 1 p 201 †1 See also 618. Peirce: CP 2.352 Fn 2 p 201 †2 I.e., it means both p and not-p. See 383. Peirce: CP 2.354 Fn 1 p 202 †1 I.e., for every individual it holds that if it is human, it is mortal; or, for all occasions it holds that what is human is mortal. Peirce: CP 2.356 Fn 1 p 203 †1 See 3.18.

Peirce: CP 2.356 Fn 2 p 203 †2 See 3.47n. Peirce: CP 2.356 Fn 1 p 204 †1 A number of mss. on this topic have been found. They contain nothing not easily derived from the discussions in volume 3 and volume 4. Peirce: CP 2.356 Fn 2 p 204 †2 I.e., it is equivalent to: not-a or not-a or not-a . . . Peirce: CP 2.357 Fn 1 p 205 †1 Dictionary of Philosophy and Psychology, vol. 2, pp. 609-10. Peirce: CP 2.357 Fn 1 p 208 †1 Formal Logic, ch. 8. Peirce: CP 2.358 Fn 2 p 208 †2 Dictionary of Philosophy and Psychology, vol. 2, pp. 325-6. Peirce: CP 2.358 Fn 3 p 208 †3 See 378-80. Peirce: CP 2.359 Fn 1 p 209 †1 Ibid., vol. 2, pp. 326-9. Peirce: CP 2.360 Fn 2 p 209 †2 305-6. Peirce: CP 2.361 Fn 1 p 211 †1 See Prantl, op. cit., III, 279. Peirce: CP 2.361 Fn 2 p 211 †2 Super Universalia Porphyrii, qu. XIV. Peirce: CP 2.361 Fn 3 p 211 †3 Ibid. Peirce: CP 2.362 Fn 1 p 212 †1 Dictionary of Philosophy and Psychology, vol. 2, pp. 410-12. Peirce: CP 2.363 Fn 2 p 212 †2 Cf. vol. 4, bk. I, no. 4. Peirce: CP 2.364 Fn 1 p 213 †1 De divisione naturae IV, 4. Peirce: CP 2.364 Fn 2 p 213 †2 Metalogicus II, xx. Peirce: CP 2.364 Fn 3 p 213 †3 An Outline of the Necessary Laws of Thought (1842), §§52, 54, 80. Peirce: CP 2.364 Fn 4 p 213 †4 An Elementary Treatise on Logic, (1856), I, ii, §5. Peirce: CP 2.364 Fn 5 p 213

†5 418. Peirce: CP 2.364 Fn 1 p 214 †1 See Prantl, op. cit., I, 581. Peirce: CP 2.365 Fn 1 p 215 †1 Lectures on Logic, XIII, pp. 243-48. Peirce: CP 2.365 Fn 2 p 215 †2 An Essay on the New Analytic of Logical Forms, (1850). Peirce: CP 2.366 Fn 3 p 215 †3 Syllabus of a Proposed System of Logic (1860). §21ff. See also 568. Peirce: CP 2.366 Fn 1 p 216 †1 Ibid., §165. Peirce: CP 2.367 Fn 2 p 216 †2 Dictionary of Philosophy and Psychology, vol. 2, pp. 737-40: 367-9 are by Peirce alone; 370-1, given in part only, are by Peirce and Mrs. C. Ladd-Franklin. Peirce: CP 2.367 Fn 3 p 216 †3 Summulae, Tractatus II, p. 87C. Peirce: CP 2.367 Fn 4 p 216 †4 De Praedicab. II, 1, p. 11A. Peirce: CP 2.369 Fn 5 p 216 †5 Lectures on Logic, App. V (d), (3). Peirce: CP 2.369 Fn 1 p 217 †1 Cf. Nouveaux Essais, bk. IV, ch. 9. Peirce: CP 2.370 Fn 1 p 220 †1 Cf. Nouveaux Essais, Avant-Propos. Peirce: CP 2.372 Fn 1 p 221 †1 Dictionary of Philosophy and Psychology, vol. 2, pp. 265-6, by Peirce and Mrs. C. Ladd-Franklin. Peirce: CP 2.374 Fn 1 p 222 †1 Ibid., vol. 2, pp. 408-9. Peirce: CP 2.375 Fn 1 p 223 †1 Essay, II, viii, 8. Peirce: CP 2.376 Fn 1 p 224 †1 See vol. 3, no. XX, §8. Peirce: CP 2.378 Fn 2 p 224 †2 Dictionary of Philosophy and Psychology, vol. 2, pp. 146-7, by Peirce and Mrs. C. Ladd-Franklin. Peirce: CP 2.379 Fn 1 p 225 †1 See vol. 5, Introduction. Peirce: CP 2.379 Fn 1 p 227

†1 See Dictionary of Philosophy and Psychology, vol. 2, p. 369ff. Peirce: CP 2.381 Fn 1 p 228 †1 Ibid., vol. 2, pp. 6-7. Peirce: CP 2.381 Fn 2 p 228 †2 See Prantl, op. cit., I, 693. Peirce: CP 2.381 Fn 3 p 228 †3 See e.g., Formal Logic, p. 37ff. Peirce: CP 2.381 Fn 4 p 228 †4 But cf. Wolff's Logica, §208ff. Peirce: CP 2.382 Fn 1 p 229 †1 Dictionary of Philosophy and Psychology, vol. 2, pp. 89-93. Peirce: CP 2.382 Fn 2 p 229 †2 See Prantl, op. cit., II, 158f. Peirce: CP 2.386 Fn 1 p 233 †1 Bd. II, S. 127. Peirce: CP 2.387 Fn 1 p 234 †1 Cf. Logik I, 1, §§33-35. Peirce: CP 2.389 Fn 1 p 236 †1 Logik, §31. Peirce: CP 2.389 Fn 2 p 236 †2 Logic (1883), ch. 7. Peirce: CP 2.391 Fn 1 p 237 †1 §§1-6 are "Upon Logical Comprehension and Extension," Proceedings of the American Academy of Arts and Sciences, vol. 7, November 13, 1867, pp. 416-32, with additions and corrections c. 1870 and 1893; intended as Essay III of Search for a Method and as ch. 15 of the Grand Logic. Peirce: CP 2.391 Fn P1 p 237 Cross-Ref:†† †P1 Aristotle remarks in several places that genera and differences may be regarded as parts of species and species as equally parts of genera, e.g., in the 5th Meta. (1023b 22). The commentator referred to is perhaps Alexander Aphrodiensis . . .--1893. Peirce: CP 2.391 Fn P2 p 237 Cross-Ref:†† †P2 This is quoted from Baines [Baynes'] (Port Royal Logic, 2d ed. p. xxiii), who says that he is indebted to Sir William Hamilton for the information. Peirce: CP 2.391 Fn P3 p 237 Cross-Ref:†† †P3 Lotze.--1893. Peirce: CP 2.391 Fn P1 p 238 Cross-Ref:†† †P1 Porphyry appears to refer to the doctrine as an ancient one. [Ch. 1.] Peirce: CP 2.391 Fn P2 p 238 Cross-Ref:†† †P2 They were equally diligent in the study of Boëthius, who says: (Opera, p.

645) "Genus in divisione totum est, in diffinitione pars." Peirce: CP 2.391 Fn P3 Para 1/2 p 238 Cross-Ref:†† †P3 The author of De Generibus et Speciebus opposes the integral and diffinitive wholes. John of Salisbury refers to the distinction of comprehension and extension, as something "quod fere in omnium ore celebre est, aliud scilicet esse quod appellativa significant, et aliud esse quod nominant. Nominantur singularia, sed universalia significantur." (Metalogicus, lib. 2, cap. 20. Ed. of 1620, p. 111.) [By appellativa he means adjectives and the like.--1893.] Peirce: CP 2.391 Fn P3 Para 2/2 p 238 Cross-Ref:†† Vincentius Bellovacensis (Speculum Doctrinale, Lib. III, cap. xi) has the following: "Si vero quæritur utrum hoc universale 'homo' sit in quolibet homine secundum se totum an secundum partem, dicendum est quod secundum se totum, id est secundum quamlibet sui partem diffinitivam. . . ., non autem secundum quamlibet partem subjectivam. . . ." William of Auvergne (Prantl's Geschichte, vol. 3, p. 77) speaks of "totalitatem istam, quæ est ex partibus rationis seu diffinitionis, et hae partes sunt genus et differentiæ; alio modo partes speciei individua sunt, quoniam ipsam speciem, cum de eis prædicatur, sibi invicem quodammodo partiunter." [See also Duns Scotus, Opera I, 137.] If we were to go to later authors, the examples would be endless. See any commentary in Phys. Lib. I. Peirce: CP 2.392 Fn P1 p 239 Cross-Ref:†† †P1 Part I, ch. 9. Peirce: CP 2.392 Fn P2 p 239 Cross-Ref:†† †P2 Principia, Part I, §45 et seq. Peirce: CP 2.392 Fn P3 p 239 Cross-Ref:†† †P3 Eighth [ninth?] Letter to Burnet. Gerhardt's ed., vol. 3, p. 224. Peirce: CP 2.393 Fn P4 p 239 Cross-Ref:†† †P4 But intension was in use among the Leibnizians in the same sense.--1893. Peirce: CP 2.393 Fn 1 p 240 †1 Originally, "the best." Peirce: CP 2.393 Fn P1 p 240 Cross-Ref:†† †P1 Cf. Morin, Dictionnaire, Tome I, col. 684 [685?]; Chauvin, Lexicon, both editions; Eustachius, Summa, Part I, Tr. I, qu. 6. [Aquinas, sentent 1, d. 8, q. 1, art. 1.] Peirce: CP 2.393 Fn 2 p 240 †2 Originally, "authority." Peirce: CP 2.393 Fn P2 p 240 Cross-Ref:†† †P2 And such is the humility of his disciples, that not one has dared utter protest against this tax upon his credulity.--1893. Peirce: CP 2.393 Fn 3 p 240 †3 Logic, bk. I, ch. 2, §5, note. Peirce: CP 2.393 Fn P3 p 240 Cross-Ref:†† †P3 If I understand him, he expresses himself in his usual enigmatical style.--1870.

Peirce: CP 2.393 Fn P1 p 241 Cross-Ref:†† †P1 Prantl, Geschichte, vol. 3, p. 364. Peirce: CP 2.393 Fn P2 p 241 Cross-Ref:†† †P2 Ibid., p. 134n. Scotus also uses the term. Quodlib. question 13, article 4. Peirce: CP 2.393 Fn P3 p 241 Cross-Ref:†† †P3 Summa Theologica, Part I, question 53. [This work was certainly written before 1280. Roger Bacon refers to it while saying that Albertus is still alive.--1893.] Peirce: CP 2.393 Fn P4 p 241 Cross-Ref:†† †P4 The doctrine of connotare is part of the doctrine of appellatio, for which see Petrus Hispanus.--1893. Peirce: CP 2.393 Fn P5 p 241 Cross-Ref:†† †P5 Part I, ch. 10. (Ed. of 1488, fol. 6, c.) Peirce: CP 2.393 Fn 1 p 242 †1 The last sentence in this quotation appears, in the original, nearly half a page before the rest of the quotation. Peirce: CP 2.393 Fn P1 p 242 Cross-Ref:†† †P1 Fol. 23 d. See also Tartareti Expositio in Petr. Hisp. towards the end. Ed. of 1509, fol. 91, b. Peirce: CP 2.393 Fn P2 p 242 Cross-Ref:†† †P2 . . . is simply rubbish. Civilization in England does not seem as yet to have reached the stage in which men feel shame in making positive assertions based on exceptional ignorance.--1893. Peirce: CP 2.394 Fn P3 p 242 Cross-Ref:†† †P3 Logic, p. 100 [i.e., Lect. viii, 24]. In the Summa Logices attributed to Aquinas, we read: "Omnis forma sub se habens multa, idest, quae universaliter sumitur, habet quamdam latitudinem; nam invenitur in pluribus, et dicitur de pluribus." (Tr. 1, c. 3.) Peirce: CP 2.395 Fn 2 p 242 †2 Logik (1826), I, II; i, 1, iii, and ii, 1, iv. Peirce: CP 2.395 Fn 3 p 242 †3 System der Logik, (1857), §§50, 53. Peirce: CP 2.395 Fn 4 p 242 †4 Acroasis Logica, ed. 2 (1773), §24. Peirce: CP 2.395 Fn 5 p 242 †5 The Elements of Deductive Logic (1867), Pt. I, ch. 2. Peirce: CP 2.395 Fn 6 p 242 †6 An Introduction to Logical Science (1857), §§7, 30, 31. Peirce: CP 2.395 Fn 7 p 242 †7 Elements of Logic (1864), p. 10. Peirce: CP 2.395 Fn 8 p 242

†8 A System of Logic (1862), p. 191. Peirce: CP 2.395 Fn 1 p 243 †1 The Principles of Science (1874), bk. I, ch. 2, p. 31. Peirce: CP 2.396 Fn 2 p 243 †2 Baynes' translation, I, vi. Peirce: CP 2.398 Fn 3 p 243 †3 Grundress der Logik, 2te Auf. (1822), §29. Peirce: CP 2.398 Fn 4 p 243 †4 Logik, §42. Peirce: CP 2.398 Fn 5 p 243 †5 Neue Darstellung der Logik, 2te Auf. (1851), §23. Peirce: CP 2.398 Fn 6 p 243 †6 System der Logik (1828), Erster Theil, §48. Peirce: CP 2.398 Fn 7 p 243 †7 Logische Untersuchung, 2te Auf. (1862), xv, 4. Peirce: CP 2.398 Fn 8 p 243 †8 Elements of Logic (1864), pp. 10, 39ff. Peirce: CP 2.398 Fn 1 p 244 †1 An Introduction to Logical Science (1857), §31. Peirce: CP 2.398 Fn 2 p 244 †2 Logic, or the Science of Inference (1854), p. 42. Peirce: CP 2.398 Fn 3 p 244 †3 Cf. Syllabus, §131. Peirce: CP 2.398 Fn 4 p 244 †4 The Principles of Science (1874), Bk. 1, ch. 2. Peirce: CP 2.398 Fn 5 p 244 †5 A System of Logic (1862), p. 191. Peirce: CP 2.398 Fn 6 p 244 †6 The Elements of Deductive Logic (1867), Part I, ch. 2. Peirce: CP 2.398 Fn P1 p 244 Cross-Ref:†† †P1 I adopt the admirable distinction of Scotus between actual, habitual, and virtual cognition. [Reportatei, Ed. 1853, vol. 1, p. 147a. This distinction arose from mixed Aristotelean and Neoplatonic suggestions. Aristotle, as everybody knows, distinguished actual and potential thought. Alexander Aphrodisiensis distinguished material intellect ({nous hylikos}), habitual intellect ({nous kata hexin}) and intellectus adeptus. These two distinctions have little to do with one another. Still they were confounded by the Arabians, and the confused doctrine suggested to Scotus his brilliant and philosophical division.--1893.] Peirce: CP 2.399 Fn 7 p 244 †7 Baynes' translation I, vi.

Peirce: CP 2.399 Fn 8 p 244 †8 Logick (1725), Part I, ch. 3, §3. Peirce: CP 2.399 Fn 9 p 244 †9 Lehrbuch der Logik (1838), Drittes Kap., §37. Peirce: CP 2.399 Fn 10 p 244 †10 System der Logik (1828), Erster Theil, §48. Peirce: CP 2.399 Fn 11 p 244 †11 System der Logik 2te Auf. (1830), Erster Theil, §34. Peirce: CP 2.399 Fn 12 p 244 †12 Grundsätze der allgemeinen Logik 5te Auf. (1831), §29. Peirce: CP 2.399 Fn 1 p 245 †1 Cf. Logik, her. v. G. B. Jäsche (1800), I, i, §§1-7. Peirce: CP 2.399 Fn 2 p 245 †2 Die Logik (1827), S. 115. Peirce: CP 2.399 Fn 3 p 245 †3 System der Logik, 3te Auf. (1837), §20. Peirce: CP 2.399 Fn 4 p 245 †4 System der Logik (1857), §53. Peirce: CP 2.399 Fn 5 p 245 †5 Logick (1725), Part I, ch. 3, §3. Peirce: CP 2.399 Fn 6 p 245 †6 Elements of Logic (1864), pp. 39, 40. Peirce: CP 2.399 Fn 7 p 245 †7 System der Logik (1828), Erster Theil, §48. Peirce: CP 2.399 Fn 8 p 245 †8 Logic, or the Science of Inference (1854), p. 42. Peirce: CP 2.399 Fn 9 p 245 †9 Grundsätze der allgemeinen Logik, 5te Auf. (1831), §29. Peirce: CP 2.399 Fn 10 p 245 †10 A Treatise on Logic (1864), p. 67. Peirce: CP 2.399 Fn 11 p 245 †11 Neue Dartellung der Logik, 2te Auf. (1851), §23. Peirce: CP 2.399 Fn 12 p 245 †12 Formal Logic (1847), p. 234. Peirce: CP 2.399 Fn 13 p 245 †13 Outline of the Laws of Thought, 4 ed., pp. 99-102. Peirce: CP 2.399 Fn 14 p 245 †14 Intellectual Philosophy, 2d ed. (1847), ch. 7, 8.

Peirce: CP 2.399 Fn 15 p 245 †15 Lehrbuch zur Einleitung in die Philosophie (1813), II, i, §40. Peirce: CP 2.399 Fn 16 p 245 †16 Wissenschaft der Erkenntniss (1847), II, i, 2, b. Peirce: CP 2.399 Fn 17 p 245 †17 An Introduction to Logical Science (1857), §30. Peirce: CP 2.399 Fn 18 p 245 †18 Entwurf der Logik (1846), 4tes Kap. Peirce: CP 2.399 Fn 19 p 245 †19 Abriss der Philosophischen Logik (1824), S.79. Peirce: CP 2.400 Fn P1 p 246 Cross-Ref:†† †P1 This law, algebraically stated, is that if a and b are logical terms so related that a = bx, then also b = a + y and conversely. Numbers of German logicians are capable of denying this.--1893. Peirce: CP 2.400 Fn P2 p 246 Cross-Ref:†† †P2 Hoppe reverses the law of Kant, and maintains that the wider the concept the greater its content. His idea, translated into Aristotelean phraseology, is this. He admits the second antepredicamental rule, that the differences of different genera are different (This, of itself, removes him widely from logicians for whom the distinction of comprehension and extension is the turning point of logic.) Negro is not a conception formed by the union of the two concepts man and black, but the peculiar differences of negro belong to negroes alone of all beings. This naturally carries him a step further, and he says the difference is of itself sufficient to constitute the pure concept, so that the genus is not an essential predicate. Thirdly, he finds that the characters of the narrower difference are less important (wirkungsreich) than those of the higher, and to have less important consequences is to have a smaller measure of predicates.--1893. Peirce: CP 2.401 Fn 1 p 246 †1 Neue Darstellung der Logik. 2te Auf. (1851), Anhang I. Peirce: CP 2.403 Fn 1 p 247 †1 Wissenschaft der Erkenntniss (1847), S. 104-107. Peirce: CP 2.404 Fn P1 p 247 Cross-Ref:†† †P1 System der Logik, 2te Aufl., §54. Peirce: CP 2.405 Fn P2 p 247 Cross-Ref:†† †P2 Formal Logic, p. 234. His doctrine is different in the Syllabus. Peirce: CP 2.406 Fn P1 p 248 Cross-Ref:†† †P1 Laws of Thought, 4th ed., §§52, 80. [Cf. §54.] Peirce: CP 2.406 Fn P2 p 248 Cross-Ref:†† †P2 Logic, Part I, ch. 2, §5. Peirce: CP 2.407 Fn 1 p 248 †1 This and the previous section were both numbered §4 in the original publication.

Peirce: CP 2.407 Fn P1 p 249 Cross-Ref:†† †P1 I restricted myself to terms, because at the time this chapter was first written (1867), I had not remarked that the whole doctrine of breadth and depth was equally applicable to propositions and to arguments. The breadth of a proposition is the aggregate of possible states of things in which it is true; the breadth of an argument is the aggregate of possible cases to which it applies. The depth of a proposition is the total of fact which it asserts of the state of things to which it is applied; the depth of an argument is the importance of the conclusions which it draws. In fact, every proposition and every argument can be regarded as a term.--1893. Peirce: CP 2.407 Fn P2 p 249 Cross-Ref:†† †P2 It would seem needlessly to complicate the doctrine to introduce probabilities, and therefore it is understood that the information is supposed to be accepted absolutely.--1893. Peirce: CP 2.407 Fn 1 p 249 †1 Changed in 1870 and 1893 from "those things of which there is not . . . predicable are not . . ." Peirce: CP 2.407 Fn P3 p 249 Cross-Ref:†† †P3 For the distinction of extensive and comprehensive distinctness, see Scotus, i, dist. 2, qu. 3. Peirce: CP 2.408 Fn P1 p 250 Cross-Ref:†† †P1 That is, of whatever things it is applicable to. Peirce: CP 2.409 Fn P2 p 250 Cross-Ref:†† †P2 The essence of a thing is the idea of it, the law of its being, which makes it the kind of thing it is, and which should be expressed in the definition of that kind.--1893. Peirce: CP 2.412 Fn P1 p 251 Cross-Ref:†† †P1 Negative terms are called by the logicians infinite (or, recently, infinitated). This is a translation of Aristotle's {aoristos} [De Interpr. 3, 16b, 14], which really means "without definition" {horismos}.--1893. Peirce: CP 2.415 Fn P2 p 251 Cross-Ref:†† †P2 See, for example, De Generibus et Speciebus, p. 548. Peirce: CP 2.418 Fn 1 p 253 †1 On a New List of Categories, vol. 1, bk. III, ch. 6, §1. Peirce: CP 2.418 Fn P1 p 253 Cross-Ref:†† †P1 It will be seen that I depart widely from the ordinary use of this word to mean testimony given privately. As in metaphysics, information is the connection of form and matter, so it may in logic appropriately mean the measure of predication.--1893. Peirce: CP 2.422 Fn P1 p 255 Cross-Ref:†† †P1 Ascent is the most unequivocal noun to denote the passage to a broader and less deep notion, without change of information; and other words of similar literal meaning are used in the same way. It is the decrease in depth, of course, which is directly expressed, the increase in breadth being implied. Extension, which directly expresses increase of breadth, has a somewhat different meaning. It is applied to the

discovery (by increase of information) that a predicate applies--mutatis mutandis--to subjects to which it had not occurred to us to apply it. It involves no decrease of depth. Thus, Herbert Spencer says [" The Genesis of Science," British Quarterly Review, July, 1854] that the inversion of the barometer enabled us to extend the principles of mechanics to the atmosphere. Mathematicians frequently speak of the extension of a theorem. Thus, the modification of a theorem relating to plane curves so as to make it apply to all curves in space would be called an extension of that theorem. An extended theorem asserts all that the original theorem did, and more too. Generalization in its strict sense, means the discovery, by reflection upon a number of cases, of a general description applicable to them all. This is the kind of thought movement which I have elsewhere [509] called formal hypothesis, or reasoning from definition to definitum. So understood, it is not an increase in breadth but an increase in depth. For instance, I received today a number of English books printed by Hindoos in Calcutta. The manufacture is rude, yet peculiarly pleasing. Remembering other Indian manufactures I have seen, I now get a more definite conception of the characteristic of Indian taste. This, since it is an idea derived from the comparison of a number of objects, is called generalization. Yet it is not an extension of an idea already had, but, on the contrary, an increase of definiteness of the conceptions I apply to known things. Besides this, the proper meaning of the word generalization, there are two others which, though they are in good use, ought all the more for that to be severely frowned upon by all who have precision of philosophical terminology at heart. Namely, generalization is applied, secondly, to a particular kind of extension, namely to an extension in which the change of the predicate, in order to make it applicable to a new class of subjects, is so far from obvious, that it is the part of the mental process which chiefly attracts our notice. For example, what is usually called Fermat's theorem is that if {r} be a prime number, and a be any number not divisible by {r}, then a{r}-1 leaves a remainder of 1 when divided by {r}. Now, what is called the generalized theorem of Fermat is that if {k} is any integer number, and φ{k} its totient, or the number of numbers as small as {k} and prime to it, and if a be a number prime to {k}, then aφk leaves a remainder 1 when divided by {k}. Instead of calling such process a generalization, it would be far better to call it a generalizing extension.--1893. Peirce: CP 2.427 Fn 1 p 258 †1 "Terminology" a supplement to the foregoing. Peirce: CP 2.428 Fn 1 p 259 †1 See Prantl, op. cit., III, 94. Peirce: CP 2.428 Fn 1 p 260 †1 Cf. 1.549n. Peirce: CP 2.431 Fn 1 p 262 †1 Dictionary of Philosophy and Psychology, vol. 2, pp. 528-9; 431-3 are by Peirce and Mrs. C. Ladd-Franklin. Peirce: CP 2.434 Fn 1 p 263 †1 Metalogicus, II, xx. Peirce: CP 2.434 Fn 2 p 263 †2 Lexicon Rationale. Peirce: CP 2.435 Fn 1 p 265

†1 From the "Short Logic," c. 1893, following 296. Peirce: CP 2.440 Fn P1 p 266 Cross-Ref:†† †P1 That Marra and Warra were really the same cannot be positively asserted; but the hypothesis suits the known facts remarkably well, except for the difference of names, which is perhaps not an insuperable obstacle. Peirce: CP 2.442 Fn 1 p 268 †1 See 451n. Peirce: CP 2.444 Fn P1 Para 1/2 p 269 Cross-Ref:†† †P1 To speak summarily and use a symbol of abbreviation, rather than an analytical and iconical idea, we may say that the purpose of signs--which is the purpose of thought--is to bring truth to expression. The law under which a sign must be true is the law of inference; and the signs of a scientific intelligence must, above all other conditions, be such as to lend themselves to inference. Hence, the illative relation is the primary and paramount semiotic relation. Peirce: CP 2.444 Fn P1 Para 2/2 p 269 Cross-Ref:†† It might be objected that to say that the purpose of thought is to bring the truth to expression is to say that the production of propositions, rather than that of inferences, is the primary object. But the production of propositions is of the general nature of inference, so that inference is the essential function of the cognitive mind.--From the fragment used in ch. 2, §1. Peirce: CP 2.445 Fn 1 p 273 †1 From chapter 9 of the Grand Logic, 1893. Cf. 3.162ff. Peirce: CP 2.445 Fn P1 p 273 Cross-Ref:†† †P1 To be well-read, or even fairly versed, in philosophy (no easy accomplishment) it is quite indispensible to have studied Aristotle; and the study of Aristotle may most conveniently begin with the two books of Prior Analytics, certainly the most elementary of all his writings. Two books precede these in the traditional arrangement (with which Aristotle himself probably had nothing to do). One of these, the Predicaments or Categories, is a metaphysico-logical treatise, of which only the outlines are important. The other, the Peri hermeneias, is purely logical, but difficult and confused; and the doctrine is not that of the Analytics. I should recommend every serious student of logic who can pick out easy Greek without much trouble to read the Prior Analytics at any rate, and the Posterior Analytics if he can find time. The Posterior Analytics is a splendid monument to the human intellect. Both treatises are in very easy Greek; and they have so much influenced medieval thought, and through that our own, that really a man does not understand what is said to him in the streets till he has read them. I would read them out of the Berlin edition; and if you want notes, there can be nothing better than the Greek scholia there given. Then by buying this edition, you have the advantage of having the index constantly at your hand; and it is of inestimable value, every day. Waitz's edition of the Organon is good; and Trendelenburg's Beiträge, De Anima, and little epitome [Elementa Logices Aristoteleae] are very valuable. There is a capital little epitome [Outlines of the Philosophy of Aristotle] by Wallace, and Grote's Aristotle has merit. But Grote is terribly one-sided. In fact, all modern commentators have strong leanings. Peirce: CP 2.449 Fn P1 p 275 Cross-Ref:††

†P1 An incomplete argumentation is properly called an enthymeme, which is often carelessly defined as a syllogism with a suppressed premiss, as if a sorites, or complex argumentation, could not equally give an enthymeme. The ancient definition of an enthymeme was "a rhetorical argumentation," and this is generally set down as a second meaning of the word. But it comes to the same thing. By a rhetorical argumentation was meant one not depending upon logical necessity, but upon common knowledge as defining a sphere of possibility. Such an argument is rendered logical by adding as a premiss that which it assumes as a leading principle. Peirce: CP 2.451 Fn P1 p 276 Cross-Ref:†† †P1 The latter term is more familiar to our generation, having been used by Whewell [Novum Organum Renovatum, II, iv]. But the former is the more legitimate historically. Copulatum with Aulus Gellius (XVI. viii. 10) translates the Stoical {sympeplegmenon} in this sense. Conjunctions like et are called copulative by Priscian [Institutiones Grammaticae, lib. xvi, cap. 1]. Abelard uses copulare. We might use colligation where the propositions brought together are of one nature and function. But in syllogism, this is not the case. However, if the mood Darapti be admitted, it consists merely in compounding two premisses and dropping a term from the result. This will appear below. Peirce: CP 2.451 Fn P1 p 277 Cross-Ref:†† †P1 What Kant calls an explicative, or analytical, judgment is either no judgment at all, because void of content (to use his phrase), or else it sets forth distinctly in the predicate what was only indistinctly thought (that is, not actually thought at all) in the subject. In that case, it is really synthetic, and rests on experience; only the experience on which it rests is mere internal experience--experience of our own imaginations. Association by resemblance, and association by contiguity: all lies in that great distinction. Peirce: CP 2.453 Fn 1 p 278 †1 Cf. Hobbes, Computation or Logic, ch. II, §11. Peirce: CP 2.453 Fn P1 Para 1/2 p 278 Cross-Ref:†† †P1 This does not hold in the case of a limited universe of marks. For if we are confining ourselves to a certain line of predicates, there will be nothing absurd in saying that things differ in every respect. In that case, there will be a lexis of predicates, distinct from the phasis. Certainly, if the nature of reasoning is to be explored, it is necessary to take account of cases in which we limit our thought to a particular order of predicates. Some logicians treat the subject as "extra-logical"; but that only means it is outside the scope of their own studies. If a mathematician should choose to characterize the differential calculus as "extra-mathematical," he would exhibit the same determination to keep his science small and simple that animates many of the logicians. Peirce: CP 2.453 Fn P1 Para 1/2 p 278 Cross-Ref:†† But although the limited universe of marks is not for me extra-logical, I think it is proper to exclude it from elementary syllogistic, for the reason that it is one of the simplest conceivable instances of the logic of relatives, and when that is treated this problem is virtually solved, even if it be not directly attended. Peirce: CP 2.455 Fn P1 p 279 Cross-Ref:†† †P1 It dates from Apuleius, and is more assified than golden. Universal and Particular have the same origin. Affirmative and Negative are words manufactured

by Boëthius. [See Prantl op. cit., I, 691.] Peirce: CP 2.455 Fn P1 p 280 Cross-Ref:†† †P1 From {phémi}, not {phainö}; therefore nothing to do with phase. Peirce: CP 2.457 Fn P1 p 281 Cross-Ref:†† †P1 The term universe, now in general use, was introduced by De Morgan in 1846. Cambridge Philosophical Transactions, VIII, 380. Peirce: CP 2.459 Fn 1 p 282 †1 See Prantl, op. cit., I, 687ff. Peirce: CP 2.459 Fn 2 p 282 †2 Ibid., 687. Peirce: CP 2.459 Fn 3 p 282 †3 Ibid., 661. Peirce: CP 2.459 Fn 4 p 282 †4 Ibid., 684, 692. Peirce: CP 2.461 Fn 1 p 284 †1 Proceedings of the American Academy of Arts and Sciences, vol. 7, April 9, 1867, pp. 261-87, with additions and corrections of 1893. Intended as Essay I of the Search for a Method. Peirce: CP 2.461 Fn P1 p 284 Cross-Ref:†† †P1 There can be little doubt that argumentum acquired its logical meaning in the Roman law courts; and Cicero not only uses it as above, but expressly defines it as "ratio rei dubiae faciens fidem." The definition of Boëthius, who intends to follow Cicero, makes it a medium proving a conclusion. Medium is here used in the sense of premiss; but since it usually means in logic a middle term, argument has been by many understood in that sense; and Hamilton is among those who go so far as to stigmatize the other and ancient use as improper; wishing to substitute for this the word argumentation. The substitution, however, seems to me historically wrong, contrary to common usage, and not particularly convenient. Still, to avoid reproach, I was inclined to replace argument in this essay by inference (for, as Locke well says, "to infer is nothing but, by virtue of one proposition laid down as true, to draw in another as true," and those who would restrict it to reasoning from effect to cause violate all good usage), until I reflected that to do so would have the air of admitting what I could never admit, that logic is primarily conversant with unexpressed thought and only secondarily with language.--1893. Peirce: CP 2.461 Fn P2 p 284 Cross-Ref:†† †P2 So far as that is separable from the rest.--1893. Peirce: CP 2.465 Fn P1 p 285 Cross-Ref:†† †P1 Or else made a conditional antecedent to the conclusion.--1893. Peirce: CP 2.465 Fn P2 p 285 Cross-Ref:†† †P2 To this it might be objected that if from premiss, P, we infer conclusion, C, then to infer "If P, then C," needs no premiss at all. But if the hypothetical judgment is immediately made, it is not inferential; and if not, it is requisite to begin with some premiss, though its modality be only problematical.--1893.

Peirce: CP 2.466 Fn P1 Para 1/2 p 286 Cross-Ref:†† †P1 Neither of these terms is quite satisfactory. Enthymeme is usually defined as a syllogism with a premiss suppressed. This seems to determine the same sphere as the definition I have given; but the doctrine of a suppressed premiss is objectionable. The sense of a premiss which is said to be suppressed is either conveyed in some way, or it is not. If it is, the premiss is not suppressed in any sense which concerns the logician; if it is not, it ceases to be a premiss altogether. What I mean by the distinction is this. He who is convinced that Sortes is mortal because he is a man (the latter belief not only being the cause of the former, but also being felt to be so) necessarily says to himself that all such arguments are valid. This genus of argument is either clearly or obscurely recognized. In the former case, the judgment amounts to another premiss, because the proposition (for example), "All reasoning from humanity to mortality is certain," only says in other words that every man is mortal. But if the judgment amounts merely to this, that the argument in question belongs to some genus all under which are valid, then in one sense it does, and in another it does not, contain a premiss. It does in this sense, that by an act of attention such a proposition may be shown to have been virtually involved in it; it does not in this sense, that the person making the judgment did not actually understand this premiss to be contained in it. This I express by saying that this proposition is contained in the leading principle, but is not laid down. This manner of stating the matter frees us at once from all psychological perplexities; and at the same time we lose nothing, since all that we know of thought is but a reflection of what we know of its expression. Peirce: CP 2.466 Fn P1 Para 2/2 p 286 Cross-Ref:†† These vague arguments are just such as alone are suitable to oratory or popular discourse, and they are appropriate to no other; and this fact justifies the appellation, "rhetorical argument." There is also authority for this use of the term. "Complete" and "incomplete" are adjectives which I have preferred to "perfect" and "imperfect," as being less misleading when applied to argument, although the latter are the best when syllogism is the noun to be limited. [Perhaps it is necessary further to distinguish between a complete and a logical argument. [See 474n.]--1893.] Peirce: CP 2.467 Fn 1 p 287 †1 Originally, "a proposition." Peirce: CP 2.467 Fn 2 p 287 †2 Originally, "Considered as regulating the procedure of inference, it is determinate." Peirce: CP 2.467 Fn P1 p 287 Cross-Ref:†† †P1 Any assertion means merely how we would act under given circumstances, but a logical principle does not even mean this, but only what we would infer from certain premisses.--1893. Peirce: CP 2.468 Fn 3 p 287 †3 Originally numbered §4, though following immediately after §2. Peirce: CP 2.469 Fn P2 p 287 Cross-Ref:†† †P2 The view here taken appears inadmissible, unless it be understood to bear a greatly generalized meaning. For, according to this, from two premisses, "A" and "B," but one complete argument could be formed, namely, that which concludes the conjunctive proposition "A and B"; and so, all the arguments of which this paper treats would have to be excluded. But we must not use the word "argument" in a sense

which completely annuls its utility. The mere "colligation of facts," to use Whewell's term, is a most important and difficult part of that whole operation which in its totality is called reasoning. But it is not the only part. Given the premiss that every man (living, dead, or unborn) is the son of a man, then by a process which is an important part of reasoning, we conclude that every man is a grandson (or a descendant of any order) of a man. Nothing is to be gained by excluding such an operation of thought from the number of arguments. To further illustrate this point, take any branch of pure mathematics--say the theory of numbers. What are its premisses? Some of them are mere definitions of terms, such as product, sum, and the like--terms which may be dispensed with entirely, their definitions always being substituted. The other premisses taken together define the relationship existing between integer numbers; and they may without difficulty be comprised in a single proposition, which will really be a definition of the subject-matter, number. Thus, the whole fabric of the theory of numbers, which in its posse, at least, may well be called vast, will be deduced from a single premiss. Nay, it would be very simple to make this single proposition the hypothetical antecedent of every conclusion, when it would be needed no more as a premiss. Nevertheless, it is impossible to regard reasoning otherwise than as a process; and as such, it must involve a substitution of a conclusion in place of premisses. The logic of relatives clearly shows that not all reasoning is eliminative; but so far as it is eliminative, it involves two premisses and is of the general type set forth in §4, below.--1893. Peirce: CP 2.474 Fn P1 p 289 Cross-Ref:†† †P1 Jevons is often referred to as the originator of the conception of reasoning as a substitution. He first set forth this idea in a little treatise called The Substitution of Similars [1869], which appeared after he had seen the present essay in print. But he never claimed the idea as original; nor do I. It was familiar to the Leibnizian logicians. In my opinion, the peculiar doctrine of the substitution of similars is utterly false and untenable.--1893. Peirce: CP 2.474 Fn P2 p 289 Cross-Ref:†† †P2 This treatment of the subject has the effect of excluding the dilemma altogether from the scheme of classification. It was not until the Renaissance that this formal argument ever appeared in the logical treatises; though it had been given in the rhetorics since ancient times. The stock example of the dilemma is this from Aulus Gellius: a handsome wife would be unfaithful, which is bad; and an ugly wife would confer no pleasure, which is bad; therefore, to take any wife would be bad. This is usually reduced to syllogism by assuming as a premiss a proposition which is nothing but a corollary from the principle of excluded middle; and this would be requisite to make the argument a complete one, according to the definition in the text. It would, however, have been proper to recognize, among incomplete arguments, those as logical which suppress no premiss except a mere logical principle, such as the principle of excluded middle. Such argument is a trifle less infantile than syllogism. By a dilemmatic argument I mean any argument whose validity depends upon the principle of excluded middle. Many of these are reckoned as syllogism by De Morgan [Formal Logic, p. 117ff]. I do not think anything should be a dilemma which does not depend on the same principle.--1893. Peirce: CP 2.475 Fn 1 p 290 †1 Peirce's first printed paper, of which Part II is substantially a restatement, is in the appendix to the present volume.

Peirce: CP 2.475 Fn P1 p 290 Cross-Ref:†† †P1 This operation will be termed a contraposition of the premiss and conclusion. Peirce: CP 2.476 Fn P2 p 290 Cross-Ref:†† †P2 What S is meant being generally undetermined. Peirce: CP 2.477 Fn 1 p 291 †1 This sentence was deleted and the following substituted for it in 1893. "For example, suppose one man says 'Some S's are Q,' that is, 'I can so choose from the class of S's that those I choose shall be Q'; and suppose another replies, 'Some of that some are R,' that is, 'I can so choose from among those which you will choose that (whether they be Q or not) they shall be R.' Plainly he says in effect that all S's are R. It will be the same if instead of two men, it is but one man at two different times." Peirce: CP 2.482 Fn 1 p 293 †1 When Peirce incorporated this paper in the "Search for a Method" he deleted this paragraph and wrote instead: "It is customary to enumerate six moods of the third figure instead of four; but Darapti and Felapton are omitted, because when the universal premisses are not understood to assert the existence of their subjects these moods become invalid." Peirce: CP 2.486 Fn 1 p 296 †1 On the authorship of these mood names and other mnemonic devices, cf. Prantl, op. cit., III, 11-50, passim. Peirce: CP 2.487 Fn P1 p 296 Cross-Ref:†† †P1 This inference, S is not P, .·. S is not-P, being the conclusion of an affirmative from a negative premiss, can obviously not be ordinary syllogism. The suppressed premiss is,

Not-P is not P,

which is the definition of not-P. It will appear below (Part III, §1) that this is ordinary reasoning from definition to definitum, and is essentially of the second figure.--1893. Peirce: CP 2.488 Fn P2 p 296 Cross-Ref:†† †P2 This is not reasoning from definition; for that is the substitution of a compact predicate for a complex one. The present inference is: Some S is P, .·.Some-S is P.

It draws a universal conclusion from a particular premiss, and is, therefore, not ordinary syllogism. The suppressed premiss is,

Some S is some-S;

and the inference is of the nature of reasoning from enumeration, as will be seen below. Such inference is essentially of the third figure.--1893.

Peirce: CP 2.496 Fn P1 p 299 Cross-Ref:†† †P1 Or rather, they bear only the formal meaning that consists in showing certain forms of inference to be valid.--1893. Peirce: CP 2.496 Fn P2 p 299 Cross-Ref:†† †P2 Or at most, they are but formal inferences in the sense in which this word has just been used.--1893. Peirce: CP 2.497 Fn 1 p 301 †1 This paragraph was deleted in the "Search for a Method" and the following substituted: "Now, putting aside the long reductions of the moods concluding O, or a particular negative, which are needlessly complicated and involve irrelevant conversions, in every other case the forms of inference used belong to the very same figure as the syllogism to be reduced." Peirce: CP 2.498 Fn P1 p 301 Cross-Ref:†† †P1 A formal inference is a substitution having the form of an inference. Peirce: CP 2.500 Fn 1 p 303 †1 Cf. Ladd-Franklin's "Antilogism" in "On the Algebra of Logic," Studies in Logic, p. 37ff. Peirce: CP 2.502 Fn 2 p 303 †2 See Prantl, op. cit., III, 15, 16. Peirce: CP 2.506 Fn P1 p 305 Cross-Ref:†† †P1 I leave the Theophrastean syllogism, as every logician has found it, almost entirely useless.--1893. Peirce: CP 2.506 Fn P1 p 306 Cross-Ref:†† †P1 Hypotheticals have not been considered above, the well-known opinion having been adopted that, "If A, then B," means the same as "Every state of things in which A is true is a state of things in which B is (or will be) true." Peirce: CP 2.511 Fn P1 Para 1/2 p 308 Cross-Ref:†† †P1 Positivism, apart from its theory of history and of the relations between the sciences, is distinguished from other doctrines by the manner in which it regarded hypotheses. Almost all men think that metaphysical theories are valueless, because metaphysicians differ so much among themselves; but the positivists give another reason, namely, that these theories violate the sole condition of all legitimate hypothesis. This condition is that every good hypothesis must be such as is certainly capable of subsequent verification with the degree of certainty proper to the conclusions of the branch of science to which it belongs. There is, it seems to me, a confusion here between the probability of an hypothesis in itself, and its admissibility

into any one of those bodies of doctrine which have received distinct names, or have been admitted into a scheme of the sciences, and which admit only conclusions which have a very high probability indeed. I have here to deal with the rule only so far as it is a general canon of the legitimacy of hypotheses, and not so far as it determines their relevancy to a particular science; and I shall, therefore, consider only another common statement of it; namely, "that no hypothesis is admissible which is not capable of verification by direct observation." The positivist regards an hypothesis, not as an inference, but as a device for stimulating and directing observation. But I have shown above that certain premisses will render an hypothesis probable, so that there is such a thing as legitimate hypothetic inference. It may be replied that such conclusions are not hypotheses, but inductions. That the sense in which I have used "hypothesis" is supported by good usage, I could prove by a hundred authorities. The following is from Kant: "An hypothesis is the holding for true of the judgment of the truth of a reason on account of the sufficiency of its consequents." Mill's definition (Logic, Book III, Ch. XIV §4) also nearly coincides with mine. Moreover, an hypothesis in every sense is an inference, because it is adopted for some reason, good or bad, and that reason, in being regarded as such, is regarded as lending the hypothesis some plausibility. The arguments which I term hypothetic are certainly not inductions, for induction is reasoning from particulars to generals, and this does not take place in these cases. The positivist canon for hypotheses is neither sufficient nor necessary. If it is granted that hypotheses are inferred, it will hardly be questioned that the observed facts must follow apodictically from the hypothesis without the aid of subsidiary hypotheses, and the characters of that which is predicated in the hypothesis, and from which the inference is drawn, must be taken as they occur, and not be picked out in order to make a plausible argument. That the maxim of the positivists is superfluous or worse, is shown; first, by the fact that it is not implied in the proof that hypothetic inference is valid; and next, by the absurdities to which it gives rise when strictly applied to history, which is entirely hypothetical, and is absolutely incapable of verification by direct observation. To this last argument I know of but two answers: first, that this pushes the rule further than was intended, it being considered that history has already been so verified; and second, that the positivist does not pretend to know the world as it absolutely exists, but only the world which appears to him. To the first answer, the rejoinder is that a rule must be pushed to its logical consequences in all cases, until it can be shown that some of these cases differ in some material respect from the others. To the second answer, the rejoinder is double: first, that I mean no more by "is" than the positivist by "appears" in the sense in which he uses it in saying that only what "appears" is known, so that the answer is irrelevant; second, that positivists, like the rest of the world, reject historic testimony sometimes, and in doing so distinguish hypothetically between what is and what in some other sense appears, and yet have no means of verifying the distinction by direct observation. Peirce: CP 2.511 Fn P1 Para 2/2 p 308 Cross-Ref:†† Another error in reference to hypothesis is that the antecedent probability of what is testified to cannot affect the probability of the testimony of a good witness. This is as much as to say that probable arguments can neither support nor weaken one another. Mr. Venn goes so far as to maintain the impossibility of a conflict of probabilities. The difficulty is instantly removed by admitting indeterminate probabilities. Peirce: CP 2.517 Fn 1 p 313 †1 §1 and §2 are "Note A," the Johns Hopkins Studies in Logic, edited by C.

S. Peirce, Little, Brown and Co., Boston, (1883) pp. 182-6, as rewritten in 1893, for the Grand Logic, ch. 13. Cf. 3.345ff and 3.403F ff. Peirce: CP 2.517 Fn P1 p 313 Cross-Ref:†† †P1 The term was introduced by De Morgan in 1846. Cambridge Philosophical Transactions, VIII (1849), p. 380. Peirce: CP 2.519 Fn 2 p 313 †2 Originally: "The conception of ordinary syllogism is so unclear that it would hardly be accurate to say that it supposes an unlimited universe of characters; but it comes nearer to that than to any other consistent view." Peirce: CP 2.519 Fn 1 p 314 †1 This sentence was added in 1893. Peirce: CP 2.521 Fn 2 p 314 †2 The remainder of this section and the whole of the next differ in detail from the article as first published. Peirce: CP 2.521 Fn 1 p 315 †1 See 3.345 for the interpretation of some of these in terms of "relatives." They are perhaps clearer when expressed with quantifiers, thus: 1. (s) (π) πs. 2. (Σs) (π) πs. 3. (π) (Σs)πs. 4. (πΣ) (s) πs, etc., where (Σs) means "some s" and (s) means "every s." Peirce: CP 2.524 Fn 1 p 316 †1 (Σs) (π) πs -< (π) (Σs) πs. Peirce: CP 2.526 Fn 1 p 319 †1 . . . "the conclusion is of a kind called spurious by De Morgan if, and only if, the middle term is affected by 'some' in both premisses"--in the original, p. 185. See 607; De Morgan, Formal Logic, pp. 153-4 and Syllabus, §76ff.; and B. Gilman, Johns Hopkins University Circular, August, 1882, on these "Spurious Propositions." Peirce: CP 2.532 Fn 1 p 322 †1 From ch. 10, "Extension of the Aristotelian Syllogistic," of the Grand Logic, 1893. Peirce: CP 2.533 Fn 1 p 324 †1 See also Appendix 1 to De Morgan's Formal Logic. Peirce: CP 2.534 Fn 2 p 324 †2 An Essay on the New Analytic of Logical Forms, Edinburgh, (1850). Peirce: CP 2.534 Fn 1 p 325 †1 See Lectures on Logic, VI, p. 114. Peirce: CP 2.536 Fn 1 p 326 †1 From "Universe," Dictionary of Philosophy and Psychology, vol. 2, p. 742, by Peirce and Mrs. C. Ladd-Franklin. Peirce: CP 2.536 Fn 1 p 327 †1 See An Investigation of the Laws of Thought, etc., p. 42. Peirce: CP 2.537 Fn 1 p 328 †1 Dictionary of Philosophy and Psychology, vol. 2, pp. 27-28.

Peirce: CP 2.544 Fn 1 p 329 †1 Ibid., vol. 2, p. 402. Peirce: CP 2.547 Fn 1 p 330 †1 Ibid., vol. 2, p. 219. Peirce: CP 2.548 Fn 2 p 330 †2 Ibid., vol. 1, p. 561. Peirce: CP 2.548 Fn 3 p 330 †3 See vol. 4, bk. I, No. 3. Peirce: CP 2.549 Fn 4 p 330 †4 Dictionary of Philosophy and Psychology, vol. 2, pp. 44-45. Peirce: CP 2.549 Fn 1 p 331 †1 See 579. Peirce: CP 2.550 Fn 2 p 331 †2 Mind, vol. 1 (1876), p. 424. Peirce: CP 2.551 Fn 1 p 332 †1 Dictionary of Philosophy and Psychology, vol. 2, p. 199. Peirce: CP 2.552 Fn 2 p 332 †2 Ibid., vol. 2, pp. 628-9, 633-9. Peirce: CP 2.553 Fn 1 p 333 †1 See Novum Organum Renovatum, II, iv. Peirce: CP 2.554 Fn 2 p 333 †2 §20. Peirce: CP 2.554 Fn 1 p 334 †1 Aristotle's Rhetoric with Cope's Commentary, Sandys' edition, vol. 1, p. 217. Peirce: CP 2.554 Fn P1 p 334 Cross-Ref:†† †P1 Written, as there is strong unpublished ground for thinking, 394 B. C., while Aristotle only went to Athens 368 or 367 B.C. (Grote thinks not till 362 B. C.) All the other dialogues here mentioned are subsequent to Aristotle's joining the school. Peirce: CP 2.554 Fn 2 p 334 †2 The Origin and Growth of Plato's Logic (1897), p. 203. Peirce: CP 2.554 Fn 3 p 334 †3 Ibid., p. 464. Peirce: CP 2.554 Fn P1 p 335 Cross-Ref:†† †P1 It has been argued that Aristotle may here, as it is said he often does, employ the first person plural to mean the students of Plato; and also that {proteron allo} would not exclude aid from contemporaries. The present writer, without making any pretension to philological learning, apprehends that it is quite clear that Aristotle is speaking of himself personally, and that he means that no doctrine of the syllogism,

in which he now takes the first steps ({hös ek toioutön ex archés hyparchontön echein hé methodos}) had existed before his Analytics and Topics. Such hints as he may have received from Plato cannot (the writer believes) have been in Aristotle's memory when he penned those words. But a man does not always know how he originally came by ideas which occupied him at first little, but afterwards more and more, up to almost complete absorption for many long years. Peirce: CP 2.556 Fn 1 p 335 †1 Prior Analytic, I, 1, 24b. Peirce: CP 2.558 Fn 1 p 336 †1 3.636. Peirce: CP 2.558 Fn 2 p 336 †2 Vol. 4, bk. II, ch. 2. Peirce: CP 2.558 Fn 1 p 337 †1 † is the sign of relative addition, ~y is the negative of y, $y is the converse of y, xy means x and y, and x z means x is other than z. Peirce: CP 2.559 Fn 1 p 340 †1 I.e., symmetry. Peirce: CP 2.567 Fn 1 p 345 †1 Outline of the Laws of Thought, Fourth ed., (1857), §§77ff. Peirce: CP 2.567 Fn 2 p 345 †2 An Introduction to Logical Science, (1857), §§38ff. Peirce: CP 2.580 Fn 1 p 348 †1 From article "Negative," Dictionary of Philosophy and Psychology, vol. 2, p. 148. Peirce: CP 2.581 Fn 2 p 348 †2 Ibid., vol. 2, p. 77, by Peirce and Mrs. C. Ladd-Franklin. Peirce: CP 2.582 Fn 1 p 349 †1 Ibid., vol. 2, pp. 330-31. Peirce: CP 2.583 Fn 1 p 350 †1 But see second edition (1755), London, under "Premiss" where it is said: "This word is rare in the singular." Peirce: CP 2.584 Fn 2 p 350 †2 Dictionary of Philosophy and Psychology, vol. 2, pp. 87-88. Peirce: CP 2.584 Fn 3 p 350 †3 Cf. Prantl, op. cit., III, 41. Peirce: CP 2.584 Fn 4 p 350 †4 Ibid., III, 27. Peirce: CP 2.584 Fn 5 p 350 †5 Ibid., III, 43. Peirce: CP 2.584 Fn 1 p 352

†1 Cf. Prantl, op. cit., III, 15-16. Peirce: CP 2.585 Fn 1 p 353 †1 Dictionary of Philosophy and Psychology, vol. 2, p. 435. Peirce: CP 2.586 Fn 2 p 353 †2 I.e., as not implying existence. Peirce: CP 2.587 Fn 1 p 354 †1 Vol. 3. Peirce: CP 2.588 Fn 2 p 354 †2 Dictionary of Philosophy and Psychology, vol. 2, pp. 1-2. Peirce: CP 2.590 Fn 1 p 355 †1 Ibid., vol. 2, p. 183. Peirce: CP 2.593 Fn 1 p 356 †1 Ibid., vol. 1, pp. 641-44. Peirce: CP 2.598 Fn 1 p 359 †1 I.e., the principle of contradiction asserts that "not" is an alio-relative. Peirce: CP 2.598 Fn 2 p 359 †2 I.e., the principle of excluded middle asserts that "not" is a concurrent. Peirce: CP 2.598 Fn 1 p 360 †1 Cf. 3.339. Peirce: CP 2.600 Fn 1 p 362 †1 Ch. 10, §1. Peirce: CP 2.601 Fn 2 p 362 †2 From article "Regular," Dictionary of Philosophy and Psychology, vol. 2, p. 439. Peirce: CP 2.602 Fn 3 p 362 †3 Ibid., vol. 2, p. 287. Peirce: CP 2.603 Fn 1 p 363 †1 Ibid., vol. 1, pp. 525-26. Peirce: CP 2.604 Fn 2 p 363 †2 This "logical sense of implication" corresponds to "entailment" and not to Philonian "material implication." It may be defined as the relation of a set of alternatives to the set which contains them. Peirce: CP 2.605 Fn 1 p 364 †1 Dictionary of Philosophy and Psychology, vol. 2, p. 198, by Peirce and James Mark Baldwin, the editor. Peirce: CP 2.607 Fn 2 p 364 †2 Ibid., vol. 2, p. 588. Peirce: CP 2.608 Fn 1 p 365 †1 Ibid., vol. 2, p. 206.

Peirce: CP 2.608 Fn 2 p 365 †2 Or non-symmetrical. See 3.136c. Peirce: CP 2.609 Fn 3 p 365 †3 Ibid., vol. 1, p. 529. Peirce: CP 2.612 Fn 1 p 366 †1 Ibid., vol. 2, p. 434. Peirce: CP 2.612 Fn 2 p 366 †2 Anal. pr. 25, 69a, 20. Peirce: CP 2.613 Fn 3 p 366 †3 Dictionary of Philosophy and Psychology, vol. 2, p. 181. Peirce: CP 2.613 Fn 4 p 366 †4 De Sophist. Elen. I, 5, 167a, 36. Peirce: CP 2.614 Fn 1 p 368 †1 Dictionary of Philosophy and Psychology, vol. 2, pp. 287-88, by Peirce and Mrs. C. Ladd-Franklin. Peirce: CP 2.614 Fn 2 p 368 †2 Anal. Prior., II, 16. Peirce: CP 2.615 Fn 1 p 369 †1 Dictionary of Philosophy and Psychology, vol. 2, p. 290. Peirce: CP 2.616 Fn 2 p 369 †2 Ibid., from article "Saltus," vol. 2, p. 484. Peirce: CP 2.617 Fn 3 p 369 †3 Ibid., vol. 2, p. 44. Peirce: CP 2.617 Fn 4 p 369 †4 Artis Logicae Rudimenta (1862), p. 131ff. Peirce: CP 2.618 Fn 1 p 370 †1 Dictionary of Philosophy and Psychology, vol. 1, p. 554. Peirce: CP 2.618 Fn 1 p 371 †1 For a further discussion of sophisms and insolubilia, see 5.33ff. Peirce: CP 2.618 Fn 2 p 371 †2 System der Logik. Peirce: CP 2.618 Fn 3 p 371 †3 See 3.446. Peirce: CP 2.619 Fn 1 p 372 †1 Popular Science Monthly, vol. 13, pp. 470-82 (1878); intended as Essay XIII of the Search for a Method (1893). It is the sixth and last of a series of papers on the "Illustrations of the Logic of Science," which appeared in the Popular Science Monthly. For the first and second papers, see vol. 5, bk. II, chs. 4 and 5; the third and fourth constitute chapters 6 and 7 of the present book; for the fifth paper, see vol. 6, bk. II, ch. 1.

Peirce: CP 2.631 Fn 1 p 378 †1 See, e.g., 515. Peirce: CP 2.635 Fn 1 p 380 †1 See vol. 5, bk. II, ch. 4, §5. Peirce: CP 2.638 Fn 1 p 383 †1 Novum Organum, bk. I, Aphorism X. Peirce: CP 2.639 Fn 1 p 384 †1 In his Hydrodynamica. Peirce: CP 2.640 Fn 1 p 385 †1 Cf. 511n. Peirce: CP 2.641 Fn P1 p 386 Cross-Ref:†† †P1 This division was first made in a course of lectures by the author before the Lowell Institute, Boston, in 1866, and was printed in the Proceedings of the American Academy of Arts and Sciences, for April 9, 1867. [See 508-12.] Peirce: CP 2.643 Fn 1 p 387 †1 See, e.g., the first paper, vol. 5, bk. II, ch. 4. Peirce: CP 2.643 Fn 1 p 388 †1 Cf. 712. Peirce: CP 2.644 Fn 2 p 388 †2 Cf. vol. 1, bk. II, ch. 2. Peirce: CP 2.645 Fn 1 p 389 †1 Popular Science Monthly, vol. 12, pp. 604-15 (1878) with corrections of 1893 and a note of 1910; intended as ch. 18 of the Grand Logic (1893), and as Essay X of the Search for a Method (1893), the third of a series of papers on "Illustrations of the Logic of Science." See notes to ch. 5 and 6.410. Peirce: CP 2.645 Fn P1 p 389 Cross-Ref:†† †P1 This characterization of chemistry now sounds antiquated indeed; and yet it was justified by the general state of mind of chemists at that day, as is shown by the fact that only a few months before, van't Hoff had put forth a statement of the law of mass-action as something absolutely new to science. I am satisfied by considerable search after pertinent facts that no distinction between different allied sciences can represent any truth of fact other than a difference between what habitually passes in the minds, and moves the investigations of the two general bodies of the cultivators of those sciences at the time to which the distinction refers.--1910. Peirce: CP 2.646 Fn 1 p 390 †1 Novum Organum, bk. II, Aphorism XXVII. Peirce: CP 2.646 Fn P1 p 391 Cross-Ref:†† †P1 "Or rather of an idea that continuity suggests--that of limitless intermediation; i.e., of a series between every two members of which there is another member of it"--to be substituted for the phrase "or . . . degrees."--1893. Peirce: CP 2.646 Fn P2 p 391 Cross-Ref:†† †P2 For "continuity" substitute "limitless intermediation, the business of

reasoning."--1893. Peirce: CP 2.646 Fn P3 p 391 Cross-Ref:†† †P3 "And others that are involved in that of continuity."--1893. Peirce: CP 2.646 Fn P4 p 391 Cross-Ref:†† †P4 For "neglect of" substitute "want of close study of these concepts."--1893. Peirce: CP 2.646 Fn P5 p 391 Cross-Ref:†† †P5 This mode of thought is so familiarly associated with all exact numerical consideration, that the phrase appropriate to it is imitated by shallow writers in order to produce the appearance of exactitude where none exists. Certain newspapers, which affect a learned tone, talk of "the average man," when they simply mean most men, and have no idea of striking an average. Peirce: CP 2.646 Fn 1 p 391 †1 See, e.g., 1.383. Peirce: CP 2.648 Fn 1 p 393 †1 See vol. 5, bk. II, ch. 5. Peirce: CP 2.649 Fn 2 p 393 †2 Essay, bk. IV, ch. 15, §1. Peirce: CP 2.649 Fn 3 p 393 †3 See vol. 5, bk. II, ch. 4, §2. Peirce: CP 2.650 Fn 1 p 394 †1 See vol. 5, bk. II, ch. 5, §4. Peirce: CP 2.651 Fn P1 p 395 Cross-Ref:†† †P1 The conception of probability here set forth is substantially that first developed by Mr. Venn, in his Logic of Chance. Of course, a vague apprehension of the idea had always existed, but the problem was to make it perfectly clear, and to him belongs the credit of first doing this. Peirce: CP 2.654 Fn P1 p 399 Cross-Ref:†† †P1 I do not here admit an absolutely unknowable. Evidence could show us what would probably be the case after any given lapse of time; and though a subsequent time might be assigned which that evidence might not cover, yet further evidence would cover it. Peirce: CP 2.660 Fn 1 p 404 †1 Ch. 7. Peirce: CP 2.661 Fn 2 p 404 †2 1910. Peirce: CP 2.661 Fn 3 p 404 †3 See vol. 5, bk. II, chs. 1, 2, 3, particularly 5.355. Peirce: CP 2.662 Fn 1 p 406 †1 Cf. 111, 269, 756f. Peirce: CP 2.662 Fn 1 p 407 †1 See Carrington's Eusapia Palladino, B. W. Dodge & Co., New York

(1909). Peirce: CP 2.663 Fn 1 p 408 †1 Peirce read Whately's Logic at this time. Peirce: CP 2.667 Fn P1 p 413 Cross-Ref:†† †P1 Meantime it may be remarked that, though an endless series of acts is not a habit, nor a would-be, it does present the first of an endless series of steps toward the full nature of a would-be. Compare what I wrote nineteen [thirteen!] years ago, in an article on the logic of relatives [3.526ff]. Peirce: CP 2.669 Fn 1 p 415 †1 Popular Science Monthly, vol. 12, pp. 705-18 (1878), the fourth of a series of papers on "Illustrations of the Logic of Science." See 612n. Intended as Essay XI of the Search for a Method (1893). Peirce: CP 2.674 Fn 1 p 417 †1 Cf. 3.17. Peirce: CP 2.677 Fn P1 p 421 Cross-Ref:†† †P1 Strictly we should need an infinite series of numbers each depending on the probable error of the last. Peirce: CP 2.679 Fn P1 p 422 Cross-Ref:†† †P1 "Perfect indecision, belief inclining neither way, an even chance."--De Morgan, p. 182. Peirce: CP 2.682 Fn 1 p 424 †1 Théorie des Probabilités, deuxième partie, §1. Peirce: CP 2.684 Fn 1 p 426 †1 Cf. vol. 6, bk. II, ch. 1, §2. Peirce: CP 2.684 Fn 2 p 426 †2 See 692. Peirce: CP 2.690 Fn P1 p 430 Cross-Ref:†† †P1 Logique. The same is true, according to him, of every performance of a differentiation, but not of integration. He does not tell us whether it is the supernatural assistance which makes the former process so much the easier. Peirce: CP 2.694 Fn 1 p 433 †1 The Johns Hopkins Studies in Logic, edited by C. S. Peirce, Little Brown and Co., Boston (1883), pp. 126-181; intended as Essay XIV of the Search for a Method (1893). Peirce: CP 2.694 Fn 2 p 433 †2 The headings of these sections were made by Peirce in his own copy of the Johns Hopkins Studies. Peirce: CP 2.696 Fn 1 p 436 †1 Bk. IV, ch. 15, §1. Peirce: CP 2.699 Fn P1 p 439 Cross-Ref:†† †P1 In case (n + 1)r is a whole number, q has equal values for m = (n + 1)r and for m = (n + 1)r - 1.

Peirce: CP 2.702 Fn 1 p 441 †1 There was no §3 in the original, and the present section formed part of §2. Peirce: CP 2.702 Fn 2 p 441 †2 "these" is deleted in Peirce's own copy. Peirce: CP 2.705 Fn P1 Para 1/3 p 444 Cross-Ref:†† †P1 When r = 0, the last form becomes

M has all the marks P, S has no mark of M; Hence, S has none of the marks P.

Peirce: CP 2.705 Fn P1 Para 2/3 p 444 Cross-Ref:†† When the universe of marks is unlimited (see a note appended to this paper for an explanation of this expression [519]), the only way in which two terms can fail to have a common mark is by their together filling the universe of things; and consequently this form then becomes

M is P, Every non-S is M; Hence, every non-S is P.

This is one of De Morgan's syllogisms. Peirce: CP 2.705 Fn P1 Para 3/3 p 444 Cross-Ref:†† In putting r = 0 in Form II (bis) it must be noted that, since P is simple in depth, to say that S is not P is to say that it has no mark of P. Peirce: CP 2.707 Fn 1 p 445 †1 Cf. 102. Peirce: CP 2.710 Fn P1 p 447 Cross-Ref:†† †P1 I do not here speak of Mr. Jevons, because my objection to the copula of identity is of a somewhat different kind. [See Studies in Logic, pp. 17-69 and 72-106 for Miss Ladd's and Mr. Mitchell's papers.] Peirce: CP 2.711 Fn P1 p 448 Cross-Ref:†† †P1 See my paper on "How to make our ideas clear." [Vol. 5. bk. II, ch. 5.] Peirce: CP 2.712 Fn 1 p 449 †1 Cf. 643. Peirce: CP 2.718 Fn P1 p 451 Cross-Ref:†† †P1 From apagoge, {apagögé eis to adynaton}, Aristotle's name for the

reductio ad absurdum. Peirce: CP 2.720 Fn P1 p 452 Cross-Ref:†† †P1 The conclusion of the statistical deduction is here regarded as being "the proportion r of the S's are P's," and the words "probably about" as indicating the modality with which this conclusion is drawn and held for true. It would be equally true to consider the "probably about" as forming part of the contents of the conclusion; only from that point of view the inference ceases to be probable, and becomes rigidly necessary, and its apagogical inversion is also a necessary inference presenting no particular interest. Peirce: CP 2.729 Fn P1 p 456 Cross-Ref:†† †P1 This I conceive to be all the truth there is in the doctrine of Bacon and Mill regarding different Methods of Experimental Inquiry. The main proposition of Bacon's and Mill's doctrine is, that in order to prove that all M's are P's, we should not only take random instances of the M's and examine them to see that they are P's, but we should also take instances of not-P's and examine them to see that they are not-M's. This is an excellent way of fortifying one induction by another, when it is applicable; but it is entirely inapplicable when r has any other value than 1 or 0. For, in general, there is no connection between the proportion of M's that are P's and the proportion of non-P's that are non-M's. A very small proportion of calves may be monstrosities, and yet a very large proportion of monstrosities may be calves. Peirce: CP 2.731 Fn P1 p 458 Cross-Ref:†† †P1 I say about, because the doctrine of probability only deals with approximate evaluations. Peirce: CP 2.733 Fn P1 p 460 Cross-Ref:†† †P1 That this is really a correct analysis of the reasoning can be shown by the theory of probabilities. For the expression

(p + q)! (π + {r})!

(p + π)!(q + {r})!

-------- ----------- · -------------------p!q!

π!{r}!

(p + π + q + {r})!

expresses at once the probability of two events; namely, it expresses first the probability that of p + q objects drawn without replacement from a lot consisting of p + π objects having the character R together with q + {r} not having this character, the number of those drawn having this character will be p; and second, the same expression denotes the probability that if among p + π + q + {r} objects drawn at random from an infinite class (containing no matter what proportion of R's to non-R's), it happens that p + π have the character R, then among any p + q of them, designated at random, p will have the same character. Thus we see that the chances in reference to drawing without replacement from a finite class are precisely the same as those in reference to a class which has been drawn at random from an infinite class. Peirce: CP 2.738 Fn 1 p 462 †1 Wheeler's Biographical Dictionary.

Peirce: CP 2.738 Fn P1 p 464 Cross-Ref:†† †P1 The author ought to have noted that this number is open to some doubt, since the specific gravity of this form of silicon appears to vary largely. If a different value had suited the theory better, he might have been able to find reasons for preferring that other value. But I do not mean to imply that Dr. Playfair has not dealt with perfect fairness with his facts, except as to the fallacy which I point out. Peirce: CP 2.738 Fn P1 p 465 Cross-Ref:†† †P1 As the relations of the different powers of the specific gravity would be entirely different if any other substance than water were assumed as the standard, the law is antecedently in the highest degree improbable. This makes it likely that some fallacy was committed, but does not show what it was. Peirce: CP 2.739 Fn P1 p 466 Cross-Ref:†† †P1 The physicians seem to use the maxim that you cannot reason from post hoc to propter hoc to mean (rather obscurely) that cases must not be used to prove a proposition that has only been suggested by these cases themselves. Peirce: CP 2.741 Fn 1 p 468 †1 See Theodor Gomperz, Herculanische Studien, pt. I (1865). Cf. 761. Peirce: CP 2.744 Fn P1 p 469 Cross-Ref:†† †P1 See Laplace, Théorie Analitique des Probabilités, [1812], livre ii, ch. vi. Peirce: CP 2.746 Fn P1 p 471 Cross-Ref:†† †P1 See Boole, Laws of Thought, p. 370. Peirce: CP 2.749 Fn 1 p 472 †1 See La Logique, Paris (1855), vol. 2, pp. 196-97. Peirce: CP 2.750 Fn 1 p 474 †1 See vol. 6, bk. II, ch. 1, 2. Peirce: CP 2.752 Fn P1 p 475 Cross-Ref:†† †P1 The different regions with the January rainfall are taken from Mr. Schott's work. [Tables and Results of the Precipitation in Rain and Snow in the United States, 1872.] The percentage of illiteracy is roughly estimated from the numbers given in the Report of the 1870 Census. [The maps originally published with this paper have not been considered worth reproducing.] Peirce: CP 2.753 Fn 1 p 477 †1 Cf. 1.118. Peirce: CP 2.755 Fn 1 p 478 †1 From ms. "G." c. 1905. Peirce: CP 2.756 Fn P1 p 479 Cross-Ref:†† †P1 When I say that it goes on that presumption, I merely mean to describe the presumption of the reasoning as being that the particular uniformity as to a certain matter that has attached to past experience will be maintained in the future. I shall explain below how there is a certain justification in this, though a very slender one. I do not mean to say, as some logicians do, that the force of the induction is just the same as that of a syllogism whose major premiss should be, "Future experience will not violate the uniformity of past experience." For such a syllogism being a fallacy of

the particularly atrocious kind called "logical fallacy," would have no justification whatsoever. For a sound syllogism must have a major premiss of definite meaning: otherwise it may be thrown into the form of a fallacy of undistributed middle. Now the induction in question, though weak, is by no means without justification. Peirce: CP 2.756 Fn 1 p 479 †1 Cf. 111, 269, 662, 757n. Peirce: CP 2.757 Fn P1 p 480 Cross-Ref:†† †P1 Induction is such a way of inference that if one persists in it one must necessarily be led to the truth, at last. It is true that this condition is most imperfectly fulfilled in the Pooh-pooh argument. For here the unexpected, when it comes, comes with a bang. But then, on the other hand, until the fatal day arrives, this argument causes us to anticipate just what does happen and prevents us from anticipating a thousand things that do not happen. I engage a stateroom; I purchase a letter of credit for fifty thousand dollars, and I start off determined to have a good time. On the way down the bay, my wife says to me, "Aren't you afraid the house may be struck by lightning while we are gone?" Pooh-pooh! "But aren't you afraid there will be a war and Boston will be bombarded?" Pooh-pooh! "But aren't you afraid that when we are in the heart of Hungary or somewhere you will get the Asiatic plague, and I shall be left unable to speak the language?" Pooh-pooh! On the morning of the fourth day out there is a terrific explosion and I find myself floating about on the middle of the Atlantic with my letter of credit safe in my breast pocket. I say to myself, my Pooh-pooh argument broke down that time sure enough, but after all it made my mind easy about a number of possibilities that did not occur, and even about this one for three days. So I had better be content with my lot. This little parable is intended to illustrate how even the Pooh-pooh argument, the weakest of all sound induction, does satisfy the essential condition of saving me from surprises both positive and negative; that is from the happening of things not anticipated and the non-occurrence of imaginary disasters.--From a fragment c. 1902. Peirce: CP 2.758 Fn 1 p 480 †1 Cf. The Grammar of Science (1892); The Chances of Death (1897). Peirce: CP 2.760 Fn 1 p 483 †1 "On the Natural Classification of Arguments," ch. 2. Peirce: CP 2.760 Fn 2 p 483 †2 "Some Consequences of Four Incapacities," vol. 5, bk. II, ch. 2. Peirce: CP 2.760 Fn 3 p 483 †3 "Deduction, Induction, and Hypothesis," ch. 5. Peirce: CP 2.760 Fn 4 p 483 †4 "A Theory of Probable Inference," ch. 8. Peirce: CP 2.760 Fn 5 p 483 †5 "The Doctrine of Necessity Examined," vol. 6, bk. I, ch. 2. Peirce: CP 2.760 Fn 6 p 483 †6 "Pearson's Grammar of Science," vol. 9. Peirce: CP 2.761 Fn 1 p 485 †1 Theodor Gomperz, Philodemi de ira liber (1864); Herkulanische Studien (1865-6).

Peirce: CP 2.765 Fn 1 p 489 †1 Bk. III, ch. 18. Peirce: CP 2.772 Fn 1 p 492 †1 See volumes 7 and 8. Peirce: CP 2.773 Fn 1 p 495 †1 Dictionary of Philosophy and Psychology, vol. 2, pp. 426-28. Peirce: CP 2.779 Fn 1 p 499 †1 Ibid., vol. 2, pp. 748-49, by Peirce and Mrs. C. Ladd-Franklin. Peirce: CP 2.782 Fn 1 p 502 †1 Ibid., vol. 2, p. 359. Peirce: CP 2.783 Fn 2 p 502 †2 Ibid., vol. 2, pp. 353-55. Peirce: CP 2.788 Fn 1 p 506 †1 Ibid., vol. 2, pp. 324-25. Peirce: CP 2.791 Fn 1 p 507 †1 Ibid., vol. 2, p. 337. Peirce: CP 2.792 Fn 1 p 508 †1 Privately printed and "distributed at the Lowell Institute, Nov. 1866." Cf. Bk. III, ch. 2, Part II. Peirce: CP 2.800 Fn P1 p 513 Cross-Ref:†† †P1 "Ubicunque ponitur s significatur quod propositio . . . debet converti simpiciter . . . et ubicunque ponitur m debet fieri transpositio in præmnissis."--Petrus Hisp.

Peirce: CP 3 Title-Page COLLECTED PAPERS OF CHARLES SANDERS PEIRCE

EDITED BY CHARLES HARTSHORNE AND PAUL WEISS VOLUME III EXACT LOGIC (Published Papers)

CAMBRIDGE HARVARD UNIVERSITY PRESS 1933 Peirce: CP 3 Copyright Page COPYRIGHT, 1933 BY THE PRESIDENT AND FELLOWS OF HARVARD COLLEGE PRINTED IN UNITED STATES OF AMERICA THE MURRAY PRINTING COMPANY CAMBRIDGE, MASSACHUSETTS

Peirce: CP 3 Introduction p iii INTRODUCTION

In the editing of the present volume Peirce's punctuation and spelling have, wherever possible, been retained. Titles supplied by the editors have been marked E; and their remarks and additions are enclosed in light-faced brackets. The editors' footnotes are indicated by various typographical signs, while Peirce's footnotes are indicated by numbers. Paragraphs are numbered consecutively throughout the volume. The numbers at the top of each page signify the volume and the first paragraph of that page. All references in the text and in the indices are to the numbers of the paragraphs. Peirce: CP 3 Introduction p iii The department and the editors desire to express their gratitude to Dr. Henry S. Leonard, who has assisted with the proofs, references and editorial footnotes.

HARVARD UNIVERSITY February, 1933

Peirce: CP 3 Editorial Note p v EDITORIAL NOTE

Charles Sanders Peirce was one of the most original and prolific logicians of the nineteenth century. His published papers contain important contributions to almost every phase of the subject. He radically modified, extended and transformed the Boolean algebra, making it applicable to propositions, relations, probability and arithmetic. Following De Morgan, he was one of the chief contributors to the logic of

relatives. In addition to an analysis of "second intentional" relatives and a detailed classification of the main species of "first intentional" relatives, he supplied most of the fundamental theorems and distinctions in this branch of logic, providing, incidentally, two distinct algebras in terms of which they could be treated. He indicated how arithmetic, multiple algebras and quaternions could be derived from logic, made an independent discovery of the propositional function, of material and formal implication, and invented a new kind of logical diagram. Other discoveries which he did not publish will be found in volume four. Peirce: CP 3 Editorial Note p v Peirce's symbolism and mode of procedure is somewhat antiquated and in many places his thought is difficult to grasp. The following selected list of important topics, together with the explanatory footnotes to the text, and the index at the end of the volume should, however, aid even the general reader to extract what is still living and important in Peirce's work. The items mentioned in the following list do not exhaust either the number of Peirce's contributions or the topics on which he will prove illuminating; they are offered solely as a guide through the mazes of his symbolism. For one unfamiliar with the history of the subject, or the technicalities of modern logic, the easiest approach is by way of paper No. XX, which consists of articles published in Baldwin's Dictionary of Philosophy and Psychology, to be followed by papers No. XIII to XV; after which the rest may be read in chronological order.

Peirce: CP 3 Topics p vi TOPICS OF HISTORICAL INTEREST

Basic formulae Logical addition Logical subtraction The definition of 0 Logical division The definition of 1 The transition from logical identities to arithmetical equalities The frequency theory of probability The criticism of Boole

The definition of arithmetical multiplication and independence The use of Σ to express a logical sum Some fundamental theorems at the foundation of arithmetic A difference between transfinite and finite arithmetic

Number as a class of classes The relation of class inclusion Kinds of addition and multiplication Kinds of logical terms The triple relative Extension of the logic of relatives to probability The "logical binomial" The variable Involution The logical quaternion The basis of linear associative algebras The alio-relative Classification of relatives Formulation of hypotheticals indifferently as implications, disjunctions and conjunctions Particular propositions as denials of universals Operations as triple relatives The properties of the converse The nature of logical propositions Associative and relative algebra The theory of the leading principle Formal implication The negation of implication Equivalence A, E, I, O as implications The new "square" of opposition The extension of Aristotelian logic The proof of the distributive principle The antilogism The block of relatives Second intensional relatives

Transaddition The relation of quantity Kinds of system The number system One-to-one correspondence Counting Syllogism of transposed quantity

Quaternions as relatives

Extensional definition of relatives Relative addition and multiplication The general logic of relatives Multiple quantifiers Intermediate quantifiers

The law converting a logic of classes into a logic of propositions Truth-values Symbolic representation of the syllogism Material implication Numerical quantifiers Identity Finite classes

The "propositional function"

Criticism of Schröder

The entitative graphs The logic of quantity

2n is always greater than n Kinds of multitudes

Two postulates for ordered sequence

Detailed classification of relatives

Identity and equality Meaning of probability Logical notation Logical simplicity Absolute terms as relatives Mathematical demonstration The variable Geometry and logic The nature of logical propositions The nature of inference Leading and logical principles Term, proposition and inference The principle of particularity Operations in logical algebra Second intensional relatives Arithmetical propositions Kinds of signs Multiple quantifiers Material implication Function of diagrams The "propositional function" The nature of logic and mathematics The nature of assertion

The nature of propositions The nature of relations

TOPICS OF GENERAL INTEREST

The meaning of unity The frequency theory of probability Relation of equality and identity Meaning of simplicity Impossibility of a fourth category The nature of mathematics Criticism of logical atomism Analysis of the nature of individuals Scholastic realism Nature of the variable Non-Euclidean geometry and the apriority of space The nature of the logical proposition

Physiological explanation of habit formation, belief, judgment, thought, inference The faith of the logician The nature of inference The nature of simples The propositions of arithmetic Different kinds of signs Possibility Relation Speech, meaning; diagrams A classification of the sciences Function of proof Three grades of clearness The logical function of different parts of speech Hecceity and living ideas

The proposition Doctrine of logical valency Division of possible problems Mathematical points, continuity, infinitesimals

BRYN MAWR COLLEGE December, 1932.

Peirce: CP 3 Contents p xi CONTENTS

INTRODUCTION EDITORIAL NOTE

Paper Paragraph Numbers Peirce: CP 3 Contents Paper 1 p xi I. ON AN IMPROVEMENT IN BOOLE'S CALCULUS OF LOGIC (1867)

Peirce: CP 3 Contents Paper 2 p xi II. UPON THE LOGIC OF MATHEMATICS (1867) 1. The Boolian Calculus 2. On Arithmetic

20

42

Peirce: CP 3 Contents Paper 3 p xi III. DESCRIPTION OF A NOTATION FOR THE LOGIC OF RELATIVES, RESULTING FROM AN AMPLIFICATION OF THE CONCEPTIONS OF BOOLE'S CALCULUS OF LOGIC (1870) 1. De Morgan's Notation

45

2. General Definitions of the Algebraic Signs

47

3. Application of the Algebraic Signs to Logic

62

4. General Formulæ 81

1

5. General Method of Working with this Notation 6. Properties of Particular Relative Terms

89

135

Peirce: CP 3 Contents Paper 4 p xi IV. ON THE APPLICATION OF LOGICAL ANALYSIS TO MULTIPLE ALGEBRA (1875) 150

Peirce: CP 3 Contents Paper 5 p xi V. NOTE ON GRASSMANN'S CALCULUS OF EXTENSION (1877)

152

Peirce: CP 3 Contents Paper 6 p xi VI. ON THE ALGEBRA OF LOGIC (1880) Part I. Syllogistic 1. Derivation of Logic154 2. Syllogism and Dialogism 162 3. Forms of Propositions

173

4. The Algebra of the Copula 182

Part II. The Logic of Non-Relative Terms 1. The Internal Multiplication and the Addition of Logic

198

2. The Resolution of Problems in Non-Relative Logic

204

Part III. The Logic of Relatives 1. Individual and Simple Terms 2. Relatives

214

218

3. Relatives connected by Transposition of Relate and Correlate 4. Classification of Relatives 225 5. The Composition of Relatives

236

6. Methods in the Algebra of Relatives

245

7. The General Formulæ for Relatives

248

Peirce: CP 3 Contents Paper 7 p xii VII. ON THE LOGIC OF NUMBER (1881)

223

1. Definition of Quantity 2. Simple Quantity

252

255

3. Discrete Quantity 257 4. Semi-infinite Quantity

260

5. Discrete Simple Quantity Infinite in both Directions

272

6. Limited Discrete Simple Quantity 280

Peirce: CP 3 Contents Paper 8 p xii VIII. ASSOCIATIVE ALGEBRAS (1881) 1. On the Relative Forms of the Algebras

289

2. On the Algebras in which Division is Unambiguous

297

Peirce: CP 3 Contents Paper 9 p xii IX. BRIEF DESCRIPTION OF THE ALGEBRA OF RELATIVES (1882)306

Peirce: CP 3 Contents Paper 10 p xii X. ON THE RELATIVE FORMS OF QUATERNIONS (1882)

Peirce: CP 3 Contents Paper 11 p xii XI. ON A CLASS OF MULTIPLE ALGEBRAS (1882)

Peirce: CP 3 Contents Paper 12 p xii XII. THE LOGIC OF RELATIVES (1883)

323

324

328

Peirce: CP 3 Contents Paper 13 p xiii XIII. ON THE ALGEBRA OF LOGIC: A CONTRIBUTION TO THE PHILOSOPHY OF NOTATION (1885) 1. Three Kinds of Signs

359

2. Non-Relative Logic365 3. First-Intentional Logic of Relatives 4. Second-Intentional Logic 398 5. Note403A

392

Peirce: CP 3 Contents Paper 14 p xiii XIV. THE CRITIC OF ARGUMENTS (1892) 1. Exact Thinking

404

2. The Reader is Introduced to Relatives

415

Peirce: CP 3 Contents Paper 15 p xiii XV. THE REGENERATED LOGIC (1896) 425

Peirce: CP 3 Contents Paper 16 p xiii XVI. THE LOGIC OF RELATIVES (1897) 1. Three Grades of Clearness 456 2. Of the Term Relation in its First Grade of Clearness 3. Of Relation in the Second Grade of Clearness

464

4. Of Relation in the Third Grade of Clearness

468

5. Triads, the Primitive Relatives

483

6. Relatives of Second Intention

488

458

7. The Algebra of Dyadic Relatives 492 8. General Algebra of Logic 499 9. Method of Calculating with the General Algebra 503 10. Schröder's Conception of Logical Problems

510

11. Professor Schröder's Pentagrammatical Notation520 12. Professor Schröder's Iconic Solution of x∠φx 13. Introduction to the Logic of Quantity

523

526

Peirce: CP 3 Contents Paper 17 p xiii XVII. THE LOGIC OF MATHEMATICS IN RELATION TO EDUCATION (1898) 1. Of Mathematics in General553 2. Of Pure Number

562A

Peirce: CP 3 Contents Paper 18 p xiii XVIII. INFINITESIMALS (1900) 563

Peirce: CP 3 Contents Paper 19 p xiv XIX. NOMENCLATURE AND DIVISIONS OF DYADIC RELATIONS (1903) 1. Nomenclature

571

2. First System of Divisions 578 3. Second System of Divisions

583

4. Third System of Divisions 588 5. Fourth System of Divisions

601

6. Note on the Nomenclature and Divisions of Modal Dyadic Relations

Peirce: CP 3 Contents Paper 13 p xiv XX. NOTES ON SYMBOLIC LOGIC AND MATHEMATICS (1901 and 1911) 1. Imaging

609

2. Individual 611 3. Involution 614 4. Logic (exact)

616

5. Multitude (in mathematics)626 6. Postulate

632

7. Presupposition 8. Relatives

635

636

9. Transposition

644

Peirce: CP 3 Contents Paper 20 p xiv APPENDIX. ON NONIONS 646

Peirce: CP 3.1 Cross-Ref:†† EXACT LOGIC

(Previously published papers)

ON AN IMPROVEMENT IN BOOLE'S CALCULUS OF LOGIC†1

606

1. The principal use of Boole's Calculus of Logic lies in its application to problems concerning probability. It consists, essentially, in a system of signs to denote the logical relations of classes. The data of any problem may be expressed by means of these signs, if the letters of the alphabet are allowed to stand for the classes themselves. From such expressions, by means of certain rules for transformation, expressions can be obtained for the classes (of events or things) whose frequency is sought in terms of those whose frequency is known. Lastly, if certain relations are known between the logical relations and arithmetical operations, these expressions for events can be converted into expressions for their probability. Peirce: CP 3.1 Cross-Ref:†† It is proposed, first, to exhibit Boole's system in a modified form, and second, to examine the difference between this form and that given by Boole himself. Peirce: CP 3.2 Cross-Ref:†† 2. Let the letters of the alphabet denote classes whether of things or of occurrences. It is obvious that an event may either be singular, as "this sunrise," or general, as "all sunrises." Let the sign of equality with a comma beneath it express numerical identity. Thus a =, b is to mean that a and b denote the same class--the same collection of individuals. Peirce: CP 3.3 Cross-Ref:†† 3. Let a +, b denote all the individuals contained under a and b together.†2 The operation here performed will differ from arithmetical addition in two respects: first, that it has reference to identity, not to equality; and second, that what is common to a and b is not taken into account twice over, as it would be in arithmetic. The first of these differences, however, amounts to nothing, inasmuch as the sign of identity would indicate the distinction in which it is founded; and therefore we may say that

(1)

If No a is b

a +, b =, a + b.†1

It is plain that

(2)

a +, a =, a†2

and also, that the process denoted by +, , and which I shall call the process of logical addition, is both commutative and associative. That is to say

(3)

and

a +, b =, b +, a

(4)

(a +, b) +, c =, a +, (b +, c).

Peirce: CP 3.4 Cross-Ref:†† 4. Let a, b denote the individuals contained at once under the classes a and b; those of which a and b are the common species. If a and b were independent events, a, b would denote the event whose probability is the product of the probabilities of each. On the strength of this analogy (to speak of no other), the operation indicated by the comma may be called logical multiplication. It is plain that

(5)

a, a =, a.†2

Peirce: CP 3.4 Cross-Ref:†† Logical multiplication is evidently a commutative and associative process. That is,

(6)

(7)

a,b =, b,a

(a,b),c =, a,(b,c).

Peirce: CP 3.4 Cross-Ref:†† Logical addition and logical multiplication are doubly distributive, so that

(8)

(a +, b),c =, a,c +, b,c

and

(9)

a,b +, c =, (a +, c),(b +, c).

Proof. Let a =, a'+x+y+o b =, b'+x+z+o c =, c'+y+z+o

where any of these letters may vanish. These formulæ comprehend every possible

relation of a, b and c; and it follows from them, that

a +, b =, a'+b'+x+y+z+o

(a +, b),c =, y+z+o.

But

a,c =, y+o b,c =, z+o a,c +, b,c =, y+z+o .·. (8).

So a,b =, x+o a,b +, c =, c'+x+y+z+o. But

(a +, c) = a'+c'+x+y+z+o (b +, c) =, b'+c'+x+y+z+o (a +, c),(b +, c) =, c'+x+y+z+o .·. (9).

Peirce: CP 3.5 Cross-Ref:†† 5. Let -, be the sign of logical subtraction; so defined that

(10)

If b +, x =, a

x =, a -, b.

Peirce: CP 3.5 Cross-Ref:†† Here it will be observed that x is not completely determinate. It may vary from a to a with b taken away. This minimum may be denoted by a-b.†1 It is also to be observed that if the sphere of b reaches at all beyond a, the expression a -, b is uninterpretable.†2 If then we denote the contradictory negative of a class by the letter which denotes the class itself, with a line above it,†P1 if we denote by v a wholly indeterminate class, and if we allow (0 -, 1) to be a wholly uninterpretable symbol, we have

(11)

a -, b =, v,a,b+a,~b+[0 -, 1],~a,b†3

which is uninterpretable unless

~a,b =, 0.

If we define zero by the following identities, in which x may be any class whatever,

(12)

0 =, x -, x =, x - x

then, zero denotes the class which does not go beyond any class,†1 that is nothing or nonentity. Peirce: CP 3.6 Cross-Ref:†† 6. Let a;b be read a logically divided by b, and be defined by the condition that

(13)

If b,x =, a

x =, a;b

x is not fully determined by this condition. It will vary from a to a + ~b and will be uninterpretable if a is not wholly contained under b.†2 Hence, allowing (1;0) to be some uninterpretable symbol,

(14)

a;b =, a,b+v,~a,~b+(1;0)a,~b†3

which is uninterpretable unless a,~b =, 0. Peirce: CP 3.7 Cross-Ref:†† 7. Unity may be defined by the following identities in which x may be any class whatever.

(15)

1 =, x;x =, x:x.†4

Peirce: CP 3.7 Cross-Ref:†† Then unity denotes the class of which any class is a part; that is, what is or ens. Peirce: CP 3.8 Cross-Ref:†† 8. It is plain that if for the moment we allow a:b to denote the maximum value of a;b, then

(16)

~x =, 1-x =, 0:x.†5

So that

(17)

x,(1-x) =, 0

x +, 0:x =, 1.

Peirce: CP 3.9 Cross-Ref:†† 9. The rules for the transformation of expressions involving logical subtraction and division would be very complicated. The following method is, therefore, resorted to.†1 Peirce: CP 3.9 Cross-Ref:†† It is plain that any operations consisting solely of logical addition and multiplication, being performed upon interpretable symbols, can result in nothing uninterpretable. Hence, if φ+Xx signifies such an operation performed upon symbols of which x is one, we have

φ+Xx =, a,x+b,(1-x)†2

where a and b are interpretable. Peirce: CP 3.9 Cross-Ref:†† It is plain, also, that all four operations being performed in any way upon any symbols, will, in general, give a result of which one term is interpretable and another not; although either of these terms may disappear. We have then

φx =, i,x+j,(1-x).†P1

Peirce: CP 3.9 Cross-Ref:†† We have seen that if either of these coefficients i and j is uninterpretable, the other factor of the same term is equal to nothing, or else the whole expression is uninterpretable. But

φ(1) =, i and φ(0) =, j.

Hence

(18)

φx =, φ(1),x + φ(0),(1-x)†P2

φ(x and y) =, φ(1 and 1),x,y + φ(1 and 0),x,~y + φ(0 and 1),~x,y + φ(0 and 0),~x,~y.

(18')

φx =, (φ(1)+,~x),(φ(0) +, x)2

φ(x and y) =, (φ(1 and 1) +, ~x +, ~y),(φ(1 and 0) +, ~x, +, y), (φ(0 and 1) +, x +, ~y),(φ(0 and 0) +, x +, y).

Developing by (18) x -, y, we have,

x -, y =, (1 -, 1),x,y+(1 -, 0),x,~y+(0 -, 1),~x,y+(0 -, 0),~x,~y.

So that, by (11),

(19) (1 -, 1) =, v. 1 -, 0 =, 1. 0 -, 1 =, (0 -, 1). 0 -, 0 =, 0.

Peirce: CP 3.10 Cross-Ref:†† 10. Developing x;y in the same way, we have †P1

x;y =, 1;1,x,y+1;0,x,~y+0;1,~x,y+0;0,~x,~y.

So that, by (14),

(20) 1;1 =,1 1;0 =, (1;0) 0;1 =, 0 0;0 =, v.

Boole gives (20),†1 but not (19). Peirce: CP 3.10 Cross-Ref:†† In solving identities we must remember that

(21)

(a +, b) - b =, a

(22)

(a -, b) +, b =, a.

From a -, b the value of b cannot be obtained.

(23)

(a,b) ⎫ b =, a

(24)

a;b,b =, a.

From a;b the value of b cannot be determined. Peirce: CP 3.11 Cross-Ref:†† 11. Given the identity φx =, 0. Required to eliminate x.

φ(1) =, x,φ(1)+(1-x),φ(1) φ(0) =, x,φ(0)+(1-x),φ(0).

Logically multiplying these identities, we get

φ(1),φ(0) =, x,φ(1),φ(0)+(1-x),φ(1),φ(0).

For two terms disappear because of (17). But we have, by (18),

φ(1),x+φ(0),(1-x) =, φx =, 0.

Multiplying logically by x we get

φ(1),x =, 0

and by (1-x) we get

φ(0),(1-x) =, 0.

Substituting these values above, we have

φ(1),φ(0) =, 0 when φx =, 0.

(25)

Peirce: CP 3.12 Cross-Ref:†† 12. Given φx =, 1. Required to eliminate x. φ'x =, 1-φx =, 0

Let

φ'(1),φ'(0) =, (1- φ(1)),(1-φ(0)) =, 0 1-(1-φ(1)),(1-φ(0)) =, 1.

Now, developing as in (18), only in reference to φ(1) and φ(0) instead of to x and y,

1-(1-φ(1)),(1-φ(0)) =, φ(1),φ(0)+φ(1),(1-φ(0)) + φ(0),(1-φ(1)).

But by (18) we have also,

φ(1) +, φ(0) =, φ(1),φ(0)+φ(1),(1-φ(0))+φ(0),(1-φ(1)).

So that

(26)

φ(1) +, φ(0) =, 1 when φx =, 1.

Boole gives (25),†1 but not (26). Peirce: CP 3.13 Cross-Ref:†† 13. We pass now from the consideration of identities to that of equations.†2 Peirce: CP 3.13 Cross-Ref:†† Let every expression for a class have a second meaning, which is its meaning in an equation. Namely, let it denote the proportion of individuals of that class to be found among all the individuals examined in the long run. Peirce: CP 3.13 Cross-Ref:†† Then we have

(27)

If a =, b

a=b

(28)

a + b = (a +, b) + (a,b).

Peirce: CP 3.14 Cross-Ref:†† 14. Let b[a] denote the frequency of b's among the a's. Then considered as a class, if a and b are events, b[a] denotes the fact that if a happens b happens.

(29)

a b[a] = a,b.†3

It will be convenient to set down some obvious and fundamental properties of the function b[a].

(30)

a b[a] = b a[b]

(31) φ(b[a] and c[a]) = (φ(b and c))[a]

(32)

(1-b)[a] = 1-b[a]

(33)

b[a] = b/a + b[(1-a)](1-1/a)

(34)

a[b] = 1 - (1-a/b) b[(1-a)]

(35)

(φa)[a] = (φ(1))[a].

Peirce: CP 3.14 Cross-Ref:†† The application of the system to probabilities may best be exhibited in a few simple examples, some of which I shall select from Boole's work, in order that the solutions here given may be compared with his. Peirce: CP 3.15 Cross-Ref:†† 15. Example 1. Given the proportion of days upon which it hails, and the proportion of days upon which it thunders. Required the proportion of days upon which it does both.

Let 1 =, days, p =, days when it hails, q =, days when it thunders, r =, days when it hails and thunders.

p,q =, r

Then by (29), r =, p,q = p q[p]=q p[q].

Peirce: CP 3.15 Cross-Ref:†† Answer. The required proportion is an unknown fraction of the least of the two proportions given. Peirce: CP 3.15 Cross-Ref:†† By p might have been denoted the probability of the major, and by q that of the minor premiss of a hypothetical syllogism of the following form:

If a noise is heard, an explosion always takes place; If a match is applied to a barrel of gunpowder, a noise is heard; .·. If a match is applied to a barrel of gunpowder, an explosion always takes place. Peirce: CP 3.15 Cross-Ref:†† In this case, the value given for r would have represented the probability of the conclusion. Now Boole (page 284) solves this problem by his unmodified method, and obtains the following answer:

r = p q+a(1-q)

where a is an arbitrary constant. Here, if q=1 and p=0, r=0. That is, his answer implies that if the major premiss be false and the minor be true, the conclusion must be false. That this is not really so is shown by the above example. Boole (page 286) is forced to the conclusion that "propositions which, when true, are equivalent, are not necessarily equivalent when regarded only as probable." This is absurd, because probability belongs to the events denoted, and not to forms of expression. The probability of an event is not altered by translation from one language to another. Peirce: CP 3.15 Cross-Ref:†† Boole, in fact, puts the problem into equations wrongly (an error which it is the chief purpose of a calculus of logic to prevent), and proceeds as if the problem

were as follows: Peirce: CP 3.15 Cross-Ref:†† It being known what would be the probability of Y, if X were to happen, and what would be the probability of Z, if Y were to happen; what would be the probability of Z, if X were to happen? Peirce: CP 3.15 Cross-Ref:†† But even this problem has been wrongly solved by him. For, according to his solution, where

p = Y[X] q = Z[Y] r = Z[X],

r must be at least as large as the product of p and q. But if X be the event that a certain man is a negro, Y the event that he is born in Massachusetts, and Z the event that he is a white man, then neither p nor q is zero, and yet r vanishes. Peirce: CP 3.15 Cross-Ref:†† This problem may be rightly solved as follows:

Let p' =, Y[p] =, X,Y q' =, Z[q] =, X,Z r' =, Z[r] =, X,Z.

Then, r' =, p',q';p' =, p',q';q'.

Developing these expressions by (18) we have

r' =, p',q'+r'[p',~q'](p',~q')+r'[~p',~q'](~p',~q')

=, p',q'+r'[~p',q'](~p',q')+r'[~p',~q'](p',q')

The comparison of these two identities shows that

r' =, p',q'+r'[~p',~q'](~p',~q').

Let V =, r'[~p',~q'] =, ((x,~y,z)/(~x,y,~z+~y))

Now

p',q' =, p'- p',~q' =, q'- q',~p'

~p',~q =, ~q' - p',~q' =, ~p' - q',~p'

And

p',~q' =, p' - p'[q']q' =, ~q'- ~q[~p']~p'

~p',q' =, q' - q'[p']p' =, ~p' - ~p'[~q']~q'

Then let

A =, p'[q'] =, (x,y,z)/(y,z)

B =, ~q'[~p'] =, (~x,y,~z+~x,~y,z+x,~y,z+~x,~y,~z)/(1 - x,y)

C =, ~p'[~q'] =, (~x,y,~z+~x,~y,z+x,~y,z+~x,~y,~z)/(1 - y,z)

D =, q'[p'] =, (x,y,z)/(x,y)

And we have

r = (Y/Z)p+ V(1/Z - q)-(1 + V)((Y/Z)p - A q)

= (Y/Z)p+ V(1/Z - q)-(1 + V)(1/Z - q - B((1-Y p)/Z))

= q+V((1 - Y p)/Z) - (1 + V)((1-Y p)/Z - C((1/Z) - q))

= q+V(1-Y p/Z)-(1 + V)(q - D(Y/Z)p)

Peirce: CP 3.16 Cross-Ref:†† 16. Example 2. (See Boole, page 276.) Given r and q; to find p. p =, r;q =, r+v,(1-q) because p is interpretable.

Peirce: CP 3.16 Cross-Ref:†† Answer. The required proportion lies somewhere between the proportion of days upon which it both hails and thunders, and that added to one minus the proportion of days when it thunders. Peirce: CP 3.17 Cross-Ref:†† 17. Example 3. (See Boole, page 279.) Given, out of the number of questions put to two witnesses, and answered by yes or no, the proportion that each answers truly, and the proportion of those their answers to which disagree. Required, out of those wherein they agree, the proportion they answer truly and the proportion they answer falsely.†1

Let 1 =, the questions put to both witnesses, p =, those which the first answers truly, q =, those which the second answers truly, r =, those wherein they disagree, w =, those which both answer truly, w' =, those which both answer falsely.

w =, p,q

w' =, ~p,~q

r =, p +, q - w =, ~p +, ~q - w'.

Now by (28)

p +, q = p + q - w

~p +, ~q = p - p + 1 - q - w'.

Substituting and transposing,

2w = p + q - r

2w' = 2 - p - q - r.

Now w[1-r] = (w,(1-r))/(1-r) but w(1-r) =, w.

w'[1-r] = (w',(1-r))/(1-r) but w'(1-r) =, w'.

.·. w[1-r] = (p+q-r)/(2(1-r)) w'[1-r] = (2-p-q-r)/(2(1-r)).

Peirce: CP 3.18 Cross-Ref:†† 18. The differences of Boole's system, as given by himself, from the modification of it given here, are three. Peirce: CP 3.18 Cross-Ref:†† First. Boole does not make use of the operations here termed logical addition and subtraction. The advantages obtained by the introduction of them are three, viz., they give unity to the system; they greatly abbreviate the labor of working with it; and they enable us to express particular propositions. This last point requires illustration. Let i be a class only determined to be such that only some one individual of the class a comes under it. Then a -, i, a is the expression for some a. Boole cannot properly express some a. Peirce: CP 3.18 Cross-Ref:†† Second. Boole uses the ordinary sign of multiplication for logical multiplication. This debars him from converting every logical identity into an equality of probabilities. Before the transformation can be made the equation has to be brought into a particular form, and much labor is wasted in bringing it to that form. Peirce: CP 3.18 Cross-Ref:†† Third. Boole has no such function as a[b]. This involves him in two difficulties. When the probability of such a function is required, he can only obtain it by a departure from the strictness of his system. And on account of the absence of that symbol, he is led to declare that, without adopting the principle that simple, unconditioned events whose probabilities are given are independent, a calculus of logic applicable to probabilities would be impossible. Peirce: CP 3.19 Cross-Ref:†† 19. The question as to the adoption of this principle is certainly not one of words merely. The manner in which it is answered, however, partly determines the sense in which the term "probability" is taken. Peirce: CP 3.19 Cross-Ref:†† In the propriety of language, the probability of a fact either is, or solely depends upon, the strength of the argument in its favor, supposing all relevant relations of all known facts to constitute that argument. Now, the strength of an argument is only the frequency with which such an argument will yield a true conclusion when its premisses are true. Hence probability depends solely upon the relative frequency of a specific event (namely, that a certain kind of argument yields a true conclusion from true premisses) to a generic event (namely, that that kind of argument occurs with true premisses). Thus, when an ordinary man says that it is highly probable that it will rain, he has reference to certain indications of rain--that is, to a certain kind of argument that it will rain--and means to say that there is an argument that it will rain, which is of a kind of which but a small proportion fail. "Probability," in the untechnical sense, is therefore a vague word, inasmuch as it does not indicate what one, of the numerous subordinated and coordinated genera to which every argument belongs, is the one the relative frequency of the truth of which is expressed. It is usually the case, that there is a tacit understanding upon this point, based perhaps on the notion of an infima species of argument. But an infima s pecies is a mere fiction in logic. And very often the reference is to a very wide genus. Peirce: CP 3.19 Cross-Ref:††

The sense in which the term should be made a technical one is that which will best subserve the purposes of the calculus in question. Now, the only possible use of a calculation of a probability is security in the long run. But there can be no question that an insurance company, for example, which assumed that events were independent without any reason to think that they really were so, would be subjected to great hazard. Suppose, says Mr. Venn,†1 that an insurance company knew that nine tenths of the Englishmen who go to Madeira die, and that nine tenths of the consumptives who go there get well. How should they treat a consumptive Englishman? Mr. Venn has made an error in answering the question, but the illustration puts in a clear light the advantage of ceasing to speak of probability, and of speaking only of the relative frequency of this event to that.†P1

Peirce: CP 3.20 Cross-Ref:†† II

UPON THE LOGIC OF MATHEMATICS†1

PART I †2

§1. THE BOOLIAN CALCULUSE

20. The object of the present paper is to show that there are certain general propositions from which the truths of mathematics follow syllogistically, and that these propositions may be taken as definitions of the objects under the consideration of the mathematician without involving any assumption in reference to experience or intuition. That there actually are such objects in experience or pure intuition is not in itself a part of pure mathematics. Peirce: CP 3.21 Cross-Ref:†† 21. Let us first turn our attention to the logical calculus of Boole. I have shown in a previous communication to the Academy,†3 that this calculus involves eight operations, viz., Logical Addition, Arithmetical Addition, Logical Multiplication, Arithmetical Multiplication, and the processes inverse to these.

DEFINITIONS

1. Identity. a =, b expresses the two facts that any a is b and any b is a. 2. Logical Addition. a +, b denotes a member of the class which contains under it all the a's and all the b's, and nothing else.

3. Logical Multiplication. a,b denotes only whatever is both a and b. 4. Zero denotes nothing, or the class without extent, by which we mean that if a is any member of any class, a +, 0 is a. 5. Unity denotes being, or the class without content, by which we mean that, if a is a member of any class, a is a, 1. 6. Arithmetical Addition. a+b, if a,b =, 0, is the same as a +, b, but, if a and b are classes which have any extent in common, it is not a class. 7. Arithmetical Multiplication. a b represents an event when a and b are events only if these events are independent of each other, in which case a b =, a,b. By the events being independent is meant that it is possible to take two series of terms, A[1], A[2], A[3], etc., and B[1], B[2], B[3], etc., such that the following conditions will be satisfied. (Here x denotes any individual or class, not nothing; A[m], A[n], B[m], B[n], any members of the two series of terms, and ΣA, ΣB, Σ(A,B) logical sums of some of the A[n]'s, the B[n]'s, and the (A[n],B[n])'s respectively.)

Condition 1. No A[m] is A[n]. Condition 2. No B[m] is B[n]. Condition 3. x =, Σ(A,B) Condition 4. a =, ΣA. Condition 5. b =, ΣB. Condition 6. Some A[m] is B[n].†1

Peirce: CP 3.22 Cross-Ref:†† 22. From these definitions a series of theorems follow syllogistically, the proofs of most of which are omitted on account of their ease and want of interest.

Peirce: CP 3.23 Cross-Ref:†† THEOREMS

I

23. If a =, b, then b =, a.

Peirce: CP 3.24 Cross-Ref:†† II

24. If a =, b, and b =, c, then a =, c.

Peirce: CP 3.25 Cross-Ref:†† III

25. If a +, b =, c, then b +, a =, c.

Peirce: CP 3.26 Cross-Ref:†† 26. If a +, b =, m and b +, c =, n and a +, n =, x, then m +, c =, x. Peirce: CP 3.26 Cross-Ref:†† Corollary. These last two theorems hold good also for arithmetical addition.

Peirce: CP 3.27 Cross-Ref:†† V

27. If a + b =, c and a'+b =, c, then a =, a', or else there is nothing not b. Peirce: CP 3.27 Cross-Ref:†† This theorem does not hold with logical addition. But from definition 6 it follows that

No a is b (supposing there is any a) No a' is b (supposing there is any a')

neither of which propositions would be implied in the corresponding formulæ of logical addition. Now from definitions 2 and 6,

Any a is c .·. Any a is c not b

But again from definitions 2 and 6 we have

Any c not b is a' (if there is any not b)

.·. Any a is a' (if there is any not b)

And in a similar way it could be shown that any a' is a (under the same supposition). Hence by definition 1,

a =, a' if there is anything not b.

Peirce: CP 3.27 Cross-Ref:†† Scholium. In arithmetic this proposition is limited by the supposition that b is finite.†1 The supposition here though similar to that is not quite the same.

Peirce: CP 3.28 Cross-Ref:†† VI

28. If a,b =, c, then b,a =, c.

Peirce: CP 3.29 Cross-Ref:†† VII

29. If a,b =, m and b,c =, n and a,n =, x, then m,c =, x.

Peirce: CP 3.30 Cross-Ref:†† VIII

30. If m,n =, b and a +, m =, u and a +, n =, v and a +, b =, x, then u,v =, x.

Peirce: CP 3.31 Cross-Ref:†† IX

31. If m +, n =, b and a,m =, u and a,n =, v and a,b =, x, then u +, v =, x. Peirce: CP 3.31 Cross-Ref:†† The proof of this theorem may be given as an example of the proofs of the rest. Peirce: CP 3.31 Cross-Ref:††

It is required then (by definition 3) to prove three propositions, viz. First. That any u is x. Second. That any v is x. Third. That any x not u is v.

Peirce: CP 3.31 Cross-Ref:†† FIRST PROPOSITION

Since u =, a,m, by definition 3 Any u is m, and since m +, n =, b, by definition 2 Any m is b, whenceAny u is b, But since u =, a,m, by definition 3 Any u is a, whenceAny u is both a and b, But since a,b =, x, by definition 3 Whatever is both a and b is x whenceAny u is x.

Peirce: CP 3.31 Cross-Ref:†† SECOND PROPOSITION This is proved like the first.

Peirce: CP 3.31 Cross-Ref:†† THIRD PROPOSITION Since a,m =, u, by definition 3, Whatever is both a and m is u. or

Whatever is not u is not both a and m.

or

Whatever is not u is either not a or not m.

or

Whatever is not u and is a is not m.

But since a,b =, x, by definition 3

Any x is a, whence

Any x not u is not u and is a,

whence

Any x not u is not m.

But since a,b =, x, by definition 3 Any x is b, whence

Any x not u is b, Any x not u is b, not m.

But since m +, n =, b, by definition 2 Any b not m is n, whence

Any x not u is n,

and therefore Any x not u is both a and n.†1

But since a,n =, v, by definition 3 Whatever is both a and n†2 is v, whence

Any x not u is v.

Peirce: CP 3.32 Cross-Ref:†† 32. Corollary 1. This proposition readily extends itself to arithmetical addition. Peirce: CP 3.32 Cross-Ref:†† Corollary 2. The converse propositions produced by transposing the last two identities of theorems VIII and IX are also true. Peirce: CP 3.32 Cross-Ref:†† Corollary 3. Theorems VI, VII, and IX hold also with arithmetical multiplication. This is sufficiently evident in the case of theorem VI, because by definition 7 we have an additional premiss, namely, that a and b are independent, and an additional conclusion which is the same as that premiss. Peirce: CP 3.33 Cross-Ref:†† 33. In order to show the extension of the other theorems, I shall begin with the following lemma. If a and b are independent, then corresponding to every pair of individuals, one of which is both a and b, there is just one pair of individuals one of which is a and the other b; and conversely, if the pairs of individuals so correspond, a and b are independent. For, suppose a and b independent, then, by definition 7, condition 3, every class (A[m],B[n]) is an individual. If then A[a] denotes any A[m] which is a, and B[b] any B[m] which is b, by condition 6 (A[a],B[n]) and (A[m],B[b]) both exist, and by conditions 4 and 5 the former is any individual a, and the latter any

individual b. But given this pair of individuals, both of the pair (A[a],B[b]) and (A[m],B[n]) exist by condition 6. But one individual of this pair is both a and b. Hence the pairs correspond, as stated above. Next, suppose a and b to be any two classes. Let the series of A[m]'s be a and not-a; and let the series of B[m]'s be all individuals separately. Then the first five conditions can always be satisfied. Let us suppose, then, that the sixth alone cannot be satisfied. Then A[p] and B[q] may be taken such that (A[p],B[q]) is nothing. Since A[p] and B[q] are supposed both to exist, there must be two individuals (A[p],B[n]) and (A[m],B[q]) which exist. But there is no corresponding pair (A[m],B[n]) and (A[p],B[q]). Hence, no case in which the sixth condition cannot be satisfied simultaneously with the first five is a case in which the pairs rightly correspond; or, in other words, every case in which the pairs correspond rightly is a case in which the sixth condition can be satisfied, provided the first five can be satisfied. But the first five can always be satisfied. Hence, if the pairs correspond as stated, the classes are independent. Peirce: CP 3.34 Cross-Ref:†† 34. In order to show that theorem VII may be extended to arithmetical multiplication, we have to prove that if a and b, b and c, and a and (b,c), are independent, then (a,b) and c are independent. Let s denote any individual. Corresponding to every s with (a,b,c), there is an a and (b,c). Hence, corresponding to every s with s and with (a,b,c) (which is a particular case of that pair), there is an s with a and with (b,c). But for every s with (b,c) there is a b with c; hence, corresponding to every a with s and with (b,c), there is an a with b and with c. Hence, for every s with s and with (a,b,c) there is an a with b and with c. For every a with b there is an s with (a,b); hence, for every a with b and with c, there is an s with (a,b) and c. Hence, for every s with s and with (a,b,c) there is an s with (a,b) and with c. Hence, for every s with (a,b,c) there is an (a,b) with c. The converse could be proved in the same way. Hence, etc. Peirce: CP 3.35 Cross-Ref:†† 35. Theorem IX holds with arithmetical addition of whichever sort the multiplication is. For we have the additional premiss that "No m is n"; whence since "any u is m" and "any v is n," "no u is v," which is the additional conclusion. Peirce: CP 3.35 Cross-Ref:†† Corollary 2, so far as it relates to theorem IX, holds with arithmetical addition and multiplication. For, since no m is n, every pair, one of which is a and either m or n, is either a pair, one of which is a and m, or a pair, one of which is a and n, and is not both. Hence, since for every pair one of which is a and m, there is a pair one of which is a and the other m, and since for every pair one of which is a,n there is a pair one of which is a and the other n; for every pair one of which is a and either m or n, there is either a pair one of which is a and the other m, or a pair one of which is a and the other n, and not both; or, in other words, there is a pair one of which is a and the other either m or n. Peirce: CP 3.35 Cross-Ref:†† (It would perhaps have been better to give this complicated proof in its full syllogistic form. But as my principal object is merely to show that the various theorems could be so proved, and as there can be little doubt that if this is true of those which relate to arithmetical addition it is true also of those which relate to arithmetical multiplication, I have thought the above proof (which is quite apodeictic) to be sufficient. The reader should be careful not to confound a proof which needs

itself to be experienced with one which requires experience of the object of proof.)

Peirce: CP 3.36 Cross-Ref:†† X

36. If a b =, c and a'b =, c, then a =, a', or no b exists. This does not hold with logical, but does with arithmetical multiplication. For if a is not identical with a', it may be divided thus

a =, a,a'+a,~a'

if ~a' denotes not a'. Then

a,b =, (a,a'),b + (a,~a'),b

and by the definition of independence the last term does not vanish unless (a,~a') =, 0, or all a is a'; but since a,b =, a',b =, (a,a'),b+(~a,a'),b, this term does vanish, and, therefore, only a is a', and in a similar way it could be shown that only a' is a.

Peirce: CP 3.37 Cross-Ref:†† XI

37. 1 +, a =, 1. This is not true of arithmetical addition, for since by definition 7,

1x,1 =, x1

by theorem IX

x,(1+a) =, x(1+a) =, x1 + x a =, x + x a.

Peirce: CP 3.37 Cross-Ref:†† Whence x a =, 0, while neither x nor a is zero, which, as will appear directly,

is impossible.

Peirce: CP 3.38 Cross-Ref:†† XII

38. 0,a =, 0. Proof. For call 0, a =, x. Then by definition 3

x belongs to the class zero.

.·. by definition 4

x =, 0.

Peirce: CP 3.38 Cross-Ref:†† Corollary 1. The same reasoning applies to arithmetical multiplication. Peirce: CP 3.38 Cross-Ref:†† Corollary 2. From theorem x and the last corollary it follows that if a b =, 0, either a =, 0 or b =, 0.

Peirce: CP 3.39 Cross-Ref:†† XIII

39. a,a =, a.†1

Peirce: CP 3.40 Cross-Ref:†† XIV

40. a +, a =, a.†1

These do not hold with arithmetical operations. Peirce: CP 3.41 Cross-Ref:†† 41. General Scholium. This concludes the theorems relating to the direct operations. As the inverse operations have no peculiar logical interest, they are passed over here. In order to prevent misapprehension, I will remark that I do not undertake to demonstrate the principles of logic themselves. Indeed, as I have shown in a previous

paper, these principles considered as speculative truths are absolutely empty and indistinguishable.†2 But what has been proved is the maxims of logical procedure, a certain system of signs being given. Peirce: CP 3.41 Cross-Ref:†† The definitions given above for the processes which I have termed arithmetical plainly leave the functions of these operations in many cases uninterpreted. Thus if we write

a+b =, b+a a+(b+c) =, (a+b)+c b c =, c b (a b)c =, a(b c) a(m+n) =, a m+a n

we have a series of identities whose truth or falsity is entirely undeterminable. In order, therefore, fully to define those operations, we will say that all propositions, equations, and identities which are in the general case left by the former definitions undetermined as to truth, shall be true, provided they are so in all interpretable cases.

Peirce: CP 3.42 Cross-Ref:†† §2. ON ARITHMETIC.†1

42. Equality is a relation of which identity is a species. If we were to leave equality without further defining it, then by the last scholium all the formal rules of arithmetic would follow from it. And this completes the central design of this paper, as far as arithmetic is concerned. Peirce: CP 3.43 Cross-Ref:†† 43. Still it may be well to consider the matter a little further. Imagine, then, a particular case under Boole's calculus, in which the letters are no longer terms of first intention, but terms of second intention, and that of a special kind. Genus, species, difference, property, and accident, are the well-known terms of second intention. These relate particularly to the comprehension†2 of first intentions; that is, they refer to different sorts of predication. Genus and species, however, have at least a secondary reference to the extension†2 of first intentions. Now let the letters, in the particular application of Boole's calculus now supposed, be terms of second intention which relate exclusively to the extension of first intentions.†3 Let the differences of the characters of things and events be disregarded, and let the letters signify only the differences of classes as wider or narrower. In other words, the only logical comprehension which the letters considered as terms will have is the greater or less

divisibility of the classes. Thus, n in another case of Boole's calculus might, for example, denote "New England States"; but in the case now supposed, all the characters which make these States what they are being neglected, it would signify only what essentially belongs to a class which has the same relations to higher and lower classes which the class of New England States has, -- that is, a collection of six. Peirce: CP 3.44 Cross-Ref:†† 44. In this case, the sign of identity will receive a special meaning. For, if m denotes what essentially belongs to a class of the rank of "sides of a cube," then m =, n will imply, not that every New England State is a side of a cube, and conversely, but that whatever essentially belongs to a class of the numerical rank of "New England States" essentially belongs to a class of the rank of "sides of a cube," and conversely. Identity of this particular sort may be termed equality, and be denoted by the sign =.†P1 Moreover, since the numerical rank of a logical sum depends on the identity or diversity (in first intention) of the integrant parts, and since the numerical rank of a logical product depends on the identity or diversity (in first intention) of parts of the factors, logical addition and multiplication can have no place in this system. Arithmetical addition and multiplication, however, will not be destroyed. a b = c will imply that whatever essentially belongs at once to a class of the rank of a, and to another independent class of the rank of b belongs essentially to a class of the rank of c, and conversely.†1 a + b = c implies that whatever belongs essentially to a class which is the logical sum of two mutually exclusive classes of the ranks of a and b belongs essentially to a class of the rank of c, and conversely.†1 It is plain that from these definitions the same theorems follow as from those given above. Zero and unity will, as before, denote the classes which have respectively no extension and no comprehension; only the comprehension here spoken of is, of course, that comprehension which alone belongs to letters in the system now considered, that is, this or that degree of divisibility; and therefore unity will be what belongs essentially to a class of any rank independent of its divisibility. These two classes alone are common to the two systems, because the first intentions of these alone determine, and are determined by, their second intentions. Finally, the laws of the Boolian calculus, in its ordinary form, are identical with those of this other so far as the latter apply to zero and unity, because every class, in its first intention, is either without any extension (that is, is nothing), or belongs essentially to that rank to which every class belongs, whether divisible or not. Peirce: CP 3.44 Cross-Ref:†† These considerations, together with those advanced [in 1.556], will, I hope, put the relations of logic and arithmetic in a somewhat clearer light than heretofore.

Peirce: CP 3.45 Cross-Ref:†† III

DESCRIPTION OF A NOTATION FOR THE LOGIC OF RELATIVES, RESULTING FROM AN AMPLIFICATION OF THE CONCEPTIONS OF BOOLE'S CALCULUS OF LOGIC†1

§1. DE MORGAN'S NOTATIONE

45. Relative terms usually receive some slight treatment in works upon logic, but the only considerable investigation into the formal laws which govern them is contained in a valuable paper by Mr. De Morgan in the tenth volume of the Cambridge Philosophical Transactions.†2 He there uses a convenient algebraic notation, which is formed by adding to the well-known spiculæ of that writer the signs used in the following examples. X . . LY signifies that X is some one of the objects of thought which stand to Y in the relation L, or is one of the L's of Y. X . LMY signifies that X is not an L of an M of Y. X . . (L,M)Y signifies that X is either an L or an M of Y. LM' an L of every M. L[,]M an L of none but M's. L[[-1]]Y something to which Y is L. l (small L) non-L.

This system still leaves something to be desired. Moreover, Boole's logical algebra has such singular beauty, so far as it goes, that it is interesting to inquire whether it cannot be extended over the whole realm of formal logic, instead of being restricted to that simplest and least useful part of the subject, the logic of absolute terms, which, when he wrote, was the only formal logic known. The object of this paper is to show that an affirmative answer can be given to this question. I think there can be no doubt that a calculus, or art of drawing inferences, based upon the notation I am to describe, would be perfectly possible and even practically useful in some difficult cases, and particularly in the investigation of logic. I regret that I am not in a situation to be able to perform this labor, but the account here given of the notation itself will afford the ground of a judgment concerning its probable utility. Peirce: CP 3.46 Cross-Ref:†† 46. In extending the use of old symbols to new subjects, we must of course be guided by certain principles of analogy, which, when formulated, become new and wider definitions of these symbols. As we are to employ the usual algebraic signs as far as possible, it is proper to begin by laying down definitions of the various algebraic relations and operations. The following will, perhaps, not be objected to.

Peirce: CP 3.47 Cross-Ref:†† §2. GENERAL DEFINITIONS OF THE ALGEBRAIC SIGNS

47. Inclusion in or being as small as is a transitive relation. The consequence holds that †P1

If x -< y, and

y -< z,

then

x -< z.

Peirce: CP 3.48 Cross-Ref:†† 48. Equality is the conjunction of being as small as and its converse. To say that x = y is to say that x -< y and y -< x. Peirce: CP 3.49 Cross-Ref:†† 49. Being less than is being as small as with the exclusion of its converse. To say that x < y is to say that x -< y, and that it is not true that y -< x. Peirce: CP 3.50 Cross-Ref:†† 50. Being greater than is the converse of being less than. To say that x > y is to say that y < x. Peirce: CP 3.51 Cross-Ref:†† 51. Addition is an associative operation. That is to say,†P1

(x +, y) +, z = x +, (y +, z).

Addition is a commutative operation. That is,

x +, y = y +, x.

Peirce: CP 3.52 Cross-Ref:†† 52. Invertible †1 addition is addition the corresponding inverse of which is determinative. The last two formulæ hold good for it, and also the consequence that

If x + y = z, and then

x + y' = z, y = y'.†2

Peirce: CP 3.53 Cross-Ref:†† 53. Multiplication is an operation which is doubly distributive with reference to addition. That is,

x(y +, z) = x y +, x z,

(x +, y)z = x z +, y z.

Multiplication is almost invariably an associative operation.†3

(x y)z = x(y z).

Multiplication is not generally commutative. If we write commutative †4 multiplication with a comma,†5 we have

x,y = y,x.

Peirce: CP 3.54 Cross-Ref:†† 54. Invertible †1 multiplication is multiplication whose corresponding inverse operation (division) is determinative. We may indicate this by a dot;†2 and then the consequence holds that

If x.y = z, and x.y'= z, then

y = y'.†3

Peirce: CP 3.55 Cross-Ref:†† 55. Functional multiplication †4 is the application of an operation to a function. It may be written like ordinary multiplication; but then there will generally be certain points where the associative principle does not hold. Thus, if we write (sin abc) def, there is one such point. If we write (log (base abc) def) ghi, there are two such points. The number of such points depends on the nature of the symbol of operation, and is necessarily finite. If there were many such points, in any case, it would be necessary to adopt a different mode of writing such functions from that now usually employed. We might, for example, give to "log" such a meaning that what followed it up to a certain point indicated by a † should denote the base of the system, what followed that to the point indicated by a ‡ should be the function operated on, and what followed that should be beyond the influence of the sign "log." Thus log abc † def ‡ ghi would be (log abc) ghi, the base being def. In this paper I shall adopt a notation very similar to this, which will be more conveniently described further on. Peirce: CP 3.56 Cross-Ref:†† 56. The operation of involution obeys the formula †P1

(xy)z = x(y z).

Involution, also, follows the indexical principle.

xy +, z = xy,xz.

Involution, also, satisfies the binomial theorem.†1

(x +, y)z = xz +, Σ[p]xz-p,yp +, yz,

where Σ[p] denotes that p is to have every value less than z, and is to be taken out of z in all possible ways, and that the sum of all the terms so obtained of the form xz-p,yp is to be taken. Peirce: CP 3.57 Cross-Ref:†† 57. Subtraction is the operation inverse to addition. We may write indeterminative †2 subtraction with a comma below the usual sign. Then we shall have that

(x -, y) +, y = x, (x - y) + y = x, (x + y) - y = x.

Peirce: CP 3.58 Cross-Ref:†† 58. Division is the operation inverse to multiplication. Since multiplication is not generally commutative it is necessary to have two signs for division. I shall take

(x:y)y = x, x y/x = y.

Peirce: CP 3.59 Cross-Ref:†† 59. Division inverse to that multiplication which is indicated by a comma may be indicated by a semicolon. So that

(x;y),y = x.†3

Peirce: CP 3.60 Cross-Ref:†† 60. Evolution and taking the logarithm are the operations inverse to involution.

(x√y)x = y,

xlog[x]y = y.

Peirce: CP 3.61 Cross-Ref:†† 61. These conditions are to be regarded as imperative. But in addition to them there are certain other characters which it is highly desirable that relations and operations should possess, if the ordinary signs of algebra are to be applied to them. These I will here endeavour to enumerate. Peirce: CP 3.61 Cross-Ref:†† 1. It is an additional motive for using a mathematical sign to signify a certain operation or relation that the general conception of this operation or relation should resemble that of the operation or relation usually signified by the same sign. In particular, it will be well that the relation expressed by -< should involve the conception of one member being in the other; addition, that of taking together; multiplication, that of one factor's being taken relatively to the other (as we write 3 X 2 for a triplet of pairs, and Dφ for the derivative of φ); and involution, that of the base being taken for every unit of the exponent. Peirce: CP 3.61 Cross-Ref:†† 2. In the second place, it is desirable that, in certain general circumstances, determinate numbers should be capable of being substituted for the letters operated upon, and that when so substituted the equations should hold good when interpreted in accordance with the ordinary definitions of the signs, so that arithmetical algebra should be included under the notation employed as a special case of it. For this end, there ought to be a number known or unknown, which is appropriately substituted in certain cases, for each one of, at least, some class of letters. Peirce: CP 3.61 Cross-Ref:†† 3. In the third place, it is almost essential to the applicability of the signs for addition and multiplication, that a zero and a unity should be possible. By a zero I mean a term such that

x +, 0 = x,

whatever the signification of x; and by a unity a term for which the corresponding general formula

x1 = x

holds good. On the other hand, there ought to be no term a such that ax=x, independently of the value of x. Peirce: CP 3.61 Cross-Ref:†† 4. It will also be a strong motive for the adoption of an algebraic notation, if other formulæ which hold good in arithmetic, such as

xz,yz = (x,y)z, 1x = x, x1 = x, x0 = 0,

continue to hold good; if, for instance, the conception of a differential is possible, and Taylor's Theorem holds, and †1 or (1+i)1/i plays an important part in the system, if there should be a term having the properties of †1 (3.14159), or properties similar to those of space should otherwise be brought out by the notation, or if there should be an absurd expression having the properties and uses of †1 or the square root of the negative.

Peirce: CP 3.62 Cross-Ref:†† §3. APPLICATION OF THE ALGEBRAIC SIGNS TO LOGIC

62. While holding ourselves free to use the signs of algebra in any sense conformable to the above absolute conditions, we shall find it convenient to restrict ourselves to one particular interpretation except where another is indicated. I proceed to describe the special notation which is adopted in this paper.

Peirce: CP 3.63 Cross-Ref:†† USE OF THE LETTERS

63. The letters of the alphabet will denote logical signs. Now logical terms are of three grand classes. The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as "a --." These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such

(quale); for example, as horse, tree, or man. These are absolute terms. The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms. The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of -- to --, or buyer of -- for -- from --. These may be termed conjugative terms. The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object.†1 No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.†2 Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory or not, the fact itself is made perfectly evident by the study of the logic of relatives. I shall denote absolute terms by the Roman alphabet, a, b, c, d, etc.; relative terms by italics, a, b, c, d, etc.; and conjugative terms by a kind of type called Kennerly, a, b, c, d, etc. Peirce: CP 3.63 Cross-Ref:†† I shall commonly denote individuals by capitals, and generals †3 by small letters. General symbols for numbers will be printed in black-letter, thus, a, b, c, d, etc. The Greek letters will denote operations. Peirce: CP 3.64 Cross-Ref:†† 64. To avoid repetitions, I give here a catalogue of the letters I shall use in examples in this paper, with the significations I attach to them. a. animal.

p. President of the United States Senate.

b. black.

r. rich person.

f. Frenchman. u. violinist. h. horse. m. man.

a. enemy.

v. Vice-President of the United States. w. woman.

h. husband. o. owner.

b. benefactor. l. lover. s. servant. c. conqueror. m. mother. w. wife. e. emperor.

n. not.

g. giver to--of--. b. betrayer to--of--.

w. winner over of--to--from--. t. transferrer from--to--.

Peirce: CP 3.65 Cross-Ref:†† NUMBERS CORRESPONDING TO LETTERS

65. I propose to use the term "universe" to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan's,†1 is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains.†2 Peirce: CP 3.65 Cross-Ref:†† I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (men), the number of "tooth of" would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus [t].

Peirce: CP 3.66 Cross-Ref:†† THE SIGNS OF INCLUSION, EQUALITY, ETC.

66. I shall follow Boole †3 in taking the sign of equality to signify identity. Thus, if v denotes the Vice-President of the United States, and p the President of the Senate of the United States,

v=p

means that every Vice-President of the United States is President of the Senate, and every President of the United States Senate is Vice-President. The sign "less than" is to be so taken that

f<m

means every Frenchman is a man, but there are men besides Frenchmen. Drobisch has used this sign in the same sense.†P1 It will follow from these significations of = and < that the sign -< (or ⎥, "as small as") will mean "is." Thus,

f -< m

means "every Frenchman is a man," without saying whether there are any other men or not. So,

m -< l

will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. These significations of = and < plainly conform to the indispensable conditions. Upon the transitive character of these relations the syllogism depends, for by virtue of it, from

f -< m and

m -< a,

we can infer that

f -< a;

that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal. But not only do the significations of = and < here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.†1 So, to write 5 < 7 is to say that 5 is part of 7, just as to write f < m is to say that Frenchmen are part of men. Indeed, if f < m, then the number of Frenchmen is less than the number of men, and if v = p, then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.

Peirce: CP 3.67 Cross-Ref:†† THE SIGNS FOR ADDITION

67. The sign of addition is taken by Boole,†2 so that

x+y

denotes everything denoted by x, and, besides, everything denoted by y. Thus m+w

denotes all men, and, besides, all women. This signification for this sign is needed for connecting the notation of logic with that of the theory of probabilities. But if there is anything which is denoted by both the terms of the sum, the latter no longer stands for any logical term on account of its implying that the objects denoted by one term are to be taken besides the objects denoted by the other. For example,

f+u

means all Frenchmen besides all violinists, and, therefore, considered as a logical term, implies that all French violinists are besides themselves. For this reason alone, in a paper which is published in the Proceedings of the Academy for March 17, 1867,†1 I preferred to take as the regular addition of logic a noninvertible process, such that

m +, b

stands for all men and black things, without any implication that the black things are to be taken besides the men; and the study of the logic of relatives has supplied me with other weighty reasons for the same determination. Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in a tract called Pure Logic, or the Logic of Quality, [1864]†2 had anticipated me in substituting the same operation for Boole's addition, although he rejects Boole's operation entirely and writes the new one with a + sign while withholding from it the name of addition.†P1 It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of taking together involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves--provided all the terms summed are mutually exclusive. Addition being taken in this sense, nothing is to be denoted by zero, for then

x +, 0 = x,

whatever is denoted by x; and this is the definition of zero.†1 This interpretation is given by Boole, and is very neat, on

account of the resemblance between the ordinary conception of zero and that of nothing, and because we shall thus have

[[0]] = 0

Peirce: CP 3.68 Cross-Ref:†† THE SIGNS FOR MULTIPLICATION

68. I shall adopt for the conception of multiplication the application of a relation, in such a way that, for example, lw shall denote whatever is lover of a woman. This notation is the same as that used by Mr. De Morgan, although he appears not to have had multiplication in his mind. s(m +, w) will, then, denote whatever is servant of anything of the class composed of men and women taken together. So that

s(m +, w) = sm +, sw.

(l +, s)w will denote whatever is lover or servant to a woman, and

(l +, s)w = lw +, sw.

(s l)w will denote whatever stands to a woman in the relation of servant of a lover, and

(s l)w = s(lw).

Thus all the absolute conditions of multiplication are satisfied. The term "identical with--" is a unity for this multiplication. That is to say, if we denote "identical with--" by 1 we have

x1 = x,

whatever relative term x may be. For what is a lover of something identical with anything, is the same as a lover of that thing.

Peirce: CP 3.69 Cross-Ref:†† 69. A conjugative term like giver naturally requires two correlates, one denoting the thing given, the other the recipient of the gift. We must be able to distinguish, in our notation, the giver of A to B from the giver to A of B, and, therefore, I suppose the signification of the letter equivalent to such a relative to distinguish the correlates as first, second, third, etc., so that "giver of -- to --" and "giver to -- of --" will be expressed by different letters. Let g denote the latter of these conjugative terms. Then, the correlates or multiplicands of this multiplier cannot all stand directly after it, as is usual in multiplication, but may be ranged after it in regular order, so that

gx y

will denote a giver to x of y. But according to the notation, x here multiplies y, so that if we put for x owner (o), and for y horse (h),

goh

appears to denote the giver of a horse to an owner of a horse. But let the individual horses be H, H', H'', etc. Then

h †1 = H +, H' +, H'' +, etc. goh = go(H +, H' +, H'' +, etc.) = goH +, goH' +, goH'' +, etc.

Now this last member must be interpreted as a giver of a horse to the owner of that horse, and this, therefore, must be the interpretation of goh. This is always very important. A term multiplied by two relatives shows that the same individual is in the two relations. If we attempt to express the giver of a horse to a lover of a woman, and for that purpose write

glwh,

we have written giver of a woman to a lover of her, and if we add brackets, thus, g(lw)h,

we abandon the associative principle of multiplication. A little reflection will show that the associative principle must in some form or other be abandoned at this point.

But while this principle is sometimes falsified, it oftener holds, and a notation must be adopted which will show of itself when it holds. We already see that we cannot express multiplication by writing the multiplicand directly after the multiplier; let us then affix subjacent numbers after letters to show where their correlates are to be found. The first number shall denote how many factors must be counted from left to right to reach the first correlate, the second how many more must be counted to reach the second, and so on. Then, the giver of a horse to a lover of a woman may be written

g[12]l[1]wh = g[11]l[2]hw = g[2-1]hl[1]w.

Peirce: CP 3.70 Cross-Ref:†† 70. Of course a negative number indicates that the former correlate follows the latter by the corresponding positive number. A subjacent zero makes the term itself the correlate. Thus,

l[0]

denotes the lover of that lover or the lover of himself, just as goh denotes that the horse is given to the owner of itself, for to make a term doubly a correlate is, by the distributive principle, to make each individual doubly a correlate, so that

l[0] = L[0] +, L[0]' +, L[0]'' +, etc.

A subjacent sign of infinity may indicate that the correlate is indeterminate, so that

l[∞]

will denote a lover of something. We shall have some confirmation of this presently.†1 Peirce: CP 3.70 Cross-Ref:†† If the last subjacent number is a one it may be omitted. Thus we shall have

l[1] = l,

g[11] = g[1] = g.

This enables us to retain our former expressions lw, goh, etc. Peirce: CP 3.71 Cross-Ref:†† 71. The associative principle does not hold in this counting of factors. Because it does not hold, these subjacent numbers are frequently inconvenient in practice, and I therefore use also another mode of showing where the correlate of a term is to be found. This is by means of the marks of reference, † ‡ || § PARASYMBOLQX, which are placed subjacent to the relative term and before and above the correlate. Thus, giver of a horse to a lover of a woman may be written

g†‡†l||||w‡h.

The asterisk I use exclusively to refer to the last correlate of the last relative of the algebraic term. Peirce: CP 3.72 Cross-Ref:†† 72. Now, considering the order of multiplication to be: -- a term, a correlate of it, a correlate of that correlate, etc., -- there is no violation of the associative principle. The only violations of it in this mode of notation are that in thus passing from relative to correlate, we skip about among the factors in an irregular manner, and that we cannot substitute in such an expression as goh a single letter for oh. I would suggest that such a notation may be found useful in treating other cases of non-associative multiplication. By comparing this with what was said above †1 concerning functional multiplication, it appears that multiplication by a conjugative term is functional, and that the letter denoting such a term is a symbol of operation. I am therefore using two alphabets, the Greek and Kennerly, where only one was necessary. But it is convenient to use both. Peirce: CP 3.73 Cross-Ref:†† 73. Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives. Now the absolute term "man" is really exactly equivalent to the relative term "man that is --," and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term. Then man that is black will be written

m,b.

But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one. Then

l,sw †2

will denote a lover of a woman that is a servant of that woman. The comma here after l should not be considered as altering at all the meaning of l, but as only a subjacent sign, serving to alter the arrangement of the correlates. In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates. So

m,,b,r, interpreted like

goh,

means a man that is a rich individual and is a black that is that rich individual. But this has no other meaning than

m,b,r,

or a man that is a black that is rich. Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number. If, therefore, l,,sw is not the same as l,sw (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates. And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should), we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series "that is--and is--and is--etc." Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates? Any term may be regarded as having an infinite number of factors, those at the end being ones, thus,

l,sw = l,sw,1,1,1,1,1,1,1, etc.

A subjacent number may therefore be as great as we please. But all these ones denote the same identical individual denoted by w; what then can be the subjacent numbers to be applied to s, for instance, on account of its infinite "that is" 's? What numbers can separate it from being identical with w? There are only two. The first is zero, which plainly neutralizes a comma completely, since

s,[0]w = sw,†1

and the other is infinity; for as 1∞ is indeterminate in ordinary algebra, so it will be shown hereafter to be here, so that to remove the correlate by the product of an infinite series of ones is to leave it indeterminate. Accordingly,

m,[∞]

should be regarded as expressing some man. Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros. Peirce: CP 3.73 Cross-Ref:†† "Something" may then be expressed by

1[∞].†1

I shall for brevity frequently express this by an antique figure one (I). Peirce: CP 3.73 Cross-Ref:†† "Anything" by

1[0].†2

I shall often also write a straight 1 for anything. Peirce: CP 3.74 Cross-Ref:†† 74. It is obvious that multiplication into a multiplicand indicated by a comma is commutative,†P1 that is,

s,l = l,s.

This multiplication is effectively the same as that of Boole in his logical calculus. Boole's unity is my 1, that is, it denotes whatever is. Peirce: CP 3.75 Cross-Ref:†† 75. The sum x + x generally denotes no logical term. But x,[∞] + x,[∞] may be considered as denoting some two x's. It is natural to write x+x = 2.x, and

x,[∞]+x,[∞] = 2.x,[∞],

where the dot shows that this multiplication is invertible. We may also use the antique figures so that

2.x,[∞] = 2X, just as

1[∞] = 1.

Then 2 alone will denote some two things. But this multiplication is not in general commutative, and only becomes so when it affects a relative which imparts a relation such that a thing only bears it to one thing, and one thing alone bears it to a thing. For instance, the lovers of two women are not the same as two lovers of women, that is,

l2.w and 2.lw

are unequal; but the husbands of two women are the same as two husbands of women, that is,

h2.w = 2.hw,

and in general,

x,2.y = 2.x,y.

Peirce: CP 3.76 Cross-Ref:†† 76. The conception of multiplication we have adopted is that of the application of one relation to another. So, a quaternion being the relation of one vector to another, the multiplication of quaternions is the application of one such relation to a second. Even ordinary numerical multiplication involves the same idea, for 2 X 3 is a pair of triplets, and 3 X 2 is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives. Peirce: CP 3.76 Cross-Ref:†† If we have an equation of the form

x y = z,

and there are just as many x's per y as there are per things, things of the universe, then we have also the arithmetical equation,

[[x]][[y]] = [[z]].

For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then

[[t]][[f]] = [[t f]]

holds arithmetically. So if men are just as apt to be black as things in general,

[[m,]][[b]] = [[m,b]],

where the difference between [[m]] and [[m,]] must not be overlooked. It is to be observed that

[[1]] = 1.

Peirce: CP 3.76 Cross-Ref:†† Boole was the first to show this connection between logic and probabilities.†1 He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of expectation. Peirce: CP 3.76 Cross-Ref:†† Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.

Peirce: CP 3.77 Cross-Ref:†† THE SIGN OF INVOLUTION

77. I shall take involution in such a sense that xy will denote everything which is an x for every individual of y. Thus lw will be a lover of every woman. Then (sl)w will denote whatever stands to every woman in the relation of servant of every lover of hers; and s(lw) will denote whatever is a servant of everything that is lover of a woman. So that

(sl)w = s(lw).

A servant of every man and woman will be denoted by sm +, w†1, and sm, sw will denote a servant of every man that is a servant of every woman. So that

sm +, w = sm,sw.

That which is emperor or conqueror of every Frenchman will be denoted by (e +, c)f, and ef +, Σ[p]ef-p, cp +, cf will denote whatever is emperor of every Frenchman or emperor of some Frenchmen and conqueror of all the rest, or conqueror of every Frenchman. Consequently,

(e +, c)f = ef +, Σ[p]ef-p,cp +, cf.

Indeed, we may write the binomial theorem so as to preserve all its usual coefficients; for we have

(e +, c)f = ef+,[[f]].ef-†1,c1†+,(([[f]].([[f]]-1))/2).ef-‡2,c2‡+, etc.†2

That is to say, those things each of which is emperor or conqueror of every Frenchman consist, first, of all those individuals each of which is a conqueror [emperor!] of every Frenchman; second, of a number of classes equal to the number of Frenchmen, each class consisting of everything which is an emperor of every Frenchman but some one and is a conqueror of that one; third, of a number of classes equal to half the product of the number of Frenchmen by one less than that number, each of these classes consisting of every individual which is an emperor of every Frenchman except a certain two, and is conqueror of those two, etc. This theorem holds, also, equally well with invertible addition, and either term of the binomial may be negative provided we assume

(-x)y = (-)[[y]].xy.

Peirce: CP 3.77 Cross-Ref:†† In addition to the above equations which are required to hold good by the definition of involution, the following also holds,

(s,l)w = sw,lw,†1

just as it does in arithmetic.

Peirce: CP 3.78 Cross-Ref:†† 78. The application of involution to conjugative terms presents little difficulty after the explanations which have been given under the head of multiplication. It is obvious that betrayer to every enemy should be written

ba,

just as lover of every woman is written

l w,

but b = b[11] and therefore, in counting forward as the subjacent numbers direct, we should count the exponents, as well as the factors, of the letter to which the subjacent numbers are attached. Then we shall have, in the case of a relative of two correlates, six different ways of affixing the correlates to it, thus:

bam

betrayer of a man to an †P1 enemy of him;

(ba)m betrayer of every man to some enemy of him; bam

betrayer of each man to an enemy of every man;

bam

betrayer of a †P1 man to all †P1 enemies of all men;

bam

betrayer of a man to every enemy of him;

bam

betrayer of every man to every enemy of him.

If both correlates are absolute terms, the cases are

bmw

betrayer of a woman to a man;

(bm)w betrayer of each woman to some man; bmw

betrayer of all women to a man;

bmw

betrayer of a woman to every man;†P1

bmw

betrayer of a woman to all men;

bmw

betrayer of every woman to every man.

These interpretations are by no means obvious, but I shall show that they are correct

further on.†1 Peirce: CP 3.79 Cross-Ref:†† 79. It will be perceived that the rule still holds here that

(ba)m = b(am)

that is to say, that those individuals each of which stand to every man in the relation of betrayer to every enemy of his are identical with those individuals each of which is a betrayer to every enemy of a man of that man.

Peirce: CP 3.80 Cross-Ref:†† 80. If the proportion of lovers of each woman among lovers of other women is equal to the average number of lovers which single individuals of the whole universe have, then

[[lw]] = [[lW']] [[lW'']] [[lW''']] etc.=[[l]][[w]].

Thus arithmetical involution appears as a special case of logical involution.

Peirce: CP 3.81 Cross-Ref:†† §4. GENERAL FORMULÆ

81. The formulæ which we have thus far obtained, exclusive of mere explanations of signs and of formulæ relating to the numbers of classes, are:

(1) If x -< y and y -< z, then x -< z. (2) (x +, y) +, z = x +, (y +, z). (3) x +, y = y +, x.

(Jevons) (Jevons)

(4) (x +, y)z = x z +, y z. (5) x(y +, z) = x y +, x z. (6) (x y)z = x(y z). (7) x,(y +, z) = x,y +, x,z.

(Jevons)

(8) (x,y),z = x,(y,z).

(Boole)

(9) x,y, = y,x.

(Boole)

(10) (xy)z = x(y z). (11) xy +, z = xy,xz. (12) (x +, y)z = xz +, Σ[p](xx-p,yp) +, yp = xz+,[[z]].xz-†1,y†1 +, (([[z]].[[z-1]])/2).xz-‡2,y‡2 +, (([[z]].[[z-1]].[[z-2]])/(2.3)).xz-||3,y||3 +, etc. (13) (x,y)z = xz,yz. (14) x + 0 = x.

(Boole)

(15) x1 = x. (16) (x + y) + z = x + (y + z).

(Boole)

(17) x + y = y + x.

(Boole)

(18) x + y - y = x.

(Boole)

(19) x,(y + z) = x,y + x,z.

(Boole)

(20) (x + y)z = x + [[z]].xz-†1,y†1 + etc.

Peirce: CP 3.81 Cross-Ref:†† We have also the following, which are involved implicitly in the explanations which have been given.

(21) x -< x +, y.†1

This, I suppose, is the principle of identity, for it follows from this that x = x.†2

(22) x +, x = x. (23) x,x = x.

(Jevons) (Boole)

(24) x +, y = x + y - x,y.

The principle of contradiction is

(25) x,nx = 0

where n stands for "not." The principle of excluded middle is

(26) x +, nx = 1.

It is an identical proposition, that, if φ be determinative, we have

(27) If x = y

φx = φy.

Peirce: CP 3.81 Cross-Ref:†† The six following are derivable from the formulæ already given:

(28) (x +, y),(x +, z) = x +, y,z. (29) (x - y) +, (z - w) = (x +, z)-(y +, w) + y,z,(1-w) + x,(1 - y),w.

Peirce: CP 3.81 Cross-Ref:†† In the following, φ is a function involving only the commutative operations and the operations inverse to them.

(30) φx = (φ1),x + (φ0),(1 - x).

(Boole)

(31) φx = (φ1+,(1-x)),(φ0+,x). (32) If φx = 0 (φ1),(φ0) = 0.

(Boole)

(33) If φx = 1 φ1 +, φ0 = 1.

Peirce: CP 3.81 Cross-Ref:†† The reader may wish information concerning the proofs of formulæ (30) to (33). When involution is not involved in a function nor any multiplication except that for which x,x=x, it is plain that φx is of the first degree, and therefore, since all the rules of ordinary algebra hold, we have as in that

φx = φ0 + (φ1 - φ0),x.

We shall find, hereafter, that when y has a still more general character, we have,

φx = φ0 + (φ1 - φ0)x.

The former of these equations by a simple transformation gives (30). Peirce: CP 3.81 Cross-Ref:†† If we regard (φ1), (φ0) as a function of x and develop it by (30), we have

(φ1),(φ0) = x,(φ1),(φ0) + (φ1),(φ0),(1-x).

Comparing these terms separately with the terms of the second member of (30), we see that

(φ1),(φ0) -< x.

This gives at once (32), and it gives (31) after performing the multiplication indicated in the second member of that equation and equating φx to its value as given in (30). If (φ1 +, φ0) is developed as a function of x by (31), and the factors of the second member are compared with those of the second member of (31), we get

φx -< φ1 +, φ0,

from which (33) follows immediately.

Peirce: CP 3.82 Cross-Ref:†† PROPERTIES OF ZERO AND UNITY

82. The symbolical definition of zero is

x+0 = x,

so that by (19)

x,a = x,(a+0) = x,a+x,0.

Hence, from the invertible character of this addition, and the generality of (14), we have

x,0 = 0.

By (24) we have in general,

x +, 0 = x + 0 - x,0 = x, or

x +, 0 = x.

By (4) we have

a x = (a +, 0)x = a x +, 0x.

But if a is an absurd relation, a x = 0, so that

0x = 0,

which must hold invariably. Peirce: CP 3.82 Cross-Ref:†† From (12) we have ax = (a +, 0)x = ax +, 0x +, etc., whence by (21)

0x -< ax.†1

But if a is an absurd relation, and x is not zero,

ax = 0.

And therefore, unless x=0,

0x = 0.

Peirce: CP 3.83 Cross-Ref:†† 83. Any relative x may be conceived as a sum of relatives X, X', X'', etc., such that there is but one individual to which anything is X, but one to which anything is X', etc. Thus, if x denote "cause of," X,X',X'' would denote different kinds of causes, the causes being divided according to the differences of the things they are causes of. Then we have

X y = X(y +, 0) = X y +, X0,

whatever y may be. Hence, since y may be taken so that

X y = 0,

we have

X0 = 0;

and in a similar way,

X'0 = 0,

X''0 = 0, X'''0 = 0, etc.

We have, then,

x0 = (X +, X' +, X'' +, X''' +, etc.)0 = X0 +, X'0 +, X''0 +, X'''0 +, etc. = 0.

Peirce: CP 3.84 Cross-Ref:†† 84. If the relative x be divided in this way into X,X',X'', X''', etc., so that x is that which is either X or X' or X'' or X''', etc., then non-x is that which is at once non-X and non-X' and non-X'', etc.; that is to say,

non-x = non-X, non-X', non-X'', non-X''', etc.;

where non-X is such that there is something (Z) such that everything is non-X to Z; and so with non-X', non-X'', etc. Now, non-x may be any relative whatever. Substitute for it, then, y; and for non-X, non-X', etc., Y,Y', etc. Then we have

y = Y,Y',Y'',Y''', etc.; and

Y'Z' = 1, Y''Z'' = 1,

Y'''Z''' = 1, etc.,

where Z',Z'',Z''' are individual terms which depend for what they denote on Y',Y'',Y'''.

Then we have

1 = Y'Z' = Y'Z' = Y'(Z' +, 0) = Y'Z',Y'0 = Y'Z',Y'0, or

Y'0 = 1, Y''0 = 1, Y'''0= 1, etc.

Then y0 = (Y', Y'', Y''', etc.)0 = Y'0, Y''0, Y'''0, etc. = 1. Peirce: CP 3.84 Cross-Ref:†† We have by definition, Hence, by (6),

x1 = x.

a x = (a1)x = a(1x).

Now a may express any relation whatever, but things the same way related to everything are the same. Hence,

x = 1x.

We have by definition, 1 = 1[0]. Then if X is any individual X,1 =X,1[0] = X,1X. But

1X = X.

Hence

X,1 = X,X;

and by (23)

X,1 = X;

whence if we take

x = X + X' + X'' + X''' + etc.,

where X,X' etc, denote individuals (and by the very meaning of a general term this can always be done, whatever x may be)

x,1 = (X + X' + X'' + etc.),1 = X,1 + X',1 + X'',1 + etc. = X + X' + X'' + etc. = x, or

x,1 = x.

We have by (24) x +, 1 = x + 1 - x,1 = x + 1 - x = 1, or

x +, 1 = 1.

Peirce: CP 3.85 Cross-Ref:†† 85. We may divide all relatives into limited and unlimited. Limited relatives express such relations as nothing has to everything. For example, nothing is knower of everything. Unlimited relatives express relations such as something has to everything. For example, something is as good as anything. For limited relatives, then, we may write p1 = 0.

The converse of an unlimited relative expresses a relation which everything has to something. Thus, everything is as bad as something. Denoting such a relative by q,

q1 = 1.

These formulæ remind one a little of the logical algebra of Boole; because one of them holds good in arithmetic only for zero, and the other only for unity.

Peirce: CP 3.85 Cross-Ref:†† We have by (10) 1x = (q0)x = q(0x) = q0 = 1, 1x = 1.

or

We have by (4) or by (21)

1x = (a +, 1)x = a x +, 1x,

a x -< 1x.

But everything is somehow related to x unless x is 0; hence unless x is 0, 1x = 1.

Peirce: CP 3.85 Cross-Ref:†† If a denotes "what possesses," and y "character of what is denoted by x,"

x = ay = a(y1) = (ay)1 = x1, or

x1 = x.

Peirce: CP 3.85 Cross-Ref:†† Since 1 means "identical with," l,1w denotes whatever is both a lover of and identical with a woman, or a woman who is a lover of herself. And thus, in general,

x,1 = x[0],.

Peirce: CP 3.86 Cross-Ref:†† 86. Nothing is identical with every one of a class; and therefore 1x is zero, unless x denotes only an individual when 1x becomes equal to x. But equations founded on interpretation may not hold in cases in which the symbols have no rational interpretation. Peirce: CP 3.86 Cross-Ref:†† Collecting together all the formulæ relating to zero and unity, we have (34) x +, 0 = x.

(Jevons)

(35) x +, 1 = 1.

(Jevons)

(36) x0 = 0. (37) 0x = 0. (38) x,0 = 0.

(Boole)

(39) x0 = 1. (40) 0x = 0, provided x > 0.†1 (41) 1x = x. (42) x,1 = x[0],. (43) x1 = x. (44) 1x = 0, unless x is individual, when 1x = x. (45) q1 = 1, where q is the converse of an unlimited relative. (46) 1x = 1, provided x > 0.†1 (47) x,1 = x.

(Boole)

(48) p1 = 0, where p is a limited relative. (49) 1x = 1.

These, again, give us the following:

(50) 0 +, 1 = 1

(64) 01=0

(51) 0 +, 1 = 1

(65) 1 1=1

(52) 00 = 0

(66) 1,1=1

(53) 0,0 = 0

(67) 11=1

(54) 00 = 1

(68) 11 = 1

(55) 10 = 0

(69) 1,1 = 1

(56) 01 = 0

(70) 11 = 1

(57) 0,1 = 0

(71) 11=1

(58) 01 = 0

(72) 11 = 1

(59) 10 = 1

(73) 1,1 = 1

(60) 01 = 0

(74) 11 = 1

(61) 10 = 0

(75) 11 = 0

(62) 0,1 = 0

(76) 1, = 1

(63) 10 = 1

Peirce: CP 3.86 Cross-Ref:†† From (64) we may infer that 0 is a limited relative, and from (60) that it is not the converse of an unlimited relative. From (70) we may infer that 1 is not a limited relative, and from (68) that it is the converse of an unlimited relative.

Peirce: CP 3.87 Cross-Ref:†† FORMULÆ RELATING TO THE NUMBERS OF TERMS

87. We have already seen that

(77) If x -< y, then [[x]] -< [[y]]. (78) When x,y = 0, then [[x +, y]] = [[x]] +, [[y]], (79) When [[x y]]:[[nxy]] = [[x]]:[[nx]], then [[x y]] = [[x]][[y]]. (80) When [[xny]] = [[x]][[ny]][[1]], then [[x y]] = [[x]][[y]].

Peirce: CP 3.87 Cross-Ref:†† It will be observed that the conditions which the terms must conform to, in order that the arithmetical equations shall hold, increase in complexity as we pass from the more simple relations and processes to the more complex. Peirce: CP 3.88 Cross-Ref:†† 88. We have seen that

(81) [[0]] = 0. (82) [[1]] = 1.

Most commonly the universe is unlimited, and then

(83) [[1]] = ∞;†1

and the general properties of 1 correspond with those of infinity. Thus,

x +, 1 = 1 corresponds to x + ∞ = ∞, q1 = 1 corresponds to q ∞ = ∞, 1x = 1 corresponds to ∞ x= ∞, p1 = 0 corresponds to p ∞ = 0, 1x = 1 corresponds to ∞ x = ∞.

Peirce: CP 3.88 Cross-Ref:†† The formulæ involving commutative multiplication are derived from the equation 1, = 1. But if 1 be regarded as infinite, it is not an absolute infinite; for 10 = 0. On the other hand, 11 = 0. Peirce: CP 3.88 Cross-Ref:†† It is evident, from the definition of the number of a term, that

(84) [[x,]] = [[x]]:[[1]].

Peirce: CP 3.88 Cross-Ref:†† We have, therefore, if the probability of an individual being x to any y is independent of what other y's it is x to, and if x is independent of y,

(85) [[xy,]] = [[x,]][[y]].

Peirce: CP 3.89 Cross-Ref:†† §5. GENERAL METHOD OF WORKING WITH THIS NOTATION

89. Boole's logical algebra contains no operations except our invertible addition and commutative multiplication, together With the corresponding subtraction and division. He has, therefore, only to expand expressions involving division, by means of (30), so as to free himself from all non-determinative operations, in order to be able to use the ordinary methods of algebra, which are, moreover, greatly simplified by the fact that

x,x = x.

Peirce: CP 3.90 Cross-Ref:†† 90. Mr. Jevons's modification †1 of Boole's algebra involves only non-invertible addition and commutative multiplication, without the corresponding

inverse operations. He is enabled to replace subtraction by multiplication, owing to the principle of contradiction, and to replace division by addition, owing to the principle of excluded middle. For example, if x be unknown, and we have

x +, m = a,

or what is denoted by x together with men make up animals, we can only conclude, with reference to x, that it denotes (among other things, perhaps) all animals not men; that is, that the x's not men are the same as the animals not men. Let ~m denote non-men; then by multiplication we have

x~m, +, m,~m = x,~m = a,~m,

because, by the principle of contradiction,

m,~m = 0.

Or, suppose, x being again unknown, we have given

a,x = m.

Then all that we can conclude is that the x's consist of all the m's and perhaps some or all of the non-a's, or that the x's and non-a's together make up the m's and non-a's together. If, then, ~a denote non-a, add ~a to both sides and we have

a,x +, ~a = m +, ~a. Then by (28)

(a +, ~a),(x +, ~a) = m +, ~a.

But by the principle of excluded middle,

a +, ~a = 1 and therefore

x +, ~a = m +, ~a.

I am not aware that Mr. Jevons actually uses this latter process, but it is open to him

to do so. In this way, Mr. Jevons's algebra becomes decidedly simpler even than Boole's. Peirce: CP 3.90 Cross-Ref:†† It is obvious that any algebra for the logic of relatives must be far more complicated. In that which I propose, we labor under the disadvantages that the multiplication is not generally commutative, that the inverse operations are usually indeterminative, and that transcendental equations, and even equations like

abx = cdex + fx + x,

where the exponents are three or four deep, are exceedingly common. It is obvious, therefore, that this algebra is much less manageable than ordinary arithmetical algebra. Peirce: CP 3.91 Cross-Ref:†† 91. We may make considerable use of the general formulæ already given, especially of (1), (21), and (27), and also of the following, which are derived from them:

(86) If a -< b then there is such a term x that a +, x = b. (87) If a -< b then there is such a term x that b,x = a. (88) If b,x = a then a -< b. (89) If a -< b c +, a -< c +, b. (90) If a -< b c a -< c b. (91) If a -< b a c -< b c. (92) If a -< b cb -< ca†1 (93) If a -< b ac -< bc. (94) a,b -< a

Peirce: CP 3.91 Cross-Ref:†† There are, however, very many cases in which the formulæ thus far given are of little avail. Peirce: CP 3.92 Cross-Ref:†† 92. Demonstration of the sort called mathematical is founded on suppositions of particular cases. The geometrician draws a figure; the algebraist assumes a letter to signify a single quantity fulfilling the required conditions. But while the mathematician supposes an individual case, his hypothesis is yet perfectly general, because he considers no characters of the individual case but those which must belong to every such case. The advantage of his procedure lies in the fact that the logical

laws of individual terms are simpler than those which relate to general terms, because individuals are either identical or mutually exclusive, and cannot intersect or be subordinated to one another as classes can. Mathematical demonstration is not, therefore, more restricted to matters of intuition than any other kind of reasoning. Indeed, logical algebra conclusively proves that mathematics extends over the whole realm of formal logic; and any theory of cognition which cannot be adjusted to this fact must be abandoned. We may reap all the advantages which the mathematician is supposed to derive from intuition by simply making general suppositions of individual cases. Peirce: CP 3.93 Cross-Ref:†† 93. In reference to the doctrine of individuals,†1 two distinctions should be borne in mind. The logical atom, or term not capable of logical division, must be one of which every predicate may be universally affirmed or denied. For, let A be such a term. Then, if it is neither true that all A is X nor that no A is X, it must be true that some A is X and some A is not X; and therefore A may be divided into A that is X and A that is not X, which is contrary to its nature as a logical atom. Such a term can be realized neither in thought nor in sense. Not in sense, because our organs of sense are special -- the eye, for example, not immediately informing us of taste, so that an image on the retina is indeterminate in respect to sweetness and non-sweetness. When I see a thing, I do not see that it is not sweet, nor do I see that it is sweet; and therefore what I see is capable of logical division into the sweet and the not sweet. It is customary to assume that visual images are absolutely determinate in respect to color, but even this may be doubted. I know no facts which prove that there is never the least vagueness in the immediate sensation. In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates. A logical atom, then, like a point in space, would involve for its precise determination an endless process. We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate. Such a term as "the second Philip of Macedon" is still capable of logical division--into Philip drunk and Philip sober, for example; but we call it individual because that which is denoted by it is in only one place at one time. It is a term not absolutely indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them. Such differences we habitually disregard in the logical division of substances. In the division of relations, etc., we do not, of course, disregard these differences, but we disregard some others. There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse, and if 1 be a term which in consequence of such neglect becomes indivisible in that discourse, we have in that discourse,

[[1]] = 1.

This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words individual {to atomon} and singular (to kath' hekaston); but as those who have used the word individual have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense.†P1

Peirce: CP 3.94 Cross-Ref:†† 94. The old logics distinguish between individuum signatum and individuum vagum. "Julius Cæsar" is an example of the former; "a certain man," of the latter. The individuum vagum, in the days when such conceptions were exactly investigated, occasioned great difficulty from its having a certain generality, being capable, apparently, of logical division. If we include under the individuum vagum such a term as "any individual man," these difficulties appear in a strong light, for what is true of any individual man is true of all men. Such a term is in one sense not an individual term; for it represents every man. But it represents each man as capable of being denoted by a term Which is individual; and so, though it is not itself an individual term, it stands for any one of a class of individual terms. If we call a thought about a thing in so far as it is denoted by a term, a second intention, we may say that such a term as "any individual man" is individual by second intention. The letters which the mathematician uses (whether in algebra or in geometry) are such individuals by second intention. Such individuals are one in number, for any individual man is one man; they may also be regarded as incapable of logical division, for any individual man, though he may either be a Frenchman or not, is yet altogether a Frenchman or altogether not, and not some one and some the other. Thus, all the formal logical laws relating to individuals will hold good of such individuals by second intention, and at the same time a universal proposition may at any moment be substituted for a proposition about such an individual, for nothing can be predicated of such an individual which cannot be predicated of the whole class. Peirce: CP 3.95 Cross-Ref:†† 95. There are in the logic of relatives three kinds of terms which involve general suppositions of individual cases. The first are individual terms, which denote only individuals †1; the second are those relatives whose correlatives are individual: I term these infinitesimal relatives†2; the third are individual infinitesimal relatives, and these I term elementary relatives.†3

Peirce: CP 3.96 Cross-Ref:†† INDIVIDUAL TERMS

96. The fundamental formulæ relating to individuality are two. Individuals are denoted by capitals.

(95) (96)

If x > 0

x = X +, X' +, X'' +, X''' +, etc. yX = yX.

Peirce: CP 3.96 Cross-Ref:†† We have also the following which are easily deducible from these two:

(97) (y,z)X = (y X),(z X).

(99) [[X]] = 1.

(98) X,y[0] = X,y X.

(100) 1X = X.

We have already seen that 1x = 0, provided that [[x]] > 1. Peirce: CP 3.97 Cross-Ref:†† 97. As an example of the use of the formulæ we have thus far obtained, let us investigate the logical relations between "benefactor of a lover of every servant of every woman," "that which stands to every servant of some woman in the relation of benefactor of a lover of him," "benefactor of every lover of some servant of a woman," "benefactor of every lover of every servant of every woman," etc. Peirce: CP 3.97 Cross-Ref:†† In the first place, then, we have by (95)

sw = s(W' =, W'' +, W''' +, etc.) = sW' +, sW'' +, sW''' +, etc. sw = sW' +, W'' +, W''' +, etc. = sW' +, sW'' +, sW''' +, etc.

From the last equation we have by (96)

sw = (sW'),(sW''),(sW'''), etc.

Now by (31) x' +, x'' +, etc. = x',x",x"', etc. +, etc., or

(101)

π' -< Σ',

where π' and Σ' signify that the addition and multiplication with commas †1 are to be used. From this it follows that

(102)

sw -< sw.†2

If w vanishes, this equation fails, because in that case (95) does not hold. Peirce: CP 3.97 Cross-Ref:†† From (102) we have

(l s)w -< l sw.†3

(103)

Since

a = a,b +, etc.,

b = a,b +, etc., we have

la = l(a,b +, etc.) = l(a,b) +, l (etc.),

lb = l(a,b +, etc.) = l(a,b) +, l (etc.).

Multiplying these two equations commutatively we have

(la),(lb) = l(a,b) +, etc.

or

(104)

lπ' -< π'l.†1

Now (l s)w = (s)W' +, W'' +, W''' +, etc. = π'(l s)W = π'l sW,

l sw = l sW' +, W'' +, W''' +, etc. = lπ'sW' = lπ'sW.

Hence,

(105)

l sw -< (l s)w,

or every lover of a servant of all women stands to every woman in the relation of lover of a servant of hers.

Peirce: CP 3.97 Cross-Ref:†† From (102) we have

(106)

lsw -< l sw.†2

By (95) and (96) we have

lsw = ls(W' +, W'' +, W''' etc.) = lsW' +, lsW'' +, lsW''' +, etc.

= lsW' +, lsW'' +, lsW''' + etc.

Now sW = sW'+, W'' +, W''' +, etc. = sW',sW'',sW''', etc.

So that by (94)

sw -< sW' -< sW'.

Hence by (92)

lsW' -< lsw, lsW'' -< lsw lsW''' -< lsw.

lsW' +, lsW'' +, lsW''' -< lsw;

Adding,

or

(107)

lsw -< lsw.

That is, every lover of every servant of any particular woman is a lover of every servant of all women. Peirce: CP 3.97 Cross-Ref:†† By (102) we have

(108)

lsw -< lsw.†3

Thus we have

lsw -< lsw -< lsw -< l sw -< (l s)w -< l sw.†P1

Peirce: CP 3.98 Cross-Ref:†† 98. By similar reasoning we can easily make out the relations shown in the following table. It must be remembered that the formulæ do not generally hold when exponents vanish.

[Click here to view]†1

Peirce: CP 3.99 Cross-Ref:†† 99. It appears to me that the advantage of the algebraic notation already begins to be perceptible, although its powers are thus far very imperfectly made out. At any rate, it seems to me that such a prima facie case is made out that the reader who still denies the utility of the algebra ought not to be too indolent to attempt to write down the above twenty-two terms in ordinary language With logical precision. Having done that, he has only to disarrange them and then restore the arrangement by ordinary logic, in order to test the algebra so far as it is yet developed.

Peirce: CP 3.100 Cross-Ref:†† INFINITESIMAL RELATIVES

100. We have by the binomial theorem by (49) and by (47),

(1 +x)n = 1 + Σ[p]xn-p + xn.

Now, if we suppose the number of individuals to which any one thing is x to be reduced to a smaller and smaller number, we reach as our limit x2 = 0,

Σ[p]xn-p = [[n]].1n-†1,x†1 = xn,

(1 + x)n = 1 + x n.

Peirce: CP 3.101 Cross-Ref:†† 101. If, on account of the vanishing of its powers, we call x an infinitesimal here and denote it by i, and if we put

x n = i n = y,

our equation becomes

(109)

(1 +i) y/i = 1 + y.

Putting y = 1, and denoting (1 + i)1/i by

(110)

, we have

= (1 +i)1/i = 1 + 1.

Peirce: CP 3.102 Cross-Ref:†† 102. In fact, this agrees With ordinary algebra better than it seems to do; for 1 is itself an infinitesimal, and is 1. If the higher powers of 1 did not vanish, we should get the ordinary development of . Peirce: CP 3.103 Cross-Ref:†† 103. Positive powers of we have

(111)

-x

= 1 - x.

are absurdities in our notation. For negative powers

Peirce: CP 3.104 Cross-Ref:†† 104. There are two ways of raising

-x

to the yth power.

In the first place, by the binomial theorem,

(1-x)y = 1-[[y]].1y-†1,x†1 + ([[y]].[[y-1]]/2).1y-‡2,x‡2--etc.;

and, in the second place, by (111) and (10).

-x

y = 1 - x y.†1

It thus appears that the sum of all the terms of the binomial development of (1-x)y, after the first, is -x y.†2 The truth of this may be shown by an example. Suppose the number of y's are four, viz. Y', Y'', Y''', and Y''''. Let us use x', x'', x''', and x'''' in such senses that

x Y' = x', x Y'' = x'', x Y''' = x''', x Y'''' = x''''.

Then the negatives of the different terms of the binomial developement are,

[[y]].1y-†1,x†1 = x' + x'' + x''' + x''''.

-(([[y]].[[y-1]])/2).1y-‡2,x‡2 = -x',x''-x',x'''-x',x''''-x'',x'''-x'',x''''-x''',x''''.†3

+(([[y]].[[y-1]][[y-2]])/2.3).1-||3x||3 = x',x'',x'''+ x',x'',x'''' + x',x''',x'''' + x'',x''',x''''.†4

Now, since this addition is invertible, in the first term, x' that is x'', is counted over twice, and so with every other pair. The second term subtracts each of these pairs, so that it is only counted once. But in the first term the x' that is x'' that is x''' is counted in three times only, while in the second term it is subtracted three times; namely, in (x',x''), in (x',x''') and in (x'',x'''). On the whole, therefore, a triplet would not be represented in the sum at all, were it not added by the third term. The whole quartette is included four times in the first term, is subtracted six times by the second term, and is added four times in the third term. The fourth term subtracts it once, and thus in the sum of these negative terms each combination occurs once, and once only; that is to say the sum is

x' +, x'' +, x''' +, x'''' = x(Y' +, Y'' +, Y''' +, Y'''') = x y.

Peirce: CP 3.105 Cross-Ref:†† 105. If we write (a x)3 for [[x]].[[x-1]].[[x-2]].1x-†3,a†3, that is for whatever is a to any three x's, regard being had for the order of the x's; and employ the modern numbers as exponents with this signification generally, then

1 - a x + (1/2!)(a x)2 - (1/3!)(a x)3 + etc.

is the development of (1 - a)x and consequently it reduces itself to 1 - a x. That is,

(112) x = x - (1/2!)x2 + (1/3!)x3 + (1/4!)x4 +etc.

Peirce: CP 3.106 Cross-Ref:†† 106. 1 - x denotes everything except x, that is, whatever is other than every x; so that - means "not." We shall take log x in such a sense that

log x

= x.†P1

Peirce: CP 3.107 Cross-Ref:†† 107. I define the first difference of a function by the usual formula,

(113)

{D}φx = φ(x + {D}x) - φx,

where {D}x is an indefinite relative which never has a correlate in common with x. So that

(114)

x,({D}x) = 0

x + {D}x = x +, {D}x.

Higher differences may be defined by the formulæ

(115)

{D}n·x = 0 if n > 1

{D}2.φx ={D}{D}x = φ(x+2.{D}x)-2.φ(x+{D}x)+φx,

{D}3·φx = {D}{D}2.x = φ(x+3.{D}x)-3.φ(x+2.{D}x)+3.φ(x+{D}x)-φx.

(116) {D}n·φx = φ(x+n.{D}x)-n.φ(x+(n-1).{D}x)

+ (n.(n-1))/2.φ(x+(n-2).{D}x) - etc.

Peirce: CP 3.108 Cross-Ref:†† 108. The exponents here affixed to {D} denote the number of times this operation is to be repeated, and thus have quite a different signification from that of the numerical coefficients in the binomial theorem. I have indicated the difference by putting a period after exponents significative of operational repetition. Thus, m2 may denote a mother of a certain pair, m2. a maternal grandmother. Peirce: CP 3.109 Cross-Ref:†† 109. Another circumstance to be observed is, that in taking the second difference of x, if we distinguish the two increments which x successively receives as {D}'x and {D}''x, then by (114)

({D}'x),({D}''x) = 0

If {D}x is relative to so small a number of individuals that if the number were diminished by one {D}n·φx would vanish, then I term these two corresponding differences differentials, and write them with d instead of {D}. Peirce: CP 3.110 Cross-Ref:†† 110. The difference of the invertible sum of two functions is the sum of their differences; for by (113) and (18),

(117) {D}(φx + μx) = φ(x + {D}x) + μ(x + {D}x) - φx - μx

= φ(x + {D}x) - φx + μ(x + {D}x) - μx = {D}μx + {D}μx.

If a is a constant, we have

(118) {D}aφx = a(φx +, {D}φx) - aφx = a{D}φx - (a{D}φx),aφx,

{D}2.aφx = -{D}aφx,a{D}x, etc.

{D}(φx)a = ({D}φx)a - (({D}φx)a),φxa,

{D}2.(φx)a = -{D}(φx)a, etc.

(119) φ(a,φx) = a,{D}φx.

Peirce: CP 3.110 Cross-Ref:†† Let us differentiate the successive powers of x. We have in the first place,

{D}(x2) = (x + {D}x)2 - x2 = 2.x2-†1,({D}x)†1 + ({D}x)2.

Here, if we suppose {D}x to be relative to only one individual, ({D}x)2 vanishes, and we have, with the aid of (115),

d(x2) = 2.x1,dx .

Considering next the third power, we have, for the first differential,

{D}(x3) = (x + {D}x)3 - x3 = 3.x3-†1,({D}x)†1 + 3.x3-‡2,({D}x)‡2+({D}x)3,

d(x3) = 3.x2,d(x).

To obtain the second differential, we proceed as follows:

{D}2.(x3) = (x + 2.{D}x)3 - 2.(x + {D}x)3 + x3

= x3 + 6.x3-‡1,({D}x)†1 + 12.x3-‡2,({D}x)‡2 + 8.({D}x)3 - 2.x3 - 6.x3-||1,({D}x)||1 - 6.x3-§2,({D}x)§2 - 2.({D}x)3 + x3

= 6.x3-‡2,({D}x)‡2 + 6.({D}x)3.

Here, if {D}x is relative to less than two individuals, {D}φx vanishes. Making it relative to two only, then, we have

d2.(x3) = 6.x1,(dx)2.

These examples suffice to show what the differentials of xn will be. If for the number n we substitute the logical term n, we have

{D}(xn) = (x + {D}x)n - xn = [[n]].xn-†1,({D}x)†1 + etc.

d(xn) = [[n]].xn-1,(dx).

We should thus readily find

(120) dm·(xn) = [[n]].[[n-1]].[[n-2]]....[n-m+1].xn-†m,(dx)†m.

Peirce: CP 3.110 Cross-Ref:†† Let us next differentiate lx. We have, in the first place,

{D}lx = lx +, {D}x - lx = lx,l{D}x - lx = lx,(l{D}x - 1).

The value of l{D}'x - 1 is next to be found.

We have by (111)

l{D}z - 1 = l{D}[x].

l{D}x - 1 = log l{D}x.

Hence,

But by (10)

log l{D}x = (log l){D}x.

Substituting this value of ldx - 1 in the equation lately found for dlx we have

(121) dlx = lx,(log,l) dx = lx,(l - 1) dx = -lx,(1 - l) dx.

Peirce: CP 3.111 Cross-Ref:†† 111. In printing this paper, I here make an addition which supplies an omission in the account given above †1 of involution in this algebra. We have seen that every term which does not vanish is conceivable as logically divisible into individual terms. Thus we may write

s = S'+, S'' +, S''' +, etc.

where not more than one individual is in any one of these relations to the same individual, although there is nothing to prevent the same person from being so related to many individuals.†1 Thus, "bishop of the see of" may be divided into first bishop, second bishop, etc., and only one person can be nth bishop of any one see, although the same person may (where translation is permitted) be nth bishop of several sees. Now let us denote the converse of x by Kx; thus, if s is "servant of," Ks is "master or mistress of." Then we have

Ks = KS' +, KS'' +, KS''' +, etc.;

and here each of the terms of the second member evidently expresses such a relation that the same person cannot be so related to more than one, although more than one may be so related to the same. Thus, the converse of "bishop of the see of --" is "see one of whose bishops is --," the converse of "first bishop of --" is "see whose first bishop is --," etc. Now, the same see cannot be a see whose nth bishop is more than one individual, although several sees may be so related to the same individual. Such relatives I term infinitesimal on account of the vanishing of their higher powers. Every relative has a converse, and since this converse is conceivable as divisible into individual terms, the relative itself is conceivable as divisible into infinitesimal terms. To indicate this we may write

(122)

If x > 0 x = X[,] +, X[,,] +, X[,,,] +, etc.

Peirce: CP 3.112 Cross-Ref:†† 112. As a term which vanishes is not an individual, nor is it composed of individuals, so it is neither an infinitesimal nor composed of infinitesimals.

As we write l S',l S'',l S''', etc. = ls,

so we may write

(123)

L[,]s,L[,,]s,L[,,,]s, etc. = ls,

But as the first formula is affected by the circumstance that zero is not an individual, so that lsw does not vanish on account of no woman having the particular kind of servant denoted by S'', lsw denoting merely every lover of whatever servant there is of any woman; so the second formula is affected in a similar way, so that the vanishing of L[,]s does not make ls to vanish, but this is to be interpreted as denoting everything which is a lover, in whatever way it is a lover at all, of a servant.†1 Then just as we have by (112), that

(124)

ls = 1 - (1 - l)s;†2

so we have

(125)

ls = 1 - l(1 - s).†3

Mr. De Morgan denotes ls and ls by L S[,] and L[,]S respectively,†4 and he has traced out the manner of forming the converse and negative of such functions in detail. The following table contains most of his results in my notation.†5 For the converse of m, I write w; and for that of n, u.

-----------------------------------------------x

|

Kx

-----------------------------------------------mn mn = (1-m)(1-n)

| uw | uw = (1-u)(1-w)

mn = (1-m)(1-n)

| uw = (1-u)(1-w)

-----------------------------------------------⎧-x

| K⎧-x

-----------------------------------------------(1-m)n = m(1-n)

| w(1-u) = (1-w)u

(1-m)n

| w(1-u)

m(1-n)

| (1-m)n

------------------------------------------------

Peirce: CP 3.113 Cross-Ref:†† 113. I shall term the operation by which w is changed to lw, backward involution. All the laws of this but one are the same as for ordinary involution, and the one exception is of that kind which is said to prove the rule. It is that whereas with ordinary involution we have,

(ls)w = l(sw);

in backward involution we have

(126)

l(sw) = (l s)w;

that is, the things which are lovers to nothing but things that are servants to nothing but women are the things which are lovers of servants to nothing but women. Peirce: CP 3.114 Cross-Ref:†† 114. The other fundamental formulæ of backward involution are as follows:

(127)

l +, sw = lw,sw,

or, the things which are lovers or servants to nothing but women are the things which are lovers to nothing but women and servants to nothing but women.

(128)

l(f,u) = lf,lu,

or, the things which are lovers to nothing but French violinists are the things that are lovers to nothing but Frenchmen and lovers to nothing but violinists. This is perhaps not quite axiomatic. It is proved as follows. By (125) and (30)

l(f,u) =

-l(1-f,u)

=

-(l(1-f)

+, l(1-u))

By (125), (13), and (7),

lf,lu =

-l(1-f),

-l(1-u)

=

-(l(1-f)

+, l(1-u)).

Finally, the binomial theorem holds with backward involution. For those persons who are lovers of nothing but Frenchmen and violinists consist first of those who are lovers of nothing but Frenchmen; second, of those who in some ways are lovers of nothing but Frenchmen and in all other ways of nothing but violinists, and finally of those who are lovers only of violinists. That is,

(129) l(u +, f) = lu +, Σ[p]l-pu,pf +, lf.

In order to retain the numerical coefficients, we must let {l} be the number of persons that one person is lover of. We can then write

l(u+f) = lu + {l}l-†1u,†1f+ (({l}·{l-1})/2)l-‡2u,‡2f + etc.

Peirce: CP 3.115 Cross-Ref:†† 115. We have also the following formula which combines the two involutions:

(130)

l(sw) = (ls)w;

that is, the things which are lovers of nothing but what are servants of all women are the same as the things which are related to all women as lovers of nothing but their servants. Peirce: CP 3.116 Cross-Ref:†† 116. It is worth while to mention, in passing, a singular proposition derivable from (128). Since, by (124) and (125)

xy = (1-x)(1-y),†1

and since

1-(u +, f) =

-(u +, f) =

-u,

-f =

(1-u),(1-f),

(128) gives us,

(1-l)(1-u),(1-f) = (1-l)(1-u) +, Σ[p](1-(l-p))(1-u), (1-p)(1-f) +, (1-l)(1-f).

This is, of course, as true for u and f as for (1-u) and (1-f). Making those substitutions, and taking the negative of both sides, we have, by (124)

(131) l(u,f) = (lu),π'[p]((l-p)u +, pf),(lf),

or, the lovers of French violinists are those persons who, in reference to every mode of loving whatever, either in that way love some violinists or in some other way love some Frenchmen. This logical proposition is certainly not self-evident, and its practical importance is considerable. In a similar way, from (12) we obtain

(132)

(e,c)f = π'[p](e(f-p) +, c p),

that is, to say that a person is both emperor and conqueror of the same Frenchman is the same as to say that, taking any class of Frenchmen whatever, this person is either an emperor of some one of this class, or conqueror of some one among the remaining Frenchmen. Peirce: CP 3.117 Cross-Ref:†† 117. The properties of zero and unity, with reference to backward involution, are easily derived from (125). I give them here in comparison with the corresponding formulæ for forward involution.

(133)

0x

(134)

q0 = 0

=1

x0= 1. 0r = 0,

where q is the converse of an unlimited relative, and r is greater than zero.

(135)

1x = x

x1 = x.

(136)

y1 = y

1z = z,

where y is infinitesimal, and z is individual. Otherwise, both vanish.

(137)

1s

=0

p1 = 0,

where s is less than unity and p is a limited relative.

(138)

x1 = 1

1x = 1.

Peirce: CP 3.118 Cross-Ref:†† 118. In other respects the formulæ for the two involutions are not so analogous as might be supposed; and this is owing to the dissimilarity between individuals and infinitesimals. We have, it is true, if X' is an infinitesimal and X' an individual,

(139) X[,](y,z) = X[,]y,X[,]z like (y,z)X' = y X',z X'; (140) X[,]y[0] = X[,],X[,]y (141) {X[,]} = 1

"

X',y[0] = X',y X';

" [[X']] = 1.

We also have

(142)

X[,]y -< X[,]y.

But we have not X'y = X'y, and consequently we have not sw -< sw, for this fails if there is anything which is not a servant at all, while the corresponding formula sw -< sw only fails if there is not anything which is a woman. Now, it is much more often the case that there is something which is not x, than that there is not anything which is x. We have with the backward involution, as with the forward,†1 the formulæ

(143)

If x -< y

yz -< xz;†1

(144)

If x -< y

zx -< zy;†1

The former of these gives us

(145)

l sw -< (ls)w,

or, whatever is lover to nothing but what is servant to nothing but women †2 stands to nothing but a woman in the relation of lover of every servant of hers. The following formulæ can be proved without difficulty.

(146)

lsw -< lsw,

or, every lover of somebody who is servant to nothing but a woman stands to nothing but women in the relation of lover of nothing but a servant of them.

(147)

lsw -< l(sw),

or, whatever stands to a woman in the relation of lover of nothing but a servant of hers is a lover of nothing but servants of women.

The differentials of functions involving backward involution are

(148)

dnx = {n}n-1x,dx.

(149)

dxl = xl,dx log.x.

In regard to powers of

(150)

x

=

we have

x.

Exponents with a dot may also be put upon either side of the letters which they affect. Peirce: CP 3.119 Cross-Ref:††

119. The greater number of functions of x in this algebra may be put in the form

φx = Σ[p] Σ[q] [p]A pxq [p]B[q].

For all such functions Taylor's and Maclaurin's theorems hold good in the form, ------(151) | y | | 0 | ∞ ----| ----| Σ[p] 1/p! . dp· = 1. |dx | | y | 0 -------

The symbol

---|a| ---| is used to denote that a is to be substituted |b| ----

for b in what follows. For the sake of perspicuity, I will write Maclaurin's theorem at length. ---|x | φx = ----| |dx | ----

---|0 | ----| ((1/0!).d0+(1/1!).d1+(1/2!).d2+(1/3!).d3+etc.)φx |y | ----

Peirce: CP 3.119 Cross-Ref:†† The proof of these theorems is very simple. The (p+q)th differential of pxq is the only one which does not vanish when x vanishes. This differential then becomes [[p+q]]!.p(dx)q. It is plain, therefore, that the theorems hold when the coefficients pAq and pBq are 1. But the general development, by Maclaurin's theorem, of aφx or (φx)a is in a form which (112) reduces to identity. It is very likely that the application of these theorems is not confined within the limits to which I have restricted it. We may write these theorems in the form ---|y | (152) ----| |dx | ----

---|0 | ----| ⎧d = 1, |y | ----

provided we assume that when the first differential is positive

⎧d = (I/0!)d0 + (I/I!)dI + (I/2!)d2 + etc.,

but that when the first differential is negative this becomes by (111),

⎧d = 1 + d.

Peirce: CP 3.120 Cross-Ref:†† 120. As another illustration of the use which may be made of differentiation in logic, let us consider the following problem. In a certain institution all the officers (x) and also all their common friends (f) are privileged persons (y). How shall the class of privileged persons be reduced to a minimum? Here we have

y = x + fx, dy = dx + dfx = dx - fx,(1-f)dx.

When y is at a minimum it is not diminished either by an increase or diminution of x. That is,

[[dy]] >- 0,

and when [[x]] is diminished by one,

[[dy]] -< 0 ,

Peirce: CP 3.120 Cross-Ref:†† When x is a minimum, then

[[dx-fx,(1-f)dx]] >- 0 [[dx-fx-I,(1-f)dx]] -< 0 (A) [[dx]]-[[fx,(1-f)dx >- 0 [[dx]]-[[fx-I,(1-f)dx]] - 1.

This is the general solution of the problem. If the event of a person who may be an official in the institution being a friend of a second such person is independent of and equally probable with his being a friend of any third such person, and if we take p, or the whole class of such persons, for our universe, we have,

p = 1;

[[fx,]] = [[fx]]/[[p]] = ([[f]]/[[p]])[[x]],

[[(1-f)dx]] = [[1-f]].[[dx]] = ([[p]]-[[f]]).[[dx]],

[fx,(1-f)dx] = ([[f]])/[[p]])[[x]].([[p]]-[[f]]).[[dx]]

Substituting these values in our equations marked (A) we get, by a little reduction,

[[x]] >- (log([[p]]-[[f]]))/(log[[p]]-log[[f]]),

[[x]] -< (log([[p]]-[[f]]))/(log[[p]]-log[[f]]) + 1.

The same solution would be reached through quite a different road by applying the calculus of finite differences in the usual way.

Peirce: CP 3.121 Cross-Ref:†† ELEMENTARY RELATIVES †1

121. By an elementary relative I mean one which signifies a relation which exists only between mutually exclusive pairs (or in the case of a conjugative term, triplets, or quartettes, etc.) of individuals, or else between pairs of classes in such a way that every individual of one class of the pair is in that relation to every individual of the other. If we suppose that in every school, every teacher teaches every pupil (a supposition which I shall tacitly make whenever in this paper I speak of a school), then pupil is an elementary relative. That every relative may be conceived of as a logical sum of elementary relatives is plain, from the fact that if a relation is sufficiently determined it can exist only between two individuals. Thus, a father is either father in the first ten years of the Christian era, or father in the second ten years, in the third ten years, in the first ten years, B. C., in the second ten years, or the third ten years, etc. Any one of these species of father is father for the first time or father for the second time, etc. Now such a relative as "father for the third time in the second decade of our era, of --" signifies a relation which can exist only between mutually exclusive pairs of individuals, and is therefore an elementary relative; and so the relative father may be resolved into a logical sum of elementary relatives. Peirce: CP 3.122 Cross-Ref:†† 122. The conception of a relative as resolvable into elementary relatives has the same sort of utility as the conception of a relative as resolvable into infinitesimals or of any term as resolvable into individuals. Peirce: CP 3.123 Cross-Ref:†† 123. Elementary simple relatives are connected together in systems of four. For if A:B be taken to denote the elementary relative which multiplied into B gives A, then this relation existing as elementary, we have the four elementary relatives

A:A A:B B:A B:B.

An example of such a system is--colleague: teacher: pupil: schoolmate. In the same

way, obviously, elementary conjugatives are in systems the number of members in which is (n+1)n+1 where n is the number of correlates which the conjugative has. At present, I shall consider only the simple relatives. Peirce: CP 3.124 Cross-Ref:†† 124. The existence of an elementary relation supposes the existence of mutually exclusive pairs of classes. The first members of those pairs have something in common which discriminates them from the second members, and may therefore be united in one class, while the second members are united into a second class. Thus pupil is not an elementary relative unless there is an absolute distinction between those who teach and those who are taught. We have, therefore, two general absolute terms which are mutually exclusive, "body of teachers in a school," and "body of pupils in a school." These terms are general because it remains undetermined what school is referred to. I shall call the two mutually exclusive absolute terms which any system of elementary relatives supposes, the universal extremes of that system. There are certain characters in respect to the possession of which both members of any one of the pairs, between which there is a certain elementary relation, agree. Thus, the body of teachers and the body of pupils in any school agree in respect to the country and age in which they live, etc., etc. Such characters I term scalar characters for the system of elementary relatives to which they are so related; and the relatives written with a comma which signify the possession of such characters, I term scalars for the system. Thus, supposing French teachers have only French pupils and vice versa, the relative

f,

will be a scalar for the system "colleague: teacher: pupil: schoolmate." If r is an elementary relative for which s is a scalar,

(154)

s,r = rs,.

Peirce: CP 3.125 Cross-Ref:†† 125. Let c, t, p, s, denote the four elementary relatives of any system; such as colleague, teacher, pupil, schoolmate; and let a,, b,, c,, d,, be scalars for this system. Then any relative which is capable of expression in the form

a,c + b,t + c,p + d,s

I shall call a logical quaternion. Let such relatives be denoted by q, q', q'', etc. It is plain, then, from what has been said, that any relative may be regarded as resolvable into a logical sum of logical quaternions. Peirce: CP 3.126 Cross-Ref:†† 126. The multiplication of elementary relatives of the same system follows a

very simple law. For if u and v be the two universal extremes of the system c, t, p, s, we may write

c = u:u t = u:v p = v:u s = v:v,

and then if w and w' are each either u or v, we have

(w':w)⎧-w = 0.

(155)

This gives us the following multiplication-table, where the multiplier is to be entered at the side of the table and the multiplicand at the top, and the product is found in the middle:

c (156) t p s

c t p s ---------------------------------|c |t |0 |0 ---------------------------------|0 |0 |c |t ---------------------------------|p |s |0 |0 ---------------------------------|0 |0 |p |s ----------------------------------

| | | |

Peirce: CP 3.126 Cross-Ref:†† The sixteen propositions expressed by this table are in ordinary language as follows:†1 The colleagues of the colleagues of any person are that person's colleagues; The colleagues of the teachers of any person are that person's teachers; There are no colleagues of any person's pupils; There are no colleagues of any person's schoolmates; There are no teachers of any person's colleagues; There are no teachers of any person's teachers; The teachers of the pupils of any person are that person's colleagues; The teachers of the schoolmates of any person are that person's teachers; The pupils of the colleagues of any person are that person's pupils; The pupils of the teachers of any person are that person's schoolmates;

There are no pupils of any person's pupils; There are no pupils of any person's schoolmates; There are no schoolmates of any person's colleagues; There are no schoolmates of any person's teachers; The schoolmates of the pupils of any person are that person's pupils; The schoolmates of the schoolmates of any person are that person's schoolmates.

Peirce: CP 3.126 Cross-Ref:†† This simplicity and regularity in the multiplication of elementary relatives must clearly enhance the utility of the conception of a relative as resolvable into a sum of logical quaternions. Peirce: CP 3.127 Cross-Ref:†† 127. It may sometimes be convenient to consider relatives each one of which is of the form

a,i + b,j + c,k + d,l + etc.

where a,, b,, c,, d,, etc. are scalars, and i, j, k, l, etc. are each of the form

m,u + n,v + o,w + etc.

where m,, n,, o,, etc. are scalars, and u, v, w, etc. are elementary relatives. In all such cases (155) Will give a multiplication-table for i, j, k, l, etc. For example, if we have three classes of individuals, u[1], u[2], u[3], which are related to one another in pairs, we may put

u[1]:u[1] = i u[1]:u[2] = j u[1]:u[3] = k

u[2]:u[1] = l u[2]:u[2] = m u[2]:u[3] = n

u[3]:u[1] = o u[3]:u[2] = p u[3]:u[3] = q

and by (155) we get the multiplication-table

i j k l m n o p q

i j k l m n o p q ----------------------------------------------------------------------------|i |j |k |0 |0 |0 |0 |0 |0 ----------------------------------------------------------------------------|0 |0 |0 |i |j |k |0 |0 |0 ----------------------------------------------------------------------------|0 |0 |0 |0 |0 |0 |i |j |k ----------------------------------------------------------------------------|l |m |n |0 |0 |0 |0 |0 |0 ----------------------------------------------------------------------------|0 |0 |0 |l |m |n |0 |0 |0 ----------------------------------------------------------------------------|0 |0 |0 |0 |0 |0 |l |m |n ----------------------------------------------------------------------------|o |p |q |0 |0 |0 |0 |0 |0 ----------------------------------------------------------------------------|0 |0 |0 |o |p |q |0 |0 |0 ----------------------------------------------------------------------------|0 |0 |0 |0 |0 |0 |o |p |q -----------------------------------------------------------------------------

Peirce: CP 3.128 Cross-Ref:†† 128. If we take

i = u[1]:u[2] + u[2]:u[3] + u[3]:u[4],

j = u[1]:u[3] + u[2]:u[4],

k = 2.u[1]:u[4],

we have

i j k

i j k -------------------------|j |k |0 | -------------------------|k |0 |0 | -------------------------|0 |0 |0 | --------------------------

| | | | | | | | |

Peirce: CP 3.129 Cross-Ref:†† 129. If we take

i = u[1]:u[2] + u[2]:u[3] + u[3]:u[4] +u[5]:u[6] + u[7]:u[8],

j = u[1]:u[3] +u[2]:u[4],

k = 2.u[1]:u[4],

l = u[6]:u[8] + a.u[5]:u[7] + 2b.u[1]:u[9] + u[9]:u[4] + c.u[5]:u[6],

m = u[5]:u[8],

we have

i j k l m

i j k l m ------------------------------------------| j | k | 0 | m | 0 ------------------------------------------| k | 0 | 0 | 0 | 0 ------------------------------------------| 0 | 0 | 0 | 0 | 0 ------------------------------------------| a.m | 0 | 0 |b.k+ | 0 | | | |c.m | ------------------------------------------| 0 | 0 | 0 | 0 | 0 -------------------------------------------

| | | | | |

Peirce: CP 3.130 Cross-Ref:†† 130. These multiplication-tables have been copied from Professor Peirce's monograph on Linear Associative Algebras.†P1 I can assert, upon reasonable inductive evidence, that all such algebras can be interpreted on the principles of the present notation in the same way as those given above. In other words, all such algebras are complications and modifications of the algebra of (156). It is very likely that this is true of all algebras whatever. The algebra of (156), which is of such a fundamental character in reference to pure algebra and our logical notation, has been shown by Professor Peirce †1 to be the algebra of Hamilton's quaternions.†2 In fact, if we put

1 = i + l. ______

_______

_______

i'= √1-b2 ⎭i - (√1-a2b + a b

)j + (√1-a2b -

______ )k - √1-b2 ⎭l.

ab

______

______ ______ ______ ______

j' = -b√1-c2 ⎭i+(a c-√1-a2√1-b2√1-c2-(√1-a2c ______ ______ + a√1-b2√1-c2) ______

______ ______ ______

)j-(a c-√1-a2√1-b2√1-c2

______ ______

+ (√1-a2c+a√1-b2√1-c2) ______ ______

______

)k+b√1-c2⎭l. ______

______

k' = b c⎭i + (√1-a2√1-b2c+a√1-c2+(a√1-b2c ______ ______ - √1-a2√1-c2)

______ ______

______

)j-(√1-a2√1-b2c+a√1-c2

______ -(a√1-b2c ______ ______ -√1-a2√1-c2)

)k-b c⎭l.

where a, b, c, are scalars, then 1, i', j', k' are the four fundamental factors of quaternions, the multiplication-table of which is as follows:

1 i' j' k'

1 i' j' k' ---------------------------------|1 | i' | j' | k' ---------------------------------| i' |-1 | k' |-j' ---------------------------------| j' |-k' |-1 | i' ---------------------------------| k' | j' |-i' |-1 ----------------------------------

| | | |

Peirce: CP 3.131 Cross-Ref:†† 131. It is no part of my present purpose to consider the bearing upon the philosophy of space of this occurrence, in pure logic, of the algebra which expresses all the properties of space; but it is proper to point out that one method of working with this notation would be to transform the given logical expressions into the form of Hamilton's quaternions (after representing them as separated into elementary relatives), and then to make use of geometrical reasoning. The following formulæ will assist this procesS. Take the quaternion relative

q = x i + y j + z k + w l,

where x, y, z, and w are scalars. The conditions of q being a scalar, vector, etc. (that is, being denoted by an algebraic expression which denotes a scalar, a vector, etc., in geometry), are

(157) Form of a scalar: x(i + l). (158) Form of a vector: x i+y i+z k-x l. (159) Form of a versor:

x/y((x/z)-1)-1/2i + y/x((x/z)-1)-1/2j + z/y((z/x)-1)-1/2k + y/z((z/x)-1)-1/2l.

(160) Form of zero: x i + x y j + (z/y)k + z l. (161) Scalar of q: Sq = 1/2(x + w)(i + 1). (162) Vector of q: Vq = 1/2(x-w)i + y j + z k + 1/2(w-x)l. _____ (163) Tensor of q: Tq = √x w-y z (i+l). (164) Conjugate of q: Kq = w i - y j - z k + x l.

Peirce: CP 3.132 Cross-Ref:†† 132. In order to exhibit the logical interpretations of these functions, let us consider a universe of married monogamists, in which husband and wife always have country, race, wealth, and virtue, in common. Let i denote "man that is --," j "husband of --," k "wife of --," and l "woman that is --"; x "negro that is --," y "rich person that is --," z "American that is --," and w "thief that is --." Then, q being defined as above, the q's of any class will consist of so many individuals of that class as are negro-men or women-thieves together with all persons who are rich husbands

or American wives of persons of this class. Then, 2Sq denotes, by (160),†1 all the negroes and besides all the thieves, while Sq is the indefinite term which denotes half the negroes and thieves. Now, those persons who are self-q's of any class (that is, the q's of themselves among that class) are x i + w l; add to these their spouses and we have 2Sq. In general, let us term (j + k) the "correspondent of --." Then, the double scalar of any quaternion relative, q, is that relative which denotes all self-q's, and, besides, "all correspondents of self-q's of --." (Tq)2 denotes all persons belonging to pairs of corresponding self-q's minus all persons belonging to pairs of corresponding q's of each other. Peirce: CP 3.133 Cross-Ref:†† 133. As a very simple example of the application of geometry to the logic of relatives, we may take the following. Euclid's axiom concerning parallels corresponds to the quaternion principle that the square of a vector is a scalar. From this it follows, since by (157) y z + z k [?] is a vector, that the rich husbands and American wives of the rich husbands and American wives of any class of persons are wholly contained under that class, and can be described without any discrimination of sex. In point of fact, by (156), the rich husbands and American wiveS of the rich husbands and American wives of any class of persons, are the rich Americans of that class. Peirce: CP 3.133 Cross-Ref:†† Lobatchewsky †1 has shown that Euclid's axiom concerning parallels may be supposed to be false without invalidating the propositions of spherical trigonometry. In order, then, that corresponding propositions should hold good in logic, we need not resort to elementary relatives, but need only take S and V in such senses that every relative of the class considered should be capable of being regarded as a sum of a scalar and a vector, and that a scalar multiplied by a scalar should be a scalar, while the product of a scalar and a vector is a vector. Now, to fulfill these conditions we have only to take Sq as "self-q of," and Vq as "alio-q of" (q of another, that other being --), and q may be any relative whatever. For, "lover," for example, is divisible into self-lover and alio-lover; a self-lover of a self-benefactor of personS of any class is contained under that class, and neither the self-lover of an alio-benefactor of any persons nor the alio-lover of the self-benefactor of any persons are among those persons. Suppose, then, we take the formula of spherical trigonometry,

cos a = cos b cos c + cos A sin b sin c.

In quaternion form, this is,

(165)

S(p q) = (Sp)(Sq)+ S((Vp) (Vq)).

Let p be "lover," and q be "benefactor." Then this reads, lovers of their own benefactors consist of self-lovers of self-benefactors together with alio-lovers of alio-benefactors of themselves. So the formula

sin b cos p b'= -sin a cos c cos p a' -sin c cos a cos p c'+ sin a sin c sin b cos p b,

where A', B', C', are the positive poles of the sides a, b, c, is in quaternions

(166) V(p q) = (Vp)(Sq) + (Sp)(Vq) + V((Vp)(Vq)),

and the logical interpretation of this is: lovers of benefactors of others consist of alio-lovers of self-benefactors, together with self-lovers of alio-benefactors, together with alio-lovers of alio-benefactors of others. It is a little striking that just as in the non-Euclidean or imaginary geometry of Lobatchewsky the axiom concerning parallels holds good only with the ultimate elements of space, so its logical equivalent holds good only for elementary relatives. Peirce: CP 3.134 Cross-Ref:†† 134. It follows from what has been said that for every proposition in geometry there is a proposition in the pure logic of relatives. But the method of working with logical algebra which is founded on this principle seems to be of little use. On the other hand, the fact promises to throw some light upon the philosophy of space.†P1

Peirce: CP 3.135 Cross-Ref:†† §6. PROPERTIES OF PARTICULAR RELATIVE TERMS CLASSIFICATION OF SIMPLE RELATIVES †1

135. Any particular property which any class of relative terms may have may be stated in the form of an equation, and affords us another premiss for the solution of problems in which such terms occur. A good classification of relatives is, therefore, a great aid in the use of this notation, as the notation is also an aid in forming such a classification. Peirce: CP 3.136 Cross-Ref:†† 136. The first division of relatives is, of course, into simple relatives and conjugatives. The most fundamental divisions of simple relatives are based on the distinction between elementary relatives of the form (A:A), and those of the form (A:B). These are divisions in regard to the amount of opposition between relative and correlative. Peirce: CP 3.136 Cross-Ref:†† a. Simple relatives are in this way primarily divisible into relatives all of whose elements are of the form (A:A) and those which contain elements of the form (A:B). The former express a mere agreement among things, the latter set one thing over against another, and in that sense express an opposition ({antikeisthai}); I shall therefore term the former concurrents,†P1 and the latter opponents. The distinction appears in this notation as between relatives with a comma, such as (w,), and relatives

without a comma, such as (w); and is evidently of the highest importance. The character which is signified by a concurrent relative is an absolute character, that signified by an opponent is a relative character, that is, one which cannot be prescinded from reference to a correlate. Peirce: CP 3.136 Cross-Ref:†† b. The second division of simple relatives with reference to the amount of opposition between relative and correlative is into those whose elements may be arranged in collections of squares, each square like this,

A:A

A:B

A:C

B:A

B:B

B:C

C:A

C:B

C:C

and those whose elements cannot be so arranged.†1 The former (examples of which are, "equal to --," "similar to --") may be called copulatives,†P2 the latter non-copulatives. A copulative multiplied into itself gives itself. Professor Peirce calls letters having this property, idempotents.†2 The present distinction is of course very important in pure algebra. All concurrents are copulatives. Peirce: CP 3.136 Cross-Ref:†† c. Third, relatives are divisible into those which for every element of the form (A:B) have another of the form (B:A), and those which want this symmetry. This is the old division into equiparants†P1 and disquiparants,†P2 or in Professor De Morgan's language, convertible and inconvertible relatives.†1 Equiparants are their own correlatives. All copulatives are equiparant. Peirce: CP 3.136 Cross-Ref:†† d. Fourth, simple relatives are divisible into those which contain elements of the form (A:A) and those which do not. The former express relations such as a thing may have to itself, the latter (as cousin of --, hater of --) relations which nothing can have to itself. The former may be termed self-relatives,†P3 the latter alio-relatives. All copulatives are self-relatives. Peirce: CP 3.136 Cross-Ref:†† e. The fifth division is into relatives some power (i.e. repeated product) of which contains †P4 elements of the form (A:A), and those of which this is not true.†2 The former I term cyclic, the latter non-cyclic†P5 relatives. As an example of the former, take

(A:B) +, (B:A) +, (C:D) +, (D:E) +, (E:C).

The product of this into itself is

(A:A) +, (B:B) +, (C:E) +, (D:C) +, (E:D).

The third power is

(A:B) +, (B:A) +, (C:C) +, (D:D) +, (E:E).

The fourth power is

(A:A) +, (B:B) +, (C:D) +, (D:E) +, (E:C).

The fifth power is

(A:B) +, (B:A) +, (C:E) +, (D:C) +, (E:D).

The sixth power is

(A:A) +, (B:B) +, (C:C) +, (D:D) +, (E:E).

where all the terms are of the form (A:A). Such relatives, as cousin of --, are cyclic. All equiparants are cyclic. Peirce: CP 3.136 Cross-Ref:†† f. The sixth division is into relatives no power of which is zero, and relatives some power of which is zero. The former may be termed inexhaustible, the latter exhaustible. An example of the former is "spouse of --," of the latter, "husband of --." All cyclics are inexhaustible. Peirce: CP 3.136 Cross-Ref:†† g. Seventh, simple relatives may be divided into those whose products into themselves are not zero, and those whose products into themselves are zero. The former may be termed repeating, the latter, non-repeating relatives. All inexhaustible relatives are repeating. Peirce: CP 3.136 Cross-Ref:†† h. Repeating relatives may be divided (after De Morgan) into those whose products into themselves are contained under themselves, and those of which this is not true. The former are well named by De Morgan †1 transitive, the latter intransitive. All transitives are inexhaustible; all copulatives are transitive; and all transitive equiparants are copulative. The class of transitive equiparants has a

character, that of being self-relatives, not involved in the definitions of the terms.†2 Peirce: CP 3.136 Cross-Ref:†† i. Transitives are further divisible into those whose products by themselves are equal to themselves, and those whose products by themselves are less than themselves; the former may be termed continuous,†P1 the latter discontinuous. An example of the second is found in the pure mathematics of a continuum, where if a is greater than b it is greater than something greater than b; and as long as a and b are not of the same magnitude, an intervening magnitude always exists. All concurrents are continuous. Peirce: CP 3.136 Cross-Ref:†† j. Intransitives may be divided into those the number of the powers (repeated products) of which not contained in the first is infinite, and those some power of which is contained in the first. The former may be called infinites, the latter finites. Infinite inexhaustibles are cyclic. In addition to these, the old divisions of relations into relations of reason and real relations, of the latter into aptitudinal and actual, and of the last into extrinsic and intrinsic, are often useful.†P1

"NOT"

Peirce: CP 3.137 Cross-Ref:†† 137. We have already seen that "not," or "other than," is denoted by -1. It is often more convenient to write it, n. The fundamental property of this relative has been given above (111). It is that,

-x

= 1 - x.

Two other properties are expressed by the principles of contradiction and excluded middle. They are,

x, -x = 0;†1

x +,

-x

= 1.†1

The following deduced properties are of frequent application:

(167)

-(x,y)

=

-x

+,

-y;†2

-xy

(168)

=

-xy

The former of these is the counterpart of the general formula, zx +, y = zx,zy†2, The latter enables us always to bring the exponent of the exponent of - down to the line, and make it a factor. By the former principle, objects not French violinists consist of objects not Frenchmen, together with objects not violinists; by the latter, individuals not servants of all women are the same as non-servants of some women. Another singular property of

If [[x]] > 1

-1x

- is that,

= 1.

"CASE OF THE EXISTENCE OF --," AND "CASE OF THE NON-EXISTENCE OF --."

Peirce: CP 3.138 Cross-Ref:†† 138. That which first led me to seek for the present extension of Boole's logical notation was the consideration that as he left his algebra, neither hypothetical propositions nor particular propositions could be properly expressed. It is true that Boole was able to express hypothetical propositions in a way which answered some purposes perfectly. He could, for example, express the proposition, "Either the sun will shine, or the enterprise will be postponed," by letting x denote "the truth of the proposition that the sun will shine," and y "the truth of the proposition that the enterprise will be postponed"; and writing,

x +, y = 1,

or, with the invertible addition,

x + (1 - x),y = 1.

But if he had given four letters denoting the four terms, "sun," "what is about to shine," "the enterprise," and "what is about to be postponed," he could make no use of these to express his disjunctive proposition, but would be obliged to assume others. The imperfection of the algebra here was obvious. As for particular propositions, Boole could not accurately express them at all. He did undertake to express them and wrote

Some Y's are X's: Some Y's are not X's:

v,y = v,x; v,y = v,(1-x).

The letter v is here used, says Boole, for an "indefinite class symbol."†1 This betrays a radical misapprehension of the nature of a particular proposition. To say that some Y's are X's, is not the same as saying that a logical species of Y's are X's. For the logical species need not be the name of anything existing. It is only a certain description of things fully expressed by a mere definition, and it is a question of fact whether such a thing really exist or not. St. Anselm wished to infer existence from a definition, but that argument has long been exploded. If, then, v is a mere logical species in general, there is not necessarily any such thing, and the equation means nothing. If it is to be a logical species, then, it is necessary to suppose in addition that it exists, and further that some v is y. In short, it is necessary to assume concerning it the truth of a proposition, which, being itself particular, presents the original difficulty in regard to its symbolical expression. Moreover, from

v,y = v,(1-x)

we can, according to algebraic principles, deduce successively

v,y = v - v,x v,x = v - v,y = v,(1-y).

Now if the first equation means that some Y's are not X's, the last ought to mean that some X's are not Y's; for the algebraic forms are the same, and the question is, whether the algebraic forms are adequate to the expression of particulars. It would appear, therefore, that the inference from Some Y's are not X's to Some X's are not Y's, is good; but it is not so, in fact. Peirce: CP 3.139 Cross-Ref:†† 139. What is wanted, in order to express hypotheticals and particulars analytically, is a relative term which shall denote "case of the existence of --," or "what exists only if there is any --"; or else "case of the non-existence of --," or "what exists only if there is not --." When Boole's algebra is extended to relative terms, it is easy to see what these particular relatives must be. For suppose that having expressed the propositions "it thunders," and "it lightens," we wish to express the fact that "if it lightens, it thunders." Let

A = 0 and B = 0,

be equations meaning respectively, it lightens and it thunders. Then, if φx vanishes when x does not and vice versa, whatever x may be, the formula

φA -< φB

expresses that if it lightens it thunders; for if it lightens, A vanishes; hence φA does not vanish, hence φB does not vanish, hence B vanishes, hence it thunders. It makes no difference what the function φ is, provided only it fulfills the condition mentioned. Now, 0x is such a function, vanishing when x does not, and not vanishing when x does. Zero, therefore, may be interpreted as denoting "that which exists if, and only if, there is not --." Then the equation

00 = 1

means, everything which exists, exists only if there is not anything which does not exist. So,

0x = 0

means that there is nothing which exists if, and only if, some x does not exist. The reason of this is that some x means some existing x. Peirce: CP 3.139 Cross-Ref:†† "It lightens" and "it thunders" might have been expressed by equations in the forms

A = 1,

B = 1.

In that case, in order to express that if it lightens it thunders, in the form

φA -< φB,

it would only be necessary to find a function, φx, which should vanish unless x were 1, and should not vanish if x were 1. Such a function is 1x. We must therefore interpret 1 as "that

which exists if, and only if, there is --," 1x as "that which exists if, and only if, there is nothing but x," and 1x as "that which exists if, and only if, there is some x." Then the equation

1x = 1,

means everything exists if, and only if, whatever x there is exists. Peirce: CP 3.140 Cross-Ref:†† 140. Every hypothetical proposition may be put into four equivalent forms, as follows:

If X, then Y. If not Y, then not X. Either not X or Y. Not both X and not Y.

If the propositions X and Y are A = 1 and B = 1, these four forms are naturally expressed by

1A

-< 1B,

1(1-A) -< 1(1-B),†1 1(1-A) +, B = 1, 1A,

1(1-B) = 0.

For 1x we may always substitute 0(1-x). Peirce: CP 3.141 Cross-Ref:†† 141. Particular propositions are expressed by the consideration that they are contradictory of universal propositions. Thus, as h,(1-b) = 0 means every horse is black, so 0h,(1-b) = 0 means that some horse is not black; and as h,b = 0 means that no horse is black, so 0h,b = 0 means that some horse is black. We may also write the particular affirmative 1(h,b) = 1, and the particular negative 1(h,nb) = 1. Peirce: CP 3.142 Cross-Ref:†† 142. Given the premisses, every horse is black, and every horse is an animal; required the conclusion. We have given

h -< b; h -< a.

Commutatively multiplying, we get

h -< a,b.

Then, by (92) or by (90),

0a,b -< 0h, or 1h -< 1(a,b).

Hence, by (40) or by (46),

If h > 0 0a,b = 0, or 1(a,b) = 1;

or if there are any horses, some animals are black. I think it would be difficult to reach this conclusion, by Boole's method unmodified. Peirce: CP 3.143 Cross-Ref:†† 143. Particular propositions may also be expressed by means of the signs of inequality. Thus, some animals are horses, may be written

a,h > 0;

and the conclusion required in the above problem might have been obtained in thiS form, very easily, from the product of the premisses, by (1) and (21). Peirce: CP 3.143 Cross-Ref:†† We shall presently see †1 that conditional and disjunctive propositions may also be expressed in a different way.

Peirce: CP 3.144 Cross-Ref:†† CONJUGATIVE TERMS

144. The treatment of conjugative terms presents considerable difficulty, and would no doubt be greatly facilitated by algebraic devices. I have, however, studied this part of my notation but little. Peirce: CP 3.144 Cross-Ref:†† A relative term cannot possibly be reduced to any combination of absolute terms, nor can a conjugative term be reduced to any combination of simple relatives; but a conjugative having more than two correlates can always be reduced to a combination of conjugatives of two correlates. Thus for "winner over of --, from --, to --," we may always substitute u, or "gainer of the advantage -- to --," where the first correlate is itself to be another conjugative v, or "the advantage of winning over of -from --." Then we may write,

w = u v.

It is evident that in this way all conjugatives may be expressed as production of conjugatives of two correlates. Peirce: CP 3.145 Cross-Ref:†† 145. The interpretation of such combinations as bam, etc., is not very easy. When the conjugative and its first correlative can be taken together apart from the second correlative, as in (ba)m and (ba)m and (ba)m and (ba)m, there is no perplexity, because in such cases (ba) or (ba) is a simple relative. We have, therefore, only to call the betrayer to an enemy an inimical betrayer, when we have

(ba)m = inimical betrayer of a man = betrayer of a man to an enemy of him, (ba)m = inimical betrayer of every man = betrayer of every man to an enemy of him.

And we have only to call the betrayer to every enemy an unbounded betrayer, in order to get

(ba)m = unbounded betrayer of a man = betrayer of a man to every enemy of him, (ba)m = unbounded betrayer of every man = betrayer of every man to every enemy of him. The two terms bam and bam are not quite so easily interpreted. Imagine a separated into infinitesimal relatives, A[,],A[,,],A[,,,], etc., each of which is relative to but one individual which is m. Then, because all powers of A[,],A[,,],A[,,,], etc., higher than the first, vanish, and because the number of such terms must be [[m,]] we have,

a[m] = (A[,] +, A[,,] +, A[,,,] +, etc.)m = (A[,]m),(A[,,]m),(A[,,,]m), etc.

or if M', M'', M''', etc., are the individual m's,

am = (A[,]M'),(A[,,]M''),(A[,,,]M'''), etc.

It is evident from this that bam is a betrayer to an A[,] of M', to an A[,,] of M'', to an A[,,,] of M''', etc., in short of all men to some enemy of them all. In order to interpret bam we have only to take the negative of it. This, by (124), is (1-b)am, or a non-betrayer of all men to some enemy of them. Hence, bam, or that which is not this, is a betrayer of some man to each enemy of all men. To interpret b(am) we may put it in the form (1-b)(1-a)m. This is "non-betrayer of a man to all non-enemies of all men." Now, a non-betrayer of some X to every Y, is the same as a betrayer of all X's to nothing but what is not Y; and the negative-of "non-enemy of all men," is "enemy of a man." Thus, b(am) is, "betrayer of all men to nothing but an enemy of a man." To interpret bam we may put it in the form (1-b)(1-a)m, which is, "non-betrayer of a man to every non-enemy of him." This is a logical sum of terms, each of which is "non-betrayer of an individual man M to every non-enemy of M." Each of these terms is the same as "betrayer of M to nothing but an enemy of M." The sum of them, therefore, which is bam is "betrayer of some man to nothing but an enemy of him." In the same way it is obvious that bam is "betrayer of nothing but a man to nothing but an enemy of him." We have bam = b(1-a)(1-m) or "betrayer of all non-men to a non-enemy of all non-men." This is the same as "that which stands to something which is an enemy of nothing but a man in the relation of betrayer of nothing but men to what is not it." The interpretation of bam is obviously "betrayer of nothing but a man to an enemy of him." It is equally plain that bam is "betrayer of no man to anything but an enemy of him," and that bam is "betrayer of nothing but a man to every enemy of him." By putting bam in the form b(1-a)(1-m) we find that it denotes "betrayer of something besides a man to all things which are enemies of nothing but men." When an absolute term is put in place of a, the interpretations are obtained in the same way, with greater facility. Peirce: CP 3.146 Cross-Ref:†† 146. The sign of an operation is plainly a conjugative term. Thus, our commutative multiplication might be denoted by the conjugative

1,. For we have

l,sw = 1,l,sw.

As conjugatives can all be reduced to conjugatives of two correlates, they might be expressed by an operative sign (for which a Hebrew letter might be used) put between

the symbols for the two correlates. There would often be an advantage in doing this, owing to the intricacy of the usual notation for conjugatives. If these operational signs happened to agree in their properties with any of the signs of algebra, modifications of the algebraic signs might be used in place of Hebrew letters. For instance, if were such that

x yz=

[13] y z,†1

then, if we were to substitute for

the operational sign

we have

x (y z) = (x y) z,

which is the expression of the associative principle. So, if xy=

yx

we may write, x y=y x

which is the commutative principle. If both these equations held for any conjugative, we might conveniently express it by a modified sign +. For example, let us consider the conjugative "what is denoted by a term which either denotes -- or else --." For this, the above principles obviously hold, and we may naturally denote it by `+. Then, if p denotes Protestantism, r Romanism, and f what is false,

p `+ r -< f

means either all Protestantism or all Romanism is false. In this way it is plain that all hypothetical propositions may be expressed. Moreover, if we suppose any term as "man" (m) to be separated into its individuals, M', M'', M''', etc., then,

M' `+ M'' `+ M''' `+ etc.,

means "some man." This may very naturally be written

'm'

and this gives us an improved way of writing a particular proposition; for 'x' -< y

seems a simpler way of writing "Some X is Y" than

0x,y = 0.

Peirce: CP 3.147 Cross-Ref:†† CONVERSE

147. If we separate lover into its elementary relatives, take the reciprocal of each of these, that is, change it from A:B to B:A,

and sum these reciprocals, we obtain the relative loved by. There is no such operation as this in ordinary arithmetic, but if we suppose a science of discrete quantity in quaternion form (a science of equal intervals in space), the sum of the reciprocals of the units of such a quaternion will be the conjugate-quaternion. For this reason, I express the conjugative term "what is related in the way that to -- is --, to the latter" by K. The fundamental equations upon which the properties of this term depend are

(169)

K K = 1.

(170) If x < yz then z -< (Ky)x, or

1(x,y z) = 1(z,Ky x)

We have, also,

(171)

KΣ = ΣK,

(172)

Kπ = πK,

where π denotes the product in the reverse order. Other equations will be found in Mr. De Morgan's table, given above.†1

Peirce: CP 3.148 Cross-Ref:†† CONCLUSION

148. If the question is asked, What are the axiomatic principles of this branch of logic, not deducible from others? I reply that whatever rank is assigned to the laws of contradiction and excluded middle belongs equally to the interpretations of all the general equations given under the head of "Application of the Algebraic signs to Logic," together with those relating to backward involution, and the principles expressed by equations (95), (96), (122), (142), (156), (25), (26), (14), (15). Peirce: CP 3.149 Cross-Ref:†† 149. But these axioms are mere substitutes for definitions of the universal logical relations, and so far as these can be defined, all axioms may be dispensed with. The fundamental principles of formal logic are not properly axioms, but definitions and divisions; and the only facts which it contains relate to the identity of the conceptions resulting from those processes with certain familiar ones.

Peirce: CP 3.150 Cross-Ref:†† IV

ON THE APPLICATION OF LOGICAL ANALYSIS TO MULTIPLE ALGEBRA.†1

150. The letters of an algebra express the relation of the product to the multiplicand. Thus, i A expresses the quantity which is related to A in the manner denoted by i. This being the conception of these algebras, for each of them we may imagine another "absolute" algebra, as we may call it, which shall contain letters which can only be products and multiplicands, not multipliers. Let the general expression of the absolute algebra be a I + b J + c K + d L + etc. Multiply this by any letter i of the relative algebra, and denote the product by

(A[1]a + A[2]b + A[3]c + etc.)I. + (B[1]a + B[2]b + B[3]c + etc.)J. + etc.

Peirce: CP 3.150 Cross-Ref:††

Now we may obviously enlarge the given relative algebra, so that

i = A[1]i[11] + A[2]i[12] + A[3]i[13] + etc. + B[1]i[21] + B[2]i[22] + B[3]i[23] + etc. + etc.

where i[11]i[12] etc., are such that the product of either of them into any letter of the absolute algebra shall equal some letter of that algebra. That there is no self-contradiction involved in this supposition seems axiomatic.†2 Peirce: CP 3.151 Cross-Ref:†† 151. In this way each letter of the given algebra is resolved into a sum of terms of the form a A:B, a being a scalar, and A:B such that

(A:B)(B:C) = A:C. (A:B)(C:D) = 0.

The actual resolution is usually performed with ease, but in some cases a good deal of ingenuity is required. I have not found the process facilitated by any general rules. I have actually resolved all the Double, Triple, and Quadruple algebras, and all the Quintuple ones, that appeared to present any difficulty. I give a few examples.

b i[5].†1

k j k l m

i j k l m ------------------------------------------| j | 0 | l | 0 | 0 ------------------------------------------| 0 | 0 | 0 | 0 | 0 ------------------------------------------| j+a l | 0 | 0 | 0 | bj+cl ------------------------------------------| 0 | 0 | 0 | 0 | 0 ------------------------------------------|a'j+b'l | 0 |c'j+d'l | 0 | 0 -------------------------------------------

| | | | |

i = c d'A:B+b'B:C+b'D:E. j = b'c d'A:C. k = c d'A:B+a c d'D:B+b'c2d'D:F+c d'E:C+b b'c d'A:F.

l = b'c d'D:C. m = a'c d'A:B+b'c'A:E+b'c d'D:B+b'd'D:E+b'c d'D:F+F:C.

b d[5].†2

k j k l m

i j k l m ------------------------------------------| j | 0 | l | 0 | 0 ------------------------------------------| 0 | 0 | 0 | 0 | 0 ------------------------------------------| j+rl | 0 | i+m | 0 | -j-rl ------------------------------------------| 0 | 0 | j | 0 | 0 ------------------------------------------|(r2-1)j | 0 | -l | 0 |-r2j -------------------------------------------

i = A:D+D:F+B:E+C:F.

| | | | |

j = A:F.

k = r A:B+r B:C+D:E-(1/r)D:F+E:F. l = A:E-(1/r)A:F+B:F.

m = r2A:C-A:D-B:E-C:F.

b h[6].†1

k j k l m n

i j k l m n --------------------------------------------------| i | j | k | l | m | --------------------------------------------------| j | k | | | | --------------------------------------------------| k | | | | | --------------------------------------------------| l | ak | | k | | --------------------------------------------------| | | | | | k --------------------------------------------------| n | | | | | ---------------------------------------------------

| | | | | |

i = A:A+B:B+C:C+D:D. l = a A:B+A:D+D:C. j = A:B+B:C.

m = A:E.

k = A:C.

k j k l m

n = E:C.

b r[5].†2 i j k l m ------------------------------------------| j | | | | ------------------------------------------| | | | | ------------------------------------------| l | | m | | ------------------------------------------| | | | | ------------------------------------------| | | | | -------------------------------------------

i = A:B+B:C j = A:C.

| | | | |

l = D:C. m = D:F.

k = D:B+D:E+E:F.

Peirce: CP 3.152 Cross-Ref:†† V

NOTE ON GRASSMANN'S CALCULUS OF EXTENSION.†1

152. The last Mathematische Annalen contains a paper by H. Grassmann, on the application of his calculus of extension to mechanics.†2 Peirce: CP 3.152 Cross-Ref:†† He adopts the quaternion addition of vectors. But he has two multiplications, internal and external, just as the principles of logic require. Peirce: CP 3.152 Cross-Ref:†† The internal product of two vectors, v[1] and v[2], is simply what is written in quaternions as--S.v[1]v[2]. He writes it [[v[1]|v[2]]]. So that

[[v[1]|v[2]]] = [[v[2]|v[1]]],

v[2] = (T v)2.

The external product of two vectors is the parallelogram they form, account being taken of its plane and the direction of running round it, which is equivalent to its aspect. We therefore have:

v[1] [[v[1]v[2]]] = v[1]v[2] sin < . I. v[2]

[[v[1]v[2]]] = -[[v[2]v[1]]], v2 = o,

where I is a new unit. This reminds me strongly of what is written in quaternions as -V(v[1]v[2]). But it is not the same thing in fact, because [[v[1]v[2]]]v[3] is a solid, and therefore a new kind of quantity. In truth, Grassmann has got hold (though he did not say so) of an eight-fold algebra, which may be written in my system as follows:

Three Rectangular Versors†1

i = M:A - B:Z + C:Y + X:N j = M:B - C:X + A:Z + Y:N k = M:C - A:Y + B:X + Z:N

Three Rectangular Planes

I = M:X + A:N J = M:Y + B:N K = M:Z + C:N

One Solid

V = M:N

Unity

1 = M:M + A:A + B:B + C:C + N:N + X:X + Y:Y + Z:Z

This unity might be omitted.

Peirce: CP 3.153 Cross-Ref:†† 153. The recognition †2 of the two multiplications is exceedingly interesting. The system seems to me more suitable to three dimensional space, and also more natural than that of quaternions. The simplification of mechanical formulæ is striking, but not more than quaternions would effect, that I see. Peirce: CP 3.153 Cross-Ref:†† By means of eight rotations through two-thirds of a circumference, around four symmetrically placed axes, together with unity, all distortions of a particle would be represented linearly. I have therefore thought of the nine-fold algebra thus resulting.

Peirce: CP 3.154 Cross-Ref:†† VI

ON THE ALGEBRA OF LOGIC†1

PART I.†2 SYLLOGISTIC †P1

§1. DERIVATION OF LOGIC

154. In order to gain a dear understanding of the origin of the various signs used in logical algebra and the reasons of the fundamental formulæ, we ought to begin by considering how logic itself arises. Peirce: CP 3.155 Cross-Ref:†† 155. Thinking, as cerebration, is no doubt subject to the general laws of nervous action. Peirce: CP 3.156 Cross-Ref:††

156. When a group of nerves are stimulated, the ganglions with which the group is most intimately connected on the whole are thrown into an active state, which in turn usually occasions movements of the body. The stimulation continuing, the irritation spreads from ganglion to ganglion (usually increasing meantime). Soon, too, the parts first excited begin to show fatigue; and thus for a double reason the bodily activity is of a changing kind. When the stimulus is withdrawn, the excitement quickly subsides. Peirce: CP 3.156 Cross-Ref:†† It results from these facts that when a nerve is affected, the reflex action, if it is not at first of the sort to remove the irritation, will change its character again and again until the irritation is removed; and then the action will cease. Peirce: CP 3.157 Cross-Ref:†† 157. Now, all vital processes tend to become easier on repetition. Along whatever path a nervous discharge has once taken place, in that path a new discharge is the more likely to take place. Peirce: CP 3.157 Cross-Ref:†† Accordingly, when an irritation of the nerves is repeated, all the various actions which have taken place on previous similar occasions are the more likely to take place now, and those are most likely to take place which have most frequently taken place on those previous occasions. Now, the various. actions which did not remove the irritation may have previously sometimes been performed and sometimes not; but the action which removes the irritation must have always been performed, because the action must have every time continued until it was performed. Hence, a strong habit of responding to the given irritation in this particular way must quickly be established. Peirce: CP 3.158 Cross-Ref:†† 158. A habit so acquired may be transmitted by inheritance. Peirce: CP 3.158 Cross-Ref:†† One of the most important of our habits is that one by virtue of which certain classes of stimuli throw us at first, at least, into a purely cerebral activity. Peirce: CP 3.159 Cross-Ref:†† 159. Very often it is not an outward sensation but only a fancy which starts the train of thought. In other words, the irritation instead of being peripheral is visceral. In such a case the activity has for the most part the same character; an inward action removes the inward excitation. A fancied conjuncture leads us to fancy an appropriate line of action. It is found that such events, though no external action takes. place, strongly contribute to the formation of habits of really acting in the fancied way when the fancied occasion really arises.†1 Peirce: CP 3.160 Cross-Ref:†† 160. A cerebral habit of the highest kind, which will determine what we do in fancy as well as what we do in action, is called a belief. The representation to ourselves that we have a specified habit of this kind is called a judgment. A belief-habit in its development begins by being vague, special, and meagre; it becomes more precise, general, and full, without limit. The process of this development, so far as it takes place in the imagination, is called thought. A judgment is formed; and under the influence of a belief-habit this gives rise to a new judgment, indicating an addition to belief. Such a process is called an inference; the antecedent

judgment is called the premiss; the consequent judgment, the conclusion; the habit of thought, which determined the passage from the one to the other (when formulated as a proposition), the leading principle.†P1 Peirce: CP 3.161 Cross-Ref:†† 161. At the same time that this process of inference, or the spontaneous development of belief, is continually going on within us, fresh peripheral excitations are also continually creating new belief-habits. Thus, belief is partly determined by old beliefs and partly by new experience. Is there any law about the mode of the peripheral excitations? The logician maintains that there is, namely, that they are all adapted to an end, that of carrying belief, in the long run, toward certain predestinate conclusions which are the same for all men. This is the faith of the logician. This is the matter of fact, upon which all maxims of reasoning repose. In virtue of this fact, what is to be believed at last is independent of what has been believed hitherto, and therefore has the character of reality. Hence, if a given habit, considered as determining an inference, is of such a sort as to tend toward the final result, it is correct; otherwise not. Thus, inferences become divisible into the valid and the invalid; and thus logic takes its reason of existence.

Peirce: CP 3.162 Cross-Ref:†† §2. SYLLOGISM AND DIALOGISM †1

162. The general type of inference is

P .·. C,

where .·. is the sign of illation. Peirce: CP 3.163 Cross-Ref:†† 163. The passage from the premiss (or set of premisses) P to the conclusion C takes place according to a habit or rule active within us. All the inferences which that habit would determine when once the proper premisses were admitted, form a class. The habit is logically good provided it would never (or in the case of a probable inference, seldom) lead from a true premiss to a false conclusion; otherwise it is logically bad. That is, every possible case of the operation of a good habit would either be one in which the premiss was false or one in which the conclusion would be true; whereas, if a habit of inference is bad, there is a possible case in which the premiss would be true, while the conclusion was false. When we speak of a possible case, we conceive that from the general description of cases we have struck out all those kinds which we know how to describe in general terms but which we know never will occur; those that then remain, embracing all whose nonoccurrence we are not certain of, together with all those whose non-occurrence we cannot explain on any general principle, are called possible. Peirce: CP 3.164 Cross-Ref:††

164. A habit of inference may be formulated in a proposition which shall state that every proposition c, related in a given general way to any true proposition p, is true. Such a proposition is called the leading principle of the class of inferences whose validity it implies. When the inference is first drawn, the leading principle is not present to the mind, but the habit it formulates is active in such a way that, upon contemplating the believed premiss, by a sort of perception the conclusion is judged to be true.†P1 Afterwards, when the inference is subjected to logical criticism, we make a new inference, of which one premiss is that leading principle of the former inference, according to which propositions related to one another in a certain way are fit to be premiss and conclusion of a valid inference, while another premiss is a fact of observation, namely, that the given relation does subsist between the premiss and conclusion of the inference under criticism; whence it is concluded that the inference was valid. Peirce: CP 3.165 Cross-Ref:†† 165. Logic supposes inferences not only to be drawn, but also to be subjected to criticism; and therefore we not only require the form P .·. C to express an argument, but also a form, P[i] -< C[i], to express the truth of its leading principle. Here P[i] denotes any one of the class of premisses, and C[i] the corresponding conclusion. The symbol -< is the copula, and signifies primarily that every state of things in which a proposition of the class P[i] is true is a state of things in which the corresponding propositions of the class C[i] are true. But logic also supposes some inferences to be invalid, and must have a form for denying the leading premiss [?principle]. This we shall write P[i] ~-< C[i], a dash over any symbol signifying in our notation the negative of that symbol.†P1 [Elec. ed.: A tilde (~) preceding the symbol signifies the negative in the electronic edition.] Peirce: CP 3.165 Cross-Ref:†† Thus, the form P[i] -< C[i] implies

either, 1, that it is impossible that a premiss of the class P[i] should be true,

or, 2, that every state of things in which P[i] is true is a state of things in which the corresponding C[i] is true.

Peirce: CP 3.165 Cross-Ref:†† The form P[i] ~-< C[i] implies

both, 1, that a premiss of the class P[i] is possible,

and, 2, that among the possible cases of the truth of a P[i] there is one in which the corresponding C[i] is not true.

This acceptation of the copula differs from that of other systems of syllogistic in a manner which will be explained below in treating of the negative. Peirce: CP 3.166 Cross-Ref:†† 166. In the form of inference P .·. C the leading principle is not expressed; and the inference might be justified on several separate principles. One of these, however, P[i] -< C[i], is the formulation of the habit which, in point of fact, has governed the inferences. This principle contains all that is necessary besides the premiss P to justify the conclusion. (It will generally assert more than is necessary.) We may, therefore, construct a new argument which shall have for its premisses the two propositions P and P[i] -< C[i] taken together, and for its conclusion, C. This argument, no doubt, has, like every other, its leading principle, because the inference is governed by some habit; but yet the substance of the leading principle must already be contained implicitly in the premisses, because the proposition P[i] -< C[i] contains by hypothesis all that is requisite to justify the inference of C from P. Such a leading principle, which contains no fact not implied or observable in the premisses, is termed a logical principle, and the argument it governs is termed a complete, in contradistinction to an incomplete, argument, or enthymeme. Peirce: CP 3.166 Cross-Ref:†† The above will be made clear by an example. Let us begin with the enthymeme,

Enoch was a man, .·. Enoch died.

The leading principle of this is, "All men die." Stating it, we get the complete argument,

All men die, Enoch was a man; .·. Enoch was to die.

The leading principle of this is nota notae est nota rei ipsius. Stating this as a premiss, we have the argument,

Nota notae est nota rei ipsius, Mortality is a mark of humanity, which is a mark of Enoch; .·. Mortality is a mark of Enoch.

But this very same principle of the nota notae is again active in the drawing of this last inference, so that the last state of the argument is no more complete than the last but one. Peirce: CP 3.167 Cross-Ref:†† 167. There is another way of rendering an argument complete, namely, instead of adding the leading principle P[i] -< C[i] conjunctively to the premiss P, to form a new argument, we might add its denial disjunctively to the conclusion; thus,

P .·. Either C or P[i] ~-< C[i].

Peirce: CP 3.168 Cross-Ref:†† 168. A logical principle is said to be an empty or merely formal proposition, because it can add nothing to the premisses of the argument it governs, although it is relevant; so that it implies no fact except such as is presupposed in all discourse, as we have seen in §1 that certain facts are implied. We may here distinguish between logical and extralogical validity; the former being that of a complete, the latter that of an incomplete argument. The term logical leading principle we may take to mean the principle which must be supposed true in order to sustain the logical validity of any argument. Such a principle states that among all the states of things which can be supposed without conflict with logical principles, those in which the premiss of the argument would be true would also be cases of the truth of the conclusion. Nothing more than this would be relevant to the logical leading principle, which is, therefore, perfectly determinate and not vague, as we have seen an extralogical leading principle to be. Peirce: CP 3.169 Cross-Ref:†† 169. A complete argument, with only one premiss, is called an immediate inference. Example: All crows are black birds; therefore, all crows are birds. If from the premiss of such an argument everything redundant is omitted, the state of things expressed in the premiss is the same as the state of things expressed in the conclusion, and only the form of expression is changed. Now, the logician does not undertake to enumerate all the ways of expressing facts: he supposes the facts to be already expressed in certain standard or canonical forms. But the equivalence between different ones of his own standard forms is of the highest importance to him, and thus certain immediate inferences play the great part in formal logic. Some of these will not be reciprocal inferences or logical equations, but the most important of them will have that character. Peirce: CP 3.170 Cross-Ref:†† 170. If one fact has such a relation to a different one that, if the former be true, the latter is necessarily or probably true, this relation constitutes a determinate fact; and therefore, since the leading principle of a complete argument involves no matter of fact (beyond those employed in all discourse), it follows that every complete and material (in opposition to a merely formal) argument must have at least two premisses.

Peirce: CP 3.171 Cross-Ref:†† 171. From the doctrine of the leading principle it appears that if we have a valid and complete argument from more than one premiss, we may suppress all premisses but one and still have a valid but incomplete argument. This argument is justified by the suppressed premisses; hence, from these premisses alone we may infer that the conclusion would follow from the remaining premisses. In this way, then, the original argument

PQRST

.·.C

is broken up into two, namely, 1st,

PQRS .·. T -< C

and, 2d,

T -< C T .·.C.

By repeating this process, any argument may be broken up into arguments of two premisses each. A complete argument having two premisses is called a syllogism.†P1 Peirce: CP 3.172 Cross-Ref:†† 172. An argument may also be broken up in a different way by substituting for the second constituent above, the form

T -< C .·. Either C or not T.

In this way, any argument may be resolved into arguments, each of which has one premiss and two alternative conclusions. Such an argument, when complete, may be called a dialogism.

Peirce: CP 3.173 Cross-Ref:†† §3. FORMS OF PROPOSITIONS

173. In place of the two expressions A -< B and B -< A taken together we may write A = B;†P2 in place of the two expressions A -< B and B ~-< A taken together we may write A < B or B > A; and in place of the two expressions A ~-< B and B ~-< A taken together [disjunctively] we may write A ~= B.†1 Peirce: CP 3.174 Cross-Ref:†† 174. De Morgan, in the remarkable memoir with which he opened his discussion of the syllogism (1846, p. 380,†2) has pointed out that we often carry on reasoning under an implied restriction as to what we shall consider as possible, which restriction, applying to the whole of what is said, need not be expressed. The total of all that we consider possible is called the universe of discourse, and may be very limited. One mode of limiting our universe is by considering only what actually occurs, so that everything which does not occur is regarded as impossible. Peirce: CP 3.175 Cross-Ref:†† 175. The forms A -< B, or A implies B, and A ~-< B, or A does not imply B †3, embrace both hypothetical and categorical propositions. Thus, to say that all men are mortal is the same as to say that if any man possesses any character whatever then a mortal possesses that character. To say, 'if A, then B ' is obviously the same as to say that from A, B follows, logically or extralogically. By thus identifying the relation expressed by the copula with that of illation, we identify the proposition with the inference, and the term with the proposition. This identification, by means of which all that is found true of term, proposition, or inference is at once known to be true of all three, is a most important engine of reasoning, which we have gained by beginning with a consideration of the genesis of logic.†P1 Peirce: CP 3.176 Cross-Ref:†† 176. Of the two forms A -< B and A ~-< B, no doubt the former is the more primitive, in the sense that it is involved in the idea of reasoning, while the latter is only required in the criticism of reasoning. The two kinds of proposition are essentially different, and every attempt to reduce the latter to a special case of the former must fail. Boole †1 attempts to express 'some men are not mortal,' in the form 'whatever men have a certain unknown character v are not mortal.' But the propositions are not identical, for the latter does not imply that some men have that character v; and, accordingly, from Boole's proposition we may legitimately infer that 'whatever mortals have the unknown character v are not men';†2 yet we cannot reason from 'some men are not mortal' to 'some mortals are not men.'†P2 On the other hand, we can rise to a more general form under which A -< B and A ~-< B are both included. For this purpose we write A ~-< B in the form $A -< ~B,†3 where $A is some-A and ~B is not-B. This more general form is equivocal in so far as it is left undetermined whether the proposition would be true if the subject were impossible. When the subject is general this is the case, but when the subject is particular (i.e., is subject to the modification some) it is not.†1 The general form supposes merely inclusion of the subject under the predicate. The short curved mark over the letter in the subject shows that some part of the term denoted by that letter is the subject, and that that is asserted to be in possible existence.

Peirce: CP 3.177 Cross-Ref:†† 177. The modification of the subject by the curved mark and of the predicate by the straight mark gives the old set of propositional forms, viz.:

A. a -< b Every a is b.

Universal affirmative.

E. a -< ~b No a is b.

Universal negative.

I. $a -< b Some a is b.

Particular affirmative.

O. $a -< ~b Some a is not b. Particular negative.

Peirce: CP 3.178 Cross-Ref:†† 178. There is, however, a difference between the senses in which these propositions are here taken and those which are traditional; namely, it is usually understood that affirmative propositions imply the existence of their subjects, while negative ones do not. Accordingly, it is said that there is an immediate inference from A to I and from E to O. But in the sense assumed in this paper, universal propositions do not, while particular propositions do, imply the existence of their subjects. The following figure illustrates the precise sense here assigned to the four forms A, E, I, O.

[Click here to view]

Peirce: CP 3.179 Cross-Ref:†† 179. In the quadrant marked 1 there are lines which are all vertical; in the quadrant marked 2 some lines are vertical and some not; in quadrant 3 there are lines none of which are vertical; and in quadrant 4 there are no lines. Now, taking line as subject and vertical as predicate,

A is true of quadrants 1 and 4 and false of 2 and 3. E is true of quadrants 3 and 4 and false of 1 and 2. I is true of quadrants 1 and 2 and false of 3 and 4. O is true of quadrants 2 and 3 and false of 1 and 4.

Hence, A and O precisely deny each other, and so do E and I. But any other pair of propositions may be either both true or both false or one true while the other is false.†1 Peirce: CP 3.180 Cross-Ref:†† 180. De Morgan ("On the Syllogism," No. I., 1846, p. 381) has enlarged the system of propositional forms by applying the sign of negation which first appears in A ~-< B to the subject and predicate. He thus gets

A -< B. Every A is B.†2

A is species of B.†3

A ~-< B. Some A is not B. A is exient of B. A -< ~B. No A is B.

A is external of B.

A ~-< ~B. Some A is B.

A is partient of B.

~A -< B. Everything is either A or B. A is complement of B. ~A ~-< B. There is something besides A and B.

A is coinadequate of

B.

~A -< ~B. A includes all B. A is genus of B. ~A ~-< ~B. A does not include all B. A is deficient of B.

De Morgan's table of the relations of these propositions must be modified to conform to the meanings here attached to -< and to ~- 0

means that something is a lover of something; and

π[i]Σ[j]l[i j] > 0

means that everything is a lover of something. We shall, however, naturally omit, in writing the inequalities, the > 0 which terminates them all; and the above two propositions will appear as

Σ[i]Σ[j]l[i j] and π[i]Σ[j]l[i j].

Peirce: CP 3.352 Cross-Ref:†† 352. The following are other examples:

π[i]Σ[j](l)[i j](b)[i j]

means that everything is at once a lover and a benefactor of something.

π[i]Σ[j](l)[i j](b)[j i]

means that everything is a lover of a benefactor of itself.

Σ[i]Σ[k]π[j](l[i j] + b[j k])

means that there is something which stands to something in the relation of loving everything except benefactors of it.†1 Peirce: CP 3.353 Cross-Ref:†† 353. Let α denote the triple relative "accuser to -- of --," and ε the triple relative "excuser to -- of --. Then,

Σ[i]π[j]Σ[k](α)[i j k](ε)[j k i]

means that an individual i can be found, such, that taking any individual whatever, j, it will always be possible so to select a third individual, k, that i is an accuser to j of k, and j an excuser to k of i. Peirce: CP 3.354 Cross-Ref:†† 354. Let {p} denote "preferrer to -- of --." Then,

π[i]Σ[j]Σ[k](α)[i j k](ε[j k i] + {p}[k i j])

means that, having taken any individual i whatever, it is always possible so to select two, j and k, that i is an accuser to j of k, and also is either excused by j to k or is something to which j is preferred by k. Peirce: CP 3.355 Cross-Ref:†† 355. When we have a number of premisses expressed in this manner, the conclusion is readily deduced by the use of the following simple rules. In the first place, we have

Σ[i]π[j] -< π[j]Σ[i].

In the second place, we have the formulæ

{π[i]φ(i)} {π[j]{Ps}(j)} = π[i]{φ(i)·{Ps}(i)}. {π[i]φ(i)} {Σ[j]{Ps}(j)} -< Σ[i]{φ(i)·{Ps}(i)}.

In the third place, since the numerical coefficients are all either zero or unity, the Boolian calculus is applicable to them. Peirce: CP 3.356 Cross-Ref:†† 356. The following is one of the simplest possible examples. Required to eliminate servant from these two premisses: First premiss. There is somebody who accuses everybody to everybody, unless the latter is loved by some person that is servant of all not accused to him.†1 Second premiss. There are two persons, the first of whom excuses everybody to everybody, unless the unexcused be benefited by, without the person to whom he is unexcused being a servant of, the second.

These premisses may be written thus:

Σ[h]π[i]Σ[j]π[k](α[h i k] + s[j k]l[j i]) Σ[u]Σ[v]π[x]π[y](ε[u y x]+~s[y v]b[v x]).

The second yields the immediate inference,

π[x]Σ[u]π[y]Σ[v](ε[u y x] + ~s[y v]b[v x]).

Combining this with the first, we have

Σ[x]Σ[u]Σ[y]Σ[v](ε[u y x]+~s[y v]b[v x])(a[x u v]+s[y v]l[y u]).

Finally, applying the Boolian calculus, we deduce the desired conclusion

Σ[x]Σ[u]Σ[y]Σ[v](ε[u y x]+α[x u v]+ε[u y x]l[y u]+α[x u v]b[v x]).

The interpretation of this is that either there is somebody excused by a person to whom he accuses somebody, or somebody excuses somebody to his (the excuser's) lover, or somebody accuses his own benefactor. Peirce: CP 3.357 Cross-Ref:†† 357. The procedure may often be abbreviated by the use of operations intermediate between π and Σ. Thus, we may use π', π'', etc. to mean the products for all individuals except one, except two, etc. Thus,

π[i]'π[j]''(l[i j] + b[j i])

will mean that every person except one is a lover of everybody except its benefactors, and at most two non-benefactors. In the same manner, Σ', Σ'', etc., will denote the sums of all products of two, of all products of three, etc. Thus,

Σ[i]''(l[i i])

will mean that there are at least three things in the universe that are lovers of themselves. It is plain that if m < n, we have

πm -< πn Σn -< Σm.

(π[i]mφi)(Σ[j]n{Ps}j)-< Σ[i]n-m(φi·{Ps}i)

(π[i]mφi)(π[j]n{Ps}j)-< π[i]m+n(φi·{Ps}i)

Peirce: CP 3.358 Cross-Ref:†† 358. Mr. Schlötel has written to the London Mathematical Society,†1 accusing me of having, in my Algebra of Logic, plagiarized from his writings. He has also written to me to inform me that he has read that Memoir with "heitere Ironie," and that Professor Drobisch, the Berlin Academy, and I constitute a "liederliche Kleeblatt," with many other things of the same sort. Up to the time of publishing my Memoir, I had never seen any of Mr. Schlötel's writings; I have since procured his Logik,†2 and he has been so obliging as to send me two cuttings from his papers, thinking, apparently, that I might be curious to see the passages that I had appropriated. But having examined these productions, I find no thought in them that I ever did, or ever should be likely to put forth as my own.

Peirce: CP 3.359 Cross-Ref:†† XIII

ON THE ALGEBRA OF LOGIC

A CONTRIBUTION TO THE PHILOSOPHY OF NOTATION †1

§1. THREE KINDS OF SIGNS †2

359. Any character or proposition either concerns one subject, two subjects, or a plurality of subjects. For example, one particle has mass, two particles attract one another, a particle revolves about the line joining two others. A fact concerning two subjects is a dual character or relation; but a relation which is a mere combination of two independent facts concerning the two subjects may be called degenerate, just as two lines are called a degenerate conic. In like manner a plural character or conjoint relation is to be called degenerate if it is a mere compound of dual characters. Peirce: CP 3.360 Cross-Ref:†† 360. A sign is in a conjoint relation to the thing denoted and to the mind. If this triple relation is not of a degenerate species, the sign is related to its object only in consequence of a mental association, and depends upon a habit. Such signs are always abstract and general, because habits are general rules to which the organism has become subjected. They are, for the most part, conventional or arbitrary. They include all general words, the main body of speech, and any mode of conveying a judgment. For the sake of brevity I will call them tokens.†3 Peirce: CP 3.361 Cross-Ref:†† 361. But if the triple relation between the sign, its object, and the mind, is degenerate, then of the three pairs

sign

object

sign

mind

object mind

two at least are in dual relations which constitute the triple relation. One of the connected pairs must consist of the sign and its object, for if the sign were not related to its object except by the mind thinking of them separately, it would not fulfill the function of a sign at all. Supposing, then, the relation of the sign to its object does not

lie in a mental association, there must be a direct dual relation of the sign to its object independent of the mind using the sign. In the second of the three cases just spoken of, this dual relation is not degenerate, and the sign signifies its object solely by virtue of being really connected with it. Of this nature are all natural signs and physical symptoms. I call such a sign an index, a pointing finger being the type of the class. Peirce: CP 3.361 Cross-Ref:†† The index asserts nothing; it only says "There!" It takes hold of our eyes, as it were, and forcibly directs them to a particular object, and there it stops. Demonstrative and relative pronouns are nearly pure indices, because they denote things without describing them; so are the letters on a geometrical diagram, and the subscript numbers which in algebra distinguish one value from another without saying what those values are. Peirce: CP 3.362 Cross-Ref:†† 362. The third case is where the dual relation between the sign and its object is degenerate and consists in a mere resemblance between them. I call a sign which stands for something merely because it resembles it, an icon. Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream -- not any particular existence, and yet not general. At that moment we are contemplating an icon. Peirce: CP 3.363 Cross-Ref:†† 363. I have taken pains to make my distinction †P1 of icons, indices, and tokens clear, in order to enunciate this proposition: in a perfect system of logical notation signs of these several kinds must all be employed. Without tokens there would be no generality in the statements, for they are the only general signs; and generality is essential to reasoning. Take, for example, the circles by which Euler represents the relations of terms. They well fulfill the function of icons, but their want of generality and their incompetence to express propositions must have been felt by everybody who has used them.†1 Mr. Venn †2 has, therefore, been led to add shading to them; and this shading is a conventional sign of the nature of a token. In algebra, the letters, both quantitative and functional, are of this nature. But tokens alone do not state what is the subject of discourse; and this can, in fact, not be described in general terms; it can only be indicated. The actual world cannot be distinguished from a world of imagination by any description. Hence the need of pronoun and indices, and the more complicated the subject the greater the need of them. The introduction of indices into the algebra of logic is the greatest merit of Mr. Mitchell's system.†P1 He writes F[1] to mean that the proposition F is true of every object in the universe, and F[u] to mean that the same is true of some object.†3 This distinction can only be made in some such way as this. Indices are also required to show in what manner other signs are connected together. With these two kinds of signs alone any proposition can be expressed; but it cannot be reasoned upon, for reasoning consists in the observation that where certain relations subsist certain others are found, and it accordingly requires the exhibition of the relations reasoned within an icon. It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as

rich and apparently unending a series of surprising discoveries as any observational science. Various have been the attempts to solve the paradox by breaking down one or other of these assertions, but without success. The truth, however, appears to be that all deductive reasoning, even simple syllogism, involves an element of observation; namely, deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts. For instance, take the syllogistic formula,

All M is P S is M .·. S is P.

This is really a diagram of the relations of S, M, and P. The fact that the middle term occurs in the two premisses is actually exhibited, and this must be done or the notation will be of no value. As for algebra, the very idea of the art is that it presents formulæ which can be manipulated, and that by observing the effects of such manipulation we find properties not to be otherwise discerned. In such manipulation, we are guided by previous discoveries which are embodied in general formulæ. These are patterns which we have the right to imitate in our procedure, and are the icons par excellence of algebra. The letters of applied algebra are usually tokens, but the x, y, z, etc., of a general formula, such as

(x+y)z = x z + y z,

are blanks to be filled up with tokens, they are indices of tokens. Such a formula might, it is true, be replaced by an abstractly stated rule (say that multiplication is distributive); but no application could be made of such an abstract statement without translating it into a sensible image. Peirce: CP 3.364 Cross-Ref:†† 364. In this paper, I purpose to develop an algebra adequate to the treatment of all problems of deductive logic, showing as I proceed what kinds of signs have necessarily to be employed at each stage of the development. I shall thus attain three objects. The first is the extension of the power of logical algebra over the whole of its proper realm. The second is the illustration of principles which underlie all algebraic notation. The third is the enumeration of the essentially different kinds of necessary inference; for when the notation which suffices for exhibiting one inference is found inadequate for explaining another, it is clear that the latter involves an inferential element not present to the former. Accordingly, the procedure contemplated should result in a list of categories of reasoning, the interest of which is not dependent upon the algebraic way of considering the subject. I shall not be able to perfect the algebra sufficiently to give facile methods of reaching logical conclusions: I can only give a method by which any legitimate conclusion may be reached and any fallacious one

avoided. But I cannot doubt that others, if they will take up the subject, will succeed in giving the notation a form in which it will be highly useful in mathematical work. I even hope that what I have done may prove a first step toward the resolution of one of the main problems of logic, that of producing a method for the discovery of methods in mathematics.

Peirce: CP 3.365 Cross-Ref:†† §2. NON-RELATIVE LOGIC

365. According to ordinary logic, a proposition is either true or false, and no further distinction is recognized. This is the descriptive conception, as the geometers say; the metric conception would be that every proposition is more or less false, and that the question is one of amount. At present we adopt the former view. Peirce: CP 3.366 Cross-Ref:†† 366. Let propositions be represented by quantities. Let v and f be two constant values, and let the value of the quantity representing a proposition be v if the proposition is true and be f if the proposition is false. Thus, x being a proposition, the fact that x is either true or false is written

(x - f)(v - x) = 0.†1 So

(x - f)(v - y) = 0

will mean that either x is false or y is true. This may be said to be the same as 'if x is true, y is true.' A hypothetical proposition, generally, is not confined to stating what actually happens, but states what is invariably true throughout a universe of possibility. The present proposition is, however, limited to that one individual state of things, the Actual. Peirce: CP 3.367 Cross-Ref:†† 367. We are, thus, already in possession of a logical notation, capable of working syllogism. Thus, take the premisses, 'if x is true, y is true,' and 'if y is true, z is true.' These are written

(x - f)(v - y) = 0 (y - f)(v - z) = 0.

Multiply the first by (v - z) and the second by (x - f) and add. We get

(x - f)(v - f)(v - z) = 0,

or dividing by v - f, which cannot be 0,

(x - f)(v - z) = 0;

and this states the syllogistic conclusion, "if x is true, z is true." Peirce: CP 3.368 Cross-Ref:†† 368. But this notation shows a blemish in that it expresses propositions in two distinct ways, in the form of quantities, and in the form of equations; and the quantities are of two kinds, namely those which must be either equal to f or to v, and those which are equated to zero. To remedy this, let us discard the use of equations, and perform no operations which can give rise to any values other than f and v. Peirce: CP 3.369 Cross-Ref:†† 369. Of operations upon a simple variable, we shall need but one. For there are but two things that can be said about a single proposition, by itself; that it is true and that it is false,

x = v and x = f.

The first equation is expressed by x itself, the second by any function, φ, of x, fulfilling the conditions

φv = f

φf = v.

The simplest solution of these equations is

φx = f + v - x.

A product of n factors of the two forms (x - f) and (v - y), if not zero, equals (v - f)n. Write P for the product. Then v - (P/((v - f)n-1)) is the simplest function of the variables which becomes v when the product vanishes and f when it does not. By this means any proposition relating to a single individual can be expressed. Peirce: CP 3.370 Cross-Ref:†† 370. If we wish to use algebraical signs with their usual significations, the meanings of the operations will entirely depend upon those of f and v. Boole †1 chose v = 1, f = 0. This choice gives the following forms:

f+v-x=1-x

which is best written ~x.

v - ((x-f)(v-y)/(v-f)) = 1 - x + x y = ~(x~y).

v - ((v-x)(v-y)/(v-f)) = x + y - x y†2

v - ((v-x)(v-y)(v-z)/((v-f)2)) = x+y+z-x y-x z-y z+x y z

v - ((x-f)(y-f)/(v-f)) = 1-x y = ~xy†3

Peirce: CP 3.371 Cross-Ref:†† 371. It appears to me that if the strict Boolian system is used, the sign + ought to be altogether discarded. Boole and his adherent, Mr. Venn (whom I never disagree with without finding his remarks profitable), prefer to write x+~x y in place of ~(~x~y). I confess I do not see the advantage of this, for the distributive principle holds equally well when written

~(~x~y)z = ~(~(x z)~(y z))†4 ~(~(x y)~z) = ~(~x~z).~(~y~z).†5

The choice of v = 1, f = 0, is agreeable to the received measurement of probabilities. But there is no need, and many times no advantage, in measuring probabilities in this way. I presume that Boole, in the formation of his algebra, at first considered the letters as denoting propositions or events. As he presents the subject, they are class-names; but it is not necessary so to regard them. Take, for example, the equation

t = n + h f,

which might mean that the body of taxpayers is composed of all the natives, together with householding foreigners. We might reach the signification by either of the following systems of notation, which indeed differ grammatically rather than logically.

-----------------------------------------------------------Signification. Signification. Sign. 1st System. 2d System. -----------------------------------------------------------t Taxpayer. He is a Taxpayer. n Native. He is a Native. h Householder. He is a Householder. f Foreigner. He is a Foreigner. ------------------------------------------------------------

There is no index to show who the "He" of the second system is, but that makes no difference. To say that he is a taxpayer is equivalent to saying that he is a native or is a householder and a foreigner. In this point of view, the constants 1 and 0 are simply the probabilities, to one who knows, of what is true and what is false; and thus unity is conferred upon the whole system. Peirce: CP 3.372 Cross-Ref:†† 372. For my part, I prefer for the present not to assign determinate values to f and v, nor to identify the logical operations with any special arithmetical ones, leaving myself free to do so hereafter in the manner which may be found most convenient. Besides, the whole system of importing arithmetic into the subject is artificial, and modern Boolians do not use it. The algebra of logic should be self-developed, and arithmetic should spring out of logic instead of reverting to it. Going back to the beginning, let the writing of a letter by itself mean that a certain proposition is true. This letter is a token. There is a general understanding that the actual state of things or some other is referred to. This understanding must have been established by means of an index, and to some extent dispenses with the need of other indices. The denial of a proposition will be made by writing a line over it. Peirce: CP 3.373 Cross-Ref:†† 373. I have elsewhere †1 shown that the fundamental and primary mode of relation between two propositions is that which we have expressed by the form

v - ((x-f)(v-y)/(v-f)).

We shall write this

x -< y,

which is also equivalent to (x-f)(v-y) = 0.

It is stated above that this means "if x is true, y is true." But this meaning is greatly modified by the circumstance that only the actual state of things is referred to. Peirce: CP 3.374 Cross-Ref:†† 374. To make the matter clear, it will be well to begin by defining the meaning of a hypothetical proposition, in general. What the usages of language may be does

not concern us; language has its meaning modified in technical logical formulæ as in other special kinds of discourse. The question is what is the sense which is most usefully attached to the hypothetical proposition in logic? Now, the peculiarity of the hypothetical proposition is that it goes out beyond the actual state of things and declares what would happen were things other than they are or may be. The utility of this is that it puts us in possession of a rule, say that "if A is true, B is true," such that should we hereafter learn something of which we are now ignorant, namely that A is true, then, by virtue of this rule, we shall find that we know something else, namely, that B is true. There can be no doubt that the Possible, in its primary meaning, is that which may be true for aught we know, that whose falsity we do not know.†1 The purpose is subserved, then, if throughout the whole range of possibility, in every state of things in which A is true, B is true too. The hypothetical proposition may therefore be falsified by a single state of things, but only by one in which A is true while B is false. States of things in which A is false, as well as those in which B is true, cannot falsify it. If, then, B is a proposition true in every case throughout the whole range of possibility, the hypothetical proposition, taken in its logical sense, ought to be regarded as true, whatever may be the usage of ordinary speech. If, on the other hand, A is in no case true, throughout the range of possibility, it is a matter of indifference whether the hypothetical be understood to be true or not, since it is useless. But it will be more simple to class it among true propositions, because the cases in which the antecedent is false do not, in any other case, falsify a hypothetical. This, at any rate, is the meaning which I shall attach to the hypothetical proposition in general, in this paper. Peirce: CP 3.375 Cross-Ref:†† 375. The range of possibility is in one case taken wider, in another narrower; in the present case it is limited to the actual state of things. Here, therefore, the proposition

a -< b

is true if a is false or if b is true, but is false if a is true while b is false. But though we limit ourselves to the actual state of things, yet when we find that a formula of this sort is true by logical necessity, it becomes applicable to any single state of things throughout the range of logical possibility. For example, we shall see that from x ~-< y we can infer z -< x. This does not mean that because in the actual state of things x is true and y false, therefore in every state of things either z is false or x true; but it does mean that in whatever state of things we find x true and y false, in that state of things either z is false or x is true. In that sense, it is not limited to the actual state of things, but extends to any single state of things. Peirce: CP 3.376 Cross-Ref:†† 376. The first icon of algebra is contained in the formula of identity

x -< x.

This formula does not of itself justify any transformation, any inference. It only justifies our continuing to hold what we have held (though we may, for instance, forget how we were originally justified in holding it). Peirce: CP 3.377 Cross-Ref:†† 377. The second icon is contained in the rule that the several antecedents of a consequentia may be transposed; that is, that from

x -< (y -< z) we can pass to

y -< (x -< z).

This is stated in the formula

{x -< (y -< z)} -< {y -< (x -< z)}.

Because this is the case, the brackets may be omitted, and we may write

y -< x -< z.

Peirce: CP 3.377 Cross-Ref:†† By the formula of identity

(x -< y) -< (x -< y);

and transposing the antecedents

x -< {(x -< y) -< y}

or, omitting the unnecessary brackets

x -< (x -< y) -< y.

This is the same as to say that if in any state of things x is true, and if the proposition "if x, then y" is true, then in that state of things y is true. This is the modus ponens of

hypothetical inference, and is the most rudimentary form of reasoning.†1 Peirce: CP 3.378 Cross-Ref:†† 378. To say that (x -< x) is generally true is to say that it is so in every state of things, say in that in which y is true; so that we may write

y -< (x -< x),

and then, by transposition of antecedents,

x -< (y -< x),

or from x we may infer y -< x. Peirce: CP 3.379 Cross-Ref:†† 379. The third icon is involved in the principle of the transitiveness of the copula, which is stated in the formula

(x -< y) -< (y -< z) -< x -< z.†2

According to this, if in any case y follows from x and z from y, then z follows from x.†3 This is the principle of the syllogism in Barbara. Peirce: CP 3.380 Cross-Ref:†† 380. We have already seen that from x follows y -< x. Hence, by the transitiveness of the copula, if from y -< x follows z, then from x follows z, or from

(y -< x) -< z follows or

x -< z, {(y -< x) -< z} -< x -< z.

Peirce: CP 3.381 Cross-Ref:†† 381. The original notation x -< y served without modification to express the pure formula of identity. An enlargement of the conception of the notation so as to make the terms themselves complex was required to express the principle of the transposition of antecedents; and this new icon brought out new propositions. The third icon introduces the image of a chain of consequence. We must now again enlarge the notation so as to introduce negation. We have already seen that if a is true, we can write x -< a, whatever x may be. Let b be such that we can write b -< x whatever x may be. Then b is false. We have here a fourth icon, which gives a new

sense to several formulæ. Thus the principle of the interchange of antecedents is that from

x -< (y -< z) we can infer

y -< (x -< z).

Since z is any proposition we please, this is as much as to say that if from the truth of x the falsity of y follows, then from the truth of y the falsity of x follows. Peirce: CP 3.382 Cross-Ref:†† 382. Again the formula

x -< {(x -< y) -< y}

is seen to mean that from x, we can infer that anything we please follows from that things following from x, and a fortiori from everything following from x. This is, therefore, to say that from x follows the falsity of the denial of x; which is the principle of contradiction. Peirce: CP 3.383 Cross-Ref:†† 383. Again the formula of the transitiveness of the copula, or

{x -< y} -< {(y -< z) -< (x -< z)}

is seen to justify the inference

x -< y .·. ~y -< ~x.

The same formuLa justifies the modus tollens,

x -< y ~y .·. ~x.

Peirce: CP 3.383 Cross-Ref:††

So the formula {(y -< x) -< z} -< (x -< z) shows that from the falsity of y -< x the falsity of x may be inferred. Peirce: CP 3.383 Cross-Ref:†† All the traditional moods of syllogism can easily be reduced to Barbara by this method. Peirce: CP 3.384 Cross-Ref:†† 384. A fifth icon is required for the principle of excluded middle and other propositions connected with it. One of the simplest formulæ of this kind is

{(x -< y) -< x} -< x.

This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent x being false while its antecedent (x -< y) -< x is true. If this is true, either its consequent, x, is true, when the whole formula would be true, or its antecedent x -< y is false. But in the last case the antecedent of x -< y, that is x, must be true.†P1 Peirce: CP 3.384 Cross-Ref:†† From the formula just given, we at once get

{(x -< y) -< α} -< x,

where the a is used in such a sense that (x -< y) -< α means that from (x -< y) every proposition follows. With that understanding, the formula states the principle of excluded middle, that from the falsity of the denial of x follows the truth of x. Peirce: CP 3.385 Cross-Ref:†† 385. The logical algebra thus far developed contains signs of the following kinds: First, tokens; signs of simple propositions, as t for 'He is a taxpayer,' etc. Second, the single operative sign -(1/3))~w, or more than 1/3 of the universe (the company) have not white neckties. So ~((3/4)d) = (>(1/4))~d. The combined premisses thus become

~((>(1/3))~w + (>(1/4))~d)

Now (>(1/3))~w + (>(1/4))~d gives May be (1/3 + 1/4)(~w+~d).

Thus we have and this is

May be (7/12)(~w+~d) (At least 5/12)(~w+~d),

which is the conclusion. Peirce: CP 3.393 Fn 1 p 228 †1 This is the seventh icon? Peirce: CP 3.393 Fn 2 p 228 †2 This is the eighth icon? Peirce: CP 3.396 Fn 1 p 230 †1 See Lady's and Gentleman's Diary for 1850, p. 48 for the original presentation of 'Kirkman's School-Girls Problem.' See also W. W. R. Ball, Mathematical Recreations and Essays, Ch. IX, MacMillan & Co., 5 ed. (1911), where a reference is given to Benjamin Peirce's method of solution and his enunciation of a corresponding problem. Peirce: CP 3.396 Fn 2 p 230 Cross-Ref:†† †2 Peirce defines a syntheme in the Century Dictionary, p. 6139, ed. 1889, as "a system of groups of objects comprising every one of a larger set just once, twice or other given number of times. The groups may be divided into subgroups subject to various conditions." Peirce: CP 3.396 Fn 1 p 231 †1 Obviously a misprint for Σ[i]π[j]x[i j] -< π[j]Σ[i]x[i j]. Peirce: CP 3.396 Fn 1 p 232 †1 See 403A and B at end of article. Peirce: CP 3.396 Fn 2 p 232 †2 See 403C and M. Peirce: CP 3.398 Fn 1 p 233 †1 See 403H. Peirce: CP 3.399 Fn 1 p 235 †1 See 403I. Peirce: CP 3.400 Fn 2 p 235 †2 See 403J. Peirce: CP 3.401 Fn 1 p 236 †1 Cf. 21 (7). Peirce: CP 3.401 Fn 2 p 236 †2 ~r[vαw]. Peirce: CP 3.401 Fn 3 p 236

†3 I.e., any two terms which are relates in any one-one correspondence to any term are identical; any two terms which are the correlates by one-one correspondence to any term are identical; and any case of terms related one to one is always a one-one correspondence. Peirce: CP 3.402 Fn 1 p 237 †1 Transactions Cambridge Philosophical Society, vol. 10, pp. *355-*358, (1864). Peirce: CP 3.402 Fn 2 p 237 †2 Ibid., p. *356. Peirce: CP 3.402 Fn 3 p 237 †3 See 564 and 4.103f. Peirce: CP 3.402 Fn P1 p 237 Cross-Ref:†† †P1 Another of De Morgan's examples [Formal Logic, p. 168] is this: "Suppose a person, on reviewing his purchases for the day, finds, by his counterchecks, that he has certainly drawn as many checks on his banker (and maybe more) as he has made purchases. But he knows that he paid some of his purchases in money, or otherwise than by checks. He infers then that he has drawn checks for something else except that day's purchases. He infers rightly enough." Suppose, however, that what happened was this: He bought something and drew a check for it; but instead of paying with the check, he paid cash. He then made another purchase for the same amount, and drew another check. Instead, however, of paying with that check, he paid with the one previously drawn. And thus he continued without cessation, or ad infinitum. Plainly the premisses remain true, yet the conclusion is false. Peirce: CP 3.403A Fn 1 p 239 †1 This undated note seems to have been written for publication in some issue of The American Journal of Mathematics, shortly subsequent to that in which the previous article appeared. Why it was not published is unknown. Peirce: CP 3.403G Fn 1 p 243 †1 The second formula is incorrect. The second half should read:

a~x + x b + a x~b; or a~b + x b + a~x b. Peirce: CP 3.403I Fn 2 p 243 †2 Cf. 2.517-27. Peirce: CP 3.403I Fn 1 p 244 †1 399. Peirce: CP 3.403I Fn 1 p 245 †1 I.e., every a is identical with every b. Peirce: CP 3.403K Fn 1 p 247 †1 This should be ~1[e h]. Peirce: CP 3.403L Fn 1 p 248 †1 I.e., any two terms are identical and related or they are not identical and the

said relation does not relate them. Peirce: CP 3.403L Fn 2 p 248 †2 I.e., every two terms are related by some relation. Peirce: CP 3.404 Fn 1 p 250 †1 The Open Court, vol. 6, pp. 3391-4 (1892). Peirce: CP 3.404 Fn 1 p 251 †1 Or possibly in some other Renaissance writing. My memory may deceive me; and my library is precious small. Peirce: CP 3.407 Fn 1 p 253 †1 Cf. 2.593ff. Peirce: CP 3.408 Fn 1 p 254 †1 "On the Syllogism," II, Transactions, Cambridge Philosophical Society, vol. 9, p. 104, (1851). Peirce: CP 3.409 Fn 1 p 255 †1 See 136a. Peirce: CP 3.415 Fn 1 p 257 †1 The Open Court, vol. 6, pp. 3416-8, (1892). Peirce: CP 3.415 Fn 1 p 258 †1 Nothing of the kind seems to have been published by Peirce, and there is no record of any relevant manuscript. Peirce: CP 3.416 Fn 2 p 258 †2 Page 5057, ed. of 1889; see also 571f. Peirce: CP 3.417 Fn P1 p 259 Cross-Ref:†† †P1 In this connection, see James's, Principles of Psychology, vol. 1, pp. 237-271; Briefer Course, pp. 160 et seq. James is no logician, but it is not difficult to trace a connection between the points he makes and the theory of inference. Peirce: CP 3.418 Fn 1 p 260 †1 Paper no. XVI and see vol. 4, bk. II. Peirce: CP 3.419 Fn 1 p 261 †1 Cf. discussion in vol. 2, bk. II, ch. 4, on the nature of propositions. The previous chapters of the same book should clarify what follows. Peirce: CP 3.419 Fn 1 p 262 †1 Cf. 2.287n. Peirce: CP 3.420 Fn P1 p 262 Cross-Ref:†† †P1 Nature, in connection with a picture, copy, or diagram, does not necessarily denote an object not fashioned by man, but merely the object represented, as something existing apart from the representation. Peirce: CP 3.421 Fn 2 p 262 †2 Cf. 469f, 1.289f, 1.346. Peirce: CP 3.421 Fn P2 p 262 Cross-Ref:†† †P2 Thus, CO, which appears as such a radicle in formic acid, makes of itself

a saturated compound. Peirce: CP 3.421 Fn 1 p 263 †1 Cf. 63 and 1.347. Peirce: CP 3.422 Fn 2 p 263 †2 Cf. vol. 1, bk. III. Peirce: CP 3.422 Fn 1 p 264 †1 This is the last paper on logic to be published in The Open Court. Peirce: CP 3.423 Fn P1 p 264 Cross-Ref:†† †P1 Philosophical Transactions for 1886 [pp. 1-70]. No logician should fail to study this memoir. Peirce: CP 3.423 Fn P2 p 264 Cross-Ref:†† †P2 I use this word in its proper sense, and not to mean unlike, as Mr. Kempe does. Peirce: CP 3.425 Fn 1 p 266 †1 The Monist, vol. 7, pp. 19-40, (1896). Peirce: CP 3.425 Fn P1 p 266 Cross-Ref:†† †P1 Vorlesungen über die Algebra der Logik, (Exakte Logik). Von Dr. Ernst Schröder, Ord. Professor der Mathematik an der technischen Hochschule zu Karlsruhe in Baden. Dritter Band. Algebra und Logik der Relative. Leipsic: B. G. Teubner. 1895. Price, 16M. Peirce: CP 3.426 Fn 1 p 268 †1 See 2.30. Peirce: CP 3.427 Fn 1 p 270 †1 La philosophie positive, deuxèiame leçon. Peirce: CP 3.427 Fn 2 p 270 †2 Cf. the classification of sciences in vol. 1, Bk. II. Peirce: CP 3.427 Fn 3 p 270 †3 By Peirce, p. 5397, ed. of 1889. Peirce: CP 3.430 Fn 1 p 272 †1 The first and second parts are the topics of bks. II and III of vol. 2; the third is discussed in vol. 5 and 6. Peirce: CP 3.430 Fn 2 p 272 †2 Opera Omnia Collecta, T. 1, pp. 45-76. L. Durand. Peirce: CP 3.430 Fn 3 p 272 †3 1.559. Peirce: CP 3.431 Fn 1 p 273 †1 Bd. 1, S. 118. Peirce: CP 3.432 Fn 2 p 273 †2 Cf. 2.19. Peirce: CP 3.441 Fn 1 p 279

†1 Acad. Quaest. II, 143. Peirce: CP 3.441 Fn 2 p 279 †2 E.g. Sextus Empiricus, Adv. Math. VIII, 113-17. Peirce: CP 3.441 Fn 3 p 279 †3 The Diodorians in opposition to the Philonians deny that material implication expresses what is usually meant by "if ... then ...." For contemporary discussion see G. E. Moore, Philosophical Studies, p. 276ff; C. I. Lewis, A Survey of Symbolic Logic, esp. p. 324ff, and Paul Weiss, "Relativity in Logic," The Monist, October, 1928; "The Nature of Systems," The Monist, April/July, 1929; "Entailment and the Future of Logic," Proceedings, Seventh International Congress of Philosophy; E. J. Nelson, "Intensional Relations," Mind, October, 1930. Peirce: CP 3.442 Fn 1 p 280 †1 Quæstiones in Octo libror Physicorum Aristotelis, L. 1, qu. II. Peirce: CP 3.446 Fn 1 p 282 †1 Cf. 2.352 and 2.618. Peirce: CP 3.446 Fn 2 p 282 †2 Vol. 5, bk. II, ch. 3. Peirce: CP 3.448 Fn 1 p 283 †1 182-197. Peirce: CP 3.448 Fn 2 p 283 †2 The editors have not considered it worth publishing. But see 4.277ff. and vol. 4, Bk. II. Peirce: CP 3.450 Fn 1 p 284 †1 See 472 and 2.532-5. Peirce: CP 3.450 Fn 2 p 284 †2 Symbolic Logic, p. 39ff. Peirce: CP 3.450 Fn 3 p 284 †3 The principle of duality is expressible in the formulæ: ~(a+b) = (~a~b) and ~(ab) = (~a+~b). Peirce: CP 3.453 Fn 1 p 286 †1 See vol. 2, bk. II, ch. 1, for a discussion of the "ethics of terminology." Peirce: CP 3.455 Fn 2 p 286 †2 Peirce's first contribution to 'exact' logic is published in the Appendix to vol. 2. Peirce: CP 3.456 Fn 1 p 288 †1 The Monist, vol. 7, pp. 161-217, (1897). Peirce: CP 3.456 Fn 2 p 288 †2 Cf. vol. 5, bk. II, ch. 5. Peirce: CP 3.456 Fn P1 p 288 Cross-Ref:†† †P1 Algebra und Logik der Relative. Leipsic: B. G. Teubner. 1895. Price, 16 M.

Peirce: CP 3.462 Fn 1 p 291 †1 See 42-44, 1.83, 2.227, 2.364 and 4.235. Peirce: CP 3.465 Fn P1 p 294 Cross-Ref:†† †P1 The Pythagoreans, who seem first to have used these words, probably attached a patronymic signification to the termination. A triad was derivative of three, etc. Peirce: CP 3.468 Fn 1 p 295 †1 In this section Peirce presents his "Entitative Graphs." The "Existential Graphs" are to be found in vol. 4, bk. II. Peirce: CP 3.468 Fn 2 p 295 †2 Part 1, 1886, pp. 1-70. Peirce: CP 3.469 Fn 3 p 295 †3 Cf. 1.289f., 1.346 and 421. Peirce: CP 3.470 Fn 1 p 296 †1 See J. J. Sylvester, "Chemistry and Algebra," Mathematical Papers, vol. III, No. 14; W. K. Clifford; "Remarks on the Chemico-Algebraic Theory," Mathematical Papers, No. 28. Peirce: CP 3.470 Fn 1 p 297 †1 Meyer used the volumes as abscissæ and the weights as ordinates. See Das Natürliche System der Chemischen Elemente, Meyer u. Mendejeff, Leipzig, 1895. Peirce: CP 3.470 Fn 2 p 297 †2 See 475f. Peirce: CP 3.473 Fn 1 p 299 †1 I.e., ~l†s = $l -< s. Peirce: CP 3.474 Fn 1 p 300 †1 i.e., ~l†s -< $s†$~l but not ~~s†$l; or perhaps more clearly ($l -< s) -< (~s -< $~l) but not s -< $l. Peirce: CP 3.474 Fn P1 p 300 Cross-Ref:†† †P1 Professor Schröder proposes to substitute the word "symmetry" for convertibility, and to speak of simply convertible modes of junction as "symmetrical." Such an example of wanton disregard of the admirable traditional terminology of logic, were it widely followed, would result in utter uncertainty as to what any

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Figure 6

writer on logic might mean to say, and would thus be utterly fatal to all our efforts to render logic exact. Professor Schröder denies that the mode of junction in "lover of a servant" is "symmetrical," which word in practice he makes synonymous with "commutative," applying it only to such junctions as that between "lover" and "servant" in "Adolphus is at once lover and servant of Eugenia." Commutativity depends on one or more polyadic relatives having two like blanks as shown in Fig. 6. Peirce: CP 3.479 Fn P1 p 305 Cross-Ref:†† †P1 In my method of graphs, the spots represent the relatives, their bonds the hecceities; while in Mr. Kempe's method, the spots represent the objects, whether individuals or abstract ideas, while their bonds represent the relations. Hence, my own exclusive employment of bonds between pairs of spots does not, in the least, conflict with my argument that in Mr. Kempe's method such bonds are insufficient. Peirce: CP 3.479 Fn 1 p 306 †1 I.e., evenly encircled bonds have π as their quantifier, while the others have Σ. Peirce: CP 3.481 Fn P1 p 309 Cross-Ref:†† †P1 "On the Natural Classification of Arguments." Proceedings of the American Academy of Arts and Sciences. [2.477.] Peirce: CP 3.491 Fn 1 p 312 †1 The use of one such logical constant is shown by Peirce to be sufficient for the development of Boolian Algebra. See e.g., 4.1ff. Peirce: CP 3.493 Fn 1 p 314 †1 Cf. 332-4. Peirce: CP 3.495 Fn 1 p 315 †1 Symbolic Logic, p. 39ff. Peirce: CP 3.496 Fn 1 p 316 †1 See 47n. Peirce: CP 3.503 Fn 1 p 317 †1 But see 396. Peirce: CP 3.509 Fn 1 p 319 †1 This should be y[j]. Peirce: CP 3.510 Fn 1 p 320 †1 1885, see No. XIII. Peirce: CP 3.510 Fn 2 p 320 †2 This does not seem to have been done. Peirce: CP 3.512 Fn 1 p 321 †1 See S. 296. Peirce: CP 3.512 Fn 2 p 321

†2 Peirce wrote

Σ= x u · a

u

which conforms neither to Schröder nor to the illustration in the text. Peirce: CP 3.514 Fn 1 p 322 †1 Cf. 2.316. Peirce: CP 3.516 Fn 1 p 324 †1 Compare the physiological explanation of deductions, inductions and abductions in 2.643. Peirce: CP 3.517 Fn 1 p 325 †1 See S.161ff. Peirce: CP 3.517 Fn 2 p 325 †2 See S.163. Peirce: CP 3.518 Fn 3 p 325 †3 See S.165. Peirce: CP 3.520 Fn 1 p 328 †1 This notation is not exactly that of Schröder, who writes a = 1aβ{g}, and ~a = 0~a~β~{g}1. See S. 205. Peirce: CP 3.521 Fn 1 p 329 †1 See S.231. Peirce: CP 3.522 Fn 1 p 330 †1 See S. 239. Peirce: CP 3.524 Fn 1 p 332 †1 For l[1] $~l[2], = $~l[1]

$~l[2] = $l[2]

$l[1]

Peirce: CP 3.526 Fn 2 p 332 †2 See vol. 4, bk. I, ch. 4. Peirce: CP 3.526 Fn 1 p 333 †1 See vol. 4, bk. I, ch. 7. Peirce: CP 3.526 Fn 2 p 333 †2 "Kalkül der Abzählenden Geometrie," 1879. Peirce: CP 3.526 Fn 3 p 333 †3 See 4.219ff. Peirce: CP 3.527 Fn 4 p 333 †4 See 6.450. Peirce: CP 3.527 Fn 5 p 333 †5 See e.g. 374 and 442. Peirce: CP 3.527 Fn P1 p 334 Cross-Ref:†† †P1 For the simple reason that the real world is a part of the ideal world.

namely, that part which sufficient experience would tend ultimately (and therefore definitively), to compel Reason to acknowledge as having a being independent of what he may arbitrarily, or willfully, create.--Marginal note, 1908. Peirce: CP 3.527 Fn P2 p 334 Cross-Ref:†† †P2 That is to say each is vaguely, not distinctly, possible. -- Marginal note, 1908. Peirce: CP 3.528 Fn P1 p 335 Cross-Ref:†† †P1 Or linearly [by taking the diagonals] as follows. But there the small primes come earlier: 1; 2, 3; 5; 4, 6, 9; 10, 15; 7; 8, 12, 18, 27; 20, 30, 45; 14, 21; 35; 16, 24, 36, 54, 81; 40, 60, 90, 135; 28, 42, 63; 70, 105; 11, etc." -- Marginal note, 1908. Peirce: CP 3.529 Fn 1 p 336 †1 Cf. 1.61, 1.591ff, 2.156. Peirce: CP 3.532 Fn 2 p 336 †2 Where? Peirce: CP 3.532 Fn 3 p 336 †3 See 4.552n. Peirce: CP 3.532 Fn 4 p 336 †4 Obviously a misprint for: Σ[i]π[j]~l[i j] Peirce: CP 3.532 Fn 1 p 337 †1 (1) Σ[i]π[j]~l[i j] -< π[j]Σ[i]l[i j] (2); (3) Σ[j]π[i]l[i j] -< π[i]Σ[j]l[i j] (4). (2) and (3) are contradictories; (1) and (4) are contradictories. If (1) were true (2) would be true. If (3) were true (4) would be true. If (1) and (3) were true, (2) and (3), which are contradictories, would both be true. Though (1) and (3) thus cannot both be true, they may both be false. They are related to one another as an A and an E; their contradictories (4) and (2) must be related to one another as an I and an O -- both can be and one at least must be true. Peirce: CP 3.535 Fn 2 p 337 †2 See 2.517ff. Peirce: CP 3.537 Fn P1 p 338 Cross-Ref:†† †P1 I prefer to speak of a member of a collection as a subject of it rather than as an object of it; for in this way I bring to mind the fact that the collection is virtually a quality or class-character. [A collection is a rhema or propositional function. Its members are those subjects which make it a true proposition. See 66.] Peirce: CP 3.537 Fn 1 p 339 †1 I.e., all the i's which are members of P, are related to the j's which are members of Q, and there is no k distinct from an i which has the same relation to the j's which the i's have to the j's. Peirce: CP 3.537 Fn P1 p 339 Cross-Ref:†† †P1 It must be remembered that to a person familiar with the algebra all such series of steps become evident at first glance. Peirce: CP 3.538 Fn 2 p 339 †2 Cf. 532.

Peirce: CP 3.547 Fn 1 p 343 †1 See "Ueber eine elementare Frage der Mannigfaltigkeitslehre," (1890-1), Georg Cantor Gesammelte Abhandlung, herausg. E. Zermelo, S. 278-81, Berlin, (1932) where Cantor shows that 2n is always greater than n. Peirce: CP 3.549 Fn P1 Para 1/9 p 343 Cross-Ref:†† †P1 Inasmuch as the above theorem is, as I believe, quite opposed to the opinion prevalent among students of Cantor, and they may suspect that some fallacy lurks in the reasoning about wishes, I shall here give a second proof of a part of the theorem, namely that there is an endless succession of infinite multitudes related to one another as above stated, a relation entirely different, by the way, from those of the orders of infinity used in the calculus. I shall not be able to prove by this second method, as is proved in the text, that there are no higher multitudes, and in particular no maximum multitude. The ways of distributing a collection into two houses are equal to the possible combinations of members of that collection (including zero); for these combinations are simply the aggregates of individuals put into either one of the houses in the different modes of distribution. Hence, the proposition is that the combinations of whole numbers are more multitudinous than the whole numbers, that the combinations of combinations of whole numbers are still more multitudinous, the combinations of combinations of combinations again more multitudinous, and so on without end. Peirce: CP 3.549 Fn P1 Para 2/9 p 343 Cross-Ref:†† I assume the previously proved proposition that of any two collections there is one which can be placed in one-to-one correspondence with a part or the whole of the other. This obviously amounts to saying that the members of any collection can be arranged in a linear series such that of any two different members one comes later in the series than the other. Peirce: CP 3.549 Fn P1 Para 3/9 p 343 Cross-Ref:†† A part may be equal to the whole; as the even numbers are equal in multitude to all the numbers (since every number has a double distinct from the doubles of all other numbers, and that double is an even number). Hence, it does not follow that because one collection can be placed in one-to-one correspondence to a part of another, it is less than that other, that is, that it cannot also, by a rearrangement, be placed in one-to-one correspondence with the whole. This makes an inconvenience in reasoning which can be overcome in a manner I proceed to describe. Peirce: CP 3.549 Fn P1 Para 4/9 p 343 Cross-Ref:†† Let a collection be arranged in a linear series. Then, let us speak of a section of that series, meaning the aggregate of all the members which are later than (or as late as) one assignable member and at the same time earlier than (or as early as) a second assignable member. Let us call a series simple if it cannot be severed into sections each equal in multitude to the whole. A series not simple itself. may be conceivably severed into simple sections, or it may be so arranged that it cannot be so severed (for example the series of rational fractions arranged in the order of their magnitudes). But suppose two collections to be each ranged in a linear series, and suppose one of them, A, is in one-to-one correspondence with a part of the other B. If now the latter series, B, can be severed into simple sections, in each of which it is possible to find a member at least as early in the series as any member of that section that is in correspondence with a member of the other collection A, and also a member at least as late in the series as any member of that section that is in correspondence

with any member of the other collection, and if it is also possible to find a section of the series, B, equal to the whole series, B, in which it is possible to find a member later than any member that is in correspondence with any member of the collection, A, then I say that the collection, B, is greater than the collection, A. This is so obvious that I think the demonstration may be omitted. Peirce: CP 3.549 Fn P1 Para 5/9 p 343 Cross-Ref:†† Now, imagine two infinite collections, the α's and the β's, of which the β's are the more multitudinous. I propose to prove that the possible combinations of β's are more multitudinous than the possible combinations of α's. For let the pairs of conjugate combinations (meaning by conjugate combinations a pair each of which includes every member of the whole collection which the other excludes) of the β's be arranged in a linear series; and those of the α's in another linear series. Let the order of the pairs in each of the two series be subject to the rule that if of two pairs one contains a combination composed of fewer members than either combination of the other pair, it shall precede the latter in the series. Let the order of the pairs in the series of pairs of combinations of β's be further determined by the rule that where the first rule does not decide, one of two pairs shall precede the other whose smaller combination (this rule not applying where one [?] combinations are equal) contains fewer β's which are in correspondence with α's in one fixed correspondence of all the α's with a part of the β's. Peirce: CP 3.549 Fn P1 Para 6/9 p 343 Cross-Ref:†† In this fixed correspondence each α has its β, while there is an infinitely greater multitude of β's without α's than with. Let the two series of pairs of combinations be so placed in correspondence that every pair of unequal combinations of α's is placed in correspondence with that pair of combinations of β's of which the smaller contains only the β's corresponding in the fixed correspondence to the smaller combination of α's; and let every pair of equal combinations of α's be put into correspondence with a pair of β's of which the smaller contains only the β's belonging in the fixed correspondence to one of the combinations of α's. Peirce: CP 3.549 Fn P1 Para 7/9 p 343 Cross-Ref:†† Then it is evident that each series will generally consist of an infinite multitude of simple sections. In none of these will the combinations be more multitudinous than those of the β's. In some, the combinations of α's will be equal to those of the β's; but in an infinitely greater multitude of such simple sections and each of these infinitely more multitudinous, the combinations of β's will be infinitely more multitudinous than those of the α's. Hence it is evident that the combinations of the β's will on the whole be infinitely more multitudinous than those of the α's. Peirce: CP 3.549 Fn P1 Para 8/9 p 343 Cross-Ref:†† That is if the multitude of finite numbers be α, and 2a = b, 2b = c, 2c = d, etc. a < b < c < d < etc. ad infinitum. Peirce: CP 3.549 Fn P1 Para 9/9 p 343 Cross-Ref:†† It may be remarked that the finite combinations of finite whole numbers form no larger a multitude than the finite whole numbers themselves; i.e. they are at least enumerable. But there are infinite collections of finite whole numbers; and it is these which are infinitely more numerous than those numbers themselves.

Peirce: CP 3.550 Fn 1 p 345 †1 Cf. 4113. Peirce: CP 3.552 Fn 2 p 345 †2 This is the last of the published papers on Schröder. Peirce: CP 3.553 Fn 1 p 346 †1 Educational Review, pp. 209-16, (1898). Peirce: CP 3.554 Fn 2 p 346 †2 Metaphysica 1061a 28-1061b 3; 1061b 21-25. Peirce: CP 3.554 Fn P1 p 346 Cross-Ref:†† †P1 Davidson, Aristotle and the ancient educational ideals. Appendix: The Seven Liberal Arts. (New York: Charles Scribner's Sons.) Peirce: CP 3.555 Fn P1 p 347 Cross-Ref:†† †P1 Brouillon, Proiet d'une atteinte aux événemens des rencontres du cône avec son plan, 1639. Peirce: CP 3.558 Fn P1 p 348 Cross-Ref:†† †P1 In his Linear associative algebras, [p. 97, published in the American Journal of Mathematics, vol. 4, (1881), pp. 97-229; see No. VIII.] Peirce: CP 3.559 Fn P1 p 349 Cross-Ref:†† †P1 See this well put in Thomson and Tait's Natural Philosophy, §447. Peirce: CP 3.561 Fn P1 p 352 Cross-Ref:†† †P1 The Mathematical Psychology of Boole and Gratry. Peirce: CP 3.562 Fn 1 p 352 †1 No further articles were published in the Educational Review. The following, however, was part of the original article and seems to have been omitted only because of lack of space. What follows is taken from paginated page proofs. Peirce: CP 3.562A Fn P1 p 353 Cross-Ref:†† †P1 I notice that writers of school arithmetics shrink from accepting the correct name of their art, Vulgar Arithmetic. It is a pity we have lost Chaucer's word "augrim," which, etymologically meaning "the art of the Chorasmian," is free from all objection. Still, we cannot find fault with these writers who adopt no more high sounding title for their subject than Practical Arithmetic. Peirce: CP 3.562B Fn 1 p 354 †1 This should be: C is not R'd by B. Peirce: CP 3.562B Fn 2 p 354 †2 Given the precepts I. (πaπb) - (a R b)

- (b R a)

II. (πaπbπc) a R b

bRc

cRa

through the use of the propositions of logic and the principles of substitution and inference, the following are some of the theorems that can be derived.

A. (πb) - (b R b)

[[I:(b/a)]] - (b R b)

- (b R b)

(1)

(1) -< - (b R b) B. (πaπb) a R b

bRa

[[II:(b/c)]] a R b

bRb

(A) · (1) -< a R b

bRa

(1)

bRa

C. (πaπbπc) b R a -< (b R c

c R a)

I = b R a -< - (a R b)

(1)

II = -(a R b) -< (b R c

c R a)

(1)·(2)-< [b R a -< (b R c

(2)

c R a)]

D. (πaπbπc)(b R a · c R b) -< c R a

[[I:(c/a)]] - (c R b)

- (b R c)

= c R b -< - (b R c)

(1) (2)

C = b R a -< [[-(b R c) -< c R a]] = [[b R a . -(b R c)] -< c R a

(3) (4)

(2) · (4) -< (b R a · c R b) -< c R a

E. (πaπbπc) - (b R a)

D = -(b R a)

- (c R b)

- (c R b)

[[I:(c/b)]] -(a R c)

-(a R c)

cRa

-(c R a)

(1) (2)

(1) · (2) -< -(b R a)

-(c R b)

Peirce: CP 3.562C Fn 1 p 355 †1 4 = -(c R a) -< -(b R a) -(xRm) -(nRx).

-(a R c) -(c R b), which by [[(n/c),m/a,x/b]] is -(nRm)-
(not used by Boole,

but necessary to express particular propositions) +, -, X, 1, 0. In place of these seven signs, I propose to use a single one. Peirce: CP 4.13 Cross-Ref:†† 13. I begin with the description of the notation for conditional or "secondary" propositions. The different letters signify propositions. Any one proposition written down by itself is considered to be asserted. Thus,

A

means that the proposition A is true. Two propositions written in a pair are considered to be both denied. Thus,

AB

means that the propositions A and B are both false; and

AA

means that A is false. We may have pairs of pairs of propositions and higher complications. In this case we shall make use of commas, semicolons, colons, periods, and parentheses, just as [in] chemical notation, to separate pairs which are themselves paired. These punctuation marks can no more count for distinct signs of algebra, than the parentheses of the ordinary notation. Peirce: CP 4.14 Cross-Ref:†† 14. To express the proposition: "If S then P," first write A for this proposition. But the proposition is that a certain conceivable state of things is absent from the universe of possibility. Hence instead of A we write BB Then B expresses the possibility of S being true and P false. Since, therefore, SS denies S, it follows that (SS, P) expresses B. Hence we write SS, P; SS, P.†1 Peirce: CP 4.15 Cross-Ref:†† 15. Required to express the two premisses, "If S then M" and "if M then P." Let A be the two premisses. Let B be the denial of the first and C that of the second; then in

place of A we write BC But we have just seen that B is (SS, M) and that C is (MM, P); accordingly we write SS, M; MM, P. Peirce: CP 4.16 Cross-Ref:†† 16. All the formulae of the calculus may be obtained by development or elimination. The development or elimination having reference say to the letter X, two processes are required which may be called the erasure of the Xs and the erasure of the double Xs. The erasure of the Xs is performed as follows: Peirce: CP 4.17 Cross-Ref:†† 17. Erase all the Xs and fill up each blank with whatever it is paired with. But where there is a double X this cannot be done; in this case erase the whole pair of which the double X forms a part, and fill up the space with whatever it is paired with. Go on following these rules. Peirce: CP 4.17 Cross-Ref:†† A pair of which both members are erased is to be considered as doubly erased. A pair of which either member is doubly erased is to be considered as only singly erased, without regard to the condition of the other member. Whatever is singly erased is to be replaced by the repetition of what it is paired with. Peirce: CP 4.17 Cross-Ref:†† To erase the double Xs, repeat every X and then erase the Xs.†1 Peirce: CP 4.18 Cross-Ref:†† 18. If φ be any expression, φ/x what it becomes after erasure of the Xs, and φ/x x what it becomes after erasure of the double Xs, then φ = φ/x, x; φ/x x, xx. If φ be asserted, then φ/x φ/x x, φ/x φ/xx may be asserted. Peirce: CP 4.19 Cross-Ref:†† 19. The following are examples. Required to develop X in terms of X. Erasing the Xs the whole becomes erased, and (φ/x) x = xx.

Erasing the double Xs, the whole becomes doubly erased and (φ/x x) xx is erased. If φ, then (φ/x) x, (φ/x x) x x = xx, xx.

So that X = xx, xx. Required to eliminate X from (xx, x; a). φ/x = 00, 0; a = aa φ/x x = 00, 00; 00: a = aa

.·. φ =†1 φ/x φ/xx, φ/x φ/xx = aa, aa; aa, aa = aa. Required to eliminate X from (xa, a). φ/x = 0a, a = aa, a φ/x x = 00, a; a = aa .·. φ†1 = aa, a; aa:aa, a; aa = aa, aa = a.†2 Required to develop (ax; b, xx:ab) according to X. φ/x = a0; b, 00: ab = aa, aa; ab = a, ab = a†3 φ/x x = a00; b, 0:ab = bb, bb; ab = b, ab = b†3 φ = (φ/x) x, (φ/x x) xx = ax; b, xx †3

Required to eliminate M from (SS, M; MM, P). φ/M = SS, 0; 00, P = SS, SS; SS, SS = SS φ/M M = SS, 00; 0, P = P, P; P, P = P .·. SS, M; MM, P =†1 SS, P; SS, P which is the syllogistic conclusion. Peirce: CP 4.19 Cross-Ref:†† We may now take an example in categoricals. Given the premisses "There is something besides Ss and Ms," and "There is nothing besides Ms and Ps," to find the conclusion. As the combined premisses state the existence of a non-S non-M and the non-existence of an MP,†4 they are expressed by SM, SM; MP.

To eliminate M, we have φ/M = S0, S0; 0P = SS, SS; PP = S, PP φ/M M = S; 00, S, 00:00, P = erased

.·. φ/M φ/M M, φ/M φ/M M =†1 S, PP; 0:S, PP; 0 =†1 S, PP; S, PP: S, PP; S, PP = S, PP. The conclusion therefore is that there is something which is not an S but is a P. Peirce: CP 4.20 Cross-Ref:†† 20. Of course, it is not maintained that this notation is convenient; but only that it shows for the first time the possibility of writing both universal and particular propositions with but one copula which serves at the same time as the only sign for compounding terms and which renders special signs for negation, for "what is" and for "nothing" unnecessary. It is true, that a 0 has been used, but it has only been used as the sign of an erasure.†2

Peirce: CP 4.21 Cross-Ref:†† II THE ESSENCE OF REASONINGP†1

§1. SOME HISTORICAL NOTES

21. . . . Logic having been written first in Greek had to be turned into Latin; and this was done for the most part by imitating the formation of each technical term. Thus, the Greek hypothesis, {hypothesis} was compounded of {hypo}, under and {tithenai}, to put. The preposition {hypo} was equivalent to the Latin sub -- which is from the same root (being altered from sup), and {tithenai} was translated by ponere. Hence resulted suppositio. It is a very curious fact, by the way, that in this process it was always necessary to change the root. For, whether it be that there is something analogous to Grimm's law applying to meanings, as that applies to sounds, certain it is that the roots bear so uniformly different meanings that a different one must always be taken. Thus, the root of {tithenai} is the same as that of the Latin facere, so that hypothetical is the equivalent of sufficient, which widely diverges in meaning. Ponere is po-sinere, of which the root may be sa, to sow, to strew. Peirce: CP 4.22 Cross-Ref:†† 22. The earliest Latin work in which we find logical words so transferred from the Greek is supposed to be a treatise on Rhetoric (Ad Herennium) usually printed with the works of Cicero, but supposed to be written by one Cornificius, a little older than Cicero. Cicero himself made a number of words on that plan which are now very common, such as quantity and quality.†2 Peirce: CP 4.23 Cross-Ref:†† 23. Apuleius, the author early in the second century of our era of the celebrated novel of the Golden Ass, wrote a treatise on logic which has somehow come to be arranged as the third book of his work De dogmate Platonis. The terminology of this treatise we may be pretty sure [Apuleius] did not invent, though it differs considerably from that of any other book which either preceded or for

centuries followed it. That terminology has overridden other rival systems of translating the Greek words and has become largely ours. If the reader asks me what the quality was which lent it this staying power, he will be surprised at the answer. Namely, [Apuleius] had one of the most artificial, word-playing, fantastically and elaborately nonsensical styles that the Indo-European literatures can show. It sedulously cultivates every quality which writers upon style admonish us to avoid. Peirce: CP 4.24 Cross-Ref:†† 24. . . . Towards the end of the fifth century there was one Martianus Minneus Felix Capella, who wrote a work entitled the Nuptials of Philology and of Mercury. This Martianus Capella thought that beneath the stars there was nothing so beautiful nor so worthy of emulation as the style of Apuleius. He did his very best to outdo him; and in studying him became embued with his phraseology. Now in the book the Seven Liberal Arts are invited to the celebration of the Nuptials aforesaid, and each one entertains the company with the greatest good taste by talking shop for all she is worth. The consequence is that the book contains seven short treatises upon these disciplines, of which logic is one. Now the masters of the cathedral schools which at the fall of the Western Empire had to take the place of the old Roman schools found that in an age when one copy of one book and that not too large a one, was all that one school could commonly afford, the work of Capella was well adapted to their purpose. And thus it happened that for some centuries that was the only secular book that ordinary clerks had ever laid eyes upon. Thus, its borrowed terminology became traditional. Peirce: CP 4.25 Cross-Ref:†† 25. Anicius Manlius Severinus Boëtius (more commonly called Boëthius) was the author of a book which, whatever its merits and faults, was sincere and has in fact excited a degree of admiration such as has fallen to few works. He is most respectable as a thinker, a logician of positive strength, a man of great learning, a most estimable and sympathetic character, and the courageous supporter of calamities that touch every heart. . . . Peirce: CP 4.26 Cross-Ref:†† 26. Petrus Hispanus was a noble Portuguese who, having taken degrees in all the faculties in Paris, returned to Lisbon and was appointed head of that school which ultimately developed into the University of Coimbra. Subsequently, he was head physician to Pope Gregory X, who created him Cardinal; and he was crowned Pope, September 20, 1276.†1 He began his pontificate with promise of grandeur; but a part of his palace fell upon him and he died in consequence of his injuries on May 16, 1277.†1 This man, who had he survived would surely have been reckoned among the world's great men, was according to the tradition, the author of the Summulæ logicales, the regular textbook in logic almost to the very end of scholasticism. There are, it is true not very many printed editions subsequent to 1520; but over fifty editions having by that time been printed upon substantial linen paper, copies could always be procured in plenty. Manuscript copies were also current long after printing came in. Peirce: CP 4.26 Cross-Ref:†† There is a Greek text of the book; it has been printed with the name of Michael Psellus attached to it. That name was a common one in Constantinople. Even if any MS. carries it, which has been denied, it does not prove that any particular Michael Psellus was the author; and the language, which is intermediate between Greek and a kind of Romaic, absolutely negatives the idea of its being written by any

Michael Psellus known. It is full of Latinisms, and of reminiscences of Latin authors. The Latin text on the other hand bears on its first page conclusive evidence that the author did not know Greek. Namely, we there read: "Dicitur enim dyalectica a dya, quod est duo, et logos, quod est fermo vel lexis quod est ratio." Nevertheless, some writers, especially Prantl,†2 have believed the Greek text to be the original. Charles Thurot has written ably on the other side. When the reader comes across anything about "Byzantine" logic, what is meant is that this book is supposed to be the relic of a development of logic in Constantinople, which in my opinion is an unfounded fancy of Prantl's taken up by many writers without sufficient examination, and solely because Prantl has looked into more logical books of the middle ages than anybody else. I am very grateful to him for what he has read and published in a most convenient form; but I find myself compelled to dissent from his judgment very many times. A more slap-dash historian it would be impossible to conceive. Peirce: CP 4.27 Cross-Ref:†† 27. There is a synchronism between the different periods of medieval architecture, and the different periods of logic. The great dispute between the Nominalists and Realists took place while men were building the round-arched churches, and the elaboration finally attained corresponds to the intricate character of the opinions of the later disputants in that controversy. From that style of architecture we pass to the early pointed architecture with only plate-tracery. The simplicity of it is perfectly paralleled by the simplicity of the early logics of the thirteenth century. Among these simple writings, I reckon the commentaries of Averroes and of Albertus Magnus. I would add to them the writings of the great psychologist, St. Thomas Aquinas. For Thomistic Logic, I refer to Aquinas,†1 to Lambertus de Monte †2 whose work was approved by the Doctors of Cologne, to the highly esteemed Logic of the Doctors of Coimbra,†3 and to the modern manual of [Antoine] Bensa.†4 Peirce: CP 4.28 Cross-Ref:†† 28. During the period of the Decorated Gothic, we have the writings of Duns Scotus, one of the greatest metaphysicians of all time, whose ideas are well worth careful study, and are remarkable for their subtilty, and their profound consideration of all aspects of the questions [of philosophy]. The logical upshot of the doctrine of Scotus is that real problems cannot be solved by metaphysics, but must be decided according to the evidence. As he was a theologian, that evidence was, for him, the dicta of the church. But the same system in the hands of a scientific man will lead to his insisting upon submitting everything to the test of observation. Especially, will he insist upon doing so as against so-called "experientialists," who, though they talk about experience as their guide, really reach the most important conclusions without any careful examination of experience. Whether their conclusions happen to be right or wrong, the Scotist will protest against the manner in which they are taken up. Scotus added a great deal to the language of logic. Of his invention is the word reality. For Scotistic logic I refer to Scotus,†1 Sirectus,†2 and Tartaretus.†3 Peirce: CP 4.29 Cross-Ref:†† 29. Scotus died in 1308. After him William of Ockham, who died in 1347, took up once more the nominalistic opinion and this gained ground more and more. Logic now took on a very elaborate, but fanciful and in great measure senseless development; and finally became so big and so useless, that men must have dropped it, even if a new awakening of thought had not occurred. This was during the flamboyant period of architecture in France, the perpendicular in England. The Occamists made important additions to the terminology. For Occamistic Logic, I refer

to Ockham's own elaborate treatise,†4 to the Summulæ of the Doctors of Mayence, to the commentaries of Bricot,†5 etc. Peirce: CP 4.30 Cross-Ref:†† 30. The new awakening consisted in the conviction that the classical authors had not been sufficiently studied. At the same time the reformation of the churches came. Logic once more became simple, and this time took on a rhetorical character. Ramus (Pierre de la Ramée),†6 Ludovicus Vives,†7 Laurentius Valla,†8 were the names of logicians who contributed a few things, but on the whole, rather important things to the tradition of logic. Peirce: CP 4.31 Cross-Ref:†† 31. Upon the heels of that movement came another, which has not yet expended itself, nor even quite completed its conquest of minds. It arose from the conviction that man had everything to learn from observation. The first great investigators in this line were Copernicus, Tycho Brahe, Kepler, Galileo, Harvey, and Gilbert. None of them seemed to have any interest at all in the general theory -- and that for a simple reason; namely, they knew no way of inquiry but the way of experiment; and their lives were so many experiments in regard to the efficacity of the method of experimentation. The first great writer on the theory of Induction, Francis Bacon, was no scientific man. He had no turn that way, though he wished to have, and though he came to his death by a foolish experiment; and his judgments of scientific men were uniformly mistaken. The details of his theory were equally at fault; yet as long as he remains upon the ground of generalities, his ponderous charges are excellent. His Novum Organum, like several other great works of this period upon method, is marked by complete contempt for the Aristotelian analysis of reasoning, which nevertheless has kept the field, and, on the whole, held its ground. Still, Bacon made some distinct contributions to the traditional stock of logical ideas. . . . Peirce: CP 4.32 Cross-Ref:†† 32. The works we are now coming to are of less historical interest, precisely because they have to be taken seriously. Truly to paint the ground where we ourselves are standing is an impossible problem in historical perspective. . . . Peirce: CP 4.33 Cross-Ref:†† 33. The nominalistic wing of the Lockian party, much influenced by Hobbes and Ockham, made a philosophical development, chiefly psychological, but also logical. Among their names are Hartley, Berkeley, Hume, James Mill, John Stuart Mill, Bain. Bentham's Logic I must confess I do not remember to have seen. That of Mill, which appeared in 1843, contributed some phrases, which many persons adhere to passionately without reference to their meaning, sometimes seeming to attach no meaning to them, except the general one of a party-reveil. The present writer cares nothing about social matters, and knows not what such things mean. He examines logical questions as such and question by question. He perceives that many adherents of John Stuart Mill seem to be in a passion about something. But until they can calm themselves sufficiently, any scientific discussion of the questions, which perhaps they care little about, anyway, is impossible. Peirce: CP 4.34 Cross-Ref:†† 34. All the Occamistic school, from the Venerable Inceptor down, have been more or less politicians. John Stuart Mill was hypochondrically scrupulous. Nevertheless, every man of action is, must be, and ought to be, cunning, worldly, and dishonest, or what seems so to a man of pure science. When such men dispute, the

dispute has some other object than the ascertainment of scientific truth. Men accomplish, roughly speaking, what they desire. Government may be ever so much more important than science; but only those men can advance science who desire simply to find out how things really are, without arrière-pensée. Peirce: CP 4.35 Cross-Ref:†† 35. Occamism is governed by a very judicious maxim of logic, called Ockham's razor. It runs thus: Entia non sunt multiplicanda præter necessitatem, that is, "Try the theory of fewest elements first; and only complicate it as such complication proves indispensible for the ascertainment of truth." It may seem, at the outset, that the more complicated theory is the more probable. Nevertheless, it is highly desirable to stop and carefully to examine the simpler theory, and not contenting oneself with concluding that it will not do, to note precisely what the nature of its shortcomings are. Realism can never establish itself except upon the basis of an ungrudging acceptance of that truth. The Occamists have followed out this rule in the most interesting manner, and have contributed much to human knowledge. Reasons will be given †1 for thinking that their simple theory will not answer; yet this in no wise detracts from their scientific merits, since the only satisfactory way of ascertaining the insufficiency of the theory was to push the application of it, just as they have done. But because the abandonment of the theory would imply the modification of their politics, they employ every means in their power to discredit and personally hamper those who reject it and to prevent the publication and circulation of works in which it is impartially examined. That is not the conduct of philosophers, however wise it may be from the point of view of statesmanship. . . . Peirce: CP 4.36 Cross-Ref:†† 36. As a logician [Leibniz] was a nominalist and leaned to the opinion of Raymond Lully, an absurdity here passed over as not worth mention. This very nominalism led Leibniz to an extraordinary metaphysical theory, his Monadology, of much interest. In regard to human knowledge, he put forth many ideas which had great influence, all of them rooted in nominalism, yet at the same time departing widely from the Occamistic spirit. Such were his tests of universality and necessity; and such was his principle of sufficient reason, which he regarded as one of the fundamental principles of logic. This principle is that whatever exists has a reason for existing, not a blind cause, but a reason. A reason is something essentially general, so that this seems to confer reality upon generals. Yet if realism be accepted, there is no need of any principle of sufficient reason. In that case, existing things do not need supporting reasons; for they are reasons, themselves. A great deal of the Leibnizian philosophy consists of attempts to annul the effect of nominalistic hypotheses. . . . Peirce: CP 4.37 Cross-Ref:†† 37. Immanuel Kant, who made a revolution in philosophy by his Critic of the Pure Reason, 1781, had great power as a logician. He unfortunately had the opinion that the traditional logic was perfect and that there was no room for any further development of it.†1 That opinion did not prevent his introducing a number of ideas which have indirectly more than directly affected the traditional logic. Peirce: CP 4.37 Cross-Ref:†† The merits of German philosophers since Kant as logicians have in the opinion of the present writer been small, while their errors and vagaries have been incessant.†2 At any rate, they have had little or no effect upon the ordinary logic. . . .

Peirce: CP 4.38 Cross-Ref:†† §2. THE PROPOSITION

38. Very little of the traditional logic relates to the subject of the present section. St. Thomas Aquinas †3 divides the operations of the Understanding in reference to the logical character of their products into Simple Apprehension, Judgment, and Ratiocination, or Reasoning. Prantl declares †4 the commentary on the Perihermeneias in which this occurs not to be the work of Aquinas. But he does not explain how it could possibly happen that all the other books of the commentary should be genuine, as he admits they are, and this spurious. From the manner in which such books are written it is utterly inadmissible to suppose Aquinas passed over this book without comment. Such conduct would have excited a riot the noise of which would have reached our ears. If, then, the existing commentary is spurious, how could the genuine one have been lost? Thomas Aquinas was already an object of worship living. There was no school which adhered so religiously to the tenets of their master. Prantl himself complains that there is absolutely nothing in the works of Lambertus de Monte, and other Thomists except what St. Thomas had said. How could, then, all those schools be deceived into rejecting one of the works of their holy master, and taking in its place a writing that was not his? How is it that men of such learning as the doctors of Coimbra should get no wind of the substitution? Even Duns Scotus, writing directly after Aquinas, uses in his questions expressions which he probably derived from the book which Prantl suspects. Prantl gives no reason whatever for his rejection. He seems to think his judgment will be so commended by the comparison of it with manuscripts in other cases so entirely that he is placed quite above the necessity of giving reasons for his opinions. Similar ideas are apt to get possession of Germans. Peirce: CP 4.39 Cross-Ref:†† 39. Simple Apprehension produces concepts expressed by names or terms, "man," a state, suspended existence, the character of eating canned vacuum. Peirce: CP 4.39 Cross-Ref:†† Judgment produces judgments, which are true or false, and are expressed by sentences, or propositions, as "Man is mortal," "some men may be insane." Peirce: CP 4.39 Cross-Ref:†† Ratiocination or reasoning produces inferences or reasonings, which are expressed by argumentations, as, " I think, therefore I must exist," "Enoch, being a man, must have died; and since the Bible says he did not die, not everything in the Bible can be true." Peirce: CP 4.40 Cross-Ref:†† 40. A term names something but asserts nothing; a proposition asserts. Propositions differ in modality, which is the degree of positiveness of their assertion, as in maybe, is, must be. In another respect propositions are said to be assertory, problematic, and apodictic. The old statement †1 was that propositions were either

modal or de inesse, i.e., assertoric. They may also be probable assertions; they may further be approximate and probable assertions, as "about 51 per cent of the births in any one year will be male." Propositions are divided into the Categorical and the Hypothetical. "Propositionum alia categorica alia hypothetica,"†1 says the Summulæ. A categorical proposition is one whose immediate parts are terms; or as the Summulæ of the Mayence doctors say,†2 "cathegorica est illa quæ habet subiectum et prædicatum tanguam partes principales sui." A hypothetical proposition, better called by the Stoics †3 a composite proposition, is one which is composed of other propositions: "Propositio hypothetica est illa quæ habet duas propositiones cathegoricas tanquam partes principales sui."†4 The old, and less incorrect doctrine about compound propositions was that they were of three kinds, conditional, copulative, and disjunctive.†5 A conditional proposition is one whose members are joined by an if, or its equivalent: "Conditionalis est illa in qua coniunguntur duæ cathegoricæ per hanc coniunctionem, si."†6 That is, what is asserted is that in case one proposition, called the antecedent, is true, another proposition, called the consequent, is true. But how it may be in the opposite case in which the antecedent is not true is not stated. A copulative proposition is one in which the truth of every one of several propositions is affirmed. A disjunctive proposition is one in which the truth of some one of several propositions is affirmed. This enumeration is faulty because the conditional and disjunctive do not differ from one another in the same way in which both differ from the copulative proposition. For the conditional merely (or, at least, principally) asserts that unless one proposition is true another is true, that is, either the contrary of the former is true or the latter is true; and the disjunctive implies no more than that if the contradictions of all the alternatives but one be true, that one is true. Hence, either these two classes should be joined together, or we ought to include three other kinds of compound propositions, one which declares the repugnancy of two or more given propositions so that all cannot be true, one which declares the independence of one proposition of others so that it can be false although they are all true, and one which declares that there is a possibility that all of certain propositions are false.†1 Peirce: CP 4.41 Cross-Ref:†† 41. The subject of a categorical proposition is that concerning which something is said, the predicate is that which is said of it. Most of the medieval logics teach that subject and predicate are the principal parts of the categorical proposition but that there is also a Copula which joins them together. . . . The Mayence doctors were quoted on this head, because Petrus Hispanus †2 makes the Subject, Predicate, and Copula to be all principal parts -- one of the numerous evidences that the text is not a translation from the Greek, a language in which the copula may be dispensed with. Aristotle, however, in his treatise upon forms of propositions, the De interpretatione,†3 analyzes the categorical proposition into the noun, or nominative, and the verb. Peirce: CP 4.42 Cross-Ref:†† 42. Categorical propositions are said to be divided according to their Quantity, into the universal, the particular, the indefinite, and the singular. A universal proposition was said to be a proposition whose subject is a common term determined by a universal sign. A common term was defined as one which is adapted to being predicated of several things (aptus natus prædicari de pluribus†4). The universal signs are every, no, any, etc. A particular proposition was said to be a proposition whose subject is a common term determined by a particular sign. The particular signs are, some, etc. An indefinite proposition was said †5 to be one in which the subject is

a common term without any sign, "ut homo currit." That unfortunate "indefinite" man has been running on now for so many centuries, it is fair he should have a rest and that we should revert to Aristotle's example, "Man is just."†6 A singular proposition was said to be one in which the subject is a singular term. A singular term was defined as "qui aptus natus est prædicari de uno solo,"†7 that is, it is a proper noun. Kant and other modern logicians very rightly drop the indefinite propositions which merely arise from the imperfect expression of what is meant. Singular propositions are for the purposes of formal logic equivalent to universal ones. Peirce: CP 4.43 Cross-Ref:†† 43. Propositions were further distinguished into propositions per se and propositions per accidens. But this was a complicated doctrine, which Kant very conveniently replaced by the distinction between analytic, or explicatory, and synthetic, or ampliative, propositions. Namely, the question is what we are talking about. If we are saying that some imaginable kind of thing does or does not occur in the real world, or even in any well-established world of fiction (as when we ask whether Hamlet was mad or not), then the proposition is synthetic. But when we are merely saying that such and such a verbal combination does or does not represent anything that can find a place in any self-consistent supposition, then, we are either talking nonsense, as when we say, "A woolly horse would be a horse," or else, we are, as Kant says,†1 expressing a result of inward experimentation and observation, as when I say, "Probability essentially involves the supposition that certain general conditions are fulfilled many times and that in the long run a specific circumstance accompanies them in some definite proportion of the occurrences." If such a proposition is true and we substitute for the subject what that subject means, the proposition is reduced to an identical proposition, or in Kantian terminology an empty form of judgment. But the real sense of it lies in its being only just now seen that such is the meaning of the subject, that subject having hitherto been obscurely apprehended. Peirce: CP 4.44 Cross-Ref:†† 44. Categorical propositions are further divided into affirmative and negative propositions. A negative is one which has the particle of exclusion, not, or other than attached to the copula. There is a confusing distinction between a negative proposition and an infinite, that is, an indefinite one. The former is like homo non est equis, the latter like homo est non equis. That is the negative does not imply the existence of the subject, while the affirmative does imply this. But this arrangement, as will be shown in another chapter,†2 greatly complicates the description of correct reasonings. For analytical propositions, though affirmative, cannot, as analytical, assert the real existence of anything.†1 Peirce: CP 4.45 Cross-Ref:†† 45. Ratiocination is defined by St. Thomas †2 as the operation by which reason proceeds from the known to the unknown. Inferences are of two kinds: the necessary and the probable. There are in either case (such is the traditional opinion which will be modified in this work †3) certain propositions called premisses laid down and granted; and these render another proposition, called the conclusion either necessary or probable, as the case may be. The conclusion is sometimes said to be collected from the premisses. It is also said to follow from them. The proposition that from such premisses such a conclusion follows, that is, is rendered necessary or probable, is called the logical rule, dictum, law, or principle. A necessary inference from a single premiss is called an immediate inference, from two premisses a

syllogism, from more than two a sorites. The massing of a number of premisses into one conjunctive proposition, which, in general consonance with the doctrine of immediate inference, might be considered as the inference of the conjunctive proposition from its members, though it is not so conceived traditionally, is conveniently called by Whewell †4 a colligation. It is plain that colligation is half the battle in ratiocination.†5 Peirce: CP 4.45 Cross-Ref:†† It may be mentioned that Scotus (Duns, of course, for Scotus Erigena was not a scholastic) and the later scholastics usually dealt, not with the Syllogism, but with an inferential form called a consequence. The consequence has only one expressed premiss, called the antecedent; its conclusion is called the consequent; and the proposition which asserts that in case the antecedent be true, the consequent is true, is called the consequence. . . . Peirce: CP 4.46 Cross-Ref:†† 46. Logic ought, for the realization of its germinal idea, to be l'art de penser. L'art de penser! What a sublime conception. A school to which an age can turn and here learn the most efficient method of solving its theoretical problems! Such is the idea of logic; but it manifestly asks that the logician should be head and shoulders above his age. That is not at all impossible. There are such men by the dozen in every age. Unfortunately, that is not enough. The man must not only live in realms of thought far removed from that of his fellow-citizens, and really be vastly their intellectual superior, but he must also be recognized as such; and that is a combination of events which hardly ever has happened. Aristotle, alone, by the extraordinary chance of adding to his vast powers, inherited wealth, and the close friendship of two kings the most powerful in the world, and both of them, men of gigantic intellect, came near to that ideal. That logic should really teach an age to think must be confessed impracticable. Let it aspire in each age to register the highest method of thinking to which that age actually attains, and it will be doing all that can be expected. This calls for the best minds. But in few ages has even this been done. The logicians instead of generally riding on the crest of the thought-wave, have, three-quarters waterlogged, drifted wherever the motion of thought was least. . . .

Peirce: CP 4.47 Cross-Ref:†† §3 THE NATURE OF INFERENCE

47. We now come to the proper subject of this chapter. What is the nature of inference? What says the traditional syllogism? That an inference consists of a colligation of propositions which if true render certain or probable another collected proposition. If, to get to the bottom of the matter, we ask what is the nature of a proposition, traditional logic tells us, that it consists of terms -- two terms, usually connected together by another kind of sign, a copula. Peirce: CP 4.47 Cross-Ref:†† This is tolerably explicit, and, so far, good. Peirce: CP 4.48 Cross-Ref:†† 48. The next question, in order, which we put to the traditional logic, is, how do you know that all that is true? to divide the question, tell us, first, how you know

that that analysis of the nature of assertion is correct. Peirce: CP 4.48 Cross-Ref:†† To this, the traditional logic has not one traditional word to say. It is perfectly plain, however, that the reason it thinks so, is that that seems a satisfactory analysis of a sentence. So it is of the majority of sentences in the Greek, Latin, English, German, French, Italian, Spanish, languages -- in short, in the Indo-European languages; and European grammarians, true children of Procrustes, manage to exhibit sentences in other languages forced into the same formula.†P1 But outside of that family of languages which bears somewhat the same relation to language in general as the phanerogams do to all plants, or the vertebrates to all animals -- while there are of course proper names -- it seems to me that general terms, in the logical sense, do not exist.†1 That the analysis of the proposition into subject and predicate represents tolerably the way we, Arians, think, I grant; but I deny that it is the only way to think. It is not even the clearest way nor the most effective way. Peirce: CP 4.49 Cross-Ref:†† 49. There appear to be very many languages in which the copula is quite needless. In the Old Egyptian language, which seems to come within earshot of the origin of speech, the most explicit expression of the copula is by means of a word, really the relative pronoun, which. Now to one who regards a sentence from the Indo-European point of view, it is a puzzle how "which" can possibly serve the purpose in place of "is." Yet nothing is more natural.†2 The fact that hieroglyphics came so easy to the Egyptians shows how their thought is pictorial. . . . [e.g.] "Aahmes what we write of is a soldier which what we write of is overthrown," means "Aahmes the soldier is overthrown." Are you on the whole quite sure that this is not the most effective way of analyzing the meaning of a proposition? Peirce: CP 4.50 Cross-Ref:†† 50.†3 Take, now, the other part of the question, namely, supposing the nature of assertion to be understood, what is the relation of inference to assertion, according to the traditional logic? Here we find a marked difference between the view taken down to A. D. 1300 or 1325 and the view which then gradually gained ground and became universal considerably before A. D. 1600, and remained so until long after A. D. 1800. After 250 years of contest in which it was always gaining ground, it remained for 250 years more in unchallenged possession of the field. The opinion referred to is nominalism. Ockham revived it. By the time the universities were reformed in the sixteenth century, it had gained a complete victory. Descartes, Leibniz, Locke, Hume, and Kant, the great landmarks of philosophical history, were all pronounced nominalists. Hegel first advocated realism; and Hegel unfortunately was about at the average degree of German correctness in logic. The author of the present treatise is a Scotistic realist. He entirely approved the brief statement of Dr. F. E. Abbott in his Scientific Theism that Realism is implied in modern science. In calling himself a Scotist, the writer does not mean that he is going back to the general views of 600 years back; he merely means that the point of metaphysics upon which Scotus chiefly insisted and which has since passed out of mind, is a very important point, inseparably bound up with the most important point to be insisted upon today. The author might with more reason, call himself a Hegelian; but that would be to appear to place himself among a known band of thinkers to which he does not in fact at all belong, although he is strongly drawn to them. Peirce: CP 4.51 Cross-Ref:†† 51. How, then, does Kant regard the apodictic inference? He holds that the

conclusion is thought in the premisses although indistinctly. That that is Kant's view could be shown in a few words. But let us rather listen to his general tone in talking of reasoning. In the Critic of The Pure Reason, Transcendental Dialectic, Introduction, Section II, Subsection B, [A303, B359] he speaks of the logical employment of the Reason, as follows: Peirce: CP 4.51 Cross-Ref:†† "A distinction is usual between things known immediately and things merely inferred. That in a figure bounded by three straight lines, there are three angles is known immediately; that the sum of these angles equals two right angles is a thing inferred. [When Kant wrote this no step in the modern revival of graphical geometry had been made. That three rays in a plane have three intersections, which, without any two rays coinciding, may reduce to one, is a theorem of graphics. But Kant confounds this proposition with another, namely, that if three lines, straight or not, enclose a space on a surface, those three lines must have at least three intersections. This is a corollary from the Census theorem of topology. That the sum of the three angles of the triangle equals two right angles, depends, as Lambert had clearly explained, before Kant wrote, upon a particular system of measurement which, however much it may be recommended by what we observe in nature, is not the only admissible system of measurement. Thus, what Kant says is immediately known, is fairly demonstrable; but what he says is demonstrable, is not so. This is not merely true in this case, but would be true of any example which Kant would feel to be a good one. It casts suspicion, at once, upon what he has to say, which has been the result of his generalizations of such examples]. Having an incessant need of inferring we become so accustomed to it, that at last the distinction spoken of escapes us. Even so called deceptions of the senses, where evidently it is the inferences that are at fault, we take for immediate perceptions. In every inference, there is one initial proposition, another, the consequent, which is drawn from it, and finally there is the consequence, or proposition according to which the truth of the consequent invariably accompanies the truth of the antecedent. [This is the doctrine of consequentia which is so extensively employed by philosophers of the fourteenth and fifteenth centuries.] If the concluded judgment is so contained in the initial judgment, that it can be derived without the intervention of any third idea, the consequence is called immediate. [This well-known term Kant would find in Wolff.] I would rather term it an Understanding-consequence. [This Kant seems to think an original idea, but that such a consequence was not an argument was the established doctrine.] But in case, besides the knowledge assigned as reason, still another judgment be needful, in order to draw the conclusion, the inference is called a Reason-inference. In the proposition "All men are mortal" is contained the propositions, "Some men are mortal," "Some mortals are men," "No immortal is a man." These, therefore, follow immediately from that. On the other hand, the proposition "All savans are mortal" is not contained in our assumed judgment (which does not contain the notion of savan), so that this proposition cannot be deduced from that other without a mediating judgment." [This is a slipshod analysis. Kant, out of his well-founded contempt for the scholastic method of trying to answer real questions by drawing distinctions, was led virtually to put the stamp of his condemnation upon all accurate thought. "Subtleties," he often says, "may sharpen the wits, but they are of no use at all."†P1 That was a very unfortunate opinion, which encouraged the down-at-the-heels, slouchy sort of logic to which Germans were prone enough and which has disgraced that country. To return to the present case, why does Kant consider only one kind of enthymeme and not another? Suppose the consequence to be the following -- which represents an

argument actually used by Kant against Boscovich -All particles are bodies; Ergo, All particles are extended. Will Kant tell us there is any idea contained in this consequent not contained in its antecedent? Not so: he himself says,†1 "I need not go beyond the notion connected with the noun body to find that extension belongs to it." Will he, then, say that the consequence is no argument? It is put forward as such by himself; and such a doctrine would be a novelty in the traditional logic, with which he professes himself eminently satisfied, which were it involved in his doctrine, he certainly ought to have called attention to. But this example shows that in Kant's opinion the conclusion of a complete and perfect argumentation is implicitly contained in its premisses.] Peirce: CP 4.52 Cross-Ref:†† 52. "With the explanation of synthetical Knowledge," says Kant [Analytic of Principles, Chapter 2, Section 2, [A154, B193] of the highest principles of all synthetical judgments]," general logic has absolutely nothing to do." The reason is obvious. Reasoning, according to the doctrine of that work, is regulated entirely by the principle of contradiction, which is the principle of analytical thought. The one law of demonstrative reasoning is that nothing must be said in the conclusion which is not implied in the premisses, that is, nothing must be said in the conclusion, not actually thought in the premisses, though not so clearly and consciously.†P1 The proposition that that is actually thought, though somewhat unconsciously, which is implicitly contained in what is thought, is absurd enough; but it is a psychological absurdity which may perhaps be passed over in logic. If that be true, nobody can tell by the most attentive introspection, what he thinks. For it will not be maintained that by carefully considering the few and simple premisses of the theory of numbers -- by just contemplating these propositions ever so nicely -- one could even discover the truth of Fermat's theorems. It would be impossible to adduce a single instance of the discovery of anything deserving the name of a mathematical theorem by any such means. Every mathematical discoverer knows very well that that is no way to succeed. If the implied proposition be thought, it is thought in some cryptic sense, and it in no wise tells us how it is that inference is performed, to say that in such sense the conclusion is thought as soon as the premisses are given. The distinction between analytical and synthetical judgments represents this conception of reasoning. The distinction may approximate to a just and valuable distinction; but it cannot be accepted as accurately defined. . . . Peirce: CP 4.53 Cross-Ref:†† 53.†1 A belief is a habit; but it is a habit of which we are conscious. The actual calling to mind of the substance of a belief, not as personal to ourselves, but as holding good, or true, is a judgment. An inference is a passage from one belief to another; but not every such passage is an inference. If noticing my ink is bluish, I cast my eye out of the window and my mind being awakened to color remark particularly a poppy, that is no inference. Or if without casting my eye out of the window, I call to mind the green tinge of Niagara or the blue of the Rhone, that is no inference. In inference one belief not only follows after another, but follows from it. Peirce: CP 4.54 Cross-Ref:†† 54. What does that mean? The proper method of finding the answer to this question is to compare pairs of beliefs which differ as little as possible except in that

in one pair one belief follows from the other and in the other pair only follows after it; and then note what practical difference, or difference that might become practical, there is between those two pairs. . . . Peirce: CP 4.55 Cross-Ref:†† 55. I think the upshot of reflection will be this. If a belief is produced for the first time directly after a judgment or colligation of judgments and is suggested by them, then that belief must be considered as the result of and as following from those judgments. The idea which is the matter of the belief is suggested by the idea in those judgments according to some habit of association, and the peculiar character of believing the idea really is so, is derived from the same element in the judgments. Thus, inference has at least two elements: the one is the suggestion of one idea by another according to the law of association, while the other is the carrying forward of the asserting element of judgment, the holding for true, from the first judgment to the second. That these two things suffice [to] constitute inference I do not say. . . .†1 Peirce: CP 4.56 Cross-Ref:†† 56.†2 Let us now inquire in what the assertory element of a judgment consists. What is there in an assertion which makes it more than a mere complication of ideas? What is the difference between throwing out the word speaking monkey, and averring that monkeys speak, and inquiring whether monkeys speak or not? This is a difficult question. Peirce: CP 4.56 Cross-Ref:†† In the first place, it is to be remarked that the first expression signifies nothing. The grammarians call it an "incomplete speech." But, in fact, it is no speech at all. As well call the termination ability -- or ationally an incomplete speech. It is also to be remarked that the number of languages in which such an expression is possible is very small. In most languages that have nouns and adjectives, the participial adjective follows the noun and when left without other words the combination would mean the monkey is speaking. Peirce: CP 4.56 Cross-Ref:†† In such languages you can't say "speaking monkey," and surely it is no defect in them; for after it is said, it is pure nonsense. . . . There are more than a dozen different families of languages, differing radically in their manner of thinking; and I believe it is fair to say that among these the Indo-European is only one in which words which are distinctively common nouns are numerous. And since a noun or combination of nouns by itself says nothing, I do not know why the logician should be required to take account of it at all. Even in Indo-European speech the linguists tell us that the roots are all verbs. It seems that, speaking broadly, ordinary words in the bulk of languages are assertory. They assert as soon as they are in any way attached to any object. If you write GLASS upon a case, you will be understood to mean that the case contains glass. It seems certainly the truest statement for most languages to say that a symbol is a conventional sign which being attached to an object signifies that that object has certain characters. But a symbol, in itself, is a mere dream; it does not show what it is talking about. It needs to be connected with its object. For that purpose, an index is indispensable. No other kind of sign will answer the purpose. That a word cannot in strictness of speech be an index is evident from this, that a word is general -- it occurs often, and every time it occurs, it is the same word, and if it has any meaning as a word, it has the same meaning every time it occurs; while an index is essentially an affair of here and now, its office being to bring the thought to a

particular experience, or series of experiences connected by dynamical relations. A meaning is the associations of a word with images, its dream exciting power. An index has nothing to do with meanings; it has to bring the hearer to share the experience of the speaker by showing what he is talking about. The words this and that are indicative words. They apply to different things every time they occur. Peirce: CP 4.56 Cross-Ref:†† It is the connection of an indicative word to a symbolic word which makes an assertion. Peirce: CP 4.57 Cross-Ref:†† 57.†1 The distinction between an assertion and an interrogatory sentence is of secondary importance. An assertion has its modality, or measure of assurance, and a question generally involves as part of it an assertion of emphatically low modality. In addition to that, it is intended to stimulate the hearer to make an answer. This is a rhetorical function which needs no special grammatical form. If in wandering about the country, I wish to inquire the way to town, I can perfectly do so by assertion, without drawing upon the interrogative form of syntax. Thus I may say, "This road leads, perhaps, to the city. I wish to know what you think about it." The most suitable way of expressing a question would, from a logical point of view, seem to be by an interjection: "This road leads, perhaps, to the city, eh?" Peirce: CP 4.58 Cross-Ref:†† 58. An index, then, is quite essential to a speech and a symbol equally so. We find in grammatical forms of syntax, a part of the sentence particularly appropriate to the index, another particularly appropriate to the symbol. The former is the grammatical subject, the latter the grammatical predicate. In the logical analysis of the sentence, we disregard the forms and consider the sense. Isolating the indices as well as we can, of which there will generally be a number, we term them the logical subjects, though more or less of the symbolic element will adhere to them unless we make our analysis more recondite than it is commonly worth while to do; while the purely symbolic parts, or the parts whose indicative character needs no particular notice, will be called the logical predicate. As the analysis may be more or less perfect -- and perfect analyses are very complicated -- different lines of demarcation will be possible between the two logical members.†1 In the sentence "John marries the mother of Thomas," John and Thomas are the logical subjects, marries-the-mother-of- is the logical predicate. . . . Peirce: CP 4.59 Cross-Ref:†† 59. In making general assertions it is not possible directly to indicate anything but the real world, or whatever world discourse may refer to. But it is necessary to give a general direction as to the manner in which an object intended may be found. Especially it is necessary to be able to say that any object whatever will answer the purpose, in which case the subject is said to be universal, and to be able to say that a suitable object occurs, in which case the subject is said to be particular. Peirce: CP 4.60 Cross-Ref:†† 60. If there are several subjects, some universal and some particular, it makes a difference in what order the selections of a universal and of a particular subject are made. For example, the four following statements are different: 1. Take any two things, A and C; then a thing, B, can be so chosen that if A and C are men, B is a man praised by A to C.

2. Take anything, A; then a thing, B, can be so chosen, that whatever third thing, C, be taken, if A and C are men, B is a man praised by A to C. 3. Take anything, C; then a thing, B, can be so chosen, that whatever third thing, A, be taken, if A and C are men, B is a man praised by A to C. 4. A thing, B, can be so chosen that whatever things A and C may be, if A and C are men, B is a man praised by A to C. Peirce: CP 4.60 Cross-Ref:†† We should usually express these as follows: 1. Every man praises some man or other to each man. 2. Every man praises some man to all men. 3. To every man some man is praised by all men. 4. There is a man whom all men praise to all men.†1 Peirce: CP 4.61 Cross-Ref:†† 61. . . . When we busy ourselves to find the answer to a question, we are going upon the hope that there is an answer, which can be called the answer, that is, the final answer. It may be there is none. If any profound and learned member of the German Shakespearian Society were to start the inquiry how long since Polonius had had his hair cut at the time of his death, perhaps the only reply that could be made would be that Polonius was nothing but a creature of Shakespeare's brain, and that Shakespeare never thought of the point raised. Now it is certainly conceivable that this world which we call the real world is not perfectly real but that there are things similarly indeterminate. We cannot be sure that it is not so. In reference, however, to the particular question which at any time we have in hand, we hope there is an answer, or something pretty close to an answer, which sufficient inquiry will compel us to accept. Peirce: CP 4.62 Cross-Ref:†† 62. Suppose our opinion with reference to a given question to be quite settled, so that inquiry, no matter how far pushed, has no surprises for us on this point. Then we may be said to have attained perfect knowledge about that question. True, it is conceivable that somebody else should attain to a like "perfect knowledge," which should conflict with ours. He might know something to be white, which we should know was black. This is conceivable; but it is not possible, considering the social nature of man, if we two are ever to compare notes; and if we never do compare notes, and no third party talks with both and makes the comparison, it is difficult to see what meaning there is in saying we disagree. When we come to study the principle of continuity †1 we shall gain a more ontological conception of knowledge and of reality; but even that will not shake the definition we now give. Peirce: CP 4.63 Cross-Ref:†† 63. Perhaps we may already have attained to perfect knowledge about a number of questions; but we cannot have an unshakable opinion that we have attained such perfect knowledge about any given question. That would be not only perfectly to know, but perfectly to know that we do perfectly know, which is what is called sure knowledge. No doubt, many people opine that they surely know certain things; but after they have read this book, I hope many of them will be led to see that that opinion

is not unshakable. At any rate, as they are, after all, in some measure reasonable beings, no matter how pig-headed they might be (I am only saying that pigheaded people exist, not that they are very frequently met with among my opponents), after a time, if they live long enough, reason must get the better of obstinate adherence to their opinion, and they must come to see that sure knowledge is impossible. Peirce: CP 4.64 Cross-Ref:†† 64. Nevertheless, in every state of intellectual development and of information, there are things that seem to us sure, because no little ingenuity and reflection is needed to see how anything can be false which all our previous experience seems to support; so that even though we tell ourselves we are not sure, we cannot clearly see how we fail of being so. Practically, therefore, life is not long enough for a given individual to rake up doubts about everything; and so, however strenuously he may hold to the doctrine of catalepsy, he will practically treat one proposition and another as certain. This is a state of practically perfect belief. Peirce: CP 4.65 Cross-Ref:†† 65. We have now to define the five words necessary, unnecessary, possible, impossible, and contingent. But first let me say that I use the word information to mean a state of knowledge, which may range from total ignorance of everything except the meanings of words up to omniscience; and by informational I mean relative to such a state of knowledge. Thus, by "informationally possible," I mean possible so far as we, or the persons considered, know. Then, the informationally possible is that which in a given information is not perfectly known not to be true. The informationally necessary is that which is perfectly known to be true. The informationally contingent, which in the given information remains uncertain, that is, at once possible and unnecessary. Peirce: CP 4.66 Cross-Ref:†† 66. The information considered may be our actual information. In that case, we may speak of what is possible, necessary, or contingent, for the present. Or it may be some hypothetical state of knowledge. Imagining ourselves to be thoroughly acquainted with all the laws of nature and their consequences, but to be ignorant of all particular facts, what we should then not know not to be true is said to be physically possible; and the phrase physically necessary has an analogous meaning. If we imagine ourselves to know what the resources of men are, but not what their dispositions and desires are, what we do not know will not be done is said to be practically possible; and the phrase practically necessary bears an analogous signification. Thus, the possible varies its meaning continually. We speak of things mathematically and metaphysically possible, meaning states of things which the most perfect mathematician or metaphysician does not qua mathematician or metaphysician know not to be true. Peirce: CP 4.67 Cross-Ref:†† 67. There are two meanings of the words possible and necessary which are of special interest to the logician more than to other men. These refer to the states of information in which we are supposed to know nothing, except the meanings of words, and their consequences, and in which we are supposed to know everything. These I term essential and substantial possibility, respectively: and of course necessity has similar varieties. That is essentially or logically possible which a person who knows no facts, though perfectly au fait at reasoning and well-acquainted with the words involved, is unable to pronounce untrue. The essentially or logically

necessary is that which such a person knows is true. For instance, he would not know whether there was or was not such an animal as a basilisk, or whether there are any such things as serpents, cocks, and eggs; but he would know that every basilisk there may be has been hatched by a serpent from a cock's egg. That is essentially necessary; because that is what the word basilisk means. On the other hand, the substantially possible refers to the information of a person who knows everything now existing, whether particular fact or law, together with all their consequences. This does not go so far as the omniscience of God; for those who admit Free-Will suppose that God has a direct intuitive knowledge of future events even though there be nothing in the present to determine them. That is to say, they suppose that a man is perfectly free to do or not do a given act; and yet that God already knows whether he will or will not do it. This seems to most persons flatly self-contradictory; and so it is, if we conceive God's knowledge to be among the things which exist at the present time. But it is a degraded conception to conceive God as subject to Time, which is rather one of His creatures. Literal fore-knowledge is certainly contradictory to literal freedom. But if we say that though God knows (using the word knows in a trans-temporal sense) he never did know, does not know, and never will know, then his knowledge in no wise interferes with freedom. The terms, substantial necessity and substantial possibility, however, refer to supposed information of the present in the present, including among the objects known all existing laws as well as special facts. In this sense, everything in the present which is possible is also necessary, and there is no present contingent. But we may suppose there are "future contingents." Many men are so cocksure that necessity governs everything that they deny that there is anything substantially contingent. But it will be shown in the course of this treatise that they are unwarrantably confident, that wanting omniscience we ought to presume there may be things substantially contingent, and further that there is overwhelming evidence that such things are. . . .†1 Peirce: CP 4.68 Cross-Ref:†† 68. To conclude from the above definitions that there is nothing analogous to possibility and necessity in the real world, but that these modes appertain only to the particular limited information which we possess, would be even less defensible than to draw precisely the opposite conclusion from the same premisses. It is a style of reasoning most absurd. Unfortunately, it is so common, that the moment a writer sets down these definitions nine out of ten critics will set him down as a nominalist. The question of realism and nominalism, which means the question how far real facts are analogous to logical relations, and why, is a very serious one, which has to be carefully and deliberately studied, and not decided offhand, and not decided on the ground that one or another answer to it is "inconceivable." Nothing is "inconceivable" to a man who sets seriously about the conceiving of it.†1 There are those who believe in their own existence, because its opposite is inconceivable; yet the most balsamic of all the sweets of sweet philosophy is the lesson that personal existence is an illusion and a practical joke. Those that have loved themselves and not their neighbors will find themselves April fools when the great April opens the truth that neither selves nor neighborselves were anything more than vicinities; while the love they would not entertain was the essence of every scent.†2 Peirce: CP 4.69 Cross-Ref:†† 69. A leading principle of inference which can lead from a true premiss to a false conclusion is insofar bad; but insofar as it can only lead either from a false premiss or to a true conclusion, it is satisfactory; and whether it leads from false to false, from true to true, or from false to true, it is equally satisfactory. The first part of

this theorem, that an inference from true to false is bad, [follows] from the essential characteristic of truth, which is its finality. For truth being our end and being able to endure, it can only be a false maxim which represents it as destroying itself. Indeed, I do not see how anybody can fail to admit that (other things being equal) it is a fault in a mode of inference that it can lead from truth to falsity. But it is by no means as evident that an inference from false to false is as satisfactory as an inference from true to true; still less, that such a one is as satisfactory as an inference from false to true. The Hegelian logicians seem to rate only that reasoning A1 which setting out from falsity leads to truth. But men of laboratories consider those truths as small that only an inward necessity compels. It is the great compulsion of the Experience of nature which they worship. On the other hand, the men of seminaries sneer at nature; the great truths for them are the inward ones. Their god is enthroned in the depths of the soul. How shall we decide the question? Let us rationally inquire into it, subordinating personal prepossessions in view of the fact that whichever way these prepossessions incline, we can but admit that wiser men than we, more sober-minded men than we, and humbler searchers after truth, do today embrace the opinion the opposite of our own. How, then, shall we decide the question? Yes, how to decide questions is precisely the question to be decided. One thing the laboratory-philosophers ought to grant: that when a question can be satisfactorily decided in a few moments by calculation, it would be foolish to spend much time in trying to answer it by experiment. Nevertheless, this is just what they are doing every day. The wisest-looking man I ever saw, with a vast domelike cranium and a weightiness of discourse that left Solon in the distance, once spent a month or more in dropping a stick on the floor and seeing how often it would fall on a crack; because that ratio of frequency afforded a means of ascertaining the value of {p}, though not near so close as it could be calculated in five minutes; and what he did it for was never made clear. Perhaps it was only for relaxation; though some people might have found reading Goldsmith or Voltaire fully as lively an occupation. If it were not for the example of this distinguished LL.D., I should have ventured to say that nothing is more foolish than carrying a question into a laboratory until reflection has done all that it can do towards clearing it up -- at least, all that it can do for the time being. Of course, for a seminary-philosopher, to send a question to the laboratory is to have done with it, to which he naturally has a reluctance; while the laboratory-philosopher is impatient to get a whack at it. Peirce: CP 4.70 Cross-Ref:†† 70. Suppose that, at any rate, we try applying this maxim of methodology to the question now in hand. Then the first thing that has to be remarked is that every inference proceeds according to a general rule -- and that, a comprehended rule -- so that in the very act of drawing it the reasoner thinks of there being other similar inferences to be drawn. For unless the premiss determines the conclusion according to a rule, there is no intelligible meaning in saying that it determines it at all; unless, indeed, we are prepared to say that the conclusion feels compelled but knows not how; and if it knows not how, how can it know it was the premiss which compelled it? But a conclusion is not only determined by the premiss, but rationally determined, and that implies that in drawing said conclusion we feel we are following a rule and a comprehensible rule. . . . Peirce: CP 4.71 Cross-Ref:†† 71. Descartes marks the period when Philosophy put off childish things and began to be a conceited young man. By the time the young man has grown to be an old man, he will have learned that traditions are precious treasures, while iconoclastic

inventions are always cheap and often nasty. He will learn that when one's opinion is beseiged and one is pushed by questions from one reason to another behind it, there is nothing illogical in saying at last, "Well, this is what we have always thought; this has been assumed for thousands of years without inconvenience." The childishness only comes in when tradition, instead of being respected, is treated as something infallible before which the reason of man is to prostrate itself, and which it is shocking to deny. In 1637, Descartes (aged 41) published his first work on philosophy, the Discours de la méthode pour bien conduise sa raison et chercher la verité dans les sciences. In the fourth part of this dissertation, after insisting upon the doubtfulness of everything, even the simplest propositions of mathematics, in a strain quite familiar to readers of the present work, he goes on to say how at one time "je me résolus de feindre que toutes les choses qui m'étoient jamais entrées en esprit n'étoient non plus vraies que les illusions de mes songes." Thereupon follows the grand passage: "Mais aussitôt après je pris garde que, pendant que je voulais ainsi penser que tout étoit faux, il falloit nécessairement que moi qui le pensois susse quelque chose; et, remarquant que cette vérité: je pense, donc je suis, étoit si ferme et si assurée que toutes les plus extravagantes suppositions des sceptiques n'étoient pas capable de l'ébranler, je jugeai que je pouvois la recevoir sans scrupule pour le premier principe de la philosophie que je cherchois." Peirce: CP 4.71 Cross-Ref:†† Descartes thought this "très-clair"; but it is a fundamental mistake to suppose that an idea which stands isolated can be otherwise than perfectly blind. He professes to doubt the testimony of his memory; and in that case all that is left is a vague indescribable idea. There is no warrant for putting it into the first person singular. "I think" begs the question. "There is an idea: therefore, I am," it may be contended represents a compulsion of thought; but it is not a rational compulsion. There is nothing clear in it. Here is a man who utterly disbelieves and almost denies the dicta of memory. He notices an idea, and then he thinks he exists. The ego of which he thinks is nothing but a holder together of ideas. But if memory lies there may be only one idea. If that one idea suggests a holder-together of ideas, how it can do so is a mystery. To make the reflection that many of the things which appear certain to us are probably false, and that there is not one which may not be among the errors, is very sensible. But to make believe one does not believe anything is an idle and self-deceptive pretence. Of the things which seem to us clearly true, probably the majority are approximations to the truth. We never can attain absolute certainty; but such clearness and evidence as a truth can acquire will consist in its appearing to form an integral unbroken part of the great body of truth. If we could reduce ourselves to a single belief, or to only two or three, those few would not appear reasonable or clear. Peirce: CP 4.72 Cross-Ref:†† 72. Now, then, how is truth to be inferred from falsehood? First, it may happen accidentally, from the falsehood that Alexander the Great was the great-grandson of Benjamin Franklin it may be inferred there lived a great-grandson of Benjamin Franklin named Alexander, which happens to be true. It cannot be considered as a merit of a rule that its results accidentally have any character; for an accidental result ex vi termini is not determined by the rule. Secondly, truth may follow from falsehood because no lie is altogether false. Every precept of inference which does not lead from truth to falsity, must sometimes lead from falsity to truth. For let A be a true premiss and B a conclusion from it according to such a precept. Then B must be true. But if we add to A something false, B will follow from it just the same. A mode of inference may accordingly infer a larger proportion of true

conclusions from false premisses than another simply by inferring less. But concluding falsehood from falsehood is by no means useless, provided it follows a precept which cannot conclude falsehood from truth. For it hastens the detection and rejection of the falsity. Consequently of two modes of inference neither ever leading from truth to falsity, one of which infers something false from a false premiss from which the other infers something true, the former is rather to be preferred because it infers more. Suppose for instance it is false that the sides of a triangle measure 4 inches, 5 inches, and 6 inches, then the rule of inference which deduces for the area 15/4 √7 square inches is certainly superior to a rule of inference which only concludes that the area is finite. Thirdly, truth may follow from falsehood because that falsehood is impossible and refutes itself. But in this way, only what is logically necessary can be inferred, that is only what a person ought to know independently of any particular premisses. As this is a mode of inference which infers less than any other, its value is the least that any mode can have which never leads from truth to falsity. Many persons will be inclined to dispute this, and will point to the utility of the reductio ad absurdum in geometry. But the reductio ad absurdum is not a method of inferring truth from falsity; it is only a form of statement of an inference from truth to truth. . . .†1 Peirce: CP 4.72 Cross-Ref:†† Now it may be that everything is so bound up with everything else that to understand perfectly any single fact, as it really is, would involve a knowledge of all facts. But this is not admitting that from any proposition, understood as it is understood, and not as the reality it represents ought to be understood, much can be inferred; far less that valuable truth can be deduced from falsehood. Peirce: CP 4.72 Cross-Ref:†† It thus appears that the inference of truth from falsity is never so valuable as when it is accidental, in which case its value is precisely the same as that of an inference from false to false. Peirce: CP 4.73 Cross-Ref:†† 73. The inference from true to true has precisely the same value as that from false to false. For to infer B from A involves inferring the falsity of A from the falsity of B. The two inferences are inseparable; when either is made the other is made. Now if either of these is an inference from truth to truth, the other is an inference from falsity to falsity; and conversely, if either is an inference from false to false, the other is an inference from true to true. Accordingly it is impossible to set different values upon the two modes of inference. Peirce: CP 4.74 Cross-Ref:†† 74. Leading principles are of two classes: those whose pretension it is to lead always to the truth unless from the false, and never astray; and those which only profess to lead toward the truth in the long run. This distinction separates two great branches of reasoning, the one bringing to light the dark things of the hidden recesses of the soul, the other those hidden in nature. We may, for the present, call them Imaginative and Experiential reasoning; or reasoning by diagrams and reasoning by experiments.†1 Peirce: CP 4.75 Cross-Ref:†† 75. . . . The necessity for a sign directly monstrative of the connection of premiss and conclusion is susceptible of proof. That proof is as follows. When we contemplate the premiss, we mentally perceive that that being true the conclusion is

true. I say we perceive it, because clear knowledge follows contemplation without any intermediate process. Since the conclusion becomes certain, there is some state at which it becomes directly certain. Now this no symbol can show; for a symbol is an indirect sign depending on the association of ideas. Hence, a sign directly exhibiting the mode of relation is required. This promised proof presents this difficulty: namely, it requires the reader actually to think in order to see the force of it. That is to say, he must represent the state of things considered in a direct imaginative way. Peirce: CP 4.76 Cross-Ref:†† 76. A large part of logic will consist in the study of the different monstrative signs, or icons, serviceable in reasoning. Peirce: CP 4.76 Cross-Ref:†† Suppose we reason

Enoch was a man, Then, Enoch must have died.

Let this reasoning be called in question, and the reasoner searches his mind to discover the leading principle which actuated it. He finds this in the truth (as he assumes it to be) that

Every man dies.

He now repeats his reasoning, joining this proposition to the premiss previously assumed, to make the compound premiss,

Enoch was a man, and every man dies.

This may be otherwise stated thus:

If we are talking of Enoch, what we are talking of is a man; and if we are talking of a man, what we are talking of dies.

The conclusion is

If we are talking of Enoch, what we are talking of dies.

Or we may state it thus:

From being Enoch follows being a man, and from being a man follows being subject to death; Hence, from being Enoch follows being subject to death.

Peirce: CP 4.76 Cross-Ref:†† If this reasoning is called in question, the reasoner searches his mind for the leading principle and may state it thus:

If one truth, A, makes another truth, B, certain, and if this truth, B, makes a third truth, C, certain; then, the truth, A, makes the truth, C, certain.

Peirce: CP 4.76 Cross-Ref:†† This is the logical principle called the Nota notæ, because one statement of it is, nota notæ est nota rei ipsius.†1 Peirce: CP 4.76 Cross-Ref:†† Now shall the reader add this as a premiss to the compound premiss already adopted? He gains nothing by doing so. For he cannot reason at all without a monstrative sign of illation; and this sign is not really monstrative unless it makes clear the proposition here proposed to be abstractly stated. Nor could any use of that statement be made without using the truth which it expresses. Peirce: CP 4.76 Cross-Ref:†† That if the fact A is certain evidence of the fact B and the fact B is certain evidence of the fact C, then the fact A is certain evidence of the fact C, appears to us perfectly clear. That appearance of evidence may be an argument that the proposition is probably about true; for our instincts are generally pretty well adapted to their ends. But its appearing clear will not prevent our reflecting that things that seem evident are often found to be mistakes, so that it may be the proposition is not true. Peirce: CP 4.77 Cross-Ref:†† 77. Now although the reader does not really doubt that the proposition is true, it may be instructive to feign such a doubt, and see what the nature of the source of knowledge is. Peirce: CP 4.77 Cross-Ref:†† A common form of the maxim is this: The word mortal is applicable to everything to which the word man is applicable, and the word man is applicable to everything to which the word Enoch is applicable. Hence, the word mortal is applicable to everything to which the word Enoch is applicable. This mode of representing the matter is embodied in a maxim called the Dictum de omni:†1 if A is in any relation to all to which B is in the same relation, and if B is in this relation to all to which C is in this relation, then A is in this relation to all to which C is in this relation; that is, if the things to which A is applicable are wholly included among the

things to which B is applicable, and the things to which B is applicable are wholly included among the things to which C is applicable, then the things to which A is applicable are wholly included among the things to which C is applicable. Peirce: CP 4.77 Cross-Ref:†† Here we have a mental diagram representing receptacles or spaces successively included in one another; and the question of the truth of the maxim may be divided into two parts: Peirce: CP 4.77 Cross-Ref:†† First: Is the maxim certainly true of the mental diagram; and if so how do we know it? Peirce: CP 4.77 Cross-Ref:†† Second: Does the mental diagram represent the relations of truths of nature to one another, in fact? Peirce: CP 4.77 Cross-Ref:†† As to the first question, there would seem to be no reason to doubt that we know it is true of our mental diagram, just as we know of our idea of numbers that 2 and 3 make 5. And no line can be drawn between this case and knowing that √2=1.414213562373095 except that the latter is more complicated. It would thus appear that our certainty about the mental diagram is merely due to our having gone over it many times and being confident we could not be all wrong about a matter so simple. Still, as it is easy to make a mistake in calculating the √2, and that mistake may be repeated, it is barely possible that any conclusion reached in the same way is wrong. Besides, how do I know I am not crazy and am not uttering the greatest absurdity when I enunciate the Nota notæ? Of course, it is not rational for a man to assume that he is utterly irrational. A man cannot be speaking the truth in saying that everything he says is false. For this very thing is one of the things he says; and if this be false then in what it says of itself it is true, and therefore false.†1 But this remark does not clear up the matter; and we shall leave the problem for the present, to return to it later.†2 Peirce: CP 4.77 Cross-Ref:†† As to the second question, it is important to remark that the Nota notæ does not declare that there is any infallible mark of anything, or any rule without exceptions. If, as we have seen, the Nota notæ itself is not absolutely certain, nothing else ought to be so regarded. We cannot go so far as to declare that absolutely no rule is without exceptions; for this declaration is itself a rule. Nor can we say that no rule but this is without exceptions. For this rule either has exceptions or it has not. If it has exceptions and every other rule has exceptions, it has no exceptions. But if it has no exceptions, then in accordance with its declaration it has exceptions. We are thus obliged to admit that there are rules without exceptions, or at least that the denial of it has no sense.†P1 But we ought not to suppose that we can identify any general proposition as being certainly or even probably without exceptions. The case is like the following. We say 1/2 of 1/3 is 1/6. Now we do not really think we can divide anything into precisely equal parts; but we think that, barring the possibility that we have made a mistake in doing the sum, which is excessively improbable, the nearer we can come to 1/2 of 1/3 of anything, that is, to the ideal state of things in our imagination, the nearer we shall come to 1/6 . . . Peirce: CP 4.78 Cross-Ref:††

78. In like manner, it may be nothing in the world precisely conforms to rigidity of our idea of something steady enough to be represented by a sign. The reader has had several examples of insolubilia,†1 as they are called by logicians, that is, cases in which every attempt to reason lands us in absurdity. Here are two more examples. Peirce: CP 4.78 Cross-Ref:†† In order to prove black is white, you have only to say, "Either what I am saying is false or black is white." Is that proposition false? It cannot be so; for it only says that one or other of two things is true; and if either is true the proposition is true. It cannot, therefore, be false; for that is one of the alternatives that it leaves open. The proposition is true, then. Consequently, one of the alternatives is true. But not the first; therefore the second. Hence, black is white. Peirce: CP 4.78 Cross-Ref:†† A man invented an ink containing Vanadium the like of which had never been made before. He was just about to try it for the first time, when a friend asked, "Has anything ever been written in Vanadium ink before?" "No." "Will you please write what I tell you for the first handsel of it?" "Yes." "Very well, here is a folded paper marked 'Exhibit A.' Write: What is written in exhibit A is true." He did so. "Now," said the friend, "do you know you have lied to me?" "Oh, but I only wrote that to please you. I did not mean to say it was true." "Very well; suppose it false. Then, what exhibit A says is false. Now read Exhibit A. It reads: 'Something written in Vanadium ink is false.' If that is false, what you have written must be true." "Good! So much the better!" "Not so fast, if you please. What you have written is, of course, true; and consequently exhibit A is true; and consequently something written in Vanadium ink is false. Now it is not what you have just written, for that is true; and therefore you must have lied when you told me nothing had ever been written with that ink before." Peirce: CP 4.78 Cross-Ref:†† It may be that all the propositions in the world would, if subjected to a dialectical examination, prove thus elusive. But that does not affect the truth of the Nota notæ, which only says that so far as things conform to our idea of successive inclusion, so far (unless we have blundered almost inconceivably) the Nota notæ holds. . . . Peirce: CP 4.79 Cross-Ref:†† 79. The logician does not assert anything, as the geometrician does; but there are certain assumed truths which he hopes for, relies upon, banks upon, in a way quite foreign to the arithmetician. Logic teaches us to expect some residue of dreaminess in the world, and even self-contradictions; but we do not expect to be brought face to face with any such phenomenon, and at any rate are forced to run the risk of it. The assumptions of logic differ from those of geometry, not merely in not being assertorically held, but also in being much less definite.

Peirce: CP 4.80 Cross-Ref:†† III SECOND INTENTIONAL LOGIC†1P

80. Second intentional, or, as I also call it, Objective Logic, is much the larger part of formal logic. It is also the more beautiful and the interesting subject; and in serious significance it is superior in a far higher ratio. But it is highly abstract, remote from the bread and butter of all parties, and to yield to the temptation of going into it would be to forget

That not to know at large of things remote From use, obscure and suttle, but to know That which before us lies in daily life, Is the prime Wisdom, what is more is fume, Or emptiness, or fond impertinence, And renders us in things that most concerne Unpractis'd, unprepar'd, and still to seek.

Peirce: CP 4.81 Cross-Ref:†† 81. Second intentional logic treats at length of the properties of logical conceptions. First, come such simple relations as 0, ∞, 1, T.†2 There is also an extensive doctrine concerning q, the relation of inherence. Kant, in his celebrated Appendix to the Transcendental Dialectic,†3 has set forth three sporadic propositions of this sort, whose significance can hardly be seen away from their crowd. Besides, it is more satisfactory to see these things set forth in a purely logical way and deduced mathematically, than to have them treated at their first presentation as regulative principles. As a part of this general doctrine of inherence, there is a special doctrine of the properties of relations. Of course, all logical treatises consider these things; but they do not consider them in a formal way, nor at all in the manner in which they are turned out by the machinery of this calculus. One of the questions which pertain to this branch of logic is that of the classification of relations. There are also some special relations of logical origin which have to be considered, among which is that of correspondence, which has been studied by mathematicians without much logical analysis. Peirce: CP 4.82 Cross-Ref:†† 82. A number of interesting features of the logical calculus itself emerge in the application of it to the second intentions. One of these, for example, is that the subjacent letters I call indices do not essentially differ from any other letters. Thus we may define identity as follows: π[i]π[j]π[K]π[x] {1[K]=~r[Kij] x[i] ~x[j]} That is, to say that anything whatever, K, is identity is to say that if any two things i and j are in the relation, K, the i to the j, then any proposition whatever, x, is true of i, or else that proposition is false of j. The point calling for notice is that x is put into the logisterium, although it is one of the principal letters of the Boolian.†1

Peirce: CP 4.83 Cross-Ref:†† 83. Another thing is that the forms of logisteria, themselves, become subjects of study, and certain general propositions with regard to them are expressed as if they were shops or trees; and yet these very propositions can be made use of in the calculus. Peirce: CP 4.83 Cross-Ref:†† Among the forms of logisteria which require attentive study and which are found to possess interesting properties are particularly those which are infinite series, though the very purpose of the lectical symbols is to embrace infinite series. But we find that we have to resort to logisteria of logisteria, and to their logisteria again, and so on indefinitely, and that the distinctive characters of different such infinite series of logisteria have to be discriminated. Peirce: CP 4.84 Cross-Ref:†† 84. Another point of logical interest is that when our discourse relates to a universe of possibility which virtually embraces all logical possibility, everything is true [from] which no false consequence can possibly follow; and the only way of investigating certain propositions is by proving that they cannot give rise to any contradiction; and this being proved, they are proved true. Peirce: CP 4.84 Cross-Ref:†† For example, it is required to prove that

π[a]Σ[K]π[b] g[Ka](~g[Kb] 1[ab])

Peirce: CP 4.84 Cross-Ref:†† That is, that every object, a, whether individual or general, has a quality, K, which is a quality of no object, b, unless b is identical with a. Now this can only lead to an absurd result if K be eliminated. But without "logical involution," or the compounding of the premiss with itself, K can only be eliminated in two ways; first, by eliminating g[Ka] and ~g[Kb] independently, and second, by identifying b with a. In the first way, we can only get π[a]π[b] (~1[ab] 1[ab]) which is true; and in the second way we can only get π[a]1[aa] which is equally true. We next proceed, then, to square the proposition in question, and so get π[a]Σ[K]π[b]π[c] g[Ka]·(~g[Kb] 1[ab])·(~g[Kc] 1[ac]). Unless we identified K in the factors, there could be no aid in eliminating K. But this identification forces us to identify a which is to the left of it in the logisterium. But it is easy to show that from the square so written K cannot be eliminated in any new way, nor from any higher power. Therefore, the proposition can lead to no absurdity, and is true.

Peirce: CP 4.84 Cross-Ref:†† Suppose, however, we were to subject the following (which seems hardly distinguishable from it to a loose thinker) to the same test Σ[K]π[a]π[b] g[Ka]·(~g[Kb] 1[ab]) Squaring this, we have Σ[K]π[a]π[b]π[c]π[d] g[Ka]·g[Kc]·(~g[Kb]

1[ab])·(~g[Kd]

1[cd]).

Now identifying d with a and c with b, we get π[a]π[b]1[ab] which is as much as to say that everything is identical with everything. So that in a universe of more than one object this proposition is false.

Peirce: CP 4.85 Cross-Ref:†† IV THE LOGIC OF QUANTITY†1P

§1. ARITHMETICAL PROPOSITIONS

85. Kant, in the introduction to his Critic of the Pure Reason†2, started an extremely important question about the logic of mathematics. He begins by drawing a famous distinction, as follows: Peirce: CP 4.85 Cross-Ref:†† "In judgments wherein the relation of a subject to a predicate is thought . . . this relation may be of two kinds. Either the predicate, B, belongs to the subject, A, as something covertly contained in A as a concept; or B is external to A, though connected with it. In the former case, I term the judgment analytical; in the latter synthetical. Analytical judgments, then, are those in which the connection of the predicate with the subject is thought to consist in identity, while those in which this connection is thought without identity, are to be called synthetical judgments. The former may also be called explicative, the latter ampliative judgments, since those by their predicates add nothing to the concept of the subject, which is only divided by analysis into partial concepts that were already thought in it though confusedly; while these add to the concept of the subject a predicate not thought in it at all, and not to be extracted from it by any analysis. For instance, if I say all bodies are extended, this is an analytical judgment. For I need not go out of the conception I attach to the word body, to find extension joined to it; it is enough to analyze my meaning, i.e., merely to become aware of the various things I always think in it, to find that predicate among them. On the other hand, if I say, all bodies are heavy, that predicate is quite another matter from anything I think in the mere concept of a body in general."

Peirce: CP 4.85 Cross-Ref:†† Like much of Kant's thought this is acute and rests on a solid basis, too; and yet is seriously inaccurate. The first criticism to be made upon it is, that it confuses together a question of psychology with a question of logic, and that most disadvantageously; for on the question of psychology, there is hardly any room for anybody to maintain Kant right. Kant reasons as if, in our thoughts, we made logical definitions of things we reason about! How grotesquely this misrepresents the facts, is shown by this, that there are thousands of people who, believing in the atoms of Boscovich, do not hold bodies to occupy any space. Yet it never occurred to them, or to anybody, that they did not believe in corporeal substance. It is only the scientific man, and the logician who makes definitions, or cares for them. Peirce: CP 4.86 Cross-Ref:†† 86. At the same time, the unscientific, as well as the scientific, frequently have occasion to ask whether something is consistent with their own or somebody's meaning; and that sort of question they themselves widely separate from a question of how experience, past or possible, is qualified. The Aristotelian [logicians] -- and, in fact, all men who ever have thought -- have made that distinction. It is embodied in the conjugations of some barbarous languages. What was peculiar to Kant -- it came from his thin study of syllogistic figure-was his way of putting the distinction, when he says we necessarily think the explicatory proposition although confusedly, whenever we think its subject. This is monstrous! The question whether a given thing is consistent with a hypothesis, is the question of whether they are logically compossible or not. I can easily throw all the axioms of number, which are neither numerous nor complicated, into the antecedent of a proposition -- or into its subject, if that be insisted upon -- so that the question of whether every number is the sum of three cubes, is simply a question of whether that is involved in the conception of the subject and nothing more. But to say that because the answer is involved in the conception of the subject, it is confusedly thought in it, is a great error. To be involved, is a phrase to which nobody before Kant ever gave such a psychological meaning. Everything is involved which can be evolved. But how does this evolution of necessary consequences take place? We can answer for ourselves after having worked a while in the logic of relatives. It is not by a simple mental stare, or strain of mental vision. It is by manipulating on paper, or in the fancy, formulæ or other diagrams -- experimenting on them, experiencing the thing. Such experience alone evolves the reason hidden within us and as utterly hidden as gold ten feet below ground -- and this experience only differs from what usually carries that name in that it brings out the reason hidden within and not the reason of Nature, as do the chemist's or physicist's experiments. Peirce: CP 4.87 Cross-Ref:†† 87. There is an immense distinction between the Inward and the Outward truth. I know them alike by experimentation only. But the distinction lies in this, that I can glut myself with experiments in the one case, while I find it most troublesome to obtain any that are satisfactory in the other. Over the Inward, I have considerable control, over the Outward very little. It is a question of degree only. Phenomena that inward force puts together appear similar; phenomena that outward force puts together appear contiguous. We can try experiments establishing similarity so easily, that it seems as if we could see through and through that; while contiguity strikes us as a marvel. The young chemist precipitates Prussian blue from two nearly colorless fluids a hundred times over without ceasing to marvel at it. Yet he finds no marvel in

the fact that any one precipitate when compared in color with the other seems similar every time. It is quite as much a mystery, in truth, and you can no more get at the heart of it, than you can get at the heart of an onion. Peirce: CP 4.87 Cross-Ref:†† But nothing could be more extravagant than to jump to the conclusion that because the distinction between the Inward and the Outward is merely one of how much, therefore it is unimportant; for the distinction between the unimportant and the important is itself purely one of little and much. Now, the difference between the Inward and the Outward worlds is certainly very, very great, with a remarkable absence of intermediate phenomena. Peirce: CP 4.88 Cross-Ref:†† 88. The first question, then, to ask concerning arithmetical and geometrical propositions is, whether they are logically necessary and merely relate to hypotheses, or whether they are logically contingent and relate to experiential fact. Peirce: CP 4.88 Cross-Ref:†† Beginning with the propositions of arithmetic, we have seen already †1 that arithmetical propositions may be syllogistic conclusions from ordinary particular propositions. From A ~~B and ~A ~~B,†2 taken together, or Some A is B, Some not-A is B, it follows that there are at least two B's. This inference is strictly logical, depending on the principle of contradiction, that is, on the non-identity of A and not-A. By the same principle, from Some A is B, Some not-A is B, Any B is C, Some not-B is C, taken together it follows that there are at least three C's. Peirce: CP 4.89 Cross-Ref:†† 89. Hamilton admits †3 that the arithmetical proposition, "Some B is not some-B," is so urgently called for in logic, that a special propositional form must be made for it. So, if a distributive meaning be given to "every," Every A is every A, implies that there is but one A, at most. This is what this proposition must mean, if it is to be the precise contradiction of the other. If a proposition is infra-logical in form, its denial must be admitted to be so. Peirce: CP 4.90 Cross-Ref:†† 90. It clearly belongs to logic to evolve the consequences of its own forms.

Hence, the whole of the theory of numbers belongs to logic; or rather, it would do so, were it not, as pure mathematics, prelogical, that is, even more abstract than logic.†4 Peirce: CP 4.91 Cross-Ref:†† 91. These considerations are sufficient of themselves to refute Kant's doctrine that the propositions of arithmetic are "synthetical." As for the argument of J. S. Mill,†5 or what is usually attributed to him, for what this elusive writer really meant, if he precisely meant anything, about any difficult point, it is utterly impossible to determine -- I mean the argument that because we can conceive of a world in which when two things were put together, a third should spring up, therefore arithmetical propositions are experiential, this argument proves too much. For, in the existing world, this often happens; and the fact that nobody dreams of its constituting any infringement of the truths of arithmetic shows that arithmetical propositions are not understood in any experiential sense. Peirce: CP 4.91 Cross-Ref:†† But Mill is wrong in supposing that those who maintain that arithmetical propositions are logically necessary, are therein ipso facto saying that they are verbal in their nature. This is only the same old idea that Barbara in all its simplicity represents all there is to necessary reasoning, utterly overlooking the construction of a diagram, the mental experimentation, and the surprising novelty of many deductive discoveries. Peirce: CP 4.91 Cross-Ref:†† If Mill wishes me to admit that experience is the only source of any kind of knowledge, I grant it at once, provided only that by experience he means personal history, life. But if he wants me to admit that inner experience is nothing, and that nothing of moment is found out by diagrams, he asks what cannot be granted. Peirce: CP 4.92 Cross-Ref:†† 92. The very word a priori involves the mistaken notion that the operations of demonstrative reasoning are nothing but applications of plain rules to plain cases. The really unobjectionable word is innate; for that may be innate which is very abstruse, and which we can only find out with extreme difficulty. All those Cartesians who advocated innate ideas took this ground; and only Locke failed to see that learning something from experience, and having been fully aware of it since birth, did not exhaust all possibilities. Peirce: CP 4.92 Cross-Ref:†† Kant declares that the question of his great work is "How are synthetical judgments a priori possible?" By a priori he means universal; by synthetical, experiential (i.e., relating to experience, not necessarily derived wholly from experience). The true question for him should have been, "How are universal propositions relating to experience to be justified?" But let me not be understood to speak with anything less than profound and almost unparalleled admiration for that wonderful achievement, that indispensible stepping-stone of philosophy. Peirce: CP 4.93 Cross-Ref:†† 93. To return to number, there are various ways in which arithmetic may be conceived to connect itself with and spring out of logic. Besides the path of spurious propositions [as indicated in 88], there is another which I pursued on an early paper †1 in which I defined the arithmetical operations in terms of those of the reformed Boolian calculus. In a later paper,†2 I considered quantity from the point of view of

the logic of relatives. Peirce: CP 4.93 Cross-Ref:†† I shall in the present chapter endeavor, as much as I can, to avoid tedious questions of detail and seek to make clear some of the main points of the logic of mathematics.

Peirce: CP 4.94 Cross-Ref:†† §2. TRANSITIVE AND COMPARATIVE RELATIONS

94. I have certainly written to little purpose, and so has Dr. Schröder, if we have not succeeded in making readers perceive the pervasive working of balance and symmetry in every part of logic. Now, we have seen the ubiquitous logical agency of the form l~$l.†3 Peirce: CP 4.94 Cross-Ref:†† We say that this is due to the formula l~$l -< T,†4 which is balanced by 1†5 -< l†~$l.†6 But, be it observed, that this is a kind of balance which throws all the active work upon the shoulders of the former principle, and allows the latter to moulder in innocuous desuetude. Yet really, the form l†~$l is all-important, inasmuch as it is the basis of all quantitative thought. For the relation expressed by it is a transitive relation. By a transitive relation, we mean a relation like that of the copula. If A be so related to B, and B be so related to C, then A is so related to C. In other words, if t is a transitive relation, tt -< t. Now, this is the case with l†~$l. For that (l†~$l)(l†~$l) -< l†~$l is obvious. Though the reader sees how, I will, in consideration of the importance of the matter, set down the steps: (l†~$l)(l†~$l) -< l†~$l(l†~$l) -< l†~$l l†~$l -< l†T†~$l -< l†~$l. Peirce: CP 4.94 Cross-Ref:†† This is not only a transitive relation, but it is one which includes identity under it. That is, 1 -< l†~$l

Peirce: CP 4.94 Cross-Ref:†† But it is further demonstrable that every transitive relation which includes identity under it is of the form l†~$l. For let t be such a relation that 1 -< t. Multiplying by ~$t, we get, ~$t -< 1~$t -< t~$t -< T. Hence,†1 t†~$t -< t†T -< t. On the other hand, t -< t1 .·. t -< t(t†~$t) .·. t -< tt†~$t. But because t is transitive, tt -< t .·. t -< t†~$t.

Having just found t†~$t -< t, we can write t = t†~$t, so that t may be expressed in the form l†~$l, Q. E. D. Peirce: CP 4.95 Cross-Ref:†† 95. I am now going to allow myself to be led aside out of the main channel of thought upon this subject merely to show how little interest there is in transitive relationship apart from the logical form (l†~$l). Peirce: CP 4.95 Cross-Ref:†† Let us use the zodiacal sign of Leo to signify a transitive relation, such that not everything is in that relation to itself. The inference holds, x y, y z, .·. x z. Let L be an individual that is not in this relation to itself, which we may write,

L~ L. Then, the (equivalent) inferences hold L x

x L

.·. x~ L

.·. L~ x.

We may, therefore, divide all other individuals into three classes; first, K, J, etc. such that K L J L, etc.; second, M, N, etc., such that L M L N, etc.; and third, Γ, {D}, etc., such that Γ~ L {D}~ L

L~∩Γ L~ {D}, etc.

Taking any one of the first class, K, and any one of the second, M, we have M~ K. Peirce: CP 4.95 Cross-Ref:†† Let G be a letter of the first class which is a non-Leo of itself; then G~ G, and the first class may be subdivided into three with reference to G, just as all were divided relatively to L. So, if R be a letter of the second class which is non-Leo of itself, or R~ R. Peirce: CP 4.95 Cross-Ref:†† We can then divide all possible individuals other than G, L, and R into ten classes, viz.: Peirce: CP 4.95 Cross-Ref:†† First, Those which, as B, C, etc. are Leos of G; as B G; Peirce: CP 4.95 Cross-Ref:†† Second, Those which, as r, are neither Leos of nor Leo'd by G, but are Leos of L; as Γ~ G, G~∩Γ, Γ Λ;

Peirce: CP 4.95 Cross-Ref:†† Third, Those which, as H, K, etc., are Leo'd by G and are Leos of L; as G K, K L; Peirce: CP 4.95 Cross-Ref:†† Fourth, Those which, as {D}, are not Leo'd by G, nor are Leos of L, but are Leos of R; as G~ {D}, {D}~ L, {D} R; Peirce: CP 4.95 Cross-Ref:†† Fifth, Those which, as Θ, are Leo'd by G, are neither Leos of nor Leo'd by L, and are Leos of R; as G∩Θ, L~∩Θ, Θ~ L, Θ Ρ; Peirce: CP 4.95 Cross-Ref:†† Sixth, Those which, as P, Q, etc. are Leo'd by L and are Leos of R; as L P, P R; Peirce: CP 4.95 Cross-Ref:†† Seventh, Those which as {X} are not Leo'd by G nor are Leos of R; as G~ {X}, {X}~ R; Peirce: CP 4.95 Cross-Ref:†† Eighth, Those which as π are Leo'd by G, are not Leo'd by L, and are not Leos of R; G∩π, L~∩π, π~ R; Peirce: CP 4.95 Cross-Ref:†† Ninth, Those as Σ which are Leo'd by L but are neither Leos of nor Leo'd by R; L∩Σ, R~∩Σ, Σ~ R; Peirce: CP 4.95 Cross-Ref:†† Tenth, Those as X, Y, etc. which are Leo'd by R; as R X. Peirce: CP 4.96 Cross-Ref:†† 96. The above gives some idea what the further doctrine of transitive relations not including identity would be like. It is evidently more interesting to consider further the study of relatives of the form (l†~$l) and others allied to them. Peirce: CP 4.96 Cross-Ref:†† The converse of l†~$l is ~l†$l,†1 which is, of course, also transitive. The negative is ~l$l,†2 which is not transitive, but which has the property, ~l$l -< ~l$l†~l$l. For ~l$l = ~l1$l -< ~l($l†~l)$l -< ~l$l†~l$l.

Peirce: CP 4.96 Cross-Ref:†† This is a property allied to transitiveness. Peirce: CP 4.96 Cross-Ref:†† If A is a lover of something not loved by B, which is, in its turn, a lover of something not loved by C, the conclusion is, that A is a lover of something different from something not loved by C. That is, (l~$l)(l~$l) -< lT~$l. Peirce: CP 4.96 Cross-Ref:†† The natural current of thought next carries us to the hypothesis that the relation expressed by l be such that l~$l -< l†~$l. In this case, l~$l is a transitive relation. For l~$l l~$l -< l~$l(l†~$l) -< l~$l. Such a relation may well be termed a comparative relation. If Samson can lift something Ajax cannot lift, then Samson can lift everything Ajax can lift. Such a relation underlies all measurement; and the propriety of the designation I propose will be allowed. Peirce: CP 4.96 Cross-Ref:†† With this conception, quantitative science begins. Note well how it has been suggested to us. Peirce: CP 4.96 Cross-Ref:†† A trial of strength must begin by young Ajax, the challenger, doing various things which he "stumps" the champion, Samson, to imitate. If Samson cannot perform all of Ajax's feats, that settles it. But if it seems that Samson can do all that Ajax can do then he will, in his turn, do something which he proposes that Ajax shall imitate. If Ajax cannot do all that Samson can do, that again settles it. But if it seems that each can repeat all the performances of the other, we conclude that they are equally strong. Thus equality is a complex relation.†3 Peirce: CP 4.96 Cross-Ref:†† In a universe of quantities of one dimension (where are only quantities, not quanta) things equal are identical; so that, not only, l~$l -< T, which is always true and l~$l l~$l -< T, which is true for all comparative relations, but also T -< l~$l

~l$l.†1

That is, if A and B are not identical, either A can do something that B cannot, or B can do something that A cannot. Peirce: CP 4.96 Cross-Ref:††

We have thus analyzed the conception of quantity; and we see that nothing but logical conceptions enter into its constitution. The idea of being able, especially in the broad sense in which one quantity is said to be able to do something another is unable to do, is only a modification of the idea of possibility, the precise explanation of which is given in second intentional logic. Peirce: CP 4.97 Cross-Ref:†† 97. Although I cannot, in this work, carry the student deeply into second intentional logic, yet it will be indispensible to look upon quantity somewhat in that way; for quantitative thought, like the traditional "chimæra bombinans in vacuo," feeds upon second intentions. Peirce: CP 4.97 Cross-Ref:†† That l is a comparative relation in a universe of quantity, may be expressed by the formula, 0†(1 l~$l

~l$l)†0†1

But the same thing may be expressed in another way, by throwing the relation l among the indices. Thus, let us use the three symbols, u, v, w. w[ij] means that i is an individual relation of which j is the general character, u[ij] means that i is an individual relation of which j is the first relate, v[ij] means that i is an individual relation of which j is the correlate. Peirce: CP 4.97 Cross-Ref:†† Then, we shall have l[ij] = Σ[k]u[ki]v[kj]w[kl], that is, there is an individual relation of which i is the relate, j the correlate, and l the general character. Peirce: CP 4.97 Cross-Ref:†† That being premissed, the proposition that l is a comparative relation may be written, π[h]π[k]Σ[i]Σ[j]Σ[p]π[q]π[r]Σ[s] 1[hk] u[ph]·v[pi]·w[pl]·(~u[qk] ~v[qi] ~w[ql]) ~w[rl])·u[sk]·v[sj]·w[sl];

(~u[rh] ~v[rj]

that is, of any two different objects, h and k, one or other is l to something to which the other is not l. Peirce: CP 4.97 Cross-Ref:†† More simply, if r[ijk] means that i is a relation in which j stands to k; we may write, π[h]π[k]Σ[i]Σ[j] 1[hk] r[lhi]~r[lki] r[lkj]~r[lhj] to express that l is a comparative relation.

Peirce: CP 4.98 Cross-Ref:†† 98. We are next naturally led to remark that it is a very important thing to say of a class of objects, say the A's, that there is some one relation such that, of any two A's not identical, one stands in that relation to the other, while the second does not stand in that relation to the first. This we write Σ[l]π[m]π[n] ~a[m] ~a[n] 1[mn] r[lmn]~r[lnm] r[lnm]~r[lmn] But this becomes particularly important, in case the relation l is a relation of comparison. If that be the case, we multiply in the above definition of such a relation. Peirce: CP 4.99 Cross-Ref:†† 99. The question arises, is it possible there should be a class which does not possess the property just defined? It is a difficult question, to which a good logician will be reluctant to give a negative answer. In order to answer it, we must have some way of constructing an icon of a class, in general. Now, a class may be said to comprise all of which something is true. Shall we say of the different individuals composing it that they are distinguished by having some of them qualities which the others do not possess? It seems far from evident that this is so; although, no doubt, after two instances have presented themselves, it is possible in the circumstances of the presentation to find distinguishing qualities. But supposing this difficulty surmounted, of two individuals of a class, each may have qualities the other wants. If, then, we seek to establish an order of precedence among the things, such as a relationship of comparison supposes, we must first establish some order of precedence among the qualities. We are thus brought back to the question with which we set out, whether among a collection of objects an order of precedence can always be establishable. It thus becomes clear that no contradiction can emerge from the hypothesis of a class among the members of which no thoroughgoing order of precedence can be established, and to which all quantitative conceptions are quite inapplicable. About such classes we can reason, but we cannot reason quantitatively.

Peirce: CP 4.100 Cross-Ref:†† §3. ENUMERABLE COLLECTIONS

100. But supposing we have to do with a class of things throughout which a relationship of comparison can be established, the next question that balance and symmetry suggest is, whether, as we have l~$l l~$l -< l~$l, we have also l~$l -< l~$l l~$l. Peirce: CP 4.100 Cross-Ref:†† It is clear enough, that cases can be imagined in which this shall not be true. Classes of a mixed character exist, too, where this holds in certain parts, but not in others. Such mixed cases are not, however, of much interest. The interesting cases are those in which

l~$l -< l~$l l~$l

invariably holds,†1 and those in which whatever is l~$l to anything is l~$l to something to which it is not l~$l l~$l. To speak in more familiar terms, whatever is greater than anything may or may not always be next greater than something, that is, may or may not be always greater than something else, greater than that thing. Peirce: CP 4.101 Cross-Ref:†† 101. But a further distinction immediately arises, according as, on the one hand, one or other of these propositions is true for every comparative relation, or on the other hand for some comparative relations the one proposition is true and for others the other. Peirce: CP 4.101 Cross-Ref:†† This trichotomy constitutes the most important distinction between classes in respect to their multitude. Peirce: CP 4.101 Cross-Ref:†† Let us first consider a class in which, no matter what comparative relation may be signified by l~$l, l~$l∞ -< [l~$l·(~l†$l†~l†$l)]∞.†1 That is, whatever is greater than anything is next greater than something, in every sense of being greater, that is, for every comparative relation, l~$l, for which 0†(1 l~$l

~l$l)†0.

Peirce: CP 4.102 Cross-Ref:†† 102. We have now pushed our way far enough into the theory of quantity and its complications of logic to meet with theorems. Such is the following: any class of which the conditions enunciated holds has a maximum and a minimum individual for each comparative relation, that is, one which is not l~$l to any member of the class, and one which is not ~l$l to any member. I will first show that there is a maximum. For this purpose, assume a[0] to be any member of the class of A's, and consider the relation (l~$l ~l$l)a[0]$a[0] l~$l a[0]$a[0]~l$l·l~$l a[0]$a[0]l~$l·~l$l.

~l$l a[0]$a[0]~l$l

~l$l

Peirce: CP 4.102 Cross-Ref:†† This is an aggregate of four relations, viz.: Peirce: CP 4.102 Cross-Ref:†† First, That having for its relate any A, superior (l~$l) or inferior (~l$l) to a[0], and for its correlate a[0], Peirce: CP 4.102 Cross-Ref:†† Second, That having for both relate and correlate A's superior to a[0], the relate being superior to the correlate, Peirce: CP 4.102 Cross-Ref:†† Third, That having for relate any A inferior to a[0], and for correlate any A

superior to a[0], Peirce: CP 4.102 Cross-Ref:†† Fourth, That having for both relate and correlate A's inferior to a[0], the relate being inferior to the correlate. Peirce: CP 4.102 Cross-Ref:†† This, I say, will be a quantitative relation. That is, it will be transitive, included under its own negative converse, and including negation under the aggregate of itself and its converse. That it is transitive, that is, that, if X is in this relation to Y, and Y to Z, then X is in this relation to Z, is plain; for, if Y is superior to a[0], then Z must either be a[0] (in which case, X, whatever A it may be, is in this relation to Z) or must be superior to a[0] but inferior to Y. Then, if X is superior to a[0], it is superior to Y, and consequently also to Z, and is in this relation to Z. But if X is inferior to a[0], it is in this relation to Y, which is superior. X can in no case be a[0]. If Y is inferior to a[0], Z is either a[0] (when as before X will be in this relation to it) or superior to a[0], or inferior to a[0], but superior to Y. X will be inferior to Y, and thus will be in this relation to Z. This shows that the relation is transitive. That it is included under its negative converse, that is, inconsistent with its converse, is plain; for if U could be in this relation to V and in the converse relation, too, that is, V in this relation to U, then, since the relation is transitive, U would be in this relation to itself, which, it is easy to see, the definition excludes. That of any pair of different A's, one is in this relation to the other, is easily seen by running over the definition. Peirce: CP 4.102 Cross-Ref:†† We will call this relation "second-superior." Now, I say, if the class of A's has no maximum for the relation l~$l, then that A which is next inferior to a[0] is not next second-superior to any A. Will it be objected that I have not proved that there is an A next inferior to a[0]? It is easy to supply the defect. For by hypothesis whatever A is superior is next superior to some A for every comparative relation. Now, we have only to substitute $l for l and vice versa, and next inferior becomes next superior. Therefore, to say that for every comparative relation, whatever has a superior has a next superior, is the same as to say that for every comparative relation, whatever has an inferior has a next inferior. The A next inferior to a[0] is second-superior only to a[0], to A's superior to a[0] and to A's inferior to a[0] but superior to itself. Because it is next inferior, of the last there are none. That of which it is next superior is therefore superior to a[0], and any other A superior to it is second-superior to it and second-inferior to the next inferior to a[0]. Thus, that to which the next inferior to a[0] is next second-superior, is the superior of all other A's superior to a[0]; that is, it is the maximum. The proof that there will be a minimum is altogether similar. Peirce: CP 4.102 Cross-Ref:†† Hence, any class of things in which whatever is anywise superior to another of the class is next superior to some one can be enumerated. For in enumeration, the objects of a class are singly told, and "told later than" evidently satisfies the three conditions of a quantitative relative. If there is a maximum, the telling comes to an end, the class is told out, it is enumerated. For that reason, it is convenient to term a class every member of which anywise superior to another is next superior to some, an enumerable collection. Peirce: CP 4.103 Cross-Ref:†† 103.†1 About an enumerable collection certain forms of reasoning hold which, though they had been used more or less since man began to be a reasoning animal,

were first signalized in a logical work by De Morgan in 1847,†P1 and constitute one of his claims to be considered the greatest of all formal logicians. In his Formal Logic he gives eight forms which in the Appendix to his fourth Memoir on the Syllogism are increased to 64. The eight are as follows:

For every Z there is an X that is Y, Some Z is not Y; .·.Some X is not Z.

For every Z there is an X not Y, Some Z is Y; .·.Some X is not Z.

For every non-Z there is an X that is Y, There is something besides Y's and Z's; .·.Some X is Z.

For every non-Z there is an X not Y, Some Y is not Z; .·.Some X is Z.

For every Z there is a Y not X, Some Z is not Y; .·.There is something besides X's and Z's.

For every Z there is something neither X nor Y, Some Y is Z; .·.There is something besides X's and Z's.

For every non-Z there is a Y not X, There is something besides Y's and Z's; .·.Some Z is not X.

For every non-Z there is something neither X nor Y, Some Y is not Z; .·.Some Z is not X.

Peirce: CP 4.103 Cross-Ref:†† We might also have such a reasoning as this:

For every Z there is an X that is Y, For every X not Z there is an X not Y; .·.Every Z is X.

Peirce: CP 4.103 Cross-Ref:†† This is not one of De Morgan's Forms. He gives, however such as these:

For every Z an X is Y, Every Y is Z; .·.Every Z is X.

For every not Z is an X not Y, For every X is a Y not Z; .·.Every Z is X.

Peirce: CP 4.103 Cross-Ref:†† De Morgan termed these "syllogisms of transposed quantity," because they transfer the lexis from one term to another. His point of view was this: Take Baroko,

Any M is P, Some S is not P; .·.Some S is not M.

The converse,

Some M is not S,

does not follow; but if there are as many M's as there are S's, then this does follow. "For if M's, as many as there are S's, be among the P's, and some of the S's be not among the P's, though all the rest were, there would not be enough to match all the M's, or some M's are not S's." Peirce: CP 4.103 Cross-Ref:†† The rank and file of old-fashioned logicians were not pleased with the syllogisms of transposed quantity. They belonged to that class of minds who decry originality, who dread novelty, who hate discoveries, and who will go to some trouble to inflict any personal injury on those who perpetrate them, provided they can inflict it without serious injury to themselves. They circulated an unfounded innuendo that De Morgan was a drunkard; their spitefulness was only bounded by their prudence. The idea of so lifeless a subject as formal logic -- too abstract to be philosophical -exciting such passions is laughable. Yet such was the fact. Peirce: CP 4.103 Cross-Ref:†† But they could not find anything better to say against those syllogisms of transposed quantity than that they were "extralogical." If it had only occurred to them that they were not sound reasoning, that is, not universally valid, they would have seized upon that defect with glee. Nor, singularly enough, does De Morgan himself seem to have remarked the circumstance; although it ought to have been evident from the line of thought which led him to those forms. By the logic of relatives, we at once find that the statement that such a syllogism is necessary implies that a certain collection is enumerable. Peirce: CP 4.103 Cross-Ref:†† The following is the first form De Morgan adduced:

Some X is Y, For every X there is something neither Y nor Z; Hence, something is neither X nor Z.

Let us put for X "odd number," for Y "prime," for Z "either an even number or not a number," so that its negative is "a number not even." Then, the conclusion is false though the premisses are true. Thus:

Some odd numbers are prime, Every odd number has for its square a number not even nor prime; Hence, some number not even is not odd.

Peirce: CP 4.104 Cross-Ref:†† 104. Let us enclose the description of a class in square brackets to denote the number of individuals in it. Then the premisses of the above may be arithmetically stated thus: [x·y] > 0 [x] ⎥ [~y·~z] Developing the last, we get [x·y·z] + [x·y·~z] + [x·~y·z] + [x·~y·~z] ⎥ [x·~y·~z] + [~x·~y·~z] Cancelling [x·~y·~z], we have [x·y·z] + [x·y·~z] + [x·~y·z] ⎥ [~x·~y·~z] Developing the first premiss [x·y·z] + [x·y·~z] > 0. It thus follows, not only that [~x·~z] > 0 but even that, throwing aside Y's, there are more non-X's not Z's than there are X's that are Z's. But the fallacy lies in assuming the Simple Simon proposition that every part is less than its whole. That is, because the odd squares are no fewer than all the odd numbers, we quietly reason as if they were more than a part of odd numbers; so that after taking away alike from odd squares and from odd numbers the odd primes, we should necessarily have as many odd squares left over as we have odd numbers. Of collections not enumerable it is not generally true that the part is less than the whole. Every integer has a square; and thus there are as many squares as there are integers; although the squares form but a part of all the integers. Peirce: CP 4.104 Cross-Ref:†† Take this example:

Every woman marries a man, For every man there is a woman; .·.Every man is married to a woman.

The necessity of this plainly arises from the fact that after every woman has got a husband, the collection of men is exhausted. To say this, is to imply that for every quantitative relation it would have a maximum, that is, a last reached, in any order of running it through. . . . Peirce: CP 4.105 Cross-Ref:†† 105. The commonest sort of paralogism by far among thoughtful persons

consists in reasoning as if collections were enumerable which are, in fact, inenumerable. How often do we hear one, speaking of objects in a linear series, say there must be a first or must be a last! Logic lends no color to such ideas; but, on the contrary, shows them to be pure assumptions. In metaphysics, particularly, it is frequently argued that something is analyzable into a series -- by pure abstract reasoning -- and then because "there must be a first" some consequence truly startling follows. Years of experience bring us to expect, as a matter of course, some fallacy, big or little, in every demonstration which seems to advance knowledge very much. Peirce: CP 4.106 Cross-Ref:†† 106. De Morgan's syllogism of transposed quantity does not seem very clearly or accurately to set forth precisely what the nature of the reasoning specially applicable to enumerable collections is. What does precisely describe it is, that in whatever order you pass, one by one, through the collection, you come to a last unit. But let us logically analyze this. Here there is a relation of the later-taken to the earlier-taken. The earlier is taken at a time at which the later is not taken. If l signify "taken at a time," then "taken at a time at which was not taken" is written l~$l. We see at once that this running through the collection is only a specialized way of saying that we have to do with a quantitative relation, a way of expressing it which brings in the irrelevant idea of time. The better statement is that, in reference to every quantitative relation, the enumerable collection has a last. From this it quite obviously follows that there is also a first, and that every superior of anything is next superior of something, and so also with the inferior. Another form of definition of the enumerable collection is, to say that it is a collection any part of which is less than the whole. That is, given a class, the b's, such that ∞ -< ~$b†a, or every b is a, while ∞ -< $a~b, or some a is not b; if, further, k is such a relative that $k b k -< 1, then ∞ -< $a(~$k†~b). Peirce: CP 4.106 Cross-Ref:†† This is a good statement of the kind of inference peculiar to enumerable classes. It has three premisses, involving two class terms and a relative term; and it reposes directly upon the axiom that an enumerable part is less than the whole. One of the three premisses is implicitly assumed but not stated by De Morgan; the relative is his "for every," and one of his three terms is superfluous. Thus he puts the argument about the checks into form as follows:†1

"For every memorandum of a purchase a countercheque is a transaction involving the drawing of a cheque,

"Some purchases are not transactions involving the drawing of checks; "Therefore, some countercheques are not memoranda of purchases."

But I should put it in form as follows:

Some payment for purchase was not a check, Every payment by purchase is told off against a check; No two payments by purchase are told against the same check; .·. Some checks are not payments by purchase.

Peirce: CP 4.106 Cross-Ref:†† That is, we prove the checks are not a part of the payments by purchase, because they are not less than the payments by purchase; and it is assumed they were enumerable. If they ran on endlessly, each payment by purchase might be told off against a check of a subsequent day, the purchases increasing in number day by day.

Peirce: CP 4.107 Cross-Ref:†† §4. LINEAR SEQUENCES

107. The second, or middling, grade of multitude is that of collections which have different attributes for different quantitative collections; namely, for some such relations, every member of the class superior to another member is next superior to some member, definitely designatable, while for other quantitative relations it is not so. I undertake to show that there is always some quantitative relation for which (1) the class has a minimum but no maximum, (2) for which every member of the class that is superior to another is next superior to some other, (3) and for which the partial class consisting of any two members of the class we are speaking of, together with all that are superior to one of these two members but inferior to the other, is enumerable. Let us begin by thinking of a member of this class, say a[x]. Then, considering a quantitative relation in which every a superior to an a is next superior to an a, let us think of that to which a[x] is next superior. Then, think of that to which the last is next superior. Then consider a partial class to all of which a[x] is superior, and the next inferior of each member of which is also included under it, so that either there is a minimum, which is not superior to any member of the class, or else, if a[y] is any member of the class to which a[x] is superior, and the a's at once inferior to a[x] and superior to a[y] are enumerable, it follows that a[y] is a member of the partial class. For if not, of all the a's superior to a[y] and inferior to a[x], a part belongs to the partial class, and this part of an enumerable collection, being itself (as such) enumerable, must have a minimum. But by the definition of the partial class, whatever is next inferior to any member of it also belongs to it. To this partial class, then, belongs every a inferior to a[x], so long as between it and a[z], the collection of

a's is enumerable. We do not know that there is an a next superior to a[x]. But we define a second partial class as containing the a next superior to a[x], if there be any, and as containing nothing else, except that it contains the a next superior to any a that it contains. Then, it will either contain all the a's superior to a[x] up to some maximum, which need not be the maximum of all the a's, but which has no a next superior to it, or, in the absence of such a maximum, it will contain all the a's up to and beyond any a superior to a[x], but such that the a's inferior to it and superior to a[x] form an enumerable collection. The proof of this (so plain that it hardly needs statement) is as follows: if this be not the case let a[z] be an a superior to a[x] and such that the a's inferior to a[z] but superior to a[x] form an enumerable multitude. Then, those of those which belong to the second partial class, being part of an enumerable collection. are themselves enumerable. Hence, they have a maximum, contrary to the hypothesis. Taking the first and second partial classes together, I propose to call such a series of a's a linear sequence. I will repeat its characteristics: 1. It contains a[x]. 2. It contains every a inferior to a[x] and identical with or superior to a[y], no matter what a[y] may be, so long as it is inferior to a[x], and so long as the a's superior to a[y] and inferior to a[z] form an enumerable collection. 3. It contains every a superior to a[x] and identical with or inferior to a[z], no matter what a[z], may be, so long as it is superior to a[x], and so long as the a's inferior to a[z] and superior to a[x] form an enumerable collection. 4. It has no minimum unless that minimum be the minimum of all the a's. 5. It has no maximum unless that maximum be an a which has no a next superior to it. 6. Unless the linear series happens to have a minimum and a maximum, it is itself an inenumerable collection of a's. Peirce: CP 4.108 Cross-Ref:†† 108. Having formed this linear sequence, if there be any a's not included in it, let a'[x] be one of them. We then proceed to form a second linear sequence in which a'[x] takes the place of a[x]. If, after that, there still be a's not included in the linear sequences already formed we proceed to form a next succeeding linear sequence, and so on indefinitely. The multitude of linear sequences may be inenumerable; but as necessary consequences of the rule for the formation of these sequences, the following propositions hold. Peirce: CP 4.108 Cross-Ref:†† First, The linear sequences are formed successively. If we take t to signify "already formed at a time --", then ~t$t -< ~t†$t that is "what is not yet formed at some time at which X was already formed," is included under "formed only at times at which X was already formed." Moreover 0†(1

~t$t

t~$t)†0.

That is, of two linear sequences not the same one was formed earlier than the other. Peirce: CP 4.108 Cross-Ref:†† Second, None of the linear sequences was ~t$t to the first. Peirce: CP 4.108 Cross-Ref:†† Third, Every sequence ~t$t, that is formed subsequent to another, was formed on the occasion of its having been found that some a's were left over not included in the previous sequences, and thus was not ~t$t~t$t, or was formed next subsequent to some other. Peirce: CP 4.108 Cross-Ref:†† Fourth, All the sequences formed previous to a given sequence, have a first and last, and are an enumerable collection. Peirce: CP 4.109 Cross-Ref:†† 109. It is well to remark, as a matter of language, that whenever a quantitative relation is applied to a class which has for that relation a minimum, and every member of the class superior to another in respect to this relation is next superior to some other, and the partial class consisting of all inferior to any given member is enumerable, then we can conveniently speak of the relationship as constituting an arithmoidal order in which the individuals are taken, the next superior being said to come next after, etc. Peirce: CP 4.110 Cross-Ref:†† 110. I propose now to show that all the a's can be embraced in such a serial order. But for this purpose, I must first establish a preliminary arithmoidal order in each of the linear sequences. The first sequence may have both a minimum and a maximum, and if so, it is enumerable. If it has a minimum but no maximum, the arithmoidal order will already exist, the minimum being the first. In every sequence which has a maximum, the arithmoidal order is established by simply considering the converse of the quantitative relation for which that maximum is a maximum. It remains only to establish an arithmoidal order in each of the sequences which has neither minimum nor maximum. This we do by taking arbitrarily any a of the sequence as first of the series, and for the one next after it, the a to which it is next superior, and thereafter the following rule is to be used; next after any a, as a[w], which is inferior to the a next preceding it, which we may call a[v], is to be taken the a next superior to a[v]; but next after any a, as a[u], which is superior to the a next preceding it, which we may call u[w], is to be taken the a next inferior to u[w]. The demonstration that this reduces the sequence to such a series is so easy that it may be omitted. We then take all the a's together in the following order: first, we take the whole of the first sequence if it is enumerable; next, we take the first a of the first inenumerable series; next, after any a of any inenumerable series we take the first a not already taken of the next inenumerable series, unless the last taken were the first of its series, when next after it we take the first not already taken of the first inenumerable series. The demonstration that this has the desired effect is sufficiently easy to be left to the student. Peirce: CP 4.110 Cross-Ref:†† Such an arithmoidal series is just like the series of positive whole numbers. I call it with reference to its grade of multitude, dinumerable. That is, it corresponds one to one to the numbers, yet the count of it cannot be completed. To such a series

applies the kind of reasoning called by me the Fermatian inference. This consists in proving a proposition to be true of such a series, because otherwise it must be false of an enumerable collection, such falsity, by reasoning on the principle of the part being less than the whole, being shown to be impossible. Fermat himself called it indefinite descent. He states his "manière de demonstrer," which he calls "une route tout à fait singulière" as consisting in showing that if the proposition to be proved were false of any number, it would be false of some smaller number. This statement shows a good comprehension of its nature.†1 Peirce: CP 4.111 Cross-Ref:†† 111. If we take all the whole numbers and write opposite to each the same figures in inverse order with a decimal point before them -- as, opposite 1894, for example, .4981 -- and then arrange the numbers in the order of these decimal fractions, we shall have established a quantitative relation according to which no number is next superior or next inferior to any other. A story is told of a bar of tin which being sent into Russia in the depths of winter, arrived in good order only that every atom had broken away from every other. If this tale did not serve to put money in somebody's pocket, it at least affords a pretty simile of the condition of the numbers when looked upon from that below-zero point of view. To bind them together after they are in their new order would require a multitude of new units inexpressibly more numerous relative to these numbers than the totality of them is to one. Peirce: CP 4.112 Cross-Ref:†† 112. If from the entire series of integer numbers, ranged in regular order, we imagine none, certain ones, or all to be omitted, we have what we may call a broken series, and the multitude of the entire collection of all such broken series possible is so great that they cannot be arranged by means of any quantitative relation so that whatever one is superior to another is next superior to some one. The proof which I offer of this is at bottom not mine. It seems to me sound, and if so is wonderful. In order to show that those broken series cannot themselves be arranged in an arithmoidal order, let us first arrange them in any quantitative order. This is easy, for each number may represent a place of decimals in the binary system of numeration, 1 the 1/2 place, 2 the 1/23 place, 3 the 1/23 place, and n the 1/2n place. The absence of a number being represented by zero, and its presence by 1, each series is represented by a binarial fraction, and these may be arranged in the order of their values. (Of course, the series could not be represented by integer numbers by reflection from the binarial point, because so they would all be infinite.) Now the question is whether the series of whole numbers can in any way whatever be made to correspond to these fractions. And in order that the matter may appear in a clearer light let us suppose that parallel to the series of infinite binarial fractions is ranged the entire series of values of rational fractions between 0 and 1, expressed in the same notation and set down in the same order. Let us first take a mode of correspondence which obviously will not fulfill the purpose, but which will serve to show the difference between such a series as that of the rational fractions and the series with which we are dealing. Suppose that the numbers correspond to the fractions in the order of their simplicity. Thus our first two fractions (I won't take the trouble to write them in binary notation) are: 1/3 .3333333333

1/2 0.5000000000;

between these the first two are

2/5 .4000000000

3/7 0.4285742857;

between these the first two are 7/17 .4117647059

5/12 0.4166666667;

between these the first two are 12/29 .4137931034

17/41 0.4146341463;

between these the first two are 41/99 .4141414141

29/70 0.4142857143;

between these the first two are 70/169 .4142011834 99/239 0.4142259414.

All these are found in both parallel rows; but they are converging toward 0.41421356 or √2-1 which is not a rational fraction. Peirce: CP 4.112 Cross-Ref:†† Now the fact that in this case the numbers happened to be rational fractions had nothing to do with the result. It is plain that in every case, when between two values we insert two, and between those two, two more again, and so on indefinitely, there remains a limit which is never reached; and the multitude which includes all such limits cannot be made to correspond, one to one, to any dinumerable collection.

Peirce: CP 4.113 Cross-Ref:†† §5. THE METHOD OF LIMITS

113. Let us settle the terminology as shown in this diagram

Enumerable | |

|-----------------| Dinumerable

|---v-----| Numerable

Inenumerable

| |

Innumerable

Peirce: CP 4.113 Cross-Ref:†† The relation of the Innumerable to the Dinumerable is analogous to that of the Dinumerable to the Enumerable. Dinumerable is the multitude of enumerable

numbers; innumerable is the multitude of dinumerable series. The dinumerable follows after the enumerable; but so closely after that as soon as you have passed all that is enumerable you have passed the dinumerable; so that we rightly reason such-and-such must be the character of the dinumerable, for if not there must be an enumerable which wants this character. In like manner the innumerable lies beyond the dinumerable; it is its limit; but it lies so closely beyond that we rightly reason, such-and-such must be the character of the innumerable; for if not, there must be a gap between this character and that of the dinumerable. Peirce: CP 4.113 Cross-Ref:†† All reasoning about the Innumerable derives its force from the conception of a limit. We therefore have to study this conception. But two or three prolegomena are called for. Peirce: CP 4.114 Cross-Ref:†† 114. The idea that there can be any vigorous and productive thought upon any great subject without reasoning like that of the differential calculus is a futile and pernicious idea. Some newspapers maintain that all doctrines involving such reasoning ought to be struck out of political economy because that science is of no service unless everybody, or the great majority of voters, individually comprehend it and assent to its reasonings. I do not observe that it is a fact that voters are such asses as to insist upon thinking they personally comprehend the effects of tariff-laws, etc. But whether they be so or not, it is certain that the ratio of the circumference to the diameter is 3.1415926535897932384626433832795028841971694 . . . whether the reasoning that proves it is hard or easy. That I feel sure of, although I personally have not verified the above figures; and if I had, I should not feel perceptibly more sure of the matter than I am. Certainly, if on attempting to verify them I got a slightly different result, I should feel pretty sure it was I who had committed an error. But whether people be wise or foolish, it remains that there is no possible way of establishing the true doctrines of political economy except by reasonings about limits, that is, reasoning essentially the same as that of the differential calculus. (I do not know why I should hesitate to say that the journal which I have particularly in mind is the New York Evening Post, incontestably one of the very best newspapers in the world, and especially remarkable for the sagacity of its judgments upon all questions of public policy.) Peirce: CP 4.115 Cross-Ref:†† 115. The reasoning of Ricardo about rent is this.†1 When competition is unrestrained by combination, producers will carry production to the limit at which it ceases to be profitable. Thus, a man will put fertilizers on his land, until the point is reached where, were he to add the least bit more, his little increased production would no more than just pay the increased expense. Every piece of land will be treated in this way, and every grade of land will be used down to the limit of the land upon which the product can just barely pay. Peirce: CP 4.115 Cross-Ref:†† The whole reasoning of political economy proceeds in this fashion. If we put an import duty upon any article, the price to the consumer cannot be raised by the full amount of the tax. For the price before the imposition was such as to sell a certain amount. Now, if the price is raised, less can be sold. If less can be sold, less will be produced. But production will only be diminished by the producer getting a less price; and it is this less price plus the duty that the consumer pays. Of course, we must understand by the duty, not merely what goes to the government, but what has to be

paid in consequence to brokers, bankers, and increased expenses of all kinds caused by the change of the law. Looking at the matter from this point of view (and abstracting from other considerations) the best articles on which to levy duties are those upon the production of which our demand is so influential that a small decrease in the demand will cause a relatively large fall in the price.†P1 Peirce: CP 4.116 Cross-Ref:†† 116. As another preliminary to the analysis of the conception of Limit, I now pass to a widely different topic. The student has not failed to remark how much I have insisted upon balance and symmetry in logic. It is a great point in the art of reasoning; although I do not know that one could say that logic requires it. As long ago as 1867 I spoke of a trivium of formal sciences of symbols in general. "The first," I said, "would treat of the formal conditions of symbols having meaning, and this might be called formal grammar; the second, logic, would treat of the formal conditions of the truth of symbols; and the third would treat of the formal conditions of the force of symbols, or their power of appealing to a mind, and this might be called formal rhetoric."†P1 It would be a mistake, in my opinion, to hold the last to be a matter of psychology. That which needs no further premisses for its support than the universal data of experience that we cannot suppose a man not to know and yet to be making inquiries, that I do not think can advantageously be thrown in with observational science. Each of these kinds of science is strong where the other is weak; and hence it is well to discriminate between them. Now, the Grundsatz of Formal Rhetoric is that an idea should be presented in a unitary, comprehensive, systematic shape. Hence it is that many a diagram which is intricate and incomprehensible by reason of the multitude of its lines is instantly rendered clear and simple by the addition of more lines, these additional lines being such as to show that those that were there first were merely parts of a unitary system. The mathematician knows this well. We have seen what endless difficulties there are with "some's" and "all's". The mathematician almost altogether frees himself from "some's"; for wherever something outlying and exceptional occurs, he enlarges his system so as to make it regular. I repeat that this is the prime principle of the rhetoric of self-communing. Nobody who neglects it can attain any great success in thinking. Peirce: CP 4.117 Cross-Ref:†† 117. The innumerable appears in two different shapes. In the first place, if we append to the entire series of finite integral numbers, which is a dinumerable collection, all the infinite numbers, we obtain an innumerable collection. Or, if we take the series of rational fractions, also a dinumerable collection, and add to them the limits of all infinite series of such fractions, then again we obtain an innumerable collection. In this latter case, each instance taken from the innumerable collection is a limit which may be passed through. This latter is a more balanced conception than the other; but the mathematician reduces the other to it by conceiving, that in the former case also, after passing through infinity as a mere point, we pass into a new region -- a new world. We pass off, for example, in a straight line parallel to the earth's axis northwardly: after passing through infinity we pass into an imaginary region from which after an infinite passage we re-emerge into our space at the extreme south. Or, it may be that this imaginary world reduces to nothing and that the points at infinity north and south coincide. This is the way the mathematician supplements facts in the interest of formal rhetoric. Of course, in doing so he has to take care not to misrepresent the real world; but his ideal addition to it may have any properties that simplicity dictates. This is an immense engine of thought in mathematics. It affords a little difficulty to the mind at first presentation; but that passes away very soon, and

then it is found to be greatly in the interest of comprehensibility. Every mathematician will tell you this; if you are not already aware of it. But even among mathematicians there is a trace of that human weakness, the stupidity of adhering to what ought to be obsolete; and consequently the idea that infinity is something to pass through has not been everywhere carried out. Peirce: CP 4.118 Cross-Ref:†† 118. In many mathematical treatises the limit is defined as a point that can "never" be reached. This is a violation not merely of formal rhetoric but of formal grammar. True, in the world of real experience, "never" has at least an approximate meaning. But in the Platonic world of pure forms with which mathematics is always dealing, "never" can only mean "not consistently with --." To say that a point can never be reached is to say that it cannot be reached consistently with --, and has no meaning until the blank is filled up. And thereupon, the mathematical and balanced conception must be that the point is instantly passed through. The metaphysicians have in this instance been clearer than the mathematicians -- and that upon a point of mathematics; for they have always declared that a limit was inconceivable without a region beyond it. Peirce: CP 4.118 Cross-Ref:†† I understand that Jordan has rewritten the first volume of his Cours d'analyse. I have not seen this new edition (for all my life my studies have been cruelly hampered by my inability to procure necessary books), but I can guess to some extent what the character of it will be; and it no doubt contains much, most pertinent to the subject now under our attention. It was, I presume, this work which suggested to Klein a remark which he makes in his Evanston Colloquium, to the effect that there is a distinction between the naïve and the refined geometrical intuition. "In imagining a line," he says, "we do not picture to ourselves length without breadth, but a strip of a certain width. Now such a strip has, of course, always a tangent; i.e., we can always imagine a straight strip having a small portion (element) in common with the curved strip."†1 The psychological remark seems to me incorrect. I, for my individual part, imagine a curve (even of an odd degree, which I convert into an even degree by doubling it, or by crossing it by a line) as the boundary between two regions pink and bluish grey; and I do not think I imagine the line as a strip. But it is of little consequence what individual ways of imagining may be. Klein's naïve view has a real importance far greater than his adjective imports, at which I have hinted in the Century Dictionary, under Limits, Doctrine of, where I say that this doctrine "should be understood to rest upon the general principle that every proposition must be interpreted as referring to a possible experience."†P1 What I mean is that absolute exactitude cannot be revealed by experience, and therefore every boundary of a figure which is to represent a possible experience ought to be blurred. If this is the case, it is both needless and useless to talk of infinitesimals. Still thought of this inexact kind (I mean upon these essentially inexact premisses) will be found much more intricate and difficult than the exact doctrine. Peirce: CP 4.119 Cross-Ref:†† 119. To define a limit, mathematicians usually write x[n], where x[1], x[2], x[3], etc. are supposed to successively approximate toward a value. Then they say that if after, perhaps, some scattering values, the successive x[n]'s at length come nearer and nearer to a constant which they indefinitely approach but

never reach, that quantity is the limit. By saying they never reach it, they mean that as the [n] of x[n] passes through infinity, x[n] passes through the limit. This n = ∞ of course marks the point at which the collection which n measures becomes dinumerable. At that point x[n] ceases to vary with n; else the value would be indeterminate. Peirce: CP 4.120 Cross-Ref:†† 120. I insert here a few remarks. The dinumerable is to the innumerable as logarithmic infinity is to ordinary infinity. The analogy may be traced in two ways; first the number of numbers expressible by n decimal points is, of course, bn where b is the base of the system of numeration; but the innumerable is the number of numbers expressible by dinumerable decimal points. In the second place, the innumerable is not only dinumerably more than the dinumerable but is innumerably more.

Peirce: CP 4.121 Cross-Ref:†† §6. THE CONTINUUM

121. It may be asked whether there be not a higher degree of multitude than that of the points upon a line. At first sight, the points on a surface seem to be more; but they are not so. For points on surfaces can be discriminated by two coördinates with values running to a dinumerable multitude of decimal places. Now two such numbers or any enumerable multitude of them can be expressed by one series of numbers. Thus to express two, write a number such that the succession of figures in the odd places of decimals gives one coördinate, and those in the even places, the other. Thus, u = 32.174118529821685238548599709435 . . . . will mean x = 3.141592653589793 . . . y = 2.718281828459045 . . . Peirce: CP 4.121 Cross-Ref:†† This method would break down if the number of dimensions were dinumerable; but even then another method could be found. But if the number of dimensions were innumerable, it is difficult to say without more study than I have given, how to proceed. The idea of space with innumerable dimensions does not, at first blush, at least, strike one as presenting great difficulty. Peirce: CP 4.121 Cross-Ref:†† But if bn when n is dinumerable gives a new grade of multitude, we might expect that when n was innumerable, a still higher grade would be given. Peirce: CP 4.121 Cross-Ref:†† Yet, on the other hand, looking at the matter from the point of view of the original definitions given above, the three classes of multitude seem to form a closed system. Still, nothing in those definitions prevents there being many grades of multiplicity in the third class. I leave the question open, while inclining to the belief that there are such grades.†1 Cantor's theory of manifolds appears to me to present certain difficulties; but I think they may be removed.

Peirce: CP 4.121 Cross-Ref:†† Let us now consider what is meant by saying that a line, for example, is continuous. The multitude of points, or limiting values of approximations upon it, is of course innumerable. But that does not make it continuous. Kant †2 defined its continuity as consisting in this, that between any two points upon it there are points. This is true, but manifestly insufficient, since it holds of the series of rational fractions, the multitude of which is only dinumerable. Indeed, Kant's definition applies if from such a series any two, together with all that are intermediate, be cut away; although in that case a finite gap is made. I have termed the property of infinite intermediety, or divisibility, the Kanticity of a series. It is one of the defining characters of a continuum. We had better define it in terms of the algebra of relatives. Be it remembered that continuity is not an affair of multiplicity simply (though nothing but an innumerable multitude can be continuous) but is an affair of arrangement also. We are therefore to say not merely that there can be a quantitative relation but that there is such, with reference to which the collection is continuous. Let l~$l denote this relation. Then, as quantitative, this has, as we have seen,†1 these properties: l~$l -< l†~$l, and 0†(1

l~$l

~l$l)†0.

Then the property of Kanticity consists in this: l~$l -< l~$l l~$l. Peirce: CP 4.122 Cross-Ref:†† 122. To complete the definition of a continuum, the a's, we require the following property. Namely, if there be a class of b's included among the a's but all inferior to a certain a, that is, if b -< a, 1 -< $a(l~$l†~b); and if further there be for each b another next superior to it; that is, 1 -< ~$b{l~$l·[~l†$l†(~b

~l†$l)]}†b,

then there is an a next superior to all the b's. That is, 1 -< $a{(l~$l†~b)·[~l†$l†(~l†$l)b]}. Peirce: CP 4.122 Cross-Ref:†† I call this the Aristotelicity of the series, because Aristotle seems to have had it obscurely in mind in his definition of a continuum as that whose parts have a common limit.†2 Peirce: CP 4.123 Cross-Ref:††

123. [Click here to view] If we consider a line (which, for rhetoric's sake, we will consider as returning into itself, though if it did not, it would give no difficulty further than an intolerably tedious complexity) it consists in a connection of points, such that by virtue of it, if any two points, A and , be taken on that line, the points are divided into two parts, say the a[0] and the a[∞], such that a certain continuous quantitative relation, say l[0], subsists between all the a[0]'s having A for minimum and for maximum, and another continuous quantitative relation, say l[∞], subsists between all the a[∞]'s having the same maximum and minimum. The student is invited to state this in algebraic form using π's, Σ's, and indices. He begins, for example

Σ[i]π[j]π[k] ~a[j] ~a[k] q[ij]·~q[ik] ~q[ij]·q[ik] l[0jk] l[0kj] [∞jk] l[∞kj].

To this I wish to add something, which seems to require a preliminary remark. There are certain quantitative relations between the points such that if one of them were to govern an arrangement of the points in space, it would derange their connection in a line, in this sense, that it would cause some four points which are connected in the cyclical order PQRS (=PSRQ) to be brought in to one of the two orders PQSR (=PRSQ) or PSQR (=PRQS). We will call such a relation, for short, incompatible. Of course, there is nothing to prevent its existing; only the points cannot be arranged according to this order and remain in their order in the line. I now say that by no compatible continuous quantitative order can we pass from any a[0] to any a[∞], without passing through A or . The student will do well to express this in terms of the algebra. Of course this statement requires modification in case the line forks. But for the purposes of logic it does not seem necessary to examine such details. Peirce: CP 4.124 Cross-Ref:†† 124. Pass we now to the study of the Surface. It is here that the mathematical conception of a "spread" -- as Clifford calls it in insular but expressive language -- at length displays itself. For an example of a surface, think of something irregularly round -- multiple-connected surfaces complicate the subject, without advantage. They are easily taken into account later and the modifications they require made. A surface

contains an innumerable multitude of lines. Let one of these -- a complete oval -- be marked upon it. Then the connection of points is such that this line separates those that do not lie upon it into two classes, such that it is impossible to pass from any point of one class to any point of the other by a compatible continuous quantitative order without passing through some point of the line. This is, however, perhaps not quite clear. Let us endeavor to make a better statement. Upon the oval take two points A and . Then by virtue of the connection of points on the surface there is an innumerable series of continuous quantitative orders, each beginning at A and ending at . Two of these, signified by the relative terms l[0] and l[∞], follow the two parts of the oval. These orders are such that no two of them embrace the same point, except the initial and final points which are common to them all. And all other lines (or compatible continuous quantitative orders), twice cutting the oval, cut these different lines in the same order, say l[0] . . l[n] . . l[∞] . . l[n] . . . l[0] . . . Every point on the surface lies on one of these lines.

Peirce: CP 4.125 Cross-Ref:†† §7. THE IMMEDIATE NEIGHBORHOOD

125. I wish to remark that it is a serious fault in the ordinary treatment of the fundamentals of geometry that attention is not paid to the distinction between the two sides of points on a line, lines on a surface, and surfaces in space. This is why certain theorems indubitably true are so difficult of formal proof. It is that a part of the fundamental properties of space have no expression among the Postulates of Geometry. Peirce: CP 4.125 Cross-Ref:†† I think that I have thus described the nature of the connection of points upon the surface -- and nothing need be added. Peirce: CP 4.125 Cross-Ref:†† But there are three very important ideas I have left undefined. I mean those of the simplest line (straight line on a plane, great circle on a sphere, perhaps the geodetic line on other surfaces), the immediate neighborhood, and measurement. I have also imaginary quantity still to consider. Peirce: CP 4.125 Cross-Ref:†† The easiest of these ideas seems to be that of the immediate neighborhood. It supposes that we recognize that every region stands in relation to a certain scale of quantity. We do not yet assign its quantity but we are able to say whether it is connected with an enumerable, a dinumerable, or an innumerable multitude. Two regions which are connected with quantities of the same class are said to be about alike. Suppose, then, we have a region about like the whole surface, or about like some region which we take as a standard. Suppose a thunder-bolt rends this into two parts about alike, a crack separating them. Suppose a second thunderbolt similarly rends both parts; and each successive thunderbolt rends all the parts the last left into two new parts about alike. Suppose these thunderbolts to follow at the completion of each rational fraction of a minute. Then, at the end of the minute, the region will be rent into innumerable parts about alike. These parts are neighborhoods or infinitesimals.

Peirce: CP 4.126 Cross-Ref:†† 126. It will at once be objected that there is no reason to suppose that this operation would leave any parts at all, or if it did there is no reason to suppose they would be surfaces rather than angels, or oranges, or precessions of the equinoxes; for the only reason for thinking they remained of the same genus is that no one thunderbolt would change them. Reasoning from that premiss, however, would be a Fermatian inference, and would, as such, only hold good for the dinumerable. Peirce: CP 4.126 Cross-Ref:†† But the reply is that there is no need of calling in the Fermatian inference. The minute of thunderbolts does not differ from any other minute, as far as the character of the surface goes. The parts have been moved a little, but all their mutual relations are undisturbed. Even if the operation broke it up into single points, which is an unfounded proposition, still all the cracks that have been made in no wise alter the relations of the points to one another. The space the region occupies, though interfiltrated through with another space, remains the same, and the relations of its parts the same. If this conception is too difficult, imagine that the thunderbolts do not rend the regions, but only cause a mind to imagine them rent. Peirce: CP 4.126 Cross-Ref:†† It would, however, be quite out of order to consider the question of whether these parts are single points or what their composition may be, until it first be fully admitted that the logical division of the region into innumerable parts is logically possible. But there is no room for dispute here. It has been irrefragably demonstrated that the points of a line, and a fortiori of a surface, are innumerable. Now, as no two coincide, there is nothing in logic to prevent their being drawn asunder. My definition of a continuum only prescribes that, after every innumerable series of points, there shall be a next following point, and does not forbid this to follow at the interval of a mile. That, therefore, certainly permits cracks everywhere; for there is no ordinal place in the series where such a limit point is not inserted. But if anybody thinks my definition is in error here, still it will not be maintained that that definition involves any contradiction. Hence, there is no contradiction in the separation into parts, even if I am wrong in saying that it involves no breach of continuity. There is no contradiction involved in breaking the region anywhere. But perhaps it may be said, the contradiction lies, not in breaking it anywhere, nor in breaking it into as many parts as it has points, but that the idea of an innumerable multitude involves a contradiction. That it does not can be formally demonstrated by second-intentional logic; but that part of this book having been excised, it is necessary to find other arguments. There is no difficulty about the existence of π, and therefore none in the existence of incommensurable limits. There is no more difficulty about the existence of any one number not accurately expressible in a finite number of decimals than in any other. Therefore, there is no logical contradiction in supposing all numbers to which decimals can indefinitely approximate to exist, i.e., as all the objects of mathematics exist, as abstract hypotheses. Besides, that the innumerable multitudes are logically possible is shown by the fact that many propositions (namely all that are true of the dinumerable but not of the numerable) cannot be demonstrated in a way which will stand logical examination unless it be expressly introduced as a premiss that a given multitude is numerable. Now a logically necessary proposition is of no avail as a premiss. On the whole then, there is nothing in logic to prevent a region from being divided into innumerable parts about alike.

Peirce: CP 4.126 Cross-Ref:†† Now I say that each of those parts contains innumerable points. For if that were not the case each of these parts could be so arranged that every point had another next after it; and, since a continuum has no molecular constitution, the divisions could everywhere be made between points having other points next them; and so, after rearranging the parts (no matter how the continuity might be broken up) all the points would have points next after them. But this is contrary to the fact that the points are innumerable. Besides, going back to the unanalyzed idea of continuity, it is evident that in a continuum the points are so connected that every part, irrespective of its magnitude, contains innumerable points. It may be objected that the single points are parts. But that is not properly true. The single points are parts of the collection; but they cannot be broken off by a division of parts unless they are on the outer boundary of a region, or unless they are not continuous with the rest. They can be extracted from the middle; but doing this breaks the continuity. Thus the incommensurable numbers taken by themselves do not form a continuum. Peirce: CP 4.127 Cross-Ref:††

127. [Click here to view] A drop of ink has fallen upon the paper and I have walled it round. Now every point of the area within the walls is either black or white; and no point is both black and white. That is plain. The black is, however, all in one spot or blot; it is within bounds. There is a line of demarcation between the black and the white. Now I ask about the points of this line, are they black or white? Why one more than the other? Are they (A) both black and white or (B) neither black nor white? Why A more than B, or B more than A? It is certainly true, Peirce: CP 4.127 Cross-Ref:†† First, that every point of the area is either black or white, Peirce: CP 4.127 Cross-Ref:†† Second, that no point is both black and white, Peirce: CP 4.127 Cross-Ref:†† Third, that the points of the boundary are no more white than black, and no more black than white.

Peirce: CP 4.127 Cross-Ref:†† The logical conclusion from these three propositions is that the points of the boundary do not exist. That is, they do not exist in such a sense as to have entirely determinate characters attributed to them for such reasons as have operated to produce the above premisses. This leaves us to reflect that it is only as they are connected together into a continuous surface that the points are colored; taken singly, they have no color, and are neither black nor white, none of them. Let us then try putting "neighboring part" for point. Every part of the surface is either black or white. No part is both black and white. The parts on the boundary are no more white than black, and no more black than white. The conclusion is that the parts near the boundary are half black and half white. This, however (owing to the curvature of the boundary), is not exactly true unless we mean the parts in the immediate neighborhood of the boundary. These are the parts we have described. They are the parts which must be considered if we attempt to state the properties at precise points of a surface, these points being considered, as they must be, in their connection of continuity. Peirce: CP 4.127 Cross-Ref:†† One begins to see that the phrase "immediate neighborhood," which at first blush strikes one as almost a contradiction in terms, is, after all, a very happy one. Peirce: CP 4.127 Cross-Ref:†† What is the velocity of a particle at any instant? I answer it is the ratio of space traversed to time of traversing, in the moment, or time in the immediate neighborhood, of that instant, or point of time. Some logicians object to this. They say that the velocity means nothing but the limiting value of the ratio of the space to the time when the time is indefinitely diminished. But they say they use the expression "immediate neighborhood" to mean nothing more than that, as a convenience of language. Sometimes we meet with an assertion difficult to refute because it involves several difficult logical blunders. The position just stated is an example of this. People who talk in this way do not see that what they say is a justification of the idea of a part such as the whole contains an innumerable multitude of. I do not yet say "immeasurably small," because we have not yet studied the conception of measurement. These people do not seem to have analyzed the conception of a "meaning,"†1 which is, in its primary acceptation, the translation of a sign into another system of signs, and which, in the acceptation here applicable, is a second assertion from which all that follows from the first assertion equally follows, and vice versa. It is true that, when we find with reference to a continuous motion that something would be true at the limit of a dinumerable series, it follows this is true for the part about the point considered. . . . This is as much as to say that the one assertion "means" the other. But do these people mean to say that when I think of a particle as having a velocity, I can only think, or that it is convenient to think, simply that at different times it is stationed at different points? Do they mean to say I have no direct, clear icon of a movement? If so, they are shutting their eyes to the plain truth. Remember it is by icons only that we really reason, and abstract statements are valueless in reasoning except so far as they aid us to construct diagrams. The sectaries of the opinion I am combating seem, on the contrary, to suppose that reasoning is performed with abstract "judgments," and that an icon is of use only as enabling me to frame abstract statements as premisses. Peirce: CP 4.127 Cross-Ref:†† The idea of "immediate neighborhood" is an exceedingly easy one, into which everybody is continually slipping, though he fancies it unjustifiable. Klein says of his

"refined intuition" that, strictly speaking, it is not an intuition. But, speaking as strictly as that, there is no such psychological phenomenon as an intuition.†1 The strip, which he says makes the curve in the naïve intuition, makes two parallel curves with a region between. But the simple idea is that of a blurred outline, to which we all, wise and simple, append the mental note that its breadth is such that an innumerable number would be contained in any surface. Peirce: CP 4.127 Cross-Ref:†† Those who, finding themselves betrayed into the use of the expression "immediate neighborhood" or something equivalent, seek to justify it by the exigencies of speech, are mistaken. It is not English grammar which forces these words upon them, but it is the very grammar of thought -- formal grammar -- which forces the idea upon them. The idea of supposing that they can think about motion without an image of something moving! Peirce: CP 4.127 Cross-Ref:†† We must return to this subject after having considered the nature of measurement.

Peirce: CP 4.128 Cross-Ref:†† §8. LINEAR SURFACES

128. Euclid †2 defines a straight line as a line which lies evenly between its points. This is the real Greek acuteness; it is as much as to say that if a straight line be moved, its new position intersects its old one in one point at most. This is substantially the idea of all modern geometry. Legendre,†3 it is true, defined the straight line as the shortest distance between two points, as it most indubitably is. Nor do I think that it would be fair to object that this definition is metrical, that is, supposes a definition of measurement. For all kinds of measurement known make the straight line the shortest (or the longest, sometimes, if there be a longest) distance, if there be a shortest distance. But a more serious objection to Legendre's definition is that, if that be adopted, its property of two straight lines not intersecting in two places follows as a consequence; while, if Euclid's definition be adopted, there must be a separate postulate to the effect that there is a shortest distance. Thus, Euclid's definition involves a more thorough analysis of the properties of space. Legendre conceived the other way, which wraps up as much as possible in one formula, to be the best. It certainly is not so for the purposes of logic. Peirce: CP 4.128 Cross-Ref:†† When instead of a plane we consider a roundish surface, it is difficult to say what sort of an oval best corresponds to a straight line. Most writers have assumed that it is the geodetic line which is the shortest (or longest) distance between its points. But they seem to have forgotten that a geodetic line on almost any surface but a perfect sphere generally intersects itself a dinumerable multitude of times. The discussion of this question would involve very difficult mathematics, quite out of place in this work. Peirce: CP 4.128 Cross-Ref:†† We must, therefore, confine ourselves to the plane. Now it is evident that the definition we have adopted supposes straight lines to move about in the plane without

ceasing to be straight. Hitherto, all the properties of the connection of points are such as might hold if the plane were a fluid; for though discontinuous fluid motion is conceivable, it has no place in the usual conceptions of the student of hydrokinetics. But now we propose that the straight line should move about as if it were a rigid bar. However, it is not necessary to broach the theory of elasticity, a doctrine of Satanic perplexity. We may call a straight line the path of a ray of light, or the shadow of a dark point cast from a luminous point. That is rather a pretty idea. Or going down to the roots of physics, we may define the straight line as the path of a particle, not deflected by any force. This is, so far as we can see, the origin of the importance of the straight line in the physical world. But, then, at present it is doubtful whether we are concerned at all with the physical world. We would like, if we could, to find some logical property of the straight line distinguishing it from other curves. I fear, however, there is none, if we are to leave its shortness out of account. We can perfectly well conceive of a cubic curve, such as is shown in the figure,

[Click here to view] moving about with modifications of its shape, so as in any position to cut any other position once and once only (in real space). A mathematician will easily write down the conditions for this. Namely, the equation of the serpentine is y = 1/(x+(1/x)), and that of the different cubics is x/a + {y-(1/(x+(1/x)))}/b=1. There is nothing in the plane geometry of the straight line which is not equally true, mutatis mutandis, of such a system of cubics. Peirce: CP 4.128 Cross-Ref:†† But the intersectional properties of straight lines in a plane are not exhausted in saying that any two straight lines intersect once and once only. Peirce: CP 4.129 Cross-Ref:†† 129. Let us resort to our algebra of relatives. Denote unlimited straight lines by lower case italic letters. Let capitals denote points. Let Greek minuscules denote certain marks of lines. All these letters are treated as indices; but they will be written on the line. Peirce: CP 4.129 Cross-Ref:†† Let aB (or any similar pair of letters) mean that the line a is considered as having the point B, which lies on it. If the point B is not on the line a, then ~(a B); but even if B be on a, it does not necessarily follow that B is regarded as belonging to the line a, and if not, again ~(a B). A point may belong to two lines, at once. Peirce: CP 4.129 Cross-Ref:†† Let ab (or any similar pair of letters) signify that the line b has the mark a, the

nature of which will appear in the sequel. Peirce: CP 4.129 Cross-Ref:†† Let aB, etc., signify that the point B belongs to some line that has the mark a. Peirce: CP 4.129 Cross-Ref:†† Let us now endeavor to sum up in a series of propositions the fundamental truths about the intersections of lines. Peirce: CP 4.129 Cross-Ref:†† First proposition. πAπBΣc cA·cB that is, any two points may be regarded as belonging to one straight line. Peirce: CP 4.129 Cross-Ref:†† Second proposition. παπβπc αc·βc, that is, given any two marks, an unlimited straight line having them both may be drawn. Peirce: CP 4.129 Cross-Ref:†† Third proposition. πaπbπCπD ~(a C)

~(a D)

~(b C)

~(bD)

1ab 1CD,

that is, if two points are regarded as belonging to two lines, either the two points or the two lines coincide. Peirce: CP 4.129 Cross-Ref:†† Fourth proposition. παπβπcπd ~(αc)

~(αd)

~(βc)

~(βd)

1αβ

1χδ,

τηατ ισ, ιφ τωο μαρκσ βελονγ το τωο λινεσ, ειτηερ τηε τωο μαρκσ αρε χοεξτενσιϖε ορ τηε τωο λινεσ χοινχιδε. Peirce: CP 4.129 Cross-Ref:†† Fifth proposition. πAπbΣ{g} {g}b·{g}A, that is, given any line, any point may be regarded as belonging to a line having a mark in common with the given line. Peirce: CP 4.129 Cross-Ref:†† Sixth proposition. παπbΣC αb·αC, that is, given any mark and any line, it is always possible to find a point which may be regarded as belonging to the given line and to some line having the given mark. Peirce: CP 4.129 Cross-Ref:††

Seventh proposition. παπβπcπD ~(αc)

~(αD)

~(βc)

~(βD)

cD 1αβ,

that is, if two marks belong to a given line and to lines to which a given point is regarded as belonging, that point must be regarded as belonging to that line, unless the two marks are coextensive. Peirce: CP 4.129 Cross-Ref:†† Eighth proposition. παπbπCπD ~(αC)

~(αD)

~(b C)

~(b D)Ψαb

1CD,

that is, if two points are regarded as belonging to a given line and to lines having a given mark, that line has that mark unless the two points coincide. Peirce: CP 4.129 Cross-Ref:†† Ninth proposition. παπbπcπdπE ~(αb)

~(αc)

~(αd)

~(bE)

~(cE)

~(dE)

1bc

1bd

1cd,

that is, any three lines either have no common point or no common mark. Peirce: CP 4.129 Cross-Ref:†† Tenth proposition. πbπcπdΣαΣE αb·αc·αd

bE·cE·dE †1†P1,

Peirce: CP 4.130 Cross-Ref:†† 130. . . . The student may object, at first blush, that the marks indicated by Greek letters have no meaning. This is a great mistake; they have precisely the meaning that is pertinent; but it is true they have no meaning in the sense of anything which particularly strikes ordinary attention. Reflect upon this. What people call an "interpretation" is a thing totally irrelevant, except that it may show by an example that no slip of logic has been committed.

Peirce: CP 4.131 Cross-Ref:†† 131. At this point, I should like to give some account of Schubert's calculus of enumerative geometry†2, which is the most extensive application of the Boolian algebra which has ever been made, and is of manifestly high utility. But I do not feel that I could possibly condense the elementary explanations or clarify them more than Schubert has himself done in his book. He has by no means exhausted the powers of his method. There is plenty of room for new researches; but his work will stand as the classical treatise upon geometry as viewed from the standpoint of arithmetic for an indefinite future.

Peirce: CP 4.132 Cross-Ref:†† §9. THE LOGICAL AND THE QUANTITATIVE ALGEBRA

132. Cauchy †3 first gave the correct logic of imaginaries, and very instructive

it is. But the majority of writers of textbooks, who reason by the rule of thumb, do not understand it to this day. The square of the imaginary unit, i, is -1, and therefore it may be allowable to speak of i and -i as being two square roots of -1. But to speak of them as the two square roots of -1 will not do. The algebraist sets out with a single continuous quantitative relation. But when he comes to quadratics he finds himself confronted with impossible problems. He says: "I want a square root of negative unity. Now there is no such thing in the universe: clearly, then, I must import it from abroad." Let us see how one would go about to prove there is no square root of negative unity. He would reason indirectly: that is the mathematician's recipe for everything. He would say let i be this square root if there be one. Then, whether its sign be + or -, its square will have a positive sign, contrary to the hypothesis. Then the whole impossibility depends upon this, that every quantity is supposed to be positive or negative. Suppose we make i neither positive nor negative. "But there is no such thing," some rule-of-thumb man says. Really? In that respect it is just like all the other objects the mathematician deals with. They are one and all mere figments of the brain.†1 "But to say that a quantity is neither positive nor negative means nothing," objects the thumbist. I reply, the meaning of a sign is the sign it has to be translated into. Now in mathematics, which is merely tracing out the consequences of hypotheses, to say a thing has no meaning is to say it is not included in our hypothesis. In that case, all we have to do is to enlarge the hypothesis and put it in. That is your course when you have a concrete hypothesis. That was our conduct when we called a debt, negative property. But, at present, we are dealing with algebra in the abstract. The only hypothesis we make is that our letters obey the laws of algebra. If there is one of those laws which requires a quantity to be either positive or negative, find out which it is and delete it. If you have a system of laws which is self-consistent, it will not be less so when one of them is wiped out. But let us see what the laws of algebra are and how they are affected toward a quantity whose square is negative. We have,

(1)If x = y, then x may anywhere be substituted for y. (2)x+y = y+x. (3)x+(y+z) = (x+y)+z. (4)xy = yx. (5)x(yz) = (xy)z. (6)(x+y)z = xz+yz. (7)x+0 = x. (8)x1 = x. (9)x+∞ = ∞. (10)If x+y = x+z, either y = z or x = ∞. (11)If xy = xz, either y = z, or x = 0, or x = ∞. (12)If x>y, not y>x. (13)If x>y, then there is a quantity a such that a>0 [and] a+y = x.

(14) If x>y, then x+z>y+z. (15) If x>0 and y>0, then xy>0. (16) Either x>y, or x = y, or y>x. (17) 1>0. It is plain that from these equations it is impossible to prove that x20 provided x>0. Also, if 0>x, let x+{x} = 0, by (13), where {x}>0. Then, by (6), x{x}+{x}2 = 0 But by (7), y+0 = y, and by (6), (y+0){x} = y{x}+0{x} = y{x}, and by (7), y{x}+0{x} = y{x}+0, and by (10), either 0{x} = 0 or y{x} = ∞ whatever y may be. But the last alternative is absurd; for then by (9), y0 = y(x+{x}) = yx+y{x} = yx+∞ = ∞. But if y = 1 by (8), y0 = 10 = 0. Hence we should have 0=∞ whence by (7) and (9), z=0=∞ whatever z may be. Hence by (1), If u>v v>u Hence by (12), in no case is u>v. But this contradicts (17). We have, then, 0{x} = 0.

Hence by (2) and (7), x{x}+{x}2 = 0, But by (15), {x}2>0. Hence by (14), x{x}+{x}2>x{x}+0. Hence by (7), x{x}+{x}2>x{x} or, 0>x{x}. But by (6) and (4), x(x+{x}) = x2+x{x} = x0 = 0x = 0. Hence, x2+x{x}>x{x}. Now since 0>x{x} by (13) there is a quantity a such that a>0, a+x{x} = 0 Hence, by (14), x2+x{x}+a>x{x}+a Hence, finally, by (3), x2+0>0 or by (7), x2>0. Hence, by (16), in every case x2>0 or x = 0. But it is plain that without (16) this conclusion could not be drawn, since no other of the formulæ (12)-(17) have anything to say about quantities neither greater, less, nor equal to one another. Peirce: CP 4.132 Cross-Ref:†† It thus appears that we have only to strike out (16), and the quantity i such that 2 i = -1, becomes perfectly possible, and perfectly conceivable, in the only clear sense of that word, namely, that we can write down i2+1 = 0

without conflict with any formula. If we define -x by the formula x+(-x) = 0, then, necessarily, if i2 = -1, we have also (-i)2 = -1. Ordinary algebra assumes there is no other quantity except these two whose square equals -1. Thus, if the algebraist finds x2 = y2 he at once writes x = ±y. This is because he chooses to exclude all other square roots of -1. I will return to this point shortly.†1 Peirce: CP 4.133 Cross-Ref:†† 133. Men are anxious to learn what the square root of negative unity means. It just means i2+1 = 0; precisely as -1 means 1+(-1) = 0. The algebraic system of symbols is a calculus; that is to say, it is a language to reason in. Consequently, while it is perfectly proper to define a debt as negative property, to explain what a negative quantity is, by saying that it is what debt is to property, is to put the cart before the horse and to explain the more intelligible by the less intelligible. To say that algebra means anything else than just its own forms is to mistake an application of algebra for the meaning of it.†P1 But to this statement a proviso should be attached. If an application of algebra consists in another system of diagrams having properties analogous to those of the sixteen fundamental formulæ, or to the greater number of them, and if that other system of diagrams is a good one to reason in, and may advantageously be taken as an adjunct of the algebraic system in reasoning, then such system of diagrams should be regarded as more than a subject for the application of logic, and though it is too much to say it is the meaning of the algebra, it may be conceived as a secondary, or junior-partner meaning. Such junior interpretations are especially, the logical and the geometrical. Peirce: CP 4.134 Cross-Ref:†† 134. Logical algebra ought to be entirely self-developed. Quantitative algebra, on the contrary, ought to be developed as a special case of logical algebra. I do not mean that elementary teaching should set it on that basis; but that should be recognized as the fundamental philosophy of it. The seminary logicians have often seemed to think that those who study logic algebraically entertain the opinion that logic is a branch of the science of quantity. Even if they did, the error would be a trifling one; since it would be an isolated opinion, having no influence upon the main results of their studies, which are purely formal. But with the possible exception of Boole himself, whose philosophical views have not been lauded by any of his followers, none of the algebraic logicians do hold any such opinion. For my part, I consider that the business of drawing demonstrative conclusions from assumed premisses, in cases so difficult as to call for the services of a specialist, is the sole business of the mathematician. Whether this makes mathematics a branch of logic, or whether it cuts off this business from logic, is a mere question of the classification of the sciences. I adopt the latter alternative, making the business of logic to be analysis and theory of reasoning, but not the practice of it. To show how reasoning about quantity may be facilitated by considering logical interpretations, I may instance the Enumerate Geometry of Schubert,†1 which works by means of the logical calculus,

and Mr. MacColl's †2 method of transposing the limits of multiple integrals, which is done by the Boolian algebra. Dr. Fabian Franklin has effected some difficult algebraical demonstrations by considering quantities as expressive of probabilities. I myself made two additions to the theory of multiple algebra by considering it as expressive of the logic of relatives.†3 Peirce: CP 4.135 Cross-Ref:†† 135. The idea of multiplication has been widely generalized by mathematicians in the interest of the science of quantity itself. In quaternions, and more generally in all linear associative algebra, which is the same as the theory of matrices, it is not commutative. The general idea which is found in all of these is that the product of two units is the pair of units regarded as a new unit. Now there are two senses in which a "pair" may be understood, according as BA is, or is not, regarded as the same as AB. Ordinary arithmetic makes them the same. Hence, 2X3 or the pairs consisting of one unit of a set of 2, say, I, J, and another unit of a set of 3, say X, Y, Z, the pairs IX, IY, IZ, JX, JY, JZ, are the same as the pairs formed by taking a unit of the set of 3 first, followed by a unit of the set of 2. So when we say that the area of a rectangle is equal to its length multiplied by its breadth, we mean that the area consists of all the units derived from coupling a unit of length with a unit of breadth. But in the multiplication of matrices, each unit in the Pth row and Qth column, which I write P:Q, of the multiplier coupled with a unit in the Qth row and Rth column, or Q:R gives (P:Q)(Q:R) = P:R or a unit of the Pth row and Rth column of the multiplicand. If their order be reversed, (Q:R)(P:Q) = 0, unless it happens that R = P. Peirce: CP 4.136 Cross-Ref:†† 136. In my earlier papers on the logic of relatives I made an application of the sign of involution †1 which, I am persuaded, is less special than it seems at first sight to be. Namely, I there wrote ls for the lover of every servant, while ls was the lover of some servant. lsm = (ls)m or the lover of everything that is servant to a man stands to every man in the relation of lover of every servant of his. lw m = lw·lm or the lover of everything that is either woman or man is the same as the lover of every woman and, at the same time, lover of every man. (l·b)m = lm·bm or that which is to every man at once lover and benefactor is the same as a lover of every man who is benefactor of every man. (e

c)f = ef

[f]ef-1*·c1*

([f]·([f]-1))/2·ef-2*·c2*

etc.

that is to say, those things each of which is to every Frenchman either emperor or conqueror consist first of the emperors of all Frenchmen; second, of a number of classes equal to the number of Frenchmen, each class consisting of all emperors of all Frenchmen but one who are at the same time conquerors of that one; third, of a number of classes equal to half the product of the number of Frenchmen by one less than the number of Frenchmen, each class consisting of every individual which is emperor of all Frenchmen but two and conqueror of those two; etc. Peirce: CP 4.136 Cross-Ref:†† This makes lm = l m[1]·l m[2]·l m[3]·l m[4]· etc. Of course, the ordinary idea which makes of involution the iteration of an operation, is a special case under this. Peirce: CP 4.136 Cross-Ref:†† Thus, quantitative algebra is only a special development of logical algebra. On the other hand, it is equally true that the Boolian algebra is nothing but the mathematics of numerical congruences having 2 for their modulus. Peirce: CP 4.137 Cross-Ref:†† 137. The geometrical interpretation affords great aid in reasoning, because man has, so to speak, a natural genius for geometry. Thus we see easily enough, algebraically, that _________ __________________

__________________

x+yi =√x2+y2 {√(x/y)/((x/y)+(y/x)) + √(y/x)/((x/y)+(y/x))·i}

and further that _____________________ (x+yi)(u+vi) = √(x2+y2)(u2+v2) _________________________________________________ {√((x/y)(u/v)-(y/x)(v/u))/((x/y)+(y/x))((u/v)+(v/u)) + ___________________________________________________ √((x/y)(v/u))+((y/x)(u/v))/((x/y)+(y/x))((u/v)+(v/u))·i}.

But that which is by no means obvious algebraically, but becomes obvious geometrically, is that when we plot x and y as abscissa and ordinate of rectangular coördinates and u, v as other values of the same coördinates, and the product in the same way, the angle from the axis of abscissas of the product is equal to the sum of those of the two factors. This once found out, in the geometrical way, is easily put into algebraical form. Geometry here renders a precisely similar service to that which the theory of probabilities often lends. There are several instances in which mechanical instincts have been valuable in the same way. A choice collection of such

lemmas would be interesting.

Peirce: CP 4.138 Cross-Ref:†† §10. THE ALGEBRA OF REAL QUATERNIONS

138. I now turn back to square roots of negative unity, not supposing multiplication to be commutative. That is, we do not generally have xy = yx. Suppose we have two quantities i and j, such that i2+1 = 0 j2+1 = 0 Then it is plain that (iji)(iji) = (ij)ii(ji) = -(ij)(ji) = -i·jj·i = ii = -1, so that iji and jij are also square roots of negative unity. Peirce: CP 4.138 Cross-Ref:†† Five cases may be studied:

First, iji = i Second,

iji = -i

Third, iji = j Fourth,iji = -j Fifth, iji = k (a third unit).

Peirce: CP 4.138 Cross-Ref:†† First Case. iji = i. Then, i·iji = -ji = ii -j = i.

Peirce: CP 4.138 Cross-Ref:†† Second Case. iji= -i. Then, i·iji = -ji = -ii j = i.

Peirce: CP 4.138 Cross-Ref:†† Third Case. iji = j. Then,

ijij = jj = - 1 jiji = jj = - 1 and ij and ji are also square roots of negative unity. iji·i = -ij = ji. But, (ij)i = j i(ij) = -j, equations that do not hold for ij = i nor for ij = j. Nor can we put ij = sin Θ.i+cos Θ.j For then, i.ij = -sin Θ+cos Θ.ij = -sinΘ+cosΘ sinΘ.i+cos2 Θ.j and we have cosΘ = √-1. Let us then write ij = k ji = -k Then, ki = iji = j ik = iij = -j kj = ijj = -i jk = -jji = i This is the algebra of quaternions.†1

Peirce: CP 4.138 Cross-Ref:†† Fourth Case.iji = -j. Then, iji.i = -ij = -ji orji = ij. Hence, sincei2 = j2 (i-j)(i+j) = 0 j = ±i.

Peirce: CP 4.138 Cross-Ref:†† Fifth Case.iji=k. The multiplication cannot be commutative. We may then have four infinite series of units

i[1] = i

j[1] = j

k1 = ij

l1 = ji

i[2] = iji

j[2] = jij

k2 = ijij

l2 = jiji

i[3] = ijiji

j[3] = jijij

k3 = ijijij

13 = jijiji

i[4] = ijijiji

j[4] = jijijij

k4 = ijijijij

14 = jijijiji

etc.

etc.

etc.

etc.

Here i[n] = ilm-1 = kn-1i j[n] = jkn-1 = ln-1j i[m]j[n] = km+n-1 j[m]i[n] = lm+n-1 It is possible to suppose these all different. Peirce: CP 4.138 Cross-Ref:†† If, on the other hand, any two are equal, there are but a finite number of different units. For example, if ijijiji = jij then, ijiji = jijij And all forms of more than five letters are equal to forms of quite as few as five letters. Thus, ijijij = jijij.j = -jiji. Peirce: CP 4.139 Cross-Ref:†† 139.†1 But the moment we suppose the number of linearly independent letters is finite we can reason as follows. Taking any expression, A, some power of it is a linear function of inferior powers. Hence, there is some equation Σ[m](a[m]Am)+a[0] = 0. By the theory of equations, this is resoluble into quadratic factors. One of these, then, must equal zero. Let it be (A-s)2+t2 = 0.

Then, ((A-s)/t)2 = -1 or, every expression, upon subtraction of a real number from it, can be converted, in one way only, into a square root, of a negative number. Let us call such a square root, the vector of the first expression, and the real number subtracted, the scalar of it. Peirce: CP 4.139 Cross-Ref:†† Let v2 = -1, j2 = -1, and ij = s+v, where s is scalar, and v vector. Then it is impossible to find three real numbers, a, b, c, such that v = a+bi+cj For assume this equation. Then, since ij.j = -i, -i = sj+vj = -c+(s+a)j+bij = bs-c+ab+b2i+(s+a+bc)j. Whence, b2 = -1, and b could not be real. Peirce: CP 4.139 Cross-Ref:†† Moreover, we shall have ji = r(s-v), where v is a real number. For write ji = s'+v', where s' is the scalar, and v' the vector, of ji. Let us write, too, vv' = s''+v'', where s'' and v'' are again the scalar and vector of vv'. Then, ij.ji = (s+v)(s'+v') = ss'+sv'+s'v+s''+v''. But ij.ji = 1. Hence, v'' = 1-ss'-s''-sv'-s'v. But it has just been proved that the vector of the product of two vectors is linearly independent of these vectors and of unity. Hence v'' = 0.

That is, sv' = 1-ss'-s''-s'v. Peirce: CP 4.139 Cross-Ref:†† But it has just been shown that a quantity can be separated into a scalar and a vector part in only one way. Hence sv' = -s'v s'' = 1-ss'. The former equation makes ji = (s'/s)(s-v). Let us next consider such an expression as ai+bj = S+V where S and V are the scalar and vector of the first member. Squaring the vector, we get V2 = -N = (ai+bj-S)2 = -a2-b2+S2+abs+abs'-2aSi-2bSj +ab(1-(s'/s))v or ab(1-(s'/s))v = -N+a2+b2-S2-abs-abs'+2aSi+2bSj. But since v is the vector of ij it must, as we have seen, be linearly independent of unity, i, and j. Hence the first member must vanish. But if v = 0, ij = s, whence -i = sj, contrary to hypothesis. Hence, 1-(s'/s) = 0 or s' = s. Whence, s'' = 1-s2. But s'' being the negative of the square of v is positive. Hence, s2⎥1. Peirce: CP 4.139 Cross-Ref:†† We know that ai+bj cannot be a scalar; for then a quantity could in two ways be resolved into a scalar and vector part. Now (ai+bj)2 = -a2-b2+2abs. This must therefore be negative. For (ai+bj) = S+V, and V does not vanish. Hence (ai+bj)2 = S2+V2+2SV;

and since, by comparison, it appears the vector part 2SV vanishes, it follows that S = 0, and the sum, or linear function, of two vectors is a vector. Peirce: CP 4.139 Cross-Ref:†† The same thing is evident because S2⎥1; whence -a2-b2+2abs = -p(a+b)2-(1-p)(a-b)2, where p = (1+s)/2 and 1-p = (1-s)/2, both of which are positive, or zero. Peirce: CP 4.139 Cross-Ref:†† Let us then assume a vector j, such that _______ j[1] = (si+j)/√(1-s2)

j[1]2 = (-s2-1+2s2)/(1-s2) = -1 _______

_______

ij[1] = (-s+s+v)/(√(1-s2) = v/√(1-s2) _______ j[1]i = -(v/√(1-s2)) _______ j[1] = v/√(1-s2) = j[1]ij[1] = -j[1]2i = i _______ v/√(1-s2)·j[1] = ij[1]j[1] = -i _______ i·v/√(1-s2) = iij[1] = -j[1] _______ v/√(1-s2)·i = -j[1]ii = j[1] _______ (v/√(1-s2))2 = -ij[1]j[1]i = -1 Peirce: CP 4.139 Cross-Ref:†† Writing j for j[1] and k for v/√(1-s2) and the above formulæ define the algebra of real quaternions. Peirce: CP 4.140 Cross-Ref:†† 140.†1 I will now prove that it is not possible to add to this a fourth linearly independent vector. For suppose [l] to be such a unit vector. Write jl = S'+V',

li = S''+V''. Substitute for l, l[1] = S''i+S'j+l Then, jl[1] = -S''k-S'+S'+V' = -S''k+V', l[1]j = S''k-S'+S'-V' = S''k-V', l[1]i = -S''-S'k+S''+V'' = -S''k+V'', il[1] = -S''+S'k+S''-V'' = S''k-V''. Let us further assume kl[1] = S'''+V''' Whence, l[1]k = S'''-V'''. But l[1]j = -jl[1] and l[1]i = -il[1]. Hence, kl = i.jl[1] = -i.l[1]j = -il[1].j = l[1]i.j = l[1].ij = lk. So, kl[1] = l[1]k or, kl[1]-l[1]k = 0 But we have seen that kl[1]-l[1]k = S'''+V''-S'''+V''' = 2V'''. Hence V''' = 0. Then these vectors are not linearly independent, and a fourth unit vector is impossible. Peirce: CP 4.140 Cross-Ref:†† But this proof does not apply when the multitude of linearly independent expressions is endless; such algebras are nonlinear. Peirce: CP 4.140 Cross-Ref:†† We thus see that even when we annul the commutative law of multiplication, there are but three linear algebras, real single algebra, ordinary imaginary algebra, and the algebra of real quaternions which obey all the other algebraic laws. The law which so limits the number is: Peirce: CP 4.140 Cross-Ref:†† If xy = xz, then y = z, unless x = 0 or x = ∞.

Peirce: CP 4.141 Cross-Ref:†† 141. In all other algebras this law fails and with it goes all semblance of importance for the inverse operation of division.†P1 The algebra of logic illustrates the vanity of that device for solving equations, which must on the contrary usually be solved by producing special known quantities by direct operations.

Peirce: CP 4.142 Cross-Ref:†† §11. MEASUREMENT

142. It was necessary to say something about imaginaries before coming to the subject of measurement since the modern theory of measurement (due to the researches of Cayley, Clifford, Klein, etc.) depends essentially upon imaginaries. Peirce: CP 4.142 Cross-Ref:†† Let us first consider measurement in one dimension. There is a certain absurdity in talking about measurement in one dimension. This is seen in the instance of time. Suppose we only knew the flow of our inward sensations, but nothing spread into two dimensions, how could one interval of time be compared with another? Certainly, their contents might be so alike that we should judge them equivalent. But that is not shoving a scale along. It does not enable us to compare intervals unless they happen to have similar contents. However, it is convenient to put that consideration aside, and to begin (with Klein) at unidimensional measurement. Peirce: CP 4.142 Cross-Ref:†† We are to measure, then, along a line. We will, for formal rhetoric's sake, conceive that line as returning into itself. We will, first, in order that we may apply numerical algebra, give a preliminary numbering to all the points of that line, so that every point has a number and but one number, and every real number, positive or negative, rational or surd, has a point and but one point, and so that the succession of any four numbers is the same as the succession on the line of the four corresponding points. Now, we must make a scale to shift along that line. We must imagine that we have a movable line which lies everywhere in coincidence with the fixed line, and which can be shifted. In the shifting, parts of it may become expanded or contracted, for we cannot tell whether they do or not unless we had some third standard to shift along to tell us; and then the same question would arise. But the continuity and succession of points shall not be broken in the shifting; and moreover, when the movable line has any one point brought back to coincidence with a former position, all the points shall be brought back. Now imagine all this extended to the imaginary numbers. Then, it is shown in the mathematical theory of functions, that if x be the number against which any point of the movable line falls in any one position and y be the number the same point falls against in any other position, it follows, because for each value of x there is just one value of y and for each value of y just one value of x, that x and y are connected by an equation linear in each, that is, an equation of the form xy+Ax+By+C = 0. This gives y = -(Ax+C)/(x+B).

Now this is a function which forms the subject of some very beautiful and simple algebraical studies.†P1 It is convenient to put A = B-α-β C = B2-(α+β)B+αβ. Then y = ((α+β-B)x-B2+(α+β)B-αβ)/(x+B)

= (((α+β)x+(α+β)B-αβ)/(x+B))-B

= (((α2-β2)(x+B)-αβ(α-β))/((x+B)(α-B)))-B

= (((x+B-β)α2-(x+B-α)β2))/((x+B-β)α1-(x+B-α)β1)))-B

But x = ((x+B-β)α1-(x+B-α)β1)/((x+B-β)α0-(x+B-α)β0)-B. So that the effect of the shifting has been to raise the exponents of α and β by 1. Peirce: CP 4.142 Cross-Ref:†† It is easily proved that the same operation, performed any number t times, gives ((x+B-β)αt+1-(x+B-α)βt+1)/((x+B-β)αt-(x+B-α)βt)-B If α has a modulus greater than that of β, it is easily seen that when t becomes a very large positive number, the first terms of numerator and denominator will become indefinitely greater than the second terms and the value will indefinitely approximate to α-B. But when t is a very large negative quantity, the reverse will occur, and the value will approximate toward β-B. Peirce: CP 4.143 Cross-Ref:†† 143. If we look at the field of imaginary quantity, what we shall see is shown in the diagram.

[Click here to view] Here we have a stereographic projection of the globe. At the south pole is β-B; at the north pole, α-B. The parallels are not at equal intervals of latitude but are crowded together infinitely about the pole. Now an increase of t by unity carries a point of the scale along a meridian from one parallel to the one next nearer the north pole. But an addition to t of an imaginary quantity carries the point of the scale round along a parallel. Peirce: CP 4.143 Cross-Ref:†† If the real line of the scale lies along a meridian all real shiftings of it will crowd its parts toward the north or the south pole; and the distance of either pole, as measured by the multitude of shiftings required to reach it, is infinite.†P1 The scale is limited, but immeasurable. Peirce: CP 4.143 Cross-Ref:†† But if the real line of the scale lies along a parallel, real shiftings, that is shiftings from real points to real points, will carry it round, so that a finite number of shiftings will restore it to its first position. Such is the scale of rotatory displacement. It is unlimited, but finite, or measurable. A scale of measurement, in the sense here defined, cannot be both limited and finite. We seem to have such a scale in the measurement of probabilities. But it is not so. Absolute certainty, or probability 0 or probability 1 are unattainable; and therefore, the numbers attached to probabilities do not constitute any proper scale of measurement, which can be shifted along. But it is possible to construct a true scale for the measurement of belief.†P2 It was a part of the definition of a scale that in all its shiftings it should cover the whole of the line measured. ("For every point of the line a number of the scale in every position.") Hence the shifting can never be arrested by abuttal against a limit. If there is a limit, it must be at an immeasurable distance. Peirce: CP 4.144 Cross-Ref:†† 144. But there is a special case of measurement, very different from the one considered. Namely, it may happen that the nature of the shifting is such that [given] the equation xy+Ax+By+C = 0, where A, B, C, may have any values, real or imaginary, we have C = 1/4(A+B)2. Substituting this in the expressions for A and C in terms of α, β, and B, we get

1/4(α+β)2 = αβ or α = β. This necessitates an altogether different treatment. In this case, we have y = -((Ax+(1/4)(A+B)2)/(x+B)) 2y = -(A+B)+((B-A)(2x+A+B)/((B-A)+(2x+A+B))) 2x = -(A+B)+((B-A)(2x+A+B)/((B-A)+0(2x+A+B))) And t shifts give -(A+B)+((B-A)(2x+A+B)/((B-A)+t(2x+A+B))). This gives for t = ±∞, -(A+B). The scale is in this case then unlimited and immeasurable. This is the manner in which the Euclidean geometry virtually conceives lengths to be measured; but whether this method accords precisely with measurement by a rigid bar is a question to be decided experimentally, or irrationally, or not at all. Peirce: CP 4.145 Cross-Ref:†† 145. The fixed limits of measurement are very appropriately termed by mathematicians the Absolute.†P1 It is clear that even when measurement is not practical, even when we can hardly see how it ever can become so, the very idea of measuring a quantity, considerably illuminates our ideas about it. Naturally, the first question to be asked about a continuous quantity is whether the two points of its absolute coincide; if not, a second less important, but still significant question is whether they are in the real line of the scale or not. These are ultimately questions of fact which have to be decided by experimental indications; but the answers to them will have great bearing on philosophical and especially cosmogonical problems. Peirce: CP 4.146 Cross-Ref:†† 146. The mathematician does not by any means pretend that the above reasoning flawlessly establishes the absolute in every case. It is evident that it involves a premiss in regard to the imaginary points which only indirectly relates to anything in visible geometry, and which, of course, may be supposed not true. Nevertheless, the doctrine of the mathematical absolute holds with little doubt for all cases of measurement, because the assumptions virtually made will hardly ever fail. Peirce: CP 4.147 Cross-Ref:†† 147. When we pass to measurement in several dimensions, there seems to be little difference between one number of dimensions and another; and therefore we may as well limit ourselves to studying measurement on a plane, the only spatial spread for which our intuition is altogether effortless. Peirce: CP 4.147 Cross-Ref:†† Radiating from each point of the plane is a continuity of lines. Each of these has upon it its two absolute points (possibly imaginary, and even possibly coincident); and assuming these to be continuous, they form a curve which, being cut in two points only by any one line, is of the second order. That is, it is a conic section, though it may be an imaginary or even degenerate one.

[Click here to view] Now as the foot has different lengths in different countries, so the ratios of units of lengths along different lines in the plane is somewhat arbitrary. But the measurement is so made that first, every point infinitely distant from another along a straight line is also infinitely distant along any broken line; and second, if two straight lines intersect at a point, A, on the absolute conic and respectively cut it again at B and C; and from D, any point collinear with B and C, two straight lines be drawn, the segments of the first two lines, EF and GH, which these cut off, are equal. I omit the geometrical proof that this involves no inconsistency. This proposition enables us to compare any two lengths. Peirce: CP 4.148 Cross-Ref:†† 148. We now have to consider angular magnitude. In the space of experience, the evidence is strong that, when we turn around and different landscapes pass panorama-wise before our vision we come round to the same direction, and not merely to a new world much like the old one. In fact, I know of no other theory for which the evidence is so strong as it is for this. But it is quite conceivable that this should not be so; there might be a world in which we never could get turned round but should always be turning to new objects. But certain conveniences result from assuming for the measurement of the angles between lines the same absolute conic which is assumed as the absolute of linear measure. Thus, it is assumed that two straight lines meeting at infinity have no inclination to one another, just as it is assumed that in a direction such that the opposite infinities should coincide, all other points would have no distance from one another. The latter is another way of saying that if a point is at an infinite distance from another point on a straight line, it is so on a broken line. The other assertion is that if an infinite turning is requisite to reach a line from one centre, it is equally so if you attempt to reach it by turning successively about different centres. The analogue of the proposition for which the last figure was drawn is as follows:

[Click here to view] Upon a line, a, tangent to the absolute let two points be taken from which the other tangents to the absolute are b and c. Through the intersection of b and c draw any line, d, then any two lines e and f, meeting at the intersection of a and b, make the same angle with one another as two other lines having the same intersections with d, and cutting one another at the intersection of a and c. This enables us to compare all angles. Peirce: CP 4.149 Cross-Ref:†† 149. Suppose a man to be standing upon an infinitely extended plane free from all obstructions. Would he see something like a horizon line, separating earth from sky, being the foreshortened parts of the plane at an infinite distance? If space is infinite, he would. Now suppose he sets up a plane of glass and traces upon it the projection of that horizon, from his eye as a centre. Would that projection be a straight line? Euclid virtually says, "Yes." Modern geometers say it is a question to be decided experimentally. As a logician, I say that no matter how near straight the line may seem, the presumption is that sufficiently accurate observation would show it was a conic section. We shall see the reason for this, when we come to study probable inference.

[Click here to view] Let us suppose, then, that the horizon is not a straight line upon a level with the eye, but is a small circle below that level. If then two straight lines meet at infinity, their other ends must be infinitely distant; but the angle between them is null. Hence, there may be a triangle having all its angles null, and all its sides infinite. Let us assume (what might, however, be proved) that two triangles, having all the sides and angles of the one respectively and in their order equal to those of the other, are of equal area. Then all triangles having the sum of their angles null are of equal area. Call this T. Then the area of an ordinary polygon of V vertices all on the absolute is (V-2)T. The area of the absolute is therefore infinite.

[Click here to view] If a triangle has two angles null and the third 1/N part of 180•, what is its area? Let ABD be the triangle, AB being on the absolute. Continue BD the absolute at C. Let ADE, EDF, etc., be N-1 triangles having their angles at D all equal to ADB. Then these N triangles are all equal, because their sides and angles are equal. They make a polygon of N+1 vertices on the absolute, the area of which is (N-1)T. Hence, the area of each triangle is (1-1/N)T.

[Click here to view] What is the area of a triangle having one angle zero and the others mX180• and nX180•? Let AMN be such a triangle; extend MN on the side of N to the absolute at B. Then the area of ABM is (1-m)T and that of ABN is nT. Hence the area of AMN, which is their difference, is (1-m-n)T.

[Click here to view] What is the area of a triangle having its three angles equal to lX180•, mX180•, and nX180•? Let LMN be such a triangle. Produce LM on the side of M to the absolute at A and join AN. Then if the angle MNA is put equal to x.180•, the area of ALN is (1-l-n-x)T, while that of AMN is (m-x)T. Hence, that of LMN, which is their difference, is (1-l-m-n)T. Peirce: CP 4.149 Cross-Ref:†† Thus, the area of a triangle is proportional to the amount by which the sum of its angles falls short of two right angles. Of course, this does not forbid that amount being infinitely small for all triangles whose sides are finite. Peirce: CP 4.150 Cross-Ref:†† 150.

[Click here to view] The above reasoning may appear to be fallacious because it forgets that subtraction is not applicable to infinites. But it does not fall into that error. I may remark, however, that subtraction is applicable to infinites, in case their transformations are so limited that x+y cannot equal x+z unless y = z. For instance, we have considered the triangle ABC having two vertices on the absolute. This triangle is finite. But we might perfectly well reason about the infinite sector ABC provided this sector be not allowed to vary so as to change the area of the triangle, and provided, further, that we always add to each sector, BAC, its equal vertical sector B'AC'. Peirce: CP 4.150 Cross-Ref:†† Looking at a triangle from this point of view, we see that the sum of the six sectors (two for each angle) is twice three times the triangle plus all the rest of the plane, or twice the area of the triangle plus the whole plane. The whole plane is four right angled sectors. But we have thus reckoned together with the sectors of the angles of the triangle their equal vertical sectors. Dividing by two, we find the sum of the angular sectors of a triangle is two right angles plus the area of the triangle. Peirce: CP 4.150 Cross-Ref:†† Now since the sector is proportional to its angle, and since further, for the largest possible sector the angle is zero, it follows that the sector is equal to the negative of the angle, whence we find

area {D} = 2 -Σ Angles. Peirce: CP 4.151 Cross-Ref:†† 151. . . . Such are the ideas which the mathematician is using every day. They are as logically unimpeachable as any in the world; but people, who are not sure of their logic, or who, like many men who pride themselves on their soundness of reason, are totally destitute of it, and who substitute for reasoning an associational rule of thumb, are naturally afraid of ideas that are unfamiliar, and which might lead them they know not whither. Peirce: CP 4.151 Cross-Ref:†† As compared with imaginaries, with the absolute, and with other conceptions with which the mathematician works fearlessly -- because good logicians, the Cauchys and the like,†P1 have led the way -- as compared with these, the idea of an infinitesimal is exceedingly natural and facile. Yet men are afraid of infinitesimals, and resort to the cumbrous method of limits. This timidity is a psychological phenomenon which history explains. But I will not occupy space with that here.†1 Peirce: CP 4.151 Cross-Ref:†† It was Fermat, a wonderful logical and still more wonderful mathematical genius, whose light was almost extinguished by the bread-and-butter difficulties which the secret plotting of worldlings forced upon him, who first taught men the method of reasoning which lies at the bottom of all modern science and modern wealth, the method of the differential calculus.†2 He gave a variety of instructive examples, did this lawyer, this "conseiller de minimis," as the jealous Descartes was base enough to call him, joining himself to the "born missionaries" who were determined to "head off" this hope of mankind. But the first and simplest of them is the solution of the problem to divide a number, a, into two parts so that their product shall be a maximum. Let the parts be x and a-x. Let e be a quantity such that a+e is "adequal" or {parisos}, say perequal to a. Then, the product being a maximum is at the point when increase of x ceases to cause it to increase. Hence Fermat writes x(a-x) = (x+e)(a-x-e) which gives 0 = -xe+e(a-x-e) or 0 = e(a-2x-e) Fermat now divides both sides by e (which assumes e is not zero). Whence 0 = a-2x-e. But a-2x-e is "adequal" to a-2x; and the e may consequently be "elided." Thus we get 0 = a-2x, or x = (1/2)a. Peirce: CP 4.151 Cross-Ref:††

The peculiar properties of e, which we now call, after Leibniz, the infinitesimal, are: Peirce: CP 4.151 Cross-Ref:†† First, that if pe = qe, then p = q, contrary to the property of zero; while Peirce: CP 4.151 Cross-Ref:†† Second, that, under certain circumstances, we treat e as if zero, writing p+e = p. Peirce: CP 4.151 Cross-Ref:†† Of course, we cannot adopt the last equation without reservation. For it would follow that e = 0, whence, since 4X0 = 5X0, 4e = 5e, and then by the first property, 4 = 5. Peirce: CP 4.151 Cross-Ref:†† The method of indivisibles †P1 had recognized that infinitely large numbers may have definite ratios, so that division is applicable to them. Peirce: CP 4.152 Cross-Ref:†† 152. The simplest way of defending the algebraical device is to say that e represents a quantity immeasurably small, that is, so small that the Fermatian inference does not hold from these quantities to any that are assignable. That no contradiction is involved in this has been shown in the former part of this chapter.†1 In the sense of measurement, then, p+e = p, while from a formally logical point of view, it is assumed that e>0. This is the most natural way, a perfectly logical way, and the way the most consonant with modern mathematics. Peirce: CP 4.152 Cross-Ref:†† It is also possible to conceive the reasoning to represent the following. (The problem is the same as above.) Let x be the unknown. Then, since x(a-x) is a maximum, x(a-x)>(x+e)(a-x-e) for all neighboring values of e. That is 0>e(a-2x-e). Then the sign of a-2x-e is opposite to that of e no matter what the value of e. It follows that 2x differs from a by less than any assignable quantity. Peirce: CP 4.152 Cross-Ref:†† The great body of modern mathematicians repudiate infinitesimals in the above literal sense, because it is not clear that such quantities are possible, or because

they cannot entirely satisfy themselves with that mode of reasoning. They therefore adopt the method of limits, which is a method of establishing the fundamental principles of the differential calculus. I have nothing against it, except its timidity or inability to see the logic of the simpler way. Let x be a variable quantity which takes an unlimited series of values x[1], x[2], . . . x[n], so that n will be a variable upon which x[n], depends. If, then, there be a quantity c such that x[∞] = c, that is, as the mathematicians prefer to say, in order to avoid speaking of infinity, if for every positive quantity e sufficiently small, there be a positive quantity {n} such that for all values of n greater than {n} Modulus (x[n]-c)<e then c is said to be the limit of x. Peirce: CP 4.152 Cross-Ref:†† Upon this definition is raised quite an imposing theory about limits which I can only regard with admiration, when it is erected with modern accuracy. Only, I wish to point out that the need for such a definition is not limited to its application to n = ∞, nor because infinity presents peculiar difficulties. It is only because ∞ is not an assignable number with which we can perform arithmetical processes. Let the function x[n] = n2, then the same difficulty arises when n = π, and the same definition of a limit is called for. Peirce: CP 4.152 Cross-Ref:†† The differential calculus deals with continuity, and in some shape or other, it is necessary to define continuity. I accept the above definition, with unimportant modifications, as a good definition of continuity. From it, as it appears to me, the idea of infinitesimals follows as a consequence; but, if not, no matter -- so long as the algebraic expression of the infinitesimal be accepted, which is really the essential point. Infinitesimals may exist and be highly important for philosophy, as I believe they are. But I quite admit that as far as the calculus goes, we only want them to reason with, and if they be admitted into our reasoning apparatus (which is the algebra) that is all we need care for.

Peirce: CP 4.153 Cross-Ref:†† V A THEORY ABOUT QUANTITY†1P

§1. THE CARDINAL NUMERALS

153. Quantity presents certain metaphysical difficulties, to appreciate which it would be as hopeless a task to bring this generation as to bring it to a sense of sin. Medieval doctors apprehended such points of logic far more clearly. Why a mass

distant one yard from a pound of matter should gravitate toward that pound by precisely that fraction of an inch per second that it does, neither more nor less -- how it is possible that the exact value of this quantity ever should be explained and brought under the manifest governance of that unyielding and universal law which is supposed to regulate all facts, how this can be when the general properties of an inch are nowise different from a mile, is a question which those philosophers who oppose my tychism †2 would find a puzzling one, could they once be brought to understand what this question is. My present purpose, however, is not to discuss this problem in its entirety, but merely to follow out, in a rambling spirit, a pretty little opening of thought suggested by an objection to a part of my solution of that problem. Peirce: CP 4.154 Cross-Ref:†† 154. That part of my solution is that Quantity is merely the mathematician's idealization of meaningless vocables invented for the experimental testing of orders of sequence. In our experience we often have occasion to remark that something is true of two or more things -- say, for example, that one thing eats another, or that one day is pleasanter than another. A fact true of several subjects is called a "relation" between them. A fact true of a pair of subjects (as in the examples just suggested) is a "dyadic relation." Of some kinds of dyadic relation we find that if one thing, A, be so related to a second, B, while B is so related to a third thing, C, then A is always related in that same way to C. When this is so, the relation is said to be "transitive." We also call it a "succession" or "sequence." Now I hold that numbers are a mere series of vocables serving no other purpose than that of expressing such transitive relations, or, at least, no other purpose except one whose accomplishment is necessarily involved in that. I admit that our senses may inform us, not merely that A is heavier than B, but that A is a great deal more heavier than B, than C is heavier than D; and undoubtedly numbers do serve to express this verdict of sense with greater precision than sense can render it. But this, as I think, is a use of numbers which necessarily results from their primary use; the judgment, that one thing is much heavier than another, being a mere complexus of judgments each that one thing is heavier by a unit than another. Peirce: CP 4.155 Cross-Ref:†† 155. When a number is mentioned, I grant that the idea of a succession, or transitive relation, is conveyed to the mind; and insofar the number is not a meaningless vocable. But then, so is this same idea suggested by the children's gibberish "Eeny, meeny, miney, mo," Yet all the world calls these meaningless words, and rightly so. Some persons would even deny to them the title of "words," thinking, perhaps, that every word properly means something. That, however, is going too far. For not only "this" and "that," but all proper names, including such words as "yard" and "metre" (which are strictly the names of individual prototype standards), and even "I" and "you," together with various other words, are equally devoid of what Stuart Mill †1 calls "connotation." Mr. Charles Leland informs us that "eeny, meeny," etc. are gipsy numerals.†2 They are certainly employed in counting nearly as the cardinal numbers are employed. The only essential difference is that the children count on to the end of the series of vocables round and round the ring of objects counted; while the process of counting a collection is brought to an end exclusively by the exhaustion of the collection, to which thereafter the last numeral word used is applied as an adjective. This adjective thus expresses nothing more than the relation of the collection to the series of

vocables. Peirce: CP 4.156 Cross-Ref:†† 156. Still, there is a real fact of great importance about the collection itself which is at once deducible from that relation; namely, that the collection cannot be in a one-to-one correspondence with any collection to which is applicable an adjective derived from a subsequent vocable, but only to a part of it; nor can any collection to which is applicable an adjective derived from a preceding collection be in a one-to-one correspondence with this collection, but only with a part of it; while, on the other hand, this collection is in one-to-one correspondence with every collection to which the same numeral adjective is applicable. This, however, is not essentially implied as a part of the significance of the adjective. On the contrary, it is only shown by means of a theorem, called "The Fundamental Theorem of Arithmetic,"†1 that this is an attribute of the collections themselves and not an accident of the particular way in which they have been counted. Nevertheless, this is a complete justification for the statement that quantity -- in this case, multitude, or collectional quantity -- is an attribute of the collections themselves. I do not think of denying this; nor do I mean that any kind of quantity is merely subjective. I am simply not using the word quantity in that acception. I am not speaking of physical, but of mathematical, quantity. Peirce: CP 4.157 Cross-Ref:†† 157. Were I to undertake to establish the correctness of my statement that the cardinal numerals are without meaning, I should unavoidably be led into a disquisition upon the nature of language quite astray from my present purpose. I will only hint at what my defence of the statement would be by saying that, according to my view, there are three categories of being; ideas of feelings, acts of reaction, and habits.†2 Habits are either habits about ideas of feelings or habits about acts of reaction. The ensemble of all habits about ideas of feeling constitutes one great habit which is a World; and the ensemble of all habits about acts of reaction constitutes a second great habit, which is another World. The former is the Inner World, the world of Plato's forms. The other is the Outer World, or universe of existence. The mind of man is adapted to the reality of being. Accordingly, there are two modes of association of ideas: inner association, based on the habits of the inner world, and outer association, based on the habits of the universe.†1 The former is commonly called association by resemblance; but in my opinion, it is not the resemblance which causes the association, but the association which constitutes the resemblance. An idea of a feeling is such as it is within itself, without any elements or relations. One shade of red does not in itself resemble another shade of red. Indeed, when we speak of a shade of red, it is already not the idea of the feeling of which we are speaking but of a cluster of such ideas. It is their clustering together in the Inner World that constitutes what we apprehend and name as their resemblance. Our minds, being considerably adapted to the inner world, the ideas of feelings attract one another in our minds, and, in the course of our experience of the inner world, develop general concepts. What we call sensible qualities are such clusters. Associations of our thoughts based on the habits of acts of reaction are called associations by contiguity, an expression with which I will not quarrel, since nothing can be contiguous but acts of reaction. For to be contiguous means to be near in space at one time; and nothing can crowd a place for itself but an act of reaction. The mind, by its instinctive adaptation to the Outer World, represents things as being in space, which is its intuitive representation of the clustering of reactions. What we call a Thing is a cluster or habit of reactions, or, to use a more familiar phrase, is a centre of forces.†2 In consequence, of this double

mode of association of ideas, when man comes to form a language, he makes words of two classes, words which denominate things, which things he identifies by the clustering of their reactions, and such words are proper names, and words which signify, or mean, qualities, which are composite photographs of ideas of feelings, and such words are verbs or portions of verbs, such as are adjectives, common nouns, etc. Peirce: CP 4.158 Cross-Ref:†† 158. Thus, the cardinal numerals in being called meaningless are only assigned to one of the two main divisions of words. But within this great class the cardinal numerals possess the unique distinction of being mere instruments of experimentation. "This" and "that" are words designed to stimulate the person addressed to perform an act of observation; and many other words have that character; but these words afford no particular help in making the observation. At any rate, any such use is quite secondary. But the sole uses of the cardinal numbers are, first, to count with them, and second, to state the results of such counts. Peirce: CP 4.159 Cross-Ref:†† 159. Of course, it is impossible to count anything but clusters of acts, i.e., events and things (including persons); for nothing but reaction-acts are individual and discrete. To attempt, for example, to count all possible shades of red would be futile. True, we count the notes of the gamut; but they are not all possible pitches, but are merely those that are customarily used in music, that is, are but habits of action. But the system of numerals having been developed during the formative period of language, are taken up by the mathematician, who, generalizing upon them, creates for himself an ideal system after the following precepts.

Peirce: CP 4.160 Cross-Ref:†† §2. PRECEPTS FOR THE CONSTRUCTION OF THE SYSTEM OF ABSTRACT NUMBERSP

160. First, There is a relation, G, such that to every number, i.e., to every object of the system, a different number is G and is G to that number alone; and we may say that a number to which another is G is "G'd" by that other; Peirce: CP 4.160 Cross-Ref:†† Second, There is a number, called zero, 0, which is G to no cardinal number; Peirce: CP 4.160 Cross-Ref:†† Third, The system contains no object that it is not necessitated to contain by the first two precepts. That is to say, a given description of number only exists provided the first two precepts require the existence of a number which may be of that description. Peirce: CP 4.161 Cross-Ref:†† 161. This system is a cluster of ideas of individual things; but it is not a cluster of real things. It thus belongs to the world of ideas, or Inner World. Nor does the mathematician, though he "creates the idea for himself," create it absolutely. Whatever it may contain of [that which is] impertinent [to Mathematics] is soilure from [elsewhere]. The idea in its purity is an eternal being of the Inner World.

Peirce: CP 4.162 Cross-Ref:†† 162. This idea of discrete quantity having an absolute minimum subsequently suggests the ideas of other systems, all of which are characterized by the prominence of transitive relations. These mathematical ideas, being then applied in physics to such phenomena as present analogous relations, form the basis of systems of measurement. Throughout them all, succession is the prominent relation; and all measurement is affected by two operations. The first is the experiment of super-position, the result of which is that we say of two objects, A and B, A is (or is not) in the transitive relation, t, to B, and B is (or is not) in the relation, t, to A; while the second operation is the experiment of counting. The question "How much is A?" only calls for the statement, A has the understood transitive relation to such things, and such things have this relation to A.

Peirce: CP 4.163 Cross-Ref:†† §3. APPLICATION OF THE THEORY TO ARITHMETICP

163. According to the theory partially stated above, pure arithmetic has nothing to do with the so-called Fundamental Theorem of Arithmetic.†1 For that theorem is that a finite collection counts up to the same number in whatever order the individuals of it are counted. But pure arithmetic considers only the numbers themselves and not the application of them to counting. Peirce: CP 4.164 Cross-Ref:†† 164. In order to illustrate the theory, I will show how the leading elementary propositions of pure arithmetic are deduced, and how it is subsequently applied to counting collections. Peirce: CP 4.164 Cross-Ref:†† Corollary 1. No number is G of more than one number. For every number necessitated by the first precept is G to a single number, and the only number necessitated by the second precept, by itself, is G to no number. Hence, by the third precept, there is no number that is G to two numbers. Peirce: CP 4.164 Cross-Ref:†† Corollary 2. No number is G'd by two numbers. For were there a number to which two numbers were G, one of the latter could be destroyed without any violation of the first two precepts, since the destruction would leave no number without a G which before had one, nor would it destroy 0, since that is not G. Hence, by the third precept, there is no number which is G to a number to which another number is G. Peirce: CP 4.164 Cross-Ref:†† Corollary 3. No number is G to itself. For every number necessitated by the first precept is G to a different number, and to that alone; and the only number necessitated by the second precept, by itself, is G to no number. Peirce: CP 4.164 Cross-Ref:†† Corollary 4. Every number except zero is G of a number. For every number necessitated by the first precept is so, and the only number directly necessitated by the second is zero.

Peirce: CP 4.164 Cross-Ref:†† Corollary 5. There is no class of numbers every one of which is G of a number of that class. For were there such a class, it could be entirely destroyed without conflict with precepts 1 and 2. For such destruction could only conflict with the first precept if it destroyed the number that was G to a number without destroying the latter. But no number of such a class could be G of any number out of the class by the first corollary. Nor could zero, the only number required to exist by the second precept alone, belong to this class, since zero is G to no number. Therefore, there would be no conflict with the first two precepts, and by the third precept such a class does not exist. Peirce: CP 4.165 Cross-Ref:†† 165. The truly fundamental theorem of pure arithmetic is not the proposition usually so called, but is the Fermatian principle, which is as follows: Peirce: CP 4.165 Cross-Ref:†† Theorem I. The Fermatian Principle: Whatever character belongs to zero and also belongs to every number that is G of a number to which it belongs, belongs to all numbers. Peirce: CP 4.165 Cross-Ref:†† Proof. For were there any numbers which did not possess that character, their destruction could not conflict with the first precept, since by hypothesis no number without that character is G to a number with it. Nor would their destruction conflict with the second precept directly, since by hypothesis zero is not one of the numbers which would be destroyed. Hence, by the third precept, there are no numbers without the character. Peirce: CP 4.166 Cross-Ref:†† 166. Definition 1. Any number, M, is, or is not, said to be greater than, a number, N, and N to be, or not to be, less than M, according to [whether] the following conditions are, or are not, fulfilled: Peirce: CP 4.166 Cross-Ref:†† First, Every number G to another is greater than that other; Peirce: CP 4.166 Cross-Ref:†† Second, Every number greater than a second, itself greater than a third, is greater than that third; Peirce: CP 4.166 Cross-Ref:†† Third, No number is greater than another unless the above two conditions necessitate its being so. Peirce: CP 4.166 Cross-Ref:†† Theorem II. Every cardinal number except zero is greater than zero.†1 Peirce: CP 4.166 Cross-Ref:†† Theorem III. No cardinal number L is greater than any number, M, unless L is G to a cardinal number, N, which either is greater than [or equal to] M. Peirce: CP 4.166 Cross-Ref:†† Corollary 1. By the same reasoning (substituting everywhere less for "greater" and G'd by for "G of") no number M is less than any number L unless L be G to M or

be greater than the number that is G to M. Peirce: CP 4.166 Cross-Ref:†† Corollary 2. Hence, by the first and second conditions of the definition, if a cardinal number, L is greater than a cardinal number, M, then the number that is G to L is greater than the number that is G to M. Peirce: CP 4.166 Cross-Ref:†† Corollary 3. Zero is greater than no number. Peirce: CP 4.166 Cross-Ref:†† Corollary 4. Every number greater than a number is G of some number. Peirce: CP 4.166 Cross-Ref:†† Theorem IV. No cardinal number is both greater and less than the same cardinal number. Peirce: CP 4.166 Cross-Ref:†† Corollary 1. No number is either greater or less than itself. Peirce: CP 4.166 Cross-Ref:†† Corollary 2. No cardinal number, M, is greater than a cardinal number, N, and less than GN. Peirce: CP 4.166 Cross-Ref:†† Theorem V. Of any two different cardinal numbers, one is greater than the other. Peirce: CP 4.166 Cross-Ref:†† Corollary. If the cardinal number, GL, that is G to L, be greater than the cardinal number, GM, that is G to M, by Theorem IV it cannot be less. Hence by Corollary 2 from Theorem III, L cannot be less than M. But by the first corollary from Theorem IV, GL is not GM, and therefore L is not M. Hence L is greater than M if GL is greater than GM. Peirce: CP 4.167 Cross-Ref:†† 167. Theorem VI. (Modified Fermatian Principle.) If a character, α, be such that, taking any two cardinal numbers, A and Z, either α does not belong to both A and Z, or no cardinal number is greater than A and less than Z, or SOME cardinal number greater than A and less than Z has the character, α, then, α is also such that, taking any two cardinal numbers, B and Y, either α does not belong both to B and Y, or no cardinal number is greater than B and less than Y, or EVERY cardinal number greater than B and less than Y has the character, α. Peirce: CP 4.167 Cross-Ref:†† Proof. For were there an exception, there would be, at least, one cardinal number of a class we may call the n's fulfilling the conditions that every n is greater than B and less than Y and no number at once greater than an n and less than an n possess α. Then, by Corollary 4 of Theorem III, every n, and also every number greater than every n, would be G to some cardinal number; and by Corollary 5 from the general precepts there would be some n G to a cardinal number, not an n, which we may call M, and there would be some number greater than every n which would be G to some n which we may call N. But then GN and M would possess α, and if any n's existed, they would be greater than M and less than GN and yet there would

be no cardinal number greater than M and less than GN having α. Hence it is absurd to suppose any exception. Peirce: CP 4.168 Cross-Ref:†† 168. Definition 2. A sum of a cardinal number, M, added to a cardinal number, N, is a cardinal number which fulfills the following conditions: Peirce: CP 4.168 Cross-Ref:†† First, A sum of zero added to zero is zero; Peirce: CP 4.168 Cross-Ref:†† Second, A sum of zero added to a cardinal number which is G to any cardinal number, N, is a number which is G to a sum of zero added to N; Peirce: CP 4.168 Cross-Ref:†† Third, A sum of a cardinal number that is G to any cardinal number, M, added to any cardinal number, N, is a cardinal number which is G to a cardinal number which is a sum of M added to N; Peirce: CP 4.168 Cross-Ref:†† Fourth, No cardinal number is a sum of a cardinal number added to a cardinal number unless it is necessitated to be so by the above conditions. Peirce: CP 4.168 Cross-Ref:†† Theorem VII. There is one cardinal number, and but one, which is a sum of a cardinal number, M, added to a cardinal number, N. Peirce: CP 4.168 Cross-Ref:†† Corollary 1. Whatever cardinal numbers M and N may be, M+N>0 unless M = N = 0. Peirce: CP 4.168 Cross-Ref:†† Corollary 2. Whatever cardinal numbers M and N may be M+N>N unless M = 0, and M+N>M unless N = 0. Peirce: CP 4.168 Cross-Ref:†† Corollary 3. Whatever cardinal numbers M and N may be, M+GN = G(M+N). Peirce: CP 4.168 Cross-Ref:†† Corollary 4. Whatever cardinal number N may be, 0+N = N. Peirce: CP 4.168 Cross-Ref:†† Corollary 5. Whatever cardinal number N may be, N+0 = N. Peirce: CP 4.168 Cross-Ref:†† Corollary 6. Whatever cardinal numbers M and N may be, M+N = N+M. Peirce: CP 4.168 Cross-Ref:†† Corollary 7. Whatever cardinal numbers L, M, and N may be, L+(M+N) = (L+M)+N. Peirce: CP 4.168 Cross-Ref:†† Theorem VIII. The sum of a greater cardinal number, L, added to any cardinal number, N, is greater than the sum of a lesser cardinal number, M, added to the same cardinal number, N.

Peirce: CP 4.168 Cross-Ref:†† Corollary 1. Whatever cardinal numbers L, M, and N may be if L>M, then N+L>N+M. Peirce: CP 4.168 Cross-Ref:†† Corollary 2. Whatever cardinal numbers A, B, C, D may be, if A>C and B>D then A+B>C+D. Peirce: CP 4.168 Cross-Ref:†† Corollary 3. Whatever cardinal numbers L, M, and N may be unless L = M, L+N or N+L is not M+N or N+M. Peirce: CP 4.168 Cross-Ref:†† Corollary 4. If L+N>M+N then L>M. Peirce: CP 4.168 Cross-Ref:†† Corollary 5. If A+B>C+D, either A or B is greater than C and than D or else either C or D is less than A and than B. Peirce: CP 4.168 Cross-Ref:†† Theorem IX. Whatever cardinal numbers L and M may be, there is one and only one cardinal number, N, such that either N+M = L or N+L = M. Peirce: CP 4.168 Cross-Ref:†† Definition 3. The difference between two cardinal numbers, L and M, is such a number, N, that either N+M = L or N+L = M. It is said to be the remainder after subtracting the smaller as subtrahend, from the larger, as minuend. It is best denoted by the "minus sign" written after the larger of L and M and before the smaller. Peirce: CP 4.168 Cross-Ref:†† Corollary 1. L-M is no cardinal number unless L>M. Peirce: CP 4.168 Cross-Ref:†† Corollary 2. If L>M, then (L-M)+M = L. Peirce: CP 4.168 Cross-Ref:†† Definition 4. The product of a cardinal number, M, multiplied into a cardinal number, N, or the product of the multiplicand, N, multiplied by the multiplier, M, is a cardinal number, written MXN, or M.N, or MN, subject to the following conditions: Peirce: CP 4.168 Cross-Ref:†† First, zero is a product of any cardinal number multiplied by 0; Peirce: CP 4.168 Cross-Ref:†† Second, a product of a cardinal number, N, multiplied by the cardinal number, GM, that is G to any cardinal number, M, is the sum, N+M.N, of N added to the product of M multiplied into N; Peirce: CP 4.168 Cross-Ref:†† Third, no cardinal number is a product of cardinal numbers unless necessitated to be so by the foregoing conditions. Peirce: CP 4.168 Cross-Ref:†† A product is said to be a multiple of its multiplicand.

Peirce: CP 4.168 Cross-Ref:†† Theorem X. There is one cardinal number and but one which is M.N a product of one cardinal number, M, multiplied into a given cardinal number, N. Peirce: CP 4.168 Cross-Ref:†† Corollary 1. The product of any cardinal number, N, multiplied into zero is zero. Peirce: CP 4.168 Cross-Ref:†† Corollary 2. Whatever cardinal numbers M and N may be MXN>0 unless M = 0 or N = 0. Peirce: CP 4.168 Cross-Ref:†† Corollary 3. The product of any cardinal number, N, multiplied by the cardinal number that is G to zero (which is called 1, one) is N. Peirce: CP 4.168 Cross-Ref:†† Corollary 4. The product of any cardinal number, N, multiplied into G0, or 1, is N. Peirce: CP 4.168 Cross-Ref:†† Corollary 5. Whatever cardinal numbers M and N may be, MXGN = M+M.N. Peirce: CP 4.168 Cross-Ref:†† Corollary 6. Whatever cardinal numbers M and N may be, MXN>M unless M = 0 or N = 0 or N = G0, and MXN>N unless N = 0 or M = 0 or M = G0. Peirce: CP 4.168 Cross-Ref:†† Corollary 7. Whatever cardinal numbers L, M, and N may be, L.(M+N) = L.M+L.N. Peirce: CP 4.168 Cross-Ref:†† Corollary 8. Whatever cardinal numbers L, M, N may be L.(M.N) = (L.M).N. Peirce: CP 4.168 Cross-Ref:†† Corollary 9. Whatever cardinal numbers M and N may be M.N = N.M. Peirce: CP 4.168 Cross-Ref:†† Theorem XI. Of two products of the same multiplicand not zero, that by the greater multiplier is the greater. Peirce: CP 4.168 Cross-Ref:†† Corollary 1. If L>M, N.L>N.M unless N = 0. Peirce: CP 4.168 Cross-Ref:†† Corollary 2. If A>C and B>D, AXB>CXD in all cases. Peirce: CP 4.168 Cross-Ref:†† Corollary 3. Unless L = M, LXN is not MXN, unless N = 0. Peirce: CP 4.168 Cross-Ref:†† Corollary 4. If LXN>MXN, then L>M in all cases. Peirce: CP 4.168 Cross-Ref:†† Corollary 5. If AXB>CXD, either A or B is greater than C and than D, or C or

D is less than A and than B. Peirce: CP 4.168 Cross-Ref:†† Corollary 6. If either B or C is greater than A or than D, then B.C>A.D, unless A+D>B+C. Peirce: CP 4.168 Cross-Ref:†† Definition 5. A divisor of a cardinal number, N, is a cardinal number which multiplied by a cardinal number gives N as product. The number, N, is said to be exactly divisible by its divisor. Peirce: CP 4.168 Cross-Ref:†† Abbreviations. We may write N N' (mod M) where M is any cardinal number, not zero, to express that N and N' are cardinal numbers leaving the same remainder after division by M. We may denote the remainder and quotient of N divided by M by R[m]N and Q[m]N, respectively. Then N = R[m]N+(Q[m]N)·M. Peirce: CP 4.168 Cross-Ref:†† We may denote GG0 by Q. Peirce: CP 4.168 Cross-Ref:†† Scholium. The number Q is logically and mathematically peculiar. In old arithmetics multiplication and division by Q are considered as peculiar operations, Duplation and Mediation. We have need of an arithmetic of two, even in reasonings which do not concern quantity in the ordinary sense. Peirce: CP 4.168 Cross-Ref:†† Theorem XII. Every cardinal number, N, has with reference to every cardinal number, M, except zero, a remainder, R[m]N, and a quotient, Q[m]N; and only one number is remainder or quotient. Peirce: CP 4.168 Cross-Ref:†† Corollary 1. If the cardinal number, N, is less than the modulus, M, its remainder, R[m]N = N. Peirce: CP 4.168 Cross-Ref:†† Corollary 2. The remainder of the sum of two numbers, N and N', is the remainder of the sum of their remainders. Peirce: CP 4.168 Cross-Ref:†† Corollary 3. The remainder of the product of two numbers, N and N', is the remainder of the product of the remainders. Peirce: CP 4.168 Cross-Ref:†† Corollary 4. The quotient of the sum of two numbers is the sum of the quotient of the sum of the remainders added to the sum of the quotients of the numbers. Peirce: CP 4.168 Cross-Ref:†† Corollary 5. The quotient of the product of two numbers, N and N', is the sum of the product of N by Q[m]N', the quotient of the other, added to the quotient of the product of N by R[m]N', the remainder of the other. Or Q[m](N.N') = N.Q[m]N'+Q[m](N.R[m]N'). Peirce: CP 4.168 Cross-Ref:††

Corollary 6. Given any cardinal number, N, and any modular number, M, there is a multiple of M greater than N. For (GQ[m]N).M is such a multiple. Peirce: CP 4.168 Cross-Ref:†† Definition 6. The powers of any cardinal number, B, called the base of the powers, are a class of cardinal numbers, each having a cardinal number, E, connected with it, called its exponent; and the power is written, BE, and powers and exponents are defined by the following conditions: Peirce: CP 4.168 Cross-Ref:†† First, G0 is a power of B whose exponent is zero; Peirce: CP 4.168 Cross-Ref:†† Second, The product of the power BE, of B with exponent E, multiplied by B is BGE, a power of B with exponent GE; Peirce: CP 4.168 Cross-Ref:†† Third, No cardinal number is a power of a cardinal number, unless necessitated to be so by the foregoing conditions. Corollary. BG0 = BXB0 = BXG0 = B. Peirce: CP 4.168 Cross-Ref:†† Theorem XIII. Given any two cardinal numbers, B and E, there is one, and but one cardinal number which is a power, BE, of B with exponent, E. Peirce: CP 4.168 Cross-Ref:†† Corollary 1. Hence, 00 = G0 and is not indeterminate. In this respect, the definition here assumed differs from the usual one, which substitutes for the first condition B1 = B and adds the condition that BE = P if BGE = B.P. But practically the present definition is just as useful, if not more so, than the usual one. Peirce: CP 4.168 Cross-Ref:†† Theorem XIV. A given exponent of two powers with the same exponent greater than zero, that with the greater base is the greater, and two powers of the same base greater than G0, that with the greater exponent is the greater. Peirce: CP 4.168 Cross-Ref:†† Definition 7. An even number is a cardinal number whose remainder, relative to GG0 as modulus, is zero. Peirce: CP 4.168 Cross-Ref:†† An odd number is a cardinal number whose remainder, relative to GG0 as modulus, is G0. Peirce: CP 4.168 Cross-Ref:†† Corollary 1. Every cardinal number, N, is even or odd; and if N be even, GN is odd and vice versa. Peirce: CP 4.168 Cross-Ref:†† Corollary 2. The double of a cardinal number, N, is N+N, the sum of N added to itself. Peirce: CP 4.168 Cross-Ref:†† Corollary 3. If a number is even, it has a cardinal number that is half of it, but

if it is odd, it has not. Peirce: CP 4.168 Cross-Ref:†† Corollary 4. If the difference of two cardinal numbers, M and N, is even, those two numbers have a cardinal number as their arithmetical mean, and the difference between this mean and either M or N is half the difference between M and N. Peirce: CP 4.169 Cross-Ref:†† 169. Theorem XV. (Binary form of the Fermatian Principle.) If any character belongs to every power of GG0 and also to the mean of any two numbers having a mean, if it belongs to the numbers themselves, then it belongs to every cardinal number except 0.

Peirce: CP 4.170 Cross-Ref:†† VI MULTITUDE AND NUMBER†1P

§1. THE ENUMERABLE

170. Let us consider the relation of a constituent unit to the collective whole of which it forms a part. Suppose A to be such a unit and B to be such a whole. Then in order to avoid the circumlocution of saying that A is a constituent unit of B as the collective whole of which it is a unit, I shall simply say A is a unit of B, and shall write "A is a u of B"; or I may reverse the order in which A and B are mentioned by writing "B is u'd by A." Peirce: CP 4.170 Cross-Ref:†† The only logical peculiarities of this relation are as follows: Peirce: CP 4.170 Cross-Ref:†† First, Whatever is u of anything is u'd by itself and by nothing else. Hence, if anything is u'd by anything not itself, it is not itself u of anything; and consequently nothing that is u'd by anything but itself is u'd by itself. Peirce: CP 4.170 Cross-Ref:†† Second, Whatever is not u'd by anything does not exist. Peirce: CP 4.171 Cross-Ref:†† 171. By a collection, I mean anything which is u'd by whatever has a certain quality, or general description, and by nothing else. That is, if C is a collection, there is some quality, α, such that taking anything whatever, say x, either x possesses the quality of α and is a unit of C, or else it neither possesses the quality α nor is a unit of C. On the other hand, if C is not a collection, no matter what quality or general description, β may be taken, there is either something possessing the quality β without being a unit of C, or there is some unit of C which does not possess the

quality, β. Peirce: CP 4.171 Cross-Ref:†† It will be perceived, therefore, that there is a collection corresponding to every common noun or general description. Corresponding to the common noun "man" there is a collection of men; and corresponding to the common noun "fairy" there is a collection of fairies. It is true that this last collection does not exist, or as we say, the total number of fairies is zero. But though it does not exist, that does not prevent it from being of the nature of a collection, any more than the non-existence of fairies deprives them of their distinguishing characteristics. . . . Peirce: CP 4.172 Cross-Ref:†† 172. Whether the constituent individuals or units of a collection have each of them a distinct identity of its own or not, depends upon the nature of the universe of discourse. If the universe of discourse is a matter of objective and completed experience, since experience is the aggregate of mental effect which the course of life has forced upon a man, by a brute bearing down of any will to resist it, each such act of brute force is destitute of anything reasonable (and therefore of the element of generality, or continuity, for continuity and generality are the same thing), and consequently the units will be individually distinct. It is such collections that I desire first to call your attention. I put aside then, for the present, such collections as the drops of water in the sea; and assume that the units are of such a kind that they may be absolutely distinguished from one another. Then, I say, as long as the discourse relates to a common objective and completed experience, those units retain each its distinct identity. If you and I talk of the great tragedians who have acted in New York within the last ten years, a definite list can be drawn up of them, and each of them has his or her proper name. But suppose we open the question of how far the general influences of the theatrical world at present favor the development of female stars rather than of male stars. In order to discuss that, we have to go beyond our completed experience, which may have been determined by accidental circumstances, and have to consider the possible or probable stars of the immediate future. We can no longer assign proper names to each. The individual actors to which our discourse now relates become largely merged into general varieties; and their separate identities are partially lost. Again, statisticians can tell us pretty accurately how many people in the city of New York will commit suicide in the year after next. None of these persons have at present any idea of doing such a thing, and it is very doubtful whether it can properly be said to be determinate now who they will be, although their number is approximately fixed. There is an approach to a want of distinct identity in the individuals of the collection of persons who are to commit suicide in the year 1899. When we say that of all possible throws of a pair of dice one thirty-sixth part will show sixes, the collection of possible throws which have not been made is a collection of which the individual units have no distinct identity. It is impossible so to designate a single one of those possible throws that have not been thrown that the designation shall be applicable to only one definite possible throw; and this impossibility does not spring from any incapacity of ours, but from the fact that in their own nature those throws are not individually distinct. The possible is necessarily general; and no amount of general specification can reduce a general class of possibilities to an individual case. It is only actuality, the force of existence, which bursts the fluidity of the general and produces a discrete unit. Since Kant it has been a very wide-spread idea that it is time and space which introduce continuity into nature. But this is an anacoluthon. Time and space are continuous because they embody conditions of possibility, and the possible is general, and continuity and generality are

two names for the same absence of distinction of individuals. Peirce: CP 4.172 Cross-Ref:†† When the universe of discourse relates to a common experience, but this experience is of something imaginary, as when we discuss the world of Shakespeare's creation in the play of Hamlet, we find individual distinction existing so far as the work of imagination has carried it, while beyond that point there is vagueness and generality. So, in the discussion of the consequences of a mathematical hypothesis, as long as we keep to what is distinctly posited and its positive implications, we find discrete elements, but when we pass to mere possibilities, the individuals merge together. This remark will be fully illustrated in the sequel. Peirce: CP 4.173 Cross-Ref:†† 173. A part of a collection called its whole is a collection such that whatever is u of the part is u of the whole, but something that is u of the whole is not u of the part. Peirce: CP 4.174 Cross-Ref:†† 174. It is convenient to use this locution; namely, instead of saying A is in the relation, r, to B, we may say A is an r to B, or of B; or, if we wish to reverse the order of mentioning A and B, we may say B is r'd by A. Peirce: CP 4.174 Cross-Ref:†† If a relation, r, is such that nothing is r to two different things, and nothing is r'd by two different things, so that some things in the universe are perhaps r to nothing while all the rest are r, each to its own distinct correlate, and there are some things perhaps to which nothing is r, but all the rest have each a single thing that is r to it, then I call r a one-to-one relation. If there be a one-to-one relation, r, such that every unit of one collection is r to a unit of a second collection, while every unit of the second collection is r'd by a unit of the first collection, those two collections are commonly said to be in a one-to-one correspondence with one another.†1. . . Peirce: CP 4.175 Cross-Ref:†† 175. I shall use the word multitude to denote that character of a collection by virtue of which it is greater than some collections and less than others, provided the collection is discrete, that is, provided the constituent units of the collection are or may be distinct. But when the units lose their individual identity because the collection exceeds every positive existence of the universe, the word multitude ceases to be applicable. I will take the word multiplicity to mean the greatness of any collection discrete or continuous. Peirce: CP 4.176 Cross-Ref:†† 176. We have to note the precise meaning of saying that a relation of a given description exists. A relation of the kind here considered has been called an ens rationis; but it cannot be said that because nobody has ever constructed it -- perhaps never will -- it exists any the less on that account. Its existence consists in the fact that, if it were constructed, it would involve no contradiction. An easy dilemma will show that to suppose three things to be in one-to-one correspondence with individuals of a pair involves contradiction. But it is much more difficult to prove that a given hypothesis involves no contradiction. In mathematics, such propositions are usually replaced by so-called "problems." That is to say, a construction shows how the thing in question can take place. When we know how it can take place, we know, of course, that it is possible. Cases are rare in mathematics in which anything is shown to be

possible without its being shown how. But when we come to philosophical questions, such a construction is generally practically beyond our powers; and it becomes necessary to examine the principles of logic in order to discover a general method of proving that a given hypothesis involves no contradiction. Without a thorough mastery of the principles of logic such a search must be fruitless. Peirce: CP 4.176 Cross-Ref:†† Mathematics never has hypotheses forced upon it that are perplexing from [their] seemingly irresoluble mistiness -- which is the aspect of such a question of philosophical possibility, at first sight. Mathematics does not need to take up any hypothesis that is not crystal-clear. Unfortunately, philosophy cannot choose its first principles at will, but has to accept them as they are. Peirce: CP 4.177 Cross-Ref:†† 177. For example, the relations of equality and excess of multitude having been defined after Cantor, philosophy can not avoid the question which instantly springs up: must every two collections be either equal or the one greater than the other, or can they be so multitudinous that the units of neither can be in one-to-one relation to units of the other? Peirce: CP 4.177 Cross-Ref:†† To say that the collection of M's and the collection of N's are equal is to say: Peirce: CP 4.177 Cross-Ref:†† There is a one-to-one relation, c, such that every M is c to an N; and there is a one-to-one relation, d, such that every N is d to an M. Peirce: CP 4.177 Cross-Ref:†† To say that the collection of M's is less than the collection of N's is to say: Peirce: CP 4.177 Cross-Ref:†† There is a one-to-one relation, c, such that every M is c to an N; but whatever one-to-one relation d may be, some N is not d to any M. Peirce: CP 4.177 Cross-Ref:†† To say the collection of M's is greater than the collection of N's is to say: Peirce: CP 4.177 Cross-Ref:†† Whatever one-to-one relation c may be, some M is not c to any N; but there is a one-to-one relation d such that every N is d to an M. Peirce: CP 4.177 Cross-Ref:†† Now, formal logic suggests the fourth relation: Peirce: CP 4.177 Cross-Ref:†† Whatever one-to-one relation c may be, some M is not c to any N, and whatever one-to-one relation d may be, some N is not d to any M. Peirce: CP 4.177 Cross-Ref:†† Or this last may be stated more simply thus: Peirce: CP 4.177 Cross-Ref:†† Whatever one-to-one relation c may be, some M is not c to any N and some N is not c'd by any M.

Peirce: CP 4.177 Cross-Ref:†† How shall we proceed in order to find out whether this last relation is a possible one, or not? . . . Peirce: CP 4.178 Cross-Ref:†† 178. In the first place, it must not be supposed that even if a collection is so great that the constituent units lose their individual identity, a one-to-one relation necessarily becomes impossible. If such a relation implied an actual operation performed, it would indeed be impossible, I suppose. But this is not the case. As the collection enlarges and the individual distinctions are little by little merged, it also passes out of the realm of brute force into the realm of ideas which is governed by rules. This sounds vague, because until I have shown you how to develop the idea of such a collection, I can offer you no example. But it is not necessary actually to construct the correspondence. It suffices to suppose that a certain number of units of the two collections having been brought into such a relation (and, in fact, they always are in such relations), then the general rules of the genesis of the two collections necessitate the falling of all the other individuals into their places in the correspondence. All this will become quite clear in the sequel. Peirce: CP 4.179 Cross-Ref:†† 179. That difficulty, then, having been removed, we have two collections, the M's and the N's; and the question is whether there is, no matter what these collections may be, always either some one-to-one relation, c, such that any M is c to an N or else some one-to-one relation, d, such that every N is d'd by an M. To begin with, there are vast multitudes of relations such that taking any one of them, r, every M is r to an N and every N is r'd by an M. For example, the relations of coexistence, maker of, non-husband of, etc. In general, each M can have any set of N's whatever as its correlates, except that there must be one of the M's that shall have among its correlates all those N's that are not r'd by any other M. And all those sets of N's for each M can be combined in any way whatever. In order to make our ideas more clear, let us for the moment suppose that the M's are equal to the finite number, {m}, and the N's are equal to the finite number, {n}. Then, for each M except one there are 2{n}-1 different sets of N's, any one of which can be its correlates. Hence, there are (2{n}-1){m}-1 different forms of the relation r, without taking account of the variety of different sets of correlates which the remaining M may have. Suppose we had a diagram of each of those relations, each diagram showing the collection of M's above and the collection of N's below, with lines drawn from each M to all the N's of which it was r. Each of that stupendous multitude of relations may be modified so as to reduce it [to] what we may call a one-to-x relation, by running through the N's and cutting away the connection of each N with every M but one; and each of the r relations could be thus cut down in a vast multitude of different ways. Call any such resulting relation, s. Then, every N would be s'd by a single M. Each one of the r relations could also be so modified as to reduce it to what we may call an x-to-one relation, by running through the M's and cutting off the connection of each M with every N but one. Call such a resulting relation, t. Then, every M would be t to a single N. Suppose we had a collection of diagrams showing all the ways in which every r relation could thus be reduced to an s relation or a t relation, that is, be reduced to a one-to-x relation or to an x-to-one relation. The question is, could the multitudes of M and N, be such that there would not be a single one-to-one relation among all those one-to-x relations [which each M has to an N] and x-to-one relations [which each N has to an M]? If among the diagrams of the one-to-x relations there were not one

where the one-to-x relation was a one-to-one relation, it would be because in each case there was some M which was s [i.e., one-to-x] to two or more N's. If, then, there were any of these diagrams in which some M was not s to any N, those diagrams could be thrown out of consideration, because there was no necessity for a pluralism of lines to one M, as long as there were M's to which no line ran; and since there was no necessity for it, there is no need of modifying those diagrams so as to take away plural lines from some of the M's so as to give lines to all the M's, because, since there is a diagram for every possible modification changing an r [x-x] relation to an s [one-x] relation, there must already be a diagram remedying this fault. There must, therefore, be among the diagrams, some diagrams in which every M is s to an N -unless indeed there is a diagram where the s is a one-to-one relation. Taking, then, any diagram in which every M is s to an N, all it is necessary to do is to erase all the lines but one which go to each M, and the relation so resulting, which we may call u, is such that every M is u to an N, no N is u'd by two M's (for no N is s'd by two M's and the erasures cannot increase the relates of any N), and no M is u of two N's. In other words, u is a one-to-one relation, and every M is u of an N. Q.E.D. Peirce: CP 4.179 Cross-Ref:†† Is this demonstration sound? It may be doubted; at any rate I can show you how by a very small modification it would certainly become unsound; and thus direct your attention to the point which requires scrutiny. If, instead of casting aside those diagrams of s relations which showed some M's that are not s to any N, I had proposed to cure them by changing the course of lines from M's having two or more lines to M's having none, until there were either no M's left without any lines or no M's left with pluralities of lines, I should have fallen into a gross petitio principii. For I should be assuming that, of those two classes of M's, the whole of one (whichever it might be) could be put into a one-to-one relation with the whole or a part of the other; and whether or not this is always possible is the very question at issue. Peirce: CP 4.179 Cross-Ref:†† But the true argument is this: Nothing can force all of the s diagrams to show pluralities of lines to M's except the fact that some of them show lines to all the M's. For since all possibilities are represented in the diagrams, if all the diagrams show pluralities of lines to M's, there must be a logical necessity for this, so that the conditions would be contradicted if it were not so. Now the only logical necessity there can be in making some lines terminate at M's, that already have lines, is that there are no M's that have not already lines. Hence, in some cases, at least, all the M's must have lines. Peirce: CP 4.179 Cross-Ref:†† The gist of this argument is that it considers in what way contradiction can arise, and thus shows that the only circumstance which could render the one-to-one correspondence impossible in one way, necessarily renders it possible in another way. Peirce: CP 4.180 Cross-Ref:†† 180. I will now prove two general theorems of great importance. The first is, that the collection of possible sets of units (including the set that includes no units at all) which can be taken from discrete collections is always greater than the collection of units.†1 . . . Peirce: CP 4.180 Cross-Ref:†† The other theorem, which gives great importance to the first, is that if a collection is not too great to be discrete, that is, to have all its units individually

distinct, neither is the collection of sets of units that can be generally formed from that collection too great to be discrete. Peirce: CP 4.180 Cross-Ref:†† For we may suppose the units of the smaller collection to be independent characters, and the larger collection to consist of individuals possessing the different possible combinations of those characters. Then, any two units of the larger collection will be distinguished by the different combinations of characters they possess, and being so distinguished from one another they must be distinct individuals. Peirce: CP 4.180 Cross-Ref:†† On those two theorems, I build the whole doctrine of collections. Peirce: CP 4.181 Cross-Ref:†† 181. I will now run over the different grades of multitude of discrete collections, and point out the most remarkable properties of those multitudes. Peirce: CP 4.181 Cross-Ref:†† The lowest grade of multitude is that of a collection which does not exist, or the multitude of none. A collection of this multitude has obvious logical peculiarities. Namely, nothing asserted of it can be false. For of it alone contradictory assertions are true. It is a collection and it is not a collection. Given the premisses that all the X's are black and that all the X's are pure white, what is the conclusion? Simply that the multitude of the X's is zero. Peirce: CP 4.181 Cross-Ref:†† The least difference by which one multitude can exceed another is by a single unit. But I do not say that the multitude next greater than a given multitude always exceeds it by a single unit. Peirce: CP 4.181 Cross-Ref:†† The multitude of ways of distributing nothing into two abodes is one. This is the next grade of multitude. This again has certain logical peculiarities. Namely, in order to prove that every individual of it possesses one character, it suffices to prove that every individual of it does not possess the negative of that character. Peirce: CP 4.181 Cross-Ref:†† The multitude of ways of distributing a single individual into two houses is two. This is the next grade of multitude. This again has certain logical peculiarities which have been noted in Schröder's Logik. Peirce: CP 4.181 Cross-Ref:†† The multitude of combinations of two things is four, which is not the next grade of multitude. The multitude of combinations of four things is 16. The multitude of combinations of 16 things is 65,536. The multitude of combinations of 65,536 things is large. It is written by 20,036 followed by 19,725 other figures. The multitude of combinations of that many things is a number to write which would require over 600,000 thousand trimillibicentioctagentiseptillions of figures on the so-called English system of numeration. What the number itself would be called it would need a multimillionaire to say. But I suppose the word trimillillillion might mean a million to the trimillillionth power; and a trimillillion would be a million to the three thousandth power. But the multitude considered is far greater than a trimillillillion. It is safe to say that it far exceeds the number of chemical atoms in the gallactic cluster. Yet this is one of the early terms of a series which is confined entirely to finite

collections and never reaches the really interesting division of multitudes, which comprises these that are infinite. Peirce: CP 4.182 Cross-Ref:†† 182. The finite collections, however, or, as I prefer to call them, the enumerable collections, have several interesting properties. The first thing to be considered is, how shall an enumerable multitude be defined? If we say that it is a multitude which can be reached by starting at 0, the lowest grade of multitude, and successively increasing it by one, we shall express the right idea. The difficulty is that this is not a clear and distinct statement. As long as we discuss the subject in ordinary language, the defect of distinctness is not felt. But it is one of the advantages of the algebra which is now used by all exact logicians, that such a statement cannot be expressed in that logical algebra until we have carefully thought out what it really means. An enumerable multitude is said to be one which can be constructed from zero by "successive" additions of unity. What does "successive," here, mean? Does it allow us to make innumerable additions of unity? If so, we certainly should get beyond the enumerable multitudes. But if we say that by "successive" additions we mean an enumerable multitude of additions, we fall into a circulus in definiendo. A little reflection will show that what we do mean is, that the enumerable multitudes are those multitudes which are necessarily reached, provided we start at zero, and provided that, any given multitude being reached, we go on to reach another multitude next greater than that. The only fault of this statement is, that it is logically inelegant. It sounds as if there were some special significance in the "reaching," which by the principles of logic there cannot be. For the enumerable multitudes are defined as those which are necessarily so reached. Now the kind of necessity to which this "necessarily" plainly refers is logical necessity. But the perfect logical necessity of a result never depends upon the material character of the predicate. If it is necessary for one predicate, it is equally so for any other. Accordingly, what is meant is that the enumerable multitudes are those multitudes every one of which possess any character whatsoever which is, in the first place, possessed by zero and, in the second place, if it is possessed by any multitude, M, whatsoever, is likewise possessed by the multitude next greater than M. We, thus, find that the definition of enumerable multitude is of this nature, that it asserts that that famous mode of reasoning which was invented by Fermat †1 applies to the succession of those multitudes. The enumerable multitudes are defined by a logical property of the whole collection of those multitudes. Peirce: CP 4.183 Cross-Ref:†† 183. Since the whole collection of enumerable multitudes has this logical property it follows a fortiori that every single enumerable multitude has the same property. Peirce: CP 4.184 Cross-Ref:†† 184. But it further follows from the same definition that every single enumerable collection has a further logical property. Peirce: CP 4.184 Cross-Ref:†† This property is, that if an enumerable collection be counted, the counting process eventually comes to an end by the exhaustion of the collection. This property follows from the other, in this sense, that it is true of the zero collection, and if it be true of any collection whatever, it is equally true of every collection that is greater than that by one individual. Hence, it is true of all enumerable collections, by Fermatian reasoning.

Peirce: CP 4.185 Cross-Ref:†† 185. You may ask why I should call this a logical property. It does not at first sight appear to be of that nature. But that is because it is not distinctly expressed. In place of "coming next after in the count," we may substitute any relation, r, such that not more than one individual (at least of the collection in question) is an r to any one. Then, the property is that if the M's form an enumerable collection, then and only then, if every M is r to an M [say, L], then every M is r'd by an M [say, N]. For example, in a count no M is immediately preceded by more than one M, hence it cannot be that every M immediately precedes an M (so that the collection is never exhausted) unless every M is immediately preceded by an M (in which case, the count would have no beginning). Because this is a logical necessity, the property is a logical property and is the foundation of that mode of inference for which De Morgan first gave the logical rules, under the name of the syllogism of transposed quantity. He, however, overlooked the fact that this mode of reasoning is only valid of enumerable collections.†1. . . Peirce: CP 4.186 Cross-Ref:†† 186. A remarkable and important property of enumerable collections is, that every finite part is less than a whole. If the finite part is measured, the multitude of units it contains is enumerable; and if it is incommensurable with the unit, the unit can be changed so as to make the finite part commensurable. Thus, to say that a finite part is less than its whole is the same as to say that an enumerable collection which is part of another is less than that other. There are two cases: first, when the whole is enumerable; and second, when the whole is inenumerable. Let us consider the first case. Let the M's be contained among the N's (which form an enumerable collection). Suppose however that the collection of M's is not less than that of the N's. Then, by the definition of equality, there is such a one-to-one relation d, that every N is d'd by an M. Then, since this M is an N, every N is d'd by an N. But d being a one-to-one relation, there are not two N's that are d'd by the same N. Hence, by the syllogism of transposed quantity, every N is d of an N. But the N's are, by their equality to the M's, d'd by nothing but M's. Hence, every N is an M. That is, we have shown that if the N's form an enumerable collection, the only collection at once contained in that collection and equal to that collection is the collection itself, and is not a part of the collection. That is, no part of an enumerable collection is equal to the collection. But the relation of inclusion is a one-to-one relation of every unit of the part to a unit of the whole. Hence, the part cannot be greater than the whole, and must be less than the whole. Peirce: CP 4.186 Cross-Ref:†† We now take up the second case. But we can go further, and show that every inenumerable collection is greater than any enumerable collection. It is to be shown that it is absurd to suppose that every unit of an inenumerable collection, the N's, is in a one-to-one relation, c, to a unit of any one enumerable collection, the M's. Let r be such a one-to-one relation that every M except one is r to an M. Then, by the syllogism of transposed quantity, every M except just one is r'd by an M. (For if every M were r to an M, every M would be r'd by an M; and since r is a one-to-one [relation], if there is a single one of the connections or relations between pairs of individuals, which is excluded from r, it leaves just one M not r to an M and just one M not r'd by an M.) This is so whatever one-to-one relation r may be. Hence, were every N c to an M, it would follow that every N but one would be c to an M that was r to an M that was c'd by an N; and this compound relation of being 'c to an r of something c'd by' would be a one-to-one relation, being compounded of one-to-one

relations. And invariably, whatever one-to-one relation r might be, one N would be the last in a count of the N's which should proceed from each N, say N[i], to that N, say N[j], such that N[i] was c of that M that was r of that M which was c'd by N[j]. In every such mode of counting, I say, some N would be the last N completing the count. And the M's being equal to the N's, and the one collection tied to the other by the relative, for every possible order of counting of the N's there would be some r relation among the M's; and thus in every possible counting of the N's there would be a last N, contrary to the hypothesis that the N's form an inenumerable collection. Thus, it is shown to be impossible that an inenumerable collection should be no greater than an enumerable collection, and the demonstration that a finite part is less than its whole is complete. Peirce: CP 4.186 Cross-Ref:†† Now it is singular that every time Euclid reasons that a part is less than its whole, he falls into some fallacy, even though the part he is speaking of be finite.†P1 I can only account for it by supposing that owing to the falsity of his axiom, he learned to think that very wonderful things could be proved by its aid, things that he would know could never be proved by any other axiom; for when a man appeals to an axiom he is pretty sure to be reasoning fallaciously. And thus he was prevented from suspecting and thoroughly criticizing those places in his reasoning. . . . Peirce: CP 4.187 Cross-Ref:†† 187. It is a curious illustration of how even that part of mankind who reason for themselves more than any others -- I mean the mathematicians -- yet how even they follow phrases and forget their meanings, that while everybody is in the habit of calling the proposition that a part is less than its whole an axiom, yet when this proposition is stated in another form of words -- for the transformation amounts to little more -- we always speak of it as the fundamental theorem of arithmetic. The statement is that if in counting a collection with the cardinal numerals the count of a collection comes to a stop from the exhaustion of the individuals it always comes to a stop at the same numeral. I say that this amounts pretty much to saying that an enumerable part can not equal its whole. For to say that the same collection can in one order of counting count 16 and in another order of counting count 15 would be the same as to say that the first 16 numerals could (through the identity of the objects counted) be put into a one-to-one correspondence with the first 15 numerals; and this, by the definition of equality, would be to say that the collection of the first 15 numerals was equal to the collection of the first 16 numerals, although the former collection is an enumerable part of the latter. Peirce: CP 4.187 Cross-Ref:†† It is generally understood to be very difficult to demonstrate this theorem logically, and so it is somewhat so if the principles of logic are not attended to. At any rate several of the proposed demonstrations egregiously beg the question.†P1

Peirce: CP 4.188 Cross-Ref:†† §2. THE DENUMERABLE

188. But I have lingered too long among enumerable multitudes. Let us go on to inquire what is the smallest possible multitude which is inenumerable?

Peirce: CP 4.188 Cross-Ref:†† Take the collection of M's. If this collection be such that taking any one-to-one relation r whatever, if every M is r to an M it necessarily follows that every M is r'd by an M, the collection of M's thereby fulfills the definition of an enumerable collection. We can substitute a phrase for the letter r in this statement and say that to call the collection of M's enumerable is the same as to assert that if every M, in any order of arrangement, is immediately succeeded by another M, and that an M which does not so immediately succeed any other of the M's, then every M immediately succeeds another M, and there is some ring arrangement without any first. To say that if there be no last there can be no first, is to say the collection spoken of is enumerable. Peirce: CP 4.188 Cross-Ref:†† To deny that the M's are enumerable is, then, as much as to assert that there is a possible arrangement in which each M is immediately followed by another M which so follows no third M, and yet there is an absolutely first M which does not follow any M. If now we deny that the collection of M's is enumerable but, at the same time, restrict it to including no individual that need not be included to make the collection inenumerable, we shall plainly have a collection of the lowest order of multitude which any inenumerable collection can have. Such a collection I call denumerable. To say, then, that the collection of M's is denumerable, is the same as to assert that it contains nothing except one particular object and except what is implied in the fact that there is a one-to-one relation r such that every M is r to an M. This is a logical character; for it is the same as to say that the syllogism of transposed quantity does not hold good of it but that the Fermatian inference does. That is, if the collection of M's is denumerable, every character which is true of a certain M, say M[0] and is also true of every M which is in a certain one-to-one relation to an M of which it is true, is necessarily true of every M of the collection. Peirce: CP 4.188 Cross-Ref:†† For example, the entire collection of whole numbers forms a denumerable collection. For zero is a whole number, which is not greater by one than any number, there is a number greater by one than any given whole number, and there is no number or numbers which could be struck out of the collection and still leave it true that zero belonged to the collection and that there was a number of the collection greater by one than each number of the collection. Peirce: CP 4.189 Cross-Ref:†† 189. I have already shown by the example of the even numbers that a part of a denumerable collection may be equal to the whole collection. I will now prove that all denumerable collections are equal. For suppose that the M's and the N's are two denumerable collections. Then, a certain M can be found which we may call M[0] such that taking a certain one-to-one relation, r, every M except M[0] is r to an M, and there is an r to every M; and in like manner there is a one-to-one relation, s, such that every N except one, N[0], is s to an N, and every N is s'd by an N. Then, I say, that the relation, c, can be so defined that every M is c to an N, and every N is c'd by an M. For let M[0] be c to N[0] and to nothing else; and let N[0] be c'd by nothing but M[0] and if anything, X, is c to anything, Y, let the r to X (and it alone) be c to the s of Y and to nothing else. Then, evidently c is a one-to-one relation. But every M is c to an N, because M[0] is c to an N (namely to N[0]) and if any M is c to an N, then the r of that M is c to an N (for it is, by the definition of c, c to the s of the N to which

the former M is c). And in like manner every N is c'd by an M, because N[0] is c'd by an M (namely by M[0]), and if any N is c'd by an M, then the s of that N is c'd by an M (for it is, by the definition of c, c'd by the r of the M by which the former N is c'd). Q. E. D. Peirce: CP 4.189 Cross-Ref:†† Accordingly, there is but a single grade of denumerable multitude. So it is to be noted as a defect in my nomenclature, which I unfortunately did not remark when I first published it,†1 that enumerable and denumerable, which sound so much alike, denote, the one a whole category of grades of multitude and the other a simple grade like, zero, or twenty-three. Peirce: CP 4.190 Cross-Ref:†† 190. It will be convenient to make here a few remarks about arithmetical operations upon multitude. Please observe that I have not said one word as yet about number, and I do not propose even to explain at all what numbers are until I have fully considered the subject of multitude, which is a radically different thing. Arithmetical operations can be performed upon both multitudes and upon numbers, just as they can be performed upon the terms of logic, the vectors of quaternions, the operations of the calculus of functions, and other subjects. What I ask you at this moment to consider is, not at all the addition and multiplication of numbers, for you do not know what I mean by numbers -- it is safe to say so, since the word bears so many different meanings -- but the addition and multiplication of multitudes. Peirce: CP 4.190 Cross-Ref:†† Addition in general differs from aggregation inasmuch as a unit is increased by being added to itself but not by being aggregated to itself. When mutually exclusive terms are aggregated, that is the same as the addition of them. Addition might, therefore, be defined as the aggregation of the positings of terms. Two positings of the same term being different positings, their aggregate is different from a single positing of the term. The sum of two multitudes is the multitude of the aggregate of two mutually exclusive collections of those multitudes. The aggregate of a collection of collections of units may be defined as that collection of units, every unit of which is a unit of one of those collections, and which has every unit of any of those collections among its units. Peirce: CP 4.191 Cross-Ref:†† 191. It is easily proved that the sum of an enumerable collection of enumerable multitudes is an enumerable multitude.†2 . . . Peirce: CP 4.192 Cross-Ref:†† 192. The sum of an enumerable multitude and the denumerable multitude is denumerable. The proof is excessively simple; for we have only to count the enumerable collection in linear series, first. The count of that has to end; and then the denumerable series may follow in its primal order. Peirce: CP 4.193 Cross-Ref:†† 193. That the denumerable multitude added to itself gives itself is made plain by zigzagging through two denumerable series. But this comes more properly under the head of multiplication of multitudes, which I propose to consider. Peirce: CP 4.193 Cross-Ref:†† Mathematicians seem to be satisfied so far to generalize the conception of multiplication as to make it the application of one operation to the result of another.

But the conception may be still further generalized, and in being further generalized it returns more closely to its primitive type. The more general conception of multiplication to which I allude is expressed in the following definition: Multiplication is the pairing of every unit of one quantity with every unit of another quantity so as to make a new unit. Since there are two acceptions of the term pair -the ordered acception, according to which AB and BA are different pairs, and the unordered acception -- there are two varieties of multiplication, the non-commutative and the commutative. Multiplication may further be distinguished into the free and the dominated. In free multiplication the idea of pairing remains in all its purity and generality. In dominated multiplication, the product of two units is that which results from the special mode of pairing which is of preëminent importance with reference to the particular kind of units that are paired. Thus, in reference to length and breadth the pairing of their units in units of area is preëminently important; in reference to an operator and its operant the pairing of their units in units of the result is preëminently important; in logic, in reference to two general terms, the pairing of their units in identical units which reunite their essential characters is preëminently important, etc. In the multiplication of multitudes we have one of the very rare instances of free multiplication. The product of a collection of multitudes called its factors may be defined as the multitude of possible sets of units any one of which could be formed out of units taken one from each of a collection of mutually exclusive collections of units having severally the multitudes of the factors. For example, to multiply 2 and 3, we take a collection of two objects, as A and B, and a distinct collection of three objects, as X, Y, and Z, and form the pairs AX, AY, AZ, BX, BY, BZ, which are all the sets that can be formed each from one unit of each collection. Then, since the multitude of these pairs is 6, the product of the multitudes, 2 and 3, is the multitude of 6. Peirce: CP 4.194 Cross-Ref:†† 194. The same general idea affords us a definition of involution. Involution is the formation of a new quantity a power from two quantities, a base, and an exponent, each unit of the power resulting from the attachment of all the units of the exponent each to some one unit of the base, without reference to how many units of the exponent are attached to any one unit of the base. Thus, 3 to the 2 power is the multitude of different ways in which both of two units, A and B, can be joined each to some one of three objects, X, Y, and Z. . . . Peirce: CP 4.195 Cross-Ref:†† 195. The product of two multitudes, {m} and {n}, is equal to the multitude of units in {m} mutually exclusive collections each of {n} units. For since there is one unit and but one for each of the {n} units of each of the {m} collections, these units are in one-to-one correspondence with the possible descriptions of single units each of which pairs a unit of a multitude of {n} with a unit of a multitude of {m}; and the multitude of such pairs is the product of {m} and {n}. Peirce: CP 4.195 Cross-Ref:†† The {m} power of {n} is equal to the product of {m} mutually exclusive collections each of {n} units. Peirce: CP 4.195 Cross-Ref:†† The product of two enumerable multitudes is an enumerable multitude. Peirce: CP 4.195 Cross-Ref:†† The product of an enumerable multitude and the denumerable multitude is the

denumerable multitude. Peirce: CP 4.195 Cross-Ref:†† An enumerable power of the denumerable multitude is the denumerable multitude. Peirce: CP 4.196 Cross-Ref:†† 196. That the second power of the denumerable multitude is the denumerable multitude is easily seen by aggregating a denumerable series of collections, each a denumerable series of units.

[Click here to view] Now we can start at the corner and proceeding from each unit we reach to a single next one and can reach any unit whatever in time without completing the proceeding. Hence, the whole forms one denumerable series. This proof is substantially that of Cantor.†1 The proposition being proved for two factors instantly extends itself to any enumerable multitude of factors. Of course, there is not the slightest difficulty in expressing this idea so as to construct the most rigidly formal demonstration. Let ℵ denote the denumerable multitude. Then, I am to show that ℵ2 = ℵ. Let the M's be a denumerable collection. That is, suppose Peirce: CP 4.196 Cross-Ref:††

First: a certain object M[0] is an M; Peirce: CP 4.196 Cross-Ref:†† Second: there is a certain non-identical one-to-one relation, r, such that every M is r'd by an M; Peirce: CP 4.196 Cross-Ref:†† Third: whatever is not necessitated to be an M by the above statements is not an M. Peirce: CP 4.196 Cross-Ref:†† Let A and B constitute a collection of two objects not M's. Let us define the relation {s} as follows: Peirce: CP 4.196 Cross-Ref:†† First: the pair of attachments of A to M[0] and B to M[0] is {s}'d by nothing; Peirce: CP 4.196 Cross-Ref:†† Second: every pair of attachments of A to an M which we may call M[i] other than M[0] and of B to an M, which we may call M[j], is {s} to the pair of attachments of A to that M which is r'd by M[i], and of B to that M which is r of M[j]; Peirce: CP 4.196 Cross-Ref:†† Third: every pair of attachments of A to M[0] and of B to an M, which we may call M[k], is {s} to the pair of attachments of A †1 to that M which is r of M[k], and of B †2 to M[0]; Peirce: CP 4.196 Cross-Ref:†† Fourth: if one thing is not necessitated by the above rules to be {s} to another, it is not {s} to that other. Peirce: CP 4.196 Cross-Ref:†† It is evident, then, that {s} is a one-to-one relation; and it is evident that every pair of attachments of A to any M, say M[x], and of B to any M, say M[y], is {s} of another such pair of attachments, that one such pair of attachments is {s}'d by nothing, and that nothing is a pair of such attachments that is not necessitated to exist by the fact that everything is {s} of something. Hence, the multitude of those pairs of attachments is denumerable; and that is the same as to say that the second power of the denumerable multitude is the denumerable multitude. Peirce: CP 4.197 Cross-Ref:†† 197. Dr. George Cantor †1 first substantially showed that between the units of any denumerable collection certain remarkable relations exist, which I call indefinitely divident relations. Namely, let the M's be any denumerable collection, and let f be any relation indefinitely divident of the collection of the M's. Then, no M is f to itself, but of any two different M's one is f to the other; and if an M is f to another it is f to every M that is f'd by that other; and if an M is f'd by another it is f'd by every M that is f to that other. And now comes the remarkable feature: If one M is f to another, it is f to an M that is not f'd by that other; whence, necessarily if one M is f'd by another, it is f'd by an M that is not f to that other. Peirce: CP 4.197 Cross-Ref:†† There are vast multitudes of such indefinitely divident relations. I will instance a single one. If we take the whole series of vulgar fractions, those of the same denominator being taken immediately following one another in the increasing

order of the numerators and those of different denominators in the increasing order of the denominators, 1/2 1/3 2/3 1/4 2/4 3/4 1/5 2/5 3/5 4/5 1/6 2/6 3/6 4/6 5/6 etc. these evidently form a denumerable collection, for they form the aggregate of a denumerable collection of enumerable collections of units. If from this collection we omit those fractions which are equal to other fractions of lower denominations we plainly have still a denumerable collection. Now for the first of these denumerable collections, that of all the vulgar fractions, an indefinitely divident relation is that of being "greater than or equal to but of higher terms than". For the second of those denumerable collections, that of all the rational quantities greater than 0 and less than 1, an indefinitely divident relation is that of being "greater than." . . . Peirce: CP 4.197 Cross-Ref:†† Numbers in themselves cannot possibly signify any magnitude other than the magnitudes of collections, or multitude; but what they principally represent is place in a serial order. Numbers do not contain the idea of the equality of parts and consequently a fraction cannot in itself signify anything involving equality of parts. They merely express the ordinal place in such a uniformly condensed series. . . .†1 Peirce: CP 4.198 Cross-Ref:†† 198. A striking difference between enumerable and denumerable collections is this, that no arrangement of an enumerable collection has any different properties from any other arrangement; for the units are or may be in all respects precisely alike, that is, have the same general characters, although they differ individually, each having its proper designation. But it is not so with regard to denumerable collections. Every such collection has a primal arrangement, according to its generating relation. There is one unit, at least, which arbitrarily belongs to the collection just as every unit of an enumerable collection belongs to that collection. But after that one unit, or some enumerable collection of units, has been arbitrarily posited as belonging to the collection, the rest belong to it by virtue of the general rule that there is in the primal arrangement one unit of the collection next after each unit of the collection. Those last units cannot be all individually designated, although any one of them may be individually designated. Nor is this merely owing to an incapacity on our part. On the contrary, it is logically impossible that they should be so designated. For were they so designated there would be no contradiction in supposing a list of them all to be made. That list would be complete, for that is the meaning of all. There would therefore be a last name on the list. But that is directly contrary to the definition of the denumerable multitude. Peirce: CP 4.198 Cross-Ref:†† The same truth may be stated thus: It is impossible that all the units of a denumerable collection should have the same general properties. For the existence of the primal arrangement is essential to it, being involved in the very definition of the denumerable collection as that of smallest multitude greater than every enumerable collection. Now, this primal arrangement is an arrangement according to a general rule, and its statement constitutes, therefore, general differences between the units of the denumerable collection. Peirce: CP 4.198 Cross-Ref:†† On the other hand any unit whatever of a denumerable collection may be individually designated, as well as all those which precede it in the primal

arrangement. And these can be all exactly alike in their general qualities. Yet there must always be a latter part of the collection which is not individually designated but is only generally described. In this part we recognize an element of ideal being as opposed to the brute and surd existence of the individual. Peirce: CP 4.198 Cross-Ref:†† The denumerable collection of whole numbers, for example, constitutes a discrete series, in the sense that there is not one which may not be distinguished completely and individually from its neighbors. Peirce: CP 4.198 Cross-Ref:†† But we cannot with any clearness of thought carry these reflections further until we are in possession of an instance of a greater collection. Peirce: CP 4.199 Cross-Ref:†† 199. The arrangement of a denumerable collection according to an indefinitely divident relation like the rational numbers -- or to take a simpler instance, like the fractions which can be written in the binary system of arithmetical notation with enumerable series of figures -- is a very recondite arrangement, not at all naturally suggested by the primal arrangement. This is shown by the fact that the world had to wait for George Cantor to inform it that the collection of rational fractions was a collection precisely like that of the whole numbers.†1 Peirce: CP 4.199 Cross-Ref:†† This remark will be found important in the sequel.

Peirce: CP 4.200 Cross-Ref:†† §3. THE PRIMIPOSTNUMERAL

200. So much, for the present, for the denumerable multitude. Let us now inquire, what is the smallest multitude which exceeds the denumerable multitude? An enumerable or denumerable multitude is a multitude such that whatever in any arrangement of an enumerable collection, in the primal arrangement of a denumerable collection, is true of the first unit, and is further true of any unit which comes next after any unit of which it is true, is true of all and every unit of the collection. . . . It has not yet been proved that there is any such minimum multitude among those which exceed the denumerable; but it is convenient to say that in fact there is. I have hitherto named this multitude, which was first clearly described by Cantor,†1 the first abnumeral multitude.†2 But I find that a name in one word is wanted. So I will hereafter name it the primipostnumeral multitude. Peirce: CP 4.201 Cross-Ref:†† 201. Suppose it to be true of a collection that in whatever way its units be arranged in a horizontal line with one unit to the extreme left, and a unit next to the right of each unit, there is something which is true of the first unit and which if true of any unit is always true of the next unit to the right, which nevertheless is not true of all the units; and suppose furthermore that the collection is no greater than it need be to bring about that state of things. Then, that collection is by definition a primipostnumeral collection. Or by the aid of the logic of relatives, we may state the matter as follows:

1. Let there be an existent collection, R; 2. Let R include no unit which is not necessitated by that condition; 3. Let r be a one-to-one relation between units; 4. Let there be a collection, the Q's, such that no Q is R; 5. Let there be a Q that is r to each unit of the collections of the Q's and R's; 6. Let the collection of the Q's include nothing not necessitated by the foregoing conditions; 7. Let h be a one-to-one relation of a unit to a collection; 8. Let there be a collection, P, such that no P is a Q or R; 9. Let there be a P which is h to every (denumerable) collection of Q's. 10. Let there be no P which is not necessitated in order to fulfill the foregoing conditions. Peirce: CP 4.201 Cross-Ref:†† Then, the collection of P's is a primipostnumeral collection. Peirce: CP 4.201 Cross-Ref:†† It would be easy to make this statement more symmetrical in appearance; but I prefer to make it perspicuous. Thus, we might make r a relation between a unit and an enumerable collection; and we might make the P's include an h for every denumerable collection of P's, Q's and R's, etc. The word "denumerable" in the ninth condition is added merely for the sake of perspicuity. Peirce: CP 4.201 Cross-Ref:†† The second, sixth, and tenth conditions are not very clear. The meaning is that the multitude is no larger than need be. Peirce: CP 4.202 Cross-Ref:†† 202. The definitions of a primipostnumeral collection just given suppose it to be constructed from a denumerable collection. But if we attempt to form a primipostnumeral collection from a denumerable collection in its primal arrangement we shall fail ignominiously. Peirce: CP 4.202 Cross-Ref:†† Let us, for example, imagine a series of dots representing, the first [dot] the position of the tortoise when Achilles began to run after him, and each successive dot the position of the tortoise at the instant when Achilles reached the position represented by the preceding dot. If there are no more dots than are necessary to fulfill this condition, the collection of dots is denumerable. If we add a dot to represent the position of the tortoise at the moment when Achilles catches up with him, the Fermatian inference seems at first sight not to hold good. For the first dot represents a position of the tortoise before Achilles had caught up with him, and if any dot represents the position of the tortoise before Achilles caught up with him, so likewise does the dot which immediately succeeds it. The Fermatian inference then would seem to be that every dot represents a position of the tortoise before Achilles had caught up with him. Yet this is not true of the last dot which represents the position of

the tortoise at the moment when Achilles caught up with him. Yet but one dot has been added to the denumerable collection, and of course, it remains denumerable. The only reason that the inference does not hold is that the dots are no longer in their primal arrangement. Put the last dot at the beginning, so as to preserve the primal arrangement, and any Fermatian inference whose premisses were true would hold good. The point I wish to make is that the denumerable collection in its primal order leads to no way of constructing or of conceiving of a primipostnumeral collection. Of course, we can say, "Let there be a dot for each denumerable collection of the tortoise-places;" but we might as well omit the tortoise-places and say, "Let there be a primipostnumeral collection of dots." The primal arrangement of the denumerable collection affords no definite places nor approximations to the places for the primipostnumeral collection. Peirce: CP 4.203 Cross-Ref:†† 203. The reason is that the latter part of the denumerable collection, which is its denumerable point, is all concentrated towards one point, whether that point be a metrically ordinary point or a point at infinity. This fault is remedied in the indefinitely divident arrangement. Here the denumerable part of the collection is spread over a line. Peirce: CP 4.203 Cross-Ref:†† In this case, if we imagine all those subdivisions to be performed which are implied by saying that the intervals resulting from each set of subdivisions are all subdivided in the next following set of subdivisions, the multitude of subdivisions is 2ℵ where ℵ is the denumerable multitude; and this is no mere algebraical form without meaning. It has a perfectly exact meaning which I explained in speaking of the effects of addition, multiplication, and involution upon multitudes. Peirce: CP 4.203 Cross-Ref:†† Moreover, you will remember that I distinctly and fully proved †1 that the multitude of possible sets of units each of which can be formed from the units of a collection always exceeds the multitude of that collection, provided it be a discrete collection. Peirce: CP 4.204 Cross-Ref:†† 204. Do you not think it possible that the stellar universe extends throughout space? If so, the whole collection of worlds is at least denumerable. At any rate, it is perfectly possible that the whole collection of intelligent beings who live, have lived, or will live anywhere is at least equal to the collection of whole numbers. It is conceivable that they are all immortal and that each one should be given each hour throughout eternity the name of one of them and he should assign that person in wish to heaven or to hell, so that in the course of eternity he would wish every one of them to heaven or to hell. Could they by all making different wishes wish among them for every possible distribution of themselves to heaven or to hell? If not, the multitude of such possible distributions is greater than the denumerable multitude. But they plainly could not wish for all possible such distributions. For if they did, some one would necessarily be perfectly satisfied with every possible distribution. But one possible distribution would consist in sending each person to the place he did not wish himself to go; and that would satisfy nobody. It was Cantor who first proved that the surd quantities form a collection exceeding the collection of rational quantities.†2 But his method was only applicable to that particular case. My method is applicable to any discrete multitude whatever and shows that 2{m}>{m} in every case in which {m} is

a discrete multitude. Peirce: CP 4.205 Cross-Ref:†† 205. I will give a few more examples of primipostnumeral collections. The collection of quantities between zero and unity, to the exact discrimination of which decimals can indefinitely approximate but never attain, is evidently 10ℵ, which of course equals 2ℵ. For 16ℵ = (24)ℵ = 2(4ℵ) = 2ℵ. Peirce: CP 4.205 Cross-Ref:†† The collection of all possible limits of convergent series. whose successive approximations are vulgar fractions, although it does not, according to any obvious rule of one-to-one correspondence, give a limit for every possible denumerable collection of vulgar fractions, does nevertheless in an obvious way correspond each limit to a denumerable collection of vulgar fractions, and to so large a part of the whole that it is primipostnumeral, as Cantor has strictly proved.†1 Peirce: CP 4.206 Cross-Ref:†† 206. Just as there is a primal arrangement of every denumerable collection, according to a generating relation, so there is a primal arrangement of every primipostnumeral collection, according to a generating arrangement. This primal arrangement of the primipostnumeral collection springs from a highly recondite arrangement of the denumerable collection. Namely, we must arrange the denumerable collection in an indefinitely divident order, and then the units, which are implied in saying that the denumerable succession of subdivisions have been completed constitute the primipostnumeral collection. But when I say that the primipostnumeral collection springs from an arrangement of the denumerable collection, I do not mean that it is formed from the denumerable collection itself; for that would not be true. On the contrary, the primipostnumeral collection can only be constructed by a method which skips the denumerable collection altogether. In order to show what I mean I will state the definition of a primipostnumeral collection in terms of relations. There are two or three trifling explanations to be made here. First an aggregate of collections is a collection of the units of those collections. It is also an aggregate of the collections, which are called its aggregants. Just as to say that Alexander cuts some knot implies that a knot exists, although to say Alexander cuts every knot, i.e., whatever knot there may be, does not imply the existence of any knot, the latter by its generality referring to an ideal being, not to a brute individual existence, so to say that a collection has a certain collection as its aggregant implies the existence of the latter collection and therefore that it contains at least one unit. I must also explain that whenever I say either one thing or another is true I never thereby mean to exclude both. Peirce: CP 4.207 Cross-Ref:†† 207. I will now describe a certain collection A, whose units I will call the P's [π's?]. Peirce: CP 4.207 Cross-Ref:†† First, The π's can be arranged in linear order. That is, there is a relation, p, such that taking as you will any π's, individually designable as π[1], π[2], and π[3], either π[3] is not p to π[2] or π[2] is not p to π[1] or (if π[3] is p to π[2] and π[2] is p to π[1]), π[3] is p to π[1]; Peirce: CP 4.207 Cross-Ref:††

Second, The line of arrangement of the π's can be taken so as not to branch. That is, taking as you will π's, individually designable as π[4] and π[5], either π[4] is p to π[5] or π[5] is p to π[4]; (of course this permits both to be true, but that I proceed to forbid). Peirce: CP 4.207 Cross-Ref:†† Third, The line of arrangement of the π's can further be so taken as not to return into itself, circularly. That is, taking as you will any π, individually designable as π[6], π[6] is not p to π[6]; Peirce: CP 4.207 Cross-Ref:†† Fourth, There are certain parts of A called "packs" of π, which are mutually exclusive. That is, taking any pack whatever and any unit of that pack, that unit is a π; and taking as you will any packs individually designable as P[7]†1 and P[8], and any π's individually designable as π[7] and π[8], either P[7] is identical with P[8] or π[7] is not a unit of P[7], or π[8] is not a unit of P[8], or else π[7] is not identical with π[8]; Peirce: CP 4.207 Cross-Ref:†† Fifth, The packs can be arranged in linear order. That is, there is a relation, s, such that taking as you will any P's, individually designable as P[1], P[2], and P[3], either P[3] is not s to P[2], or P[2] is not s to P[1], or P[3] is s to P[1]; Peirce: CP 4.207 Cross-Ref:†† Sixth, The line of arrangement of the packs can be taken so as not to branch. That is, taking as you will any P's, individually designable as P[4] and P[5], either P[4] is s to P[5] or P[5] is s to P[4]; Peirce: CP 4.207 Cross-Ref:†† Seventh, The line of arrangement of the packs can be further taken so as not to return into itself. That is, taking as you will any pack individually designable as P[6], P[6] is not s to P[6]; Peirce: CP 4.207 Cross-Ref:†† Eighth, The arrangement of the packs can further be such that each pack is immediately succeeded by a next following pack. That is, taking as you will any pack individually designable as P[9], a pack individually designable as P[10] can be found such that P[10] is s to P[9]; and such that taking thereafter as you will any pack individually designable as P[11], either P[11] is not p to P[9], or P[11] is not p'd by P[10]; Peirce: CP 4.207 Cross-Ref:†† Ninth, Such a succession of packs is not a mere idea, but actually exists if the collection A exists. That is, a certain collection, P[0], is such a pack; Peirce: CP 4.207 Cross-Ref:†† Tenth, Each pack contains a unit which, in the linear order of the π's, comes next after each unit of any of those packs which precede this pack in the linear order of the packs. That is, taking as you will any packs, individually designable as P[12] and P[13], and any unit, individually designable as π[12], a unit, individually designable as π[13], can be thereafter found such that, taking as you will any pack individually designable as P[14] and any unit individually designable as π[14], either

P[13] is not s to P[12], or π[12] is not a unit of P[12], or π[13] is p to π[12]; and either P[13] is not s to P[14] or π[14] is not a unit of P[14], or π[12] is p to π[14], or π[13] is not p to π[14]; Peirce: CP 4.207 Cross-Ref:†† Eleventh, No varieties of descriptions of π's exist than those which are necessitated by the foregoing conditions; Peirce: CP 4.207 Cross-Ref:†† Twelfth, No varieties of descriptions of packs exist than those which are necessitated by the foregoing conditions. Peirce: CP 4.207 Cross-Ref:†† This collection of π's is primipostnumeral; and you will see what I mean by saying that the construction skips the denumerable multitude, if you consider how many π's are contained in each pack. The pack P[0] is obliged by the ninth condition to exist, so that it must contain at least one π. But nothing obliges it to contain a π which is other than any π which it contains; and therefore the twelfth condition forbids it to contain [more than] one π. It consists, therefore, of a single π. If we arrange the π's in a horizontal row so that p shall be equivalent to being "further to the right than," then that P which is s to P[0], but is not s to any other pack, which pack we may call P[1], must contain one π to the right of the π of [P[0]]. It need contain no other, and therefore cannot contain any other.

[Click here to view] P0. · P1. |

·

P2. | · | · P3. | · | · | · | · P4. |·|·|·|·|·|·|·|

Peirce: CP 4.207 Cross-Ref:†† P[2] contains a π immediately to the right of that of the P[0] and another to the right of that of P[1], and after this each pack contains double the units of the preceding. Thus, P[n+1] contains 2n units. As long as n is enumerable, this is enumerable. But as soon as n becomes denumerable, it skips the denumerable multitude and becomes primipostnumeral. Peirce: CP 4.208 Cross-Ref:†† 208. In order to prove that any proposition is generally true of every member of a denumerable collection, it is always necessary -- unless it be some proposition not peculiar to such a collection -- to consider the collection either in its primal arrangement, or in reference to some relation by which the collection is generable, and then reason as follows, where r is the generating relation, and M[0] is that M which is not r to any M: M[0] is X, If any M is X then the r of M is X; .·.Every M is X. Without this Fermatian syllogism no progress would ever have been made in the mathematical doctrine of whole numbers; and though by the exercise of ingenuity we may seem to dispense with this syllogism in some cases, yet either it lurks beneath the method used, or else by a generalization the proposition is reduced to a case of a proposition not confined to the denumerable multitude. Peirce: CP 4.209 Cross-Ref:†† 209. In like manner, in order to prove that anything is true of a primipostnumeral collection, unless it is more generally true, we must consider that collection in its primal arrangement or with reference to a relation equivalent to that of its primal arrangement. The special mode of reasoning will be as follows: π[0], the unit of P[0], is X, If every π of any pack is X, then every π of the pack which is s of that pack is X; Hence, every π is X. This may be called the primipostnumeral syllogism. Peirce: CP 4.210 Cross-Ref:†† 210. Every mathematician knows that the doctrine of real quantities is in an exceedingly backward condition. It cannot be doubted by any exact logician that the reason of this is the neglect of the primipostnumeral syllogism without which it is as impossible to develop the doctrine of real quantities, as it would be to develop the theory of numbers without Fermatian reasoning. Peirce: CP 4.210 Cross-Ref:†† I do not mean to say that the primipostnumeral syllogism is altogether unknown in mathematics; for the reasoning of Ricardo †1 in his theory of rent, reasoning which is of fundamental importance in political economy, as well as much of the elementary reasoning of the differential calculus, is of that nature. But these are

only exceptions which prove the rule; for they strongly illustrate the weakness of grasp, the want of freedom and dexterity with which the mathematicians handle this tool which they seem to find so awkward that they can only employ it in a few of its manifold applications. Peirce: CP 4.211 Cross-Ref:†† 211. In the denumerable multitude we noticed the first beginnings of the phenomenon of the fusion of the units. All the units of the first part of the primal order of a denumerable multitude can be individually designated as far as we please, but those in the latter part cannot. In the primipostnumeral multitude the same phenomenon is much more marked. It is impossible to designate individually all the units in any part of a primipostnumeral multitude. Any one unit may be completely separated from all the others without the slightest disturbance of the arrangement. Peirce: CP 4.211 Cross-Ref:†† Thus, we may imagine points measured off from 0 as origin 0 .......

A

B ·

C ·------

toward A to represent the real quantities from zero toward √2. Let A be the point which according to this measurement would represent √2. But we may modify the rule of one-to-one correspondence between quantities and points, so that, for all values less than √2, the points to the left of A represent those values, while another point an inch or two to the right shall represent √2, and all quantities greater than √2 shall be represented by points as many inches or parts of an inch to the right of a third point, C, several inches to the right of B, as there are units and parts of units in the excess of those quantities over √2. This mode of representation is just as perfect as the usual unbroken correspondence. It represents all the relations of the quantities with absolute fidelity and does not disturb their arrangement in the least. Peirce: CP 4.211 Cross-Ref:†† It is, therefore, perfectly possible to set off any one unit of a primipostnumeral collection by itself, and equally possible so to set off any enumerable multitude of such units. Nor are there any singular units of the collection which resist such separation. Peirce: CP 4.211 Cross-Ref:†† I will give another illustration. It is perfectly easy to exactly describe many surd quantities simply by stating what their expressions in the Arabic system of notation would be. This may sound very false; but it is so, nevertheless. For instance, that quantity, which is expressed by a decimal point followed by a denumerable series of figures, of which every one which stands in a place appropriated to (1/10)n where n is prime shall be a figure 1, while every one which stands in a place whose logarithm n is composite shall be a cipher, is, we know, an irrational quantity. Now, I do not think there can be much doubt that, however recondite and complicated the descriptions may be, every surd quantity is capable in some such way of having its expression in decimals exactly described. Peirce: CP 4.211 Cross-Ref:†† Thus every unit of a primipostnumeral collection admits of being individually designated and exactly described in such terms as to distinguish it from every other

unit of the collection. Thus, notwithstanding a certain incipient cohesiveness between its units, it is a discrete collection, still. . . . Peirce: CP 4.212 Cross-Ref:†† 212. It is one of the effects of the deplorable neglect by mathematicians of the properties of primipostnumeral collections that we are in complete ignorance of an arrangement of such a collection, which should be related to its primal arrangement in any manner analogous to the relation of the arrangement [of] the primal arrangement of the denumerable collection to that indefinitely divident arrangement, which leads to a clear conception of the next grade of multitude. Peirce: CP 4.212 Cross-Ref:†† I have had but little time to consider this problem; but I can produce an arrangement which will be of some service. Suppose that instead of proceeding, as in the usual generation of the primipostnumeral multitude, to go through a denumerable series of operations each consisting in interpolating a unit between every pair of successive units, we go through a denumerable multitude of operations each consisting in replacing every pair by an image of the whole collection. For example, using the binary system of arithmetical notation, suppose we begin with a collection of two objects, zero and one-half. .0

.1

Each operation may consist of replacing each number by a sub-collection of [all the] numbers, each consisting of two parts, the first part being the figures of the number replaced, the second [being the figures of one of] the numbers composing the whole collection. Thus, the result of the first operation will be .00

.01

.10

.11

The result of the second operation will be .0000

.0001

.0010

.0011

.0100

.0101

.0110

.0111

.1000

.1001

.1010

.1011

.1100

.1101

.1110

.1111

The next result would be 256 numbers, the next 65,536 numbers, the next 4,294,867,296. The result of a denumerable succession of such operations will evidently be to give all the real quantities between zero and one, which is a primipostnumeral collection.

Peirce: CP 4.213 Cross-Ref:†† §4. THE SECUNDOPOSTNUMERAL AND LARGER COLLECTIONS

213. Although I have not touched upon half the questions of interest concerning the primipostnumeral multitude, I must hurry on to inquire, what is the least multitude greater than the primipostnumeral multitude? Time forbids my going through a fundamentally methodical discussion of this problem. But the speediest route to a correct solution of a difficult logical crux lies almost always through that paradox or sophism which depends upon that crux. Let us recur then for a moment to

the indefinitely divident arrangement of a primipostnumeral collection. It will be convenient to use the binary system of arithmetical notation. We begin with .0 as our π[0]. P[1] consists

.0

.1

| .01

|

.11

|

|

|

|

|

|

|

|

|.001| .011| .101| .111 | ·| | | ·| |

[Click here to view]

of a fraction equal to that but carried into the first place of secundals and of corresponding units which differ only in having a 1 in the first place of secundals. P[2] consists of fractions equal to those but carried into the second place of secundals, together with fractions differing from them only in having a 1 in the second place of secundals. And so on. Now if we use all the enumerable places of secundals, but stop before we reach any denumerable place, we shall have, among all the packs, all the fractions whose denominators are powers of 2 with enumerable exponents, and therefore we shall plainly have only a denumerable collection. But if n is the number of packs up to a given pack, then the number of fractions will be 2n-1. When we have used all the enumerable places of secundals and no others, how many packs have we used? Plainly a denumerable collection, since the multitude of enumerable whole numbers is denumerable. It would appear, then, that 2n-1, when n is denumerable, is denumerable. But on the contrary, if we consider only that pack which fills every enumerable place of secundals, since it contains the expression in secundals of every real quantity between 0 and 1, it alone is a primipostnumeral collection. Moreover,

the number of π's in P[n] is 2n and since n is denumerable for this collection, it follows that 2n is primipostnumeral. And it is impossible that the subtraction of one unit should reduce a primipostnumeral collection to a denumerable collection. Again, every pack contains a multitude of individuals only 1 more than that of all the packs that precede it in the order of the packs. How then can the former be primipostnumeral while the latter is denumerable? Peirce: CP 4.213 Cross-Ref:†† The explanation of this sophism is that it confounds two categories of characters of collections, their multitudes and their arithms. The arithm of a multitude is the multitude of multitudes less than that multitude. Thus, the arithm of 2 is 2; for the multitudes less than 2 are 0 and 1. By number in one of its senses, that in which I endeavor to restrict it in exact discussions, is meant an enumerable arithm. Thus, the arithm or number of any enumerable multitude is that multitude. The arithm of the denumerable multitude, also, is that multitude. But the arithm of the primipostnumeral multitude is the denumerable multitude. The maximum multitude of an increasing endless series that converges to a limit is the arithm of that limit, in this sense, that by the limit of an increasing endless series is meant the smallest multitude greater than all the terms of the series. If there is no such smallest multitude the series is not convergent. If, then, by the maximum multitude of an increasing series we mean the multitude of all the multitudes which would converge increasingly to the given limit, this maximum multitude is plainly the arithm of the series. Thus, the series of whole numbers is an increasing endless series. Its limit is the denumerable multitude. The arithm of this multitude is the maximum multitude of the series. If in 2n we substitute the different whole numbers for n, we get an increasing endless series whose limit is the primipostnumeral multitude. Its arithm, which is the maximum multitude of the series, is denumerable only. It is strictly true that the multitude of pack P[n], in the example to which the sophism relates, is 2n. But it is not strictly true that the multitude of π's in all preceding packs is 2n-1. It happens to be so when n is a number, that is, is enumerable. But strictly it is the multitude next smaller than the multitude of 2n. If the latter is the primipostnumeral multitude, the former can be nothing but the denumerable multitude. This is what we find to be the case, as it must be; and there is nothing paradoxical in it, when rightly understood. There is no value of n for which 2n is denumerable. Peirce: CP 4.213 Cross-Ref:†† The limit of 2n is primipostnumeral. The denumerable is skipped. But were we to reach the denumerable as we may, if we erroneously assume the sum of 2n is 2n-1, when we double that on the principle that 2n = 2X(2n-1), we, of course, only have the denumerable as the result. Peirce: CP 4.214 Cross-Ref:†† 214. Let us now consider 22n. Since 2n can never be denumerable, but skips at once from the enumerable to the primipostnumeral, when n is denumerable, it follows that 22n can never be denumerable nor primipostnumeral. For there is no value which 2n could have to make 22n denumerable; and in order that 22n should be primipostnumeral, 2n would have to be denumerable, which is impossible. Thus, 22n skips the denumerable and the denumerable [Ed. note, Burks: "primipostnumeral" should replace "denumerable" here] multitudes. But if we use square brackets to denote the arithm, so that [2]=2, [3]=3, [∞]=∞, etc., then since [2∞] is denumerable, 2[2∞] is primipostnumeral.

Peirce: CP 4.215 Cross-Ref:†† 215. When we start with .0 and .1, and repeating these varieties in the next figures, get .00 .01 .10 .11, and then repeating these varieties in the next figures, get .0000 .0001 etc., and then repeating these varieties in the next figures, get .00000000 .00000001 etc., if we say that, when this operation is carried out until the number of figures is denumerable, we get a primipostnumeral collection, we are assuming what is not true, that by continually doubling an enumerable multitude we shall ever get to a denumerable multitude. That is not true. In that process the denumerable multitude is skipped. We are assuming that because [the] multitude of all the arithmetical places which we pass by is denumerable, when the operation has been performed a denumerable multitude of times, therefore the multitude reached is denumerable. That is, we are confusing 2∞ with [2∞]. Peirce: CP 4.215 Cross-Ref:†† The function 22x is no doubt the simplest one which skips the denumerable and primipostnumeral multitudes. Therefore the multitude of this when x is denumerable is, no doubt, the smallest multitude greater than the primipostnumeral multitude. It is the secundopostnumeral multitude. Peirce: CP 4.216 Cross-Ref:†† 216. Although there can remain no doubt whatever to an exact logician of the existence in the world of mathematical ideas, of the secundopostnumeral multitude, yet I have been unable, as yet, to form any very intuitionally conception †1 of the construction of such a collection. But I must confess I have not bestowed very much thought upon this matter. I give a few constructions which have occurred to me. Peirce: CP 4.216 Cross-Ref:†† Imagine points on a line to be in one-to-one correspondence with all the different real quantities between 0 and 1. Imagine the line to be repeated over and over again in each repetition having a different set of those points marked. Then the entire collection of repetitions is a secundopostnumeral collection. Peirce: CP 4.216 Cross-Ref:†† Imagine a denumerable row of things, which we may call the B's. Let every set of B's possess some character, which we may call its crane †1 different from the crane of any other set. Imagine a collection of houses which we may call the beths †1 such that each house contains an object corresponding to each crane-character, and according as that object does or does not possess that character, the beth is said to possess or want that character. Then, the different possible varieties of beths, due to their possessing or not possessing the different cranes, form a secundopostnumeral collection. Peirce: CP 4.216 Cross-Ref:†† According to the hypothesis of Euclidean projective geometry there is a plane at infinity. That plane we virtually see when we look up at the blue spread of the sky. A straight line at infinity, although it is straight and looks straight, is called a great semi-circle of the heavens. At two opposite points of the horizon we look at the same point of the plane at infinity. Of course, we cannot look both ways at once. We measure distances on an ordinary straight line by metres and centimetres. We measure distance on a straight line in the sky by degrees and minutes. The entire circuit of the straight line is 180 degrees, and the circuits of all straight lines are equal. But in metres the measure is infinite. If by a projection we make a position of a straight line

in the sky correspond to a straight line near at hand, we perhaps make a degree correspond to a metre, although in reality a metre is to a degree in the proportion, 180 degrees to infinity. Imagine that, upon a straight filament in the sky, points are marked off metrically corresponding to all the real quantities. Then let that filament be brought down to earth. If one of those real quantities' points is at any near point, there will not be another at any finite number of kilometres from it. For were there two, when it was in the sky they would have been closer together than any finite fraction of a second of arc. If, however, when you had pulled the filament down from the sky you were to find that each of those things you took for points was really a doubly refracting crystal and that these acted quite independently of one another, so that when you looked through two others you saw four images, when you looked through three you saw eight images, and so on, then if you were to look along the filament through all the crystals, one for each real quantity, the collectum of images you would see would be a secundopostnumeral collection. Peirce: CP 4.217 Cross-Ref:†† 217. In like manner, there will be a tertiopostnumeral multitude 222ℵ, a quartopostnumeral multitude 2222ℵ and so on ad infinitum. Peirce: CP 4.217 Cross-Ref:†† All of these will be discrete multitudes although the phenomenon of the incipient cohesion of units becomes more and more marked from one to another. Peirce: CP 4.217 Cross-Ref:†† These multitudes bear no analogy to the orders of infinity of the calculus; for 1 1 ∞ X∞ = ∞2. But any of these multiplied by itself gives itself. I had intended to explain these infinites of the calculus. But I find I cannot cram so much into a single lecture. Peirce: CP 4.218 Cross-Ref:†† 218. I now inquire, is there any multitude larger than all of these? That there is a multitude greater than any of them is very evident. For every postnumeral multitude has a next greater multitude. Now suppose collections one of each postnumeral multitude, or indeed any denumerable collection of postnumeral multitudes, all unequal. As all of these are possible their aggregate is ipso facto possible. For aggregation is an existential relation, and the aggregate exists (in the only kind of existence we are talking of, existence in the world of noncontradictory ideas) by the very fact that its aggregant parts exist. But this aggregate is no longer a discrete multitude, for the formula 2n>n which I have proved holds for all discrete collections cannot hold for this. In fact writing Exp. n for 2n, (Exp.) ℵℵ is evidently so great that this formula ceases to hold and it represents a collection no longer discrete.

Peirce: CP 4.219 Cross-Ref:†† §5. CONTINUA

219. Since then there is a multiplicity or multiplicities greater than any discrete multitude, we have to examine continuous multiplicities. Considered as a mere multitude, we might be tempted to say that continuous multiplicities are incapable of discrimination. For the nature of the differences between them does not

depend upon what multitudes enter into the denumerable series of discrete multitudes out of which the continuous multiplicity may be compounded; but it depends on the manner in which they are connected. This connection does not spring from the nature of the individual units, but constitutes the mode of existence of the whole. Peirce: CP 4.219 Cross-Ref:†† The explanation of the paradoxes which arise when you undertake to consider a line or a surface as a collection of points is that, although it is true that a line is nothing but a collection of points of a particular mode of multiplicity, yet in it the individual identities of the units are completely merged, so that not a single one of them can be identified, even approximately, unless it happen to be a topically singular point, that is, either an extremity or a point of branching, in which case there is a defect of continuity at that point. This remark requires explanation, owing to the narrowness of the common ways of conceiving of geometry. Briefly to explain myself, then, geometry or rather mathematical geometry, which deals with pure hypotheses, and unlike physical geometry, does not investigate the properties of objectively valid space -- mathematical geometry, I say, consists of three branches; Topics (commonly called Topology), Graphics (or pure projective geometry), and Metrics. But metrics ought not to be regarded as pure geometry. It is the doctrine of the properties of such bodies as have a certain hypothetical property called absolute rigidity, and all such bodies are found to slide upon a certain individual surface called the Absolute. This Absolute, because it possesses individual existence, may properly be called a thing. Metrics, then, is not pure geometry; but is the study of the graphical properties of a certain hypothetical thing. But neither ought graphics to be considered as pure geometry. It is the doctrine of a certain family of surfaces called the planes. But when we ask what surfaces these planes are, we find that no other purely geometrical description can be given of them than that there is a threefold continuum of them and that every three of them have one point and one only in common. But innumerable families of surfaces can be conceived of which that is true. For imagine space to be filled with a fluid and that all the planes, or a sufficient collection of them, are marked by dark films in that fluid. Suppose the fluid to be slightly viscous, so that the different parts of it cannot break away from one another. Then give that fluid any motion. The result will be that those films will be distorted into a vast variety of shapes of all degrees of complexity, and yet any three of them will continue to possess a particle in common. The family of surfaces they then occupy will have every purely geometrical property of the family of planes; and yet they will be planes no longer. The distinguishing character of a plane is that if any particle lying in it be luminous and any filament lying in it be opaque, the shadow of that filament from that luminous particle lies wholly in the plane. Hence it is that unlimited straight lines are called rays. Graphics then is not pure geometry but is geometrical perspective. If, however, any geometer replies that the family of planes ought not to be limited to optical planes, but ought to be considered as any tridimensional continuum of surfaces, any three of which have just one point in common, then my rejoinder is that if we are to allow the planes to undergo any sort of distortion so long as the connections of the different planes of the family are preserved, then the whole doctrine of graphics is manifestly nothing but a branch of topics. For this is just what topics is. It is the study of the continuous connections and defects of continuity of loci which are free to be distorted in any way so long as the integrity of the connections and separations of all their parts is maintained. All strictly pure geometry, therefore, is topics. I now proceed to explain my remark that in a continuous locus no point has any individual identity, unless it be a topically singular point, that is, an isolating point, or either the extremity of a line, or a point from which three or more branches

of a line, or two or more sheets of a surface extend. Consider for example an oval line, and let that oval line be broken so as to make a line with two extremities. It may be said that when this happens a point of the oval bursts into two. But I say that there is no particular point of the yet unbroken oval which can be identified, even approximately, with the point which bursts. For to say that the different points of an oval move round the oval, without ever moving out of it, is a form of words entirely destitute of meaning. The points are but places; and the oval and all its parts subsist unchanged whether we regard the points as standing still or running round. In like manner, when we say the oval bursts, we introduce time with a second dimension. Considering the time, the place of the oval is a two dimensional place. This is cylindrical at the bursting and is a ribbon afterward. If one of the dimensions has a different quality from the other, the couple, consisting of a point and instant on the two dimensional continuum where the bursting takes place, has an individual identity. But it cannot be identified with any particular line in the cylindrical part of the two dimensional, even approximately. That line has no individuality. Peirce: CP 4.220 Cross-Ref:†† 220. If instead of an oval place, we consider an oval thing, say a filament, then it certainly means something to say that the parts revolve round the oval. For any one particle might be marked black and so be seen to move. And even if it were not actually marked, it would have an individuality which would make it capable of being marked. So that the filament would have a definite velocity of rotation whether it could be seen to move or not. But the reply to this is, that the marking of a single particle would be a discontinuous marking; and if the particles possess all their own individual identities, that is to suppose a discontinuity of existence everywhere, notwithstanding the continuity of place. But I go further. If those particles possess each its individual existence there is a discrete collection of them, and this collection must possess a definite multitude. Now this multitude cannot equal the multiplicity of the aggregate of all possible discrete multitudes; because it is a discrete multitude, and as such it is smaller than another possible multitude. Hence, it is not equal to the multitude of points of the oval. For that is equal to the aggregate of all possible discrete multitudes, since the line, by hypothesis, affords room for any collection of discrete points however great. Hence, if particles of the filament are distributed equally along the line of the oval, there must be, in every sensible part, continuous collections of points, that is, lines, that are unoccupied by particles. These lines may be far less than any assignable magnitudes, that is, far less than any parts into which the system of real quantities enables us to divide the line. But there is no contradiction whatever involved in that. It thus appears that true continuity is logically absolutely repugnant to the individual designation or even approximate individual designation of its units, except at points where the character of the continuity is itself not continuous. Peirce: CP 4.221 Cross-Ref:†† 221. In view of what has been said, it is not surprising that those arithmetical operations of addition and multiplication, which seemed to have lost their significance forever, now reappear in reference to continua. It is not that the points, as points, can be one more or less; but if there are defects of continuity, those discontinuities can have perfect individual identity and so be added and multiplied. Peirce: CP 4.222 Cross-Ref:†† 222. In regard to lines, there are two kinds of defects of continuity. The first is, that two or more particles moving in a line-figure may be unable to coalesce. The possible number of such non-coalescible particles may be called the chorisis of a

figure. Any kind of a geometrical figure has chorisis whether it be a point-figure, a line-figure, a surface-figure, or what. Thus the chorisis of three [not overlapping] ovals is three. The chorisis may be any discrete multitude. Peirce: CP 4.223 Cross-Ref:†† 223. The other defect of continuity that can affect a line-figure is that there may be a collection of points upon it from which a particle can move in more or fewer ways than from the generality of points of the figure. These topically singular points, as I call them, are of two kinds: those away from which a particle can move on the line in less than two ways and those from which a particle can move in the line in more than three ways. Of the first kind are, first, isolated points,†1 or topical acnodes, and extremities. Those, from which a particle can move in more than two ways, are points of branching, or topical nodes. The negative of what Listing calls the Census number of a line is, if we give a further extension to his definition, that which I would call the total singularity of the line; namely, it is half the sum of the excesses over two of the number of ways in which a particle could leave the different singular points of the line. No line can have a fractional total singularity. Peirce: CP 4.224 Cross-Ref:†† 224. In regard to surfaces, the chorisis is very simple and calls for no particular attention. Peirce: CP 4.224 Cross-Ref:†† The theory of the singular places of surfaces is somewhat complicated. The singular places may be points, and those are either isolated points or points where two or more sheets are tacked together. Or the singular places may be isolated lines, and those are either totally isolated, or they may cut the surface. Such lines can have singularities like lines generally. Or the singular places may be lines which are either bounding edges or lines of splitting of the surface, or they may be in some parts edges and in other parts lines of splitting. They have singular points at which the line need not branch. All that is necessary is that the identities of the sheets that join there should change. If such a line has an extremity or point of odd branches, an even number of the sheets which come together there must change. Peirce: CP 4.225 Cross-Ref:†† 225. In addition to that, surfaces are another kind of defect of continuity, which Listing calls their cyclosis. That is, there is room upon them for oval filaments which cannot shrink to nothing by any movement in the surface. The number of operations each of a kind calculated to destroy a simple cyclosis which have to be [employed] in order to destroy the cyclosis of a surface is the number of the cyclosis. A puncture of a surface which does not change it from a closed surface to an open surface increases the cyclosis by one. A cut from edge to edge which does not increase the chorisis diminishes the cyclosis by one. Peirce: CP 4.225 Cross-Ref:†† The cyclosis of a spherical surface is 0; that of an unlimited plane is 1; that of an anchor-ring is 2, that of a plane with a fornix (or bridge from one part to another) is 3; that of an anchor-ring with a fornix is 4, etc. Peirce: CP 4.225 Cross-Ref:†† Euler's theorem †1 concerning polyhedra is an example of the additive arithmetic of continua. Peirce: CP 4.226 Cross-Ref:††

226. The multiplicity of points upon a surface must be admitted, as it seems to me, to be the square of that of the points of a line, and so with higher dimensions. The multitude of dimensions may be of any discrete multitude.

Peirce: CP 4.227 Cross-Ref:†† VII THE SIMPLEST MATHEMATICS†1P

§1. THE ESSENCE OF MATHEMATICS 227. In this chapter, I propose to consider certain extremely simple branches of mathematics which, owing to their utility in logic, have to be treated in considerable detail, although to the mathematician they are hardly worth consideration. In Chapter 4,†2 I shall take up those branches of mathematics upon which the interest of mathematicians is centred, but shall do no more than make a rapid examination of their logical procedure. In Chapter 5,†2 I shall treat formal logic by the aid of mathematics. There can really be little logical matter in these chapters; but they seem to me to be quite indispensable preliminaries to the study of logic. Peirce: CP 4.228 Cross-Ref:†† 228. It does not seem to me that mathematics depends in any way upon logic. It reasons, of course. But if the mathematician ever hesitates or errs in his reasoning, logic cannot come to his aid. He would be far more liable to commit similar as well as other errors there. On the contrary, I am persuaded that logic cannot possibly attain the solution of its problems without great use of mathematics. Indeed all formal logic is merely mathematics applied to logic.†3 Peirce: CP 4.229 Cross-Ref:†† 229. It was Benjamin Peirce,†4 whose son I boast myself, that in 1870 first defined mathematics as "the science which draws necessary conclusions."†5 This was a hard saying at the time; but today, students of the philosophy of mathematics generally acknowledge its substantial correctness. Peirce: CP 4.230 Cross-Ref:†† 230. The common definition, among such people as ordinary schoolmasters, still is that mathematics is the science of quantity. As this is inevitably understood in English, it seems to be a misunderstanding of a definition which may be very old,†P1 the original meaning being that mathematics is the science of quantities, that is, forms possessing quantity. We perceive that Euclid was aware that a large branch of geometry had nothing to do with measurement (unless as an aid in demonstrating); and, therefore, a Greek geometer of his age (early in the third century B.C.) or later could not define mathematics as the science of that which the abstract noun quantity expresses. A line, however, was classed as a quantity, or quantum, by Aristotle †1 and his followers; so that even perspective (which deals wholly with intersections and projections, not at all with lengths) could be said to be a science of quantities, "quantity" being taken in the concrete sense. That this was what was originally meant by the definition "Mathematics is the science of quantity," is sufficiently shown by

the circumstance that those writers who first enunciate it, about A.D. 500, that is Ammonius Hermiæ†2 and Boëthius,†3 make astronomy and music branches of mathematics; and it is confirmed by the reasons they give for doing so.†P2 Even Philo of Alexandria (100 B.C.), who defines mathematics as the science of ideas furnished by sensation and reflection in respect to their necessary consequences, since he includes under mathematics, besides its more essential parts, the theory of numbers and geometry, also the practical arithmetic of the Greeks, geodesy, mechanics, optics (or projective geometry), music, and astronomy, must be said to take the word 'mathematics' in a different sense from ours. That Aristotle did not regard mathematics as the science of quantity, in the modern abstract sense, is evidenced in various ways. The subjects of mathematics are, according to him, the how much and the continuous. (See Metaph. K iii 1061 a 33). He referred the continuous to his category of quantum; and therefore he did make quantum, in a broad sense, the one object of mathematics. Peirce: CP 4.231 Cross-Ref:†† 231. Plato, in the Sixth book of the Republic,†P1 holds that the essential characteristic of mathematics lies in the peculiar kind and degree of its abstraction, greater than that of physics but less than that of what we now call philosophy; and Aristotle †1 follows his master in this definition. It has ever since been the habit of metaphysicians to extol their own reasonings and conclusions as vastly more abstract and scientific than those of mathematics. It certainly would seem that problems about God, Freedom, and Immortality are more exalted than, for example, the question how many hours, minutes, and seconds would elapse before two couriers travelling under assumed conditions will come together; although I do not know that this has been proved. But that the methods of thought of the metaphysicians are, as a matter of historical fact, in any aspect, not far inferior to those of mathematics is simply an infatuation. One singular consequence of the notion which prevailed during the greater part of the history of philosophy, that metaphysical reasoning ought to be similar to that of mathematics, only more so, has been that sundry mathematicians have thought themselves, as mathematicians, qualified to discuss philosophy; and no worse metaphysics than theirs is to be found. Peirce: CP 4.232 Cross-Ref:†† 232. Kant †2 regarded mathematical propositions as synthetical judgments a priori; wherein there is this much truth, that they are not, for the most part, what he called analytical judgments; that is, the predicate is not, in the sense he intended, contained in the definition of the subject. But if the propositions of arithmetic, for example, are true cognitions, or even forms of cognition, this circumstance is quite aside from their mathematical truth. For all modern mathematicians agree with Plato and Aristotle that mathematics deals exclusively with hypothetical states of things, and asserts no matter of fact whatever; and further, that it is thus alone that the necessity of its conclusions is to be explained.†P1 This is the true essence of mathematics; and my father's definition is in so far correct that it is impossible to reason necessarily concerning anything else than a pure hypothesis. Of course, I do not mean that if such pure hypothesis happened to be true of an actual state of things, the reasoning would thereby cease to be necessary. Only, it never would be known apodictically to be true of an actual state of things. Suppose a state of things of a perfectly definite, general description. That is, there must be no room for doubt as to whether anything, itself determinate, would or would not come under that description. And suppose, further, that this description refers to nothing occult -- nothing that cannot be summoned up fully into the imagination. Assume, then, a range of

possibilities equally definite and equally subject to the imagination; so that, so far as the given description of the supposed state of things is general, the different ways in which it might be made determinate could never introduce doubtful or occult features. The assumption, for example, must not refer to any matter of fact. For questions of fact are not within the purview of the imagination. Nor must it be such that, for example, it could lead us to ask whether the vowel OO can be imagined to be sounded on as high a pitch as the vowel EE. Perhaps it would have to be restricted to pure spatial, temporal, and logical relations. Be that as it may, the question whether in such a state of things, a certain other similarly definite state of things, equally a matter of the imagination, could or could not, in the assumed range of possibility, ever occur, would be one in reference to which one of the two answers, Yes and No, would be true, but never both. But all pertinent facts would be within the beck and call of the imagination; and consequently nothing but the operation of thought would be necessary to render the true answer. Nor, supposing the answer to cover the whole range of possibility assumed, could this be rendered otherwise than by reasoning that would be apodictic, general, and exact. No knowledge of what actually is, no positive knowledge, as we say, could result. On the other hand, to assert that any source of information that is restricted to actual facts could afford us a necessary knowledge, that is, knowledge relating to a whole general range of possibility, would be a flat contradiction in terms. Peirce: CP 4.233 Cross-Ref:†† 233. Mathematics is the study of what is true of hypothetical states of things. That is its essence and definition. Everything in it, therefore, beyond the first precepts for the construction of the hypotheses, has to be of the nature of apodictic inference. No doubt, we may reason imperfectly and jump at a conclusion; still, the conclusion so guessed at is, after all, that in a certain supposed state of things something would necessarily be true. Conversely, too, every apodictic inference is, strictly speaking, mathematics. But mathematics, as a serious science, has, over and above its essential character of being hypothetical, an accidental characteristic peculiarity -- a proprium, as the Aristotelians used to say -- which is of the greatest logical interest. Namely, while all the "philosophers" follow Aristotle in holding no demonstration to be thoroughly satisfactory except what they call a "direct" demonstration, or a "demonstration why" -- by which they mean a demonstration which employs only general concepts and concludes nothing but what would be an item of a definition if all its terms were themselves distinctly defined -- the mathematicians, on the contrary, entertain a contempt for that style of reasoning, and glory in what the philosophers stigmatize as "mere" indirect demonstrations, or "demonstrations that." Those propositions which can be deduced from others by reasoning of the kind that the philosophers extol are set down by mathematicians as "corollaries." That is to say, they are like those geometrical truths which Euclid did not deem worthy of particular mention, and which his editors inserted with a garland, or corolla, against each in the margin, implying perhaps that it was to them that such honor as might attach to these insignificant remarks was due. In the theorems, or at least in all the major theorems, a different kind of reasoning is demanded. Here, it will not do to confine oneself to general terms. It is necessary to set down, or to imagine, some individual and definite schema, or diagram -- in geometry, a figure composed of lines with letters attached; in algebra an array of letters of which some are repeated. This schema is constructed so as to conform to a hypothesis set forth in general terms in the thesis of the theorem. Pains are taken so to construct it that there would be something closely similar in every possible state of things to which the hypothetical description in the thesis would be applicable, and furthermore to construct it so that it shall have no other characters

which could influence the reasoning. How it can be that, although the reasoning is based upon the study of an individual schema, it is nevertheless necessary, that is, applicable, to all possible cases, is one of the questions we shall have to consider. Just now, I wish to point out that after the schema has been constructed according to the precept virtually contained in the thesis, the assertion of the theorem is not evidently true, even for the individual schema; nor will any amount of hard thinking of the philosophers' corollarial kind ever render it evident. Thinking in general terms is not enough. It is necessary that something should be DONE. In geometry, subsidiary lines are drawn. In algebra permissible transformations are made. Thereupon, the faculty of observation is called into play. Some relation between the parts of the schema is remarked. But would this relation subsist in every possible case? Mere corollarial reasoning will sometimes assure us of this. But, generally speaking, it may be necessary to draw distinct schemata to represent alternative possibilities. Theorematic reasoning invariably depends upon experimentation with individual schemata. We shall find that, in the last analysis, the same thing is true of the corollarial reasoning, too; even the Aristotelian "demonstration why." Only in this case, the very words serve as schemata. Accordingly, we may say that corollarial, or "philosophical" reasoning is reasoning with words; while theorematic, or mathematical reasoning proper, is reasoning with specially constructed schemata. Peirce: CP 4.234 Cross-Ref:†† 234. Another characteristic of mathematical thought is the extraordinary use it makes of abstractions. Abstractions have been a favorite butt of ridicule in modern times. Now it is very easy to laugh at the old physician who is represented as answering the question, why opium puts people to sleep, by saying that it is because it has a dormative virtue. It is an answer that no doubt carries vagueness to its last extreme. Yet, invented as the story was to show how little meaning there might be in an abstraction, nevertheless the physician's answer does contain a truth that modern philosophy has generally denied: it does assert that there really is in opium something which explains its always putting people to sleep. This has, I say, been denied by modern philosophers generally. Not, of course, explicitly; but when they say that the different events of people going to sleep after taking opium have really nothing in common, but only that the mind classes them together -- and this is what they virtually do say in denying the reality of generals -- they do implicitly deny that there is any true explanation of opium's generally putting people to sleep. Peirce: CP 4.235 Cross-Ref:†† 235. Look through the modern logical treatises, and you will find that they almost all fall into one or other of two errors, as I hold them to be; that of setting aside the doctrine of abstraction (in the sense in which an abstract noun marks an abstraction) as a grammatical topic with which the logician need not particularly concern himself; and that of confounding abstraction, in this sense, with that operation of the mind by which we pay attention to one feature of a percept to the disregard of others. The two things are entirely disconnected. The most ordinary fact of perception, such as "it is light," involves precisive abstraction, or prescission.†1 But hypostatic abstraction, the abstraction which transforms "it is light" into "there is light here," which is the sense which I shall commonly attach to the word abstraction (since prescission will do for precisive abstraction) is a very special mode of thought. It consists in taking a feature of a percept or percepts (after it has already been prescinded from the other elements of the percept), so as to take propositional form in a judgment (indeed, it may operate upon any judgment whatsoever), and in conceiving this fact to consist in the relation between the subject of that judgment and

another subject, which has a mode of being that merely consists in the truth of propositions of which the corresponding concrete term is the predicate. Thus, we transform the proposition, "honey is sweet," into "honey possesses sweetness." "Sweetness" might be called a fictitious thing, in one sense. But since the mode of being attributed to it consists in no more than the fact that some things are sweet, and it is not pretended, or imagined, that it has any other mode of being, there is, after all, no fiction. The only profession made is that we consider the fact of honey being sweet under the form of a relation; and so we really can. I have selected sweetness as an instance of one of the least useful of abstractions. Yet even this is convenient. It facilitates such thoughts as that the sweetness of honey is particularly cloying; that the sweetness of honey is something like the sweetness of a honeymoon; etc. Abstractions are particularly congenial to mathematics. Everyday life first, for example, found the need of that class of abstractions which we call collections. Instead of saying that some human beings are males and all the rest females, it was found convenient to say that mankind consists of the male part and the female part. The same thought makes classes of collections, such as pairs, leashes, quatrains, hands, weeks, dozens, baker's dozens, sonnets, scores, quires, hundreds, long hundreds, gross, reams, thousands, myriads, lacs, millions, milliards, milliasses, etc. These have suggested a great branch of mathematics.†P1 Again, a point moves: it is by abstraction that the geometer says that it "describes a line." This line, though an abstraction, itself moves; and this is regarded as generating a surface; and so on. So likewise, when the analyst treats operations as themselves subjects of operations, a method whose utility will not be denied, this is another instance of abstraction. Maxwell's notion of a tension exercised upon lines of electrical force, transverse to them, is somewhat similar. These examples exhibit the great rolling billows of abstraction in the ocean of mathematical thought; but when we come to a minute examination of it, we shall find, in every department, incessant ripples of the same form of thought, of which the examples I have mentioned give no hint. Peirce: CP 4.236 Cross-Ref:†† 236. Another characteristic of mathematical thought is that it can have no success where it cannot generalize. One cannot, for example, deny that chess is mathematics, after a fashion; but, owing to the exceptions which everywhere confront the mathematician in this field -- such as the limits of the board; the single steps of king, knight, and pawn; the finite number of squares; the peculiar mode of capture by pawns; the queening of pawns; castling -- there results a mathematics whose wings are effectually clipped, and which can only run along the ground. Hence it is that a mathematician often finds what a chess-player might call a gambit to his advantage; exchanging a smaller problem that involves exceptions for a larger one free from them. Thus, rather than suppose that parallel lines, unlike all other pairs of straight lines in a plane, never meet, he supposes that they intersect at infinity. Rather than suppose that some equations have roots while others have not, he supplements real quantity by the infinitely greater realm of imaginary quantity. He tells us with ease how many inflexions a plane curve of any description has; but if we ask how many of these are real, and how many merely fictional, he is unable to say. He is perplexed by three-dimensional space, because not all pairs of straight lines intersect, and finds it to his advantage to use quaternions which represent a sort of four-fold continuum, in order to avoid the exception. It is because exceptions so hamper the mathematician that almost all the relations with which he chooses to deal are of the nature of correspondences; that is to say, such relations that for every relate there is the same number of correlates, and for every correlate the same number of relates.

Peirce: CP 4.237 Cross-Ref:†† 237. Among the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility. It is easy to speak with precision upon a general theme. Only, one must commonly surrender all ambition to be certain. It is equally easy to be certain. One has only to be sufficiently vague. It is not so difficult to be pretty precise and fairly certain at once about a very narrow subject. But to reunite, like mathematics, perfect exactitude and practical infallibility with unrestricted universality, is remarkable. But it is not hard to see that all these characters of mathematics are inevitable consequences of its being the study of hypothetical truth. Peirce: CP 4.238 Cross-Ref:†† 238. It is difficult to decide between the two definitions of mathematics; the one by its method, that of drawing necessary conclusions; the other by its aim and subject matter, as the study of hypothetical states of things. The former makes or seems to make the deduction of the consequences of hypotheses the sole business of the mathematician as such. But it cannot be denied that immense genius has been exercised in the mere framing of such general hypotheses as the field of imaginary quantity and the allied idea of Riemann's surface, in imagining non-Euclidian measurement, ideal numbers, the perfect liquid. Even the framing of the particular hypotheses of special problems almost always calls for good judgment and knowledge, and sometimes for great intellectual power, as in the case of Boole's logical algebra. Shall we exclude this work from the domain of mathematics? Perhaps the answer should be that, in the first place, whatever exercise of intellect may be called for in applying mathematics to a question not propounded in mathematical form [it] is certainly not pure mathematical thought; and in the second place, that the mere creation of a hypothesis may be a grand work of poietic †1 genius, but cannot be said to be scientific, inasmuch as that which it produces is neither true nor false, and therefore is not knowledge. This reply suggests the further remark that if mathematics is the study of purely imaginary states of things, poets must be great mathematicians, especially that class of poets who write novels of intricate and enigmatical plots. Even the reply, which is obvious, that by studying imaginary states of things we mean studying what is true of them, perhaps does not fully meet the objection. The article Mathematics in the ninth edition of the Encyclopaedia Britannica†2 makes mathematics consist in the study of a particular sort of hypotheses, namely, those that are exact, etc., as there set forth at some length. The article is well worthy of consideration. Peirce: CP 4.239 Cross-Ref:†† 239. The philosophical mathematician, Dr. Richard Dedekind,†3 holds mathematics to be a branch of logic. This would not result from my father's definition, which runs, not that mathematics is the science of drawing necessary conclusions -which would be deductive logic -- but that it is the science which draws necessary conclusions. It is evident, and I know as a fact, that he had this distinction in view. At the time when he thought out this definition, he, a mathematician, and I, a logician, held daily discussions about a large subject which interested us both; and he was struck, as I was, with the contrary nature of his interest and mine in the same propositions. The logician does not care particularly about this or that hypothesis or its consequences, except so far as these things may throw a light upon the nature of

reasoning. The mathematician is intensely interested in efficient methods of reasoning, with a view to their possible extension to new problems; but he does not, quâ mathematician, trouble himself minutely to dissect those parts of this method whose correctness is a matter of course. The different aspects which the algebra of logic will assume for the two men is instructive in this respect. The mathematician asks what value this algebra has as a calculus. Can it be applied to unravelling a complicated question? Will it, at one stroke, produce a remote consequence? The logician does not wish the algebra to have that character. On the contrary, the greater number of distinct logical steps, into which the algebra breaks up an inference, will for him constitute a superiority of it over another which moves more swiftly to its conclusions. He demands that the algebra shall analyze a reasoning into its last elementary steps. Thus, that which is a merit in a logical algebra for one of these students is a demerit in the eyes of the other. The one studies the science of drawing conclusions, the other the science which draws necessary conclusions. Peirce: CP 4.240 Cross-Ref:†† 240. But, indeed, the difference between the two sciences is far more than that between two points of view. Mathematics is purely hypothetical: it produces nothing but conditional propositions. Logic, on the contrary, is categorical in its assertions. True, it is not merely, or even mainly, a mere discovery of what really is, like metaphysics. It is a normative science. It thus has a strongly mathematical character, at least in its methodeutic division; for here it analyzes the problem of how, with given means, a required end is to be pursued. This is, at most, to say that it has to call in the aid of mathematics; that it has a mathematical branch. But so much may be said of every science. There is a mathematical logic, just as there is a mathematical optics and a mathematical economics. Mathematical logic is formal logic. Formal logic, however developed, is mathematics. Formal logic, however, is by no means the whole of logic, or even its principal part. It is hardly to be reckoned as a part of logic proper. Logic has to define its aim; and in doing so is even more dependent upon ethics,†1 or the philosophy of aims, by far, than it is, in the methodeutic branch, upon mathematics. We shall soon come to understand how a student of ethics might well be tempted to make his science a branch of logic; as, indeed, it pretty nearly was in the mind of Socrates. But this would be no truer a view than the other. Logic depends upon mathematics; still more intimately upon ethics; but its proper concern is with truths beyond the purview of either. Peirce: CP 4.241 Cross-Ref:†† 241. There are two characters of mathematics which have not yet been mentioned, because they are not exclusive characteristics of it. One of these, which need not detain us, is that mathematics is distinguished from all other sciences †2 except only ethics, in standing in no need of ethics. Every other science, even logic -logic, especially -- is in its early stages in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an anachrioid [?] film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be. Peirce: CP 4.242 Cross-Ref:†† 242. The other character -- and of particular interest it is to us just now -- is that mathematics, along with ethics and logic alone of the sciences, has no need of any appeal to logic. No doubt, some reader may exclaim in dissent to this, on first hearing it said. Mathematics, they may say, is preëminently a science of reasoning. So it is; preëminently a science that reasons. But just as it is not necessary, in order to

talk, to understand the theory of the formation of vowel sounds, so it is not necessary, in order to reason, to be in possession of the theory of reasoning. Otherwise, plainly, the science of logic could never be developed. The contrary objection would have more excuse, that no science stands in need of logic, since our natural power of reason is enough. Make of logic what the majority of treatises in the past have made of it, and a very common class of English and French books still make of it -- that is to say, mainly formal logic, and that formal logic represented as an art of reasoning -and in my opinion this objection is more than sound, for such logic is a great hindrance to right reasoning. It would, however, be aside from our present purpose to examine this objection minutely. I will content myself with saying that undoubtedly our natural power of reasoning is enough, in the same sense that it is enough, in order to obtain a wireless transatlantic telegraph, that men should be born. That is to say, it is bound to come sooner or later. But that does not make research into the nature of electricity needless for gaining such a telegraph. So likewise if the study of electricity had been pursued resolutely, even if no special attention had ever been paid to mathematics, the requisite mathematical ideas would surely have been evolved. Faraday, indeed, did evolve them without any acquaintance with mathematics. Still it would be far more economical to postpone electrical researches, to study mathematics by itself, and then to apply it to electricity, which was Maxwell's way. In this same manner, the various logical difficulties which arise in the course of every science except mathematics, ethics, and logic, will, no doubt, get worked out after a time, even though no special study of logic be made. But it would be far more economical to make first a systematic study of logic. If anybody should ask what are these logical difficulties which arise in all the sciences, he must have read the history of science very irreflectively. What was the famous controversy concerning the measure of force but a logical difficulty? What was the controversy between the uniformitarians and the catastrophists but a question of whether or not a given conclusion followed from acknowledged premisses? This will fully appear in the course of our studies in the present work.†1 Peirce: CP 4.243 Cross-Ref:†† 243. But it may be asked whether mathematics, ethics, and logic have not encountered similar difficulties. Are the doctrines of logic at all settled? Is the history of ethics anything but a history of controversy? Have no logical errors been committed by mathematicians? To that I reply, first, as to logic, that not only have the rank and file of writers on the subject been, as an eminent psychiatrist, Maudsley, declares, men of arrested brain-development, and not only have they generally lacked the most essential qualification for the study, namely mathematical training, but the main reason why logic is unsettled is that thirteen different opinions are current as to the true aim of the science.†1 Now this is not a logical difficulty but an ethical difficulty; for ethics is the science of aims. Secondly, it is true that pure ethics has been, and always must be, a theatre of discussion, for the reason that its study consists in the gradual development of a distinct recognition of a satisfactory aim. It is a science of subtleties, no doubt; but it is not logic, but the development of the ideal, which really creates and resolves the problems of ethics. Thirdly, in mathematics errors of reasoning have occurred, nay, have passed unchallenged for thousands of years. This, however, was simply because they escaped notice. Never, in the whole history of the science, has a question whether a given conclusion followed mathematically from given premisses, when once started, failed to receive a speedy and unanimous reply. Very few have been even the apparent exceptions; and those few have been due to the fact that it is only within the last half century that mathematicians have come to have a perfectly clear recognition of what is

mathematical soil and what foreign to mathematics. Perhaps the nearest approximation to an exception was the dispute about the use of divergent series. Here neither party was in possession of sufficient pure mathematical reasons covering the whole ground; and such reasons as they had were not only of an extra-mathematical kind, but were used to support more or less vague positions. It appeared then, as we all know now, that divergent series are of the utmost utility.†P1 Peirce: CP 4.243 Cross-Ref:†† Struck by this circumstance, and making an inference, of which it is sufficient to say that it was not mathematical, many of the old mathematicians pushed the use of divergent series beyond all reason. This was a case of mathematicians disputing about the validity of a kind of inference that is not mathematical. No doubt, a sound logic (such as has not hitherto been developed) would have shown clearly that that non-mathematical inference was not a sound one. But this is, I believe, the only instance in which any large party in the mathematical world ever proposed to rely, in mathematics, upon unmathematical reasoning. My proposition is that true mathematical reasoning is so much more evident than it is possible to render any doctrine of logic proper -- without just such reasoning -- that an appeal in mathematics to logic could only embroil a situation. On the contrary, such difficulties as may arise concerning necessary reasoning have to be solved by the logician by reducing them to questions of mathematics. Upon those mathematical dicta, as we shall come clearly to see, the logician has ultimately to repose. Peirce: CP 4.244 Cross-Ref:†† 244. So a double motive induces me to devote some preliminary chapters to mathematics. For, in the first place, in studying the theory of reasoning, we are concerned to acquaint ourselves with the methods of that prior science of which acts of reasoning form the staple. In the second place, logic, like any other science, has its mathematical department, and of that, a large portion, at any rate, may with entire convenience be studied as soon as we take up the study of logic, without any propedeutic. That portion is what goes by the name of Formal Logic.†P1 It so happens that the special kind of mathematics needed for formal logic, which, therefore, we need to study in detail, as we need not study other branches of mathematics, is so excessively simple as neither to have much mathematical interest, nor to display the peculiarities of mathematical reasoning. I shall, therefore, devote the present chapter -- a very dull one, I am sorry to say, it must be -- to this kind of mathematics. Chapter 4 will treat of the more truly mathematical mathematics; and Chapter 5 will apply the results of the present chapter to the study of Formal Logic.†1

Peirce: CP 4.245 Cross-Ref:†† §2. DIVISION OF PURE MATHEMATICSP

245. We have to make a rapid survey of pure mathematics, in so far as it interests us as students of logic. Each branch of mathematics will have to be reconnoitered and its methods examined. Those parts of the calculus of which use has to be made in the study of reasoning must receive a fuller treatment. Finally, having so collected some information about mathematics, we may venture upon some useful generalizations concerning the nature of mathematical thought. But this plan calls for a preliminary dissection of mathematics into its several branches.

Peirce: CP 4.246 Cross-Ref:†† 246. Each branch of mathematics sets out from a general hypothesis of its own. I mean by its general hypothesis the substance of its postulates and axioms, and even of its definitions, should they be contaminated with any substance, instead of being the pure verbiage they ought to be. We have to make choice, then, between a division of mathematics according to the matter of its hypotheses, or according to the forms of the schemata of which it avails itself. These latter are either geometrical or algebraical. Geometrical schemata are linear figures with letters attached; the perfect imaginability, on the one hand, and the extreme familiarity, on the other hand, of spatial relations are taken advantage of, to enable us to see what will necessarily be true under supposed conditions. The algebraical schemata are arrays of characters, sometimes in series, sometimes in blocks, with which are associated certain rules of permissible transformation. With these rules the algebraist has perfectly to familiarize himself. By virtue of these rules, become habits of association, when one array has been written or assumed to be permissibly scriptible, the mathematician just as directly perceives that another array is permissibly scriptible, as he perceives that a person talking in a certain tone is angry, or [is] using certain words in such and such a sense. Peirce: CP 4.247 Cross-Ref:†† 247. The primary division of mathematics into algebra and geometry is the usual one. But, in all departments, it appears both a priori and a posteriori, that divisions according to differences of purpose should be given a higher rank than divisions according to different methods of attaining that purpose.†1 The division of pure mathematics into algebra and geometry was first adopted before the modern conception of pure mathematics had been distinctly prescinded, and when geometry and algebra seemed to deal with different subjects. It remains, a vestige of that old unclearness and a witness that not even mathematicians are able entirely to shake off the sequelæ of exploded ideas. For now that everybody knows that any mathematical subject, from the theory of numbers to topical geometry, may be treated either algebraically or geometrically, one cannot fail to see that so to divide mathematics is to make twice over the division according to fundamental hypotheses, to which one must come, at last. This duplication is worse than useless, since the geometrical and algebraical methods are by many writers continually mixed. No such inconvenience attends the other plan of classification; for two sets of fundamental hypotheses could not, properly speaking, be mixed without self-contradiction. Peirce: CP 4.248 Cross-Ref:†† 248. Let us, then, divide mathematics according to the nature of its general hypotheses, taking for the ground of primary division the multitude of units, or elements, that are supposed; and for the ground of subdivision that mode of relationship between the elements upon which the hypotheses focus the attention. Peirce: CP 4.249 Cross-Ref:†† 249. From a logician's point of view this plan of classification would seem to call for a preliminary analysis of what is meant by multitude. But to execute this analysis satisfactorily, considerable studies of logic would be indispensable preliminaries. Besides, it is not at all in the spirit of mathematics to analyze the ideas with which it works farther than is needful for using them in deducing consequences, nor sooner than that need comes to be felt. It is true that we, as students of logic, are not bound to embrace the mathematical ways of thought as far as that, but the other circumstance, that it is, at the present stage of our studies, impossible to make the

analysis, must be conclusive.

Peirce: CP 4.250 Cross-Ref:†† §3. THE SIMPLEST BRANCH OF MATHEMATICSP

250. Were nothing at all supposed, mathematics would have no ground at all to go upon. Were the hypothesis merely that there was nothing but one unit, there would not be a possibility of a question, since only one answer would be possible. Consequently, the simplest possible hypothesis is that there are two objects, which we may denote by v and f. Then the first kind of problem of this algebra will be, given certain data concerning an unknown object, x, required to know whether it is v or f. Or similar problems may arise concerning several unknowns, x, y, etc. Or when the last problem cannot be resolved, we may ask whether, supposing x to be v, will y be v or f? And similarly, supposing x to be f. Again, given certain data concerning x, we may ask, what else needs to be known in order to compel x to be v or to be f. Or again, given certain information about x, y, and z, what relations between x and z remain unchanged whether y be v or f? Peirce: CP 4.251 Cross-Ref:†† 251. Let us call v and f the two possible values, one of which must be attached to any unknown. For the form of reasoning will be the same whether we talk of identity or attachment. The attachment may be of any kind so long as each unknown must be, or be attached to, v or f, but cannot be or be attached to, both v and f. This idea of a system of values is one of the most fundamental abstractions of the algebraic method of mathematics. An object of the universe, whose value is generally unknown, though it may in special cases be known -- that is to say, an object which, to phrase the matter differently, is one of the values, though perhaps we do not know which -- is called, when we speak of it as "having" a value, a quantity. For example, suppose the problem under consideration be to determine, upon a certain hypothesis, the numerical definition of the instant, or, as we may say, to determine the exact date, at which two couriers will meet. This date is some one of the series of numbers each of which is expressible, at least to any predesignate degree of approximation, in our usual method of numeral notation. That series of numbers will be the system of values; and the number we want is one of them. But we find it convenient to use a different phrase, and to say that the date is defined to be the date at which the couriers meet, that this fixes its identity, and that what we seek to know is what value becomes attached to it in consequence of the conditions the problem supposes. It will be convenient to conceive of this statement as a "mere" variation of phraseology, although, as we shall learn, the word "mere" in such cases is often inappropriate, since great mathematical results are attainable by such means. Dichotomic algebra can be applied wherever there are just two possible alternatives. Thus, we might call the v the truth, and f falsity. Then, in regard to a given proposition we may seek to know whether it is true or false; that is, whether it is or is not a partial description of the real universe, or say, whether what it means is identical with the existent truth or identical with nothing. Looking at the matter in a different way, or phrasing it differently, we say that a proposition has one or other of two values, being either true and good for something, or false and good for nothing. The point of view of mathematics is the point of view which looks upon those two points of view as no more than different phrases for the same fact.

Peirce: CP 4.252 Cross-Ref:†† 252. There is another little group of algebraical words which must now be defined in the imperfect way in which they can be defined for dichotomic mathematics. In the first place, there are the pair of terms, constant, or constant quantity, and variable, or variable quantity. These words were introduced by the Marquis d'Hôpital †1 in 1696. Suppose two couriers to set out, at the same instant, from two points 12 miles apart and to travel toward one another, the one at the rate of 7 miles an hour, the other at the rate of 8 miles an hour: when will they meet? They evidently approach one another at the rate of 7 plus 8, or 15 miles an hour; and they will reduce the distance of 12 miles to nothing in 12/15 of an hour, or 12 times 4, or 48, minutes. But suppose we find the distance was wrongly given; that it is 12 1/2 miles. Then, the date, or numerical designation of the instant of meeting, becomes different. But if we choose to say that the quantity sought is defined as the time of meeting, and that it remains the same quantity, having the same definition, but that its value only is altered, then that quantity is said to be variable. A quantity is said to be variable when we propose to consider it as taking different values in different states of things; or, to phrase the matter differently, when we consider a group of questions together, as one general question, the single questions having different values for their answers. The most usual case is where we suppose the quantity to take all possible values under different circumstances. A quantity is called constant when the hypothesis includes no states of things in which its value changes. The difference between an unknown quantity and a variable quantity is trifling. The unknown quantity is variable at first; but special hypotheses being adopted, it is restricted to certain values, perhaps to a single value. Peirce: CP 4.253 Cross-Ref:†† 253. The word function (a sort of semi-synonym of "operation") was first used in something like its present mathematical sense in 1692, by a writer who was doubtless Leibniz.†1 It soon came into use with the circle of analysts of whom Leibniz was the centre. But the first attempt at a definition of it was by John Bernouilli,†2 in 1718. There has since been much discussion as to what precise meaning can most advantageously be applied to it; but the most general definition, that of Dirichlet,†3 is confined to a system of numerical values. Since I wish to apply the word to all sorts of algebra, I shall, under these circumstances, take the liberty of generalizing the meaning in the manner which seems to me to be called for. I shall say then, that, given two ordered sets of the same number of quantities, x[1], x[2], x[3], . . . x[n], and y[1], y[2], y[3] . . . y[n], any quantity, say x[2], of the one set is the same function of the other quantities of that same set, which are called its arguments, that the corresponding quantity, y[2], according to the order of arrangement of the other set, is of the remaining quantities of that set, if and only if every set of values which either set of quantities, in their order, can take, can likewise be taken by the other set. Thus, to say that a quantity is a given function of certain quantities as arguments is simply to say that its value stands in a given relation to theirs; or that a given proposition is true of the whole set of values in their order. To say simply that one quantity is some function of certain others is to say nothing; since of every set of values something is true. But this no more renders function a useless word than the fact, that it means nothing to say of a set of things that there is some relation between them, renders relation a useless word. Peirce: CP 4.253 Cross-Ref:†† I may mention that the old and usual expression is "a function of variables";

but the word argument here is not unusual and is more explicit. The function is also called the dependent variable; the arguments, the independent variables. Of course, any one of the whole set of quantities composed of the function and its arguments is just as dependent as any other. It is a mere way of referring to them. The function is often conceived, very conveniently, as resulting from an operation performed on the arguments, which are then called operands. The idea is that the definition of the same function implies a rule which permits such sets of values as may conform to its conditions and excludes others; and the operation is the operation of actually applying this rule, when the values of all the quantities but one are given, in order to ascertain what the value of the remaining quantity can be. Peirce: CP 4.254 Cross-Ref:†† 254. Among functions, or operations, there is one extensive class which is of particular importance. I call it the class of correspondential functions, or operations. Namely, if all the variables but one, independent and dependent, have a set of values assigned to them, then, if the relation between them is a correspondence, the number of different values which the remaining variable can have, is generally the same, whatever the particular set of assigned values may be; although this number is not necessarily the same when different quantities are thus left over to the last. I say generally the same, because there may be peculiar isolated exceptions, though this limitation can have no significance in dichotomic mathematics. A function which is in correspondence with its arguments may be called a correspondential function. It may be remarked that it is not the habit of mathematicians, in general statements, to pay attention to isolated exceptions; and when a mathematician uses the phrase "in general" he means to be understood as not considering possible peculiar cases. Thus, I have known a great mathematician to enunciate a proposition concerning multiple algebra to be true "in general" when the state of the case was that there were just two instances of its being true against an infinity of instances of its being false. Peirce: CP 4.255 Cross-Ref:†† 255. A function which has but one value for any one set of values of the arguments is called monotropic. A function which, when all the arguments except a certain one take any fixed values, always changes its value with a change of that one, may be called distinctive for that argument. Peirce: CP 4.256 Cross-Ref:†† 256. If the relation between a function and its arguments is such that one of the latter may take any value for every set of the values of the others without altering the function, the function may be said to be invariable with that argument. If the function can take any value, whatever values be assigned to the arguments, it may be said to be independent of the arguments. In either of these cases, the function may be called a degenerate function. Peirce: CP 4.257 Cross-Ref:†† 257. With this lexical preface, we come down to our dichotomic mathematics, which I shall treat algebraically. The first thing to be done is to fix upon a sign to show that any quantity, say x, has the value v, and upon another to signify that it has the value f. The simplest suggestion is that universally used since man began to keep accounts; namely, to appropriate a place in which we are to write whatever is v, say the upper of two lines, the lower of which is appropriated to quantities whose value is f. That is, we open one account for v, and another for f. In doing this, we put v and f in a radically different category from the other letters, very much as two opposite

qualities, say good and bad, are attributed to concrete objects. I do not mean that there is any other analogy than that the values, v and f, are made to be of a different nature from the quantities, x, y, z, etc. One or other of the values, but not both, is connected, in some definite sense, and it matters not what the sense may be, so long as it is definite, with each quantity. But here an important remark obtrudes. Non-connection in any definite way is only another equally definite mode of connection; especially in a strictly dichotomic state of things. If, for example, every man either does good and eschews evil, or does evil and eschews good, then the former is thereby connected with evil by eschewing it, as he is connected with good in the mode of connection called doing it. Note how the perfect balance of our initial dichotomy generates new dichotomies: first, two categories, those of value and of quantity; then, two modes of connection between a value and a quantity. Peirce: CP 4.258 Cross-Ref:†† 258. Let us modify our mode of signifying the attachment of a quantity to a value, so as to show its contrary attachment to the opposite value. For this purpose, x f ----------·----------- v y

O

from a centre, O, let us draw a horizontal arm to the right, which we will call the v-radius, and another to the left, which we will call the f-radius. Now, then, any quantity x put in the upper or v account, will be so situated that a right-handed, or clock-wise, revolution around O will bring it first to the v-radius; as it will bring a quantity, y, in the f account, to the f-radius; while a left-handed, or counter-clock-wise, turn around O will carry the quantities each to the other radius. This diagram suggests another way of signifying the value of a quantity. Let a heavy line, representing the horizontal bar of the diagram, be drawn under the sign of a quantity, thus, x_, to signify that its value is v; and the same bar be drawn above it, thus, ~y, to signify that its value is f. Peirce: CP 4.259 Cross-Ref:†† 259. It may be mentioned that this mode of indicating the value by a bar has a historical appropriateness. For although the two values f and v are, at present, merely distinguished, without any definite difference between them being admitted -- and mathematically they do stand upon a precise par, and will continue to do so -- yet when dichotomic algebra comes to be applied to logic, it will be found necessary to call one of them verity and the other falsity; and the letters v and f were chosen with a view to that. We shall find it impossible later to prevent this affecting our purest practicable mathematics, in some measure. Now it has been the practice, from antiquity, to draw a heavy line under that whose truth it was desired to emphasize. On the other hand, the obelus, or spit, is already mentioned by Lucian, in the second century A. D., as the sign of denial; and that is why it is frequently even now used in several European countries to denote an n, for non, or the other nasal letter m. Peirce: CP 4.259 Cross-Ref:†† The Greek word {obelos} means a spit, (for example, {pempöbelos} is a five-pronged fork) so that the original notion was that that which is beneath it was transfixed; just as it used to be usual to nail false coins to the counter. Peirce: CP 4.260 Cross-Ref:††

260. There is a small theorem about multitude that it will be convenient to have stated, and the reader will do well to fix it in his memory correctly, with the "each" number as exponent. If each of a set of m objects be connected with some one of a set of n objects, the possible modes of connection of the sets will number nm. Now an assertion concerning the value of a quantity either admits as possible or else excludes each of the values v and f. Thus, v and f form the set of m objects each connected with one only of n objects, admission and exclusion. Hence there are, nm, or 22, or 4, different possible assertions concerning the value of any quantity, x. Namely, one assertion will simply be a form of assertion without meaning, since it admits either value. It is represented by the letter, x. Another assertion will violate the hypothesis of dichotomies by excluding both values. It may be represented by ~x_.

[Click here to view] Of the remaining two, one will admit v and exclude f, namely, x_; the other will admit f and exclude v, namely ~x. Peirce: CP 4.261 Cross-Ref:†† 261. Now, let us consider assertions concerning the values of two quantities, x and y. Here there are two quantities, each of which has one only of two values; so that there are 22, or 4, possible states of things, as shown in this diagram. Peirce: CP 4.261 Cross-Ref:†† Above the line, slanting upward to the right, are placed the cases in which x is v; below it, those in which x is f. Above the line but slanting downward to the right, are placed the cases in which y is v; below it, those in which y is f. Now in each possible assertion each of these states of things is either admitted or excluded; but not both. Thus, m will be 22, while n will be 2; and there will be nm, or 24, or 16, possible assertions. They may be represented by drawing the lines of the diagram between x and y and closing over the compartments for the excluded sets of values. . . .†1 Peirce: CP 4.262 Cross-Ref:††

262. Of three quantities, there are 23, or 8, possible sets of values, and consequently 28, or 256, different forms of propositions. Of these, there are only 38 which can fairly be said to be expressible by the signs [used in a logic of two quantities]. It is true that a majority of the others might be expressed by two or more propositions. But we have not, as yet, expressly adopted any sign for the operation of compounding propositions. Besides, a good many propositions concerning three quantities cannot be expressed even so. Such, for example, is the statement which admits the following sets of values: x y z -----------------------v v v v f f f v f f f v Peirce: CP 4.262 Cross-Ref:†† Moreover, if we were to introduce signs for expressing [each of] these, of which we should need 8, even allowing the composition of assertions, still 16 more would be needed to express all propositions concerning 4 quantities, 32 for 5, and so on, ad infinitum. Peirce: CP 4.263 Cross-Ref:†† 263. The remedy for this state of things lies in simply giving the values v and f to propositions; that is, in admitting them to the universe of quantities. Here I will make an observation, by the way. Although formal logic is nothing but mathematics applied to logic, yet not a few of those who have cultivated it have had distinctly unmathematical minds. Indeed, in man's first steps in mathematics, he always draws back from mathematical conceptions. To first make v represent, let us say, Julius Caesar, and f, Pompey, since they may represent any subjects that are individual and definite, and thereupon further to propose to make every proposition either v or f, shocks the lower order of formal logicians. Such a mind will say, "If we have to distinguish propositions into two categories, let us denote their values by accented, or otherwise modified, letters, say v' and f', and not call them Caesar and Pompey, which is absurd." But I reply that that sort of stickling for usage bars the progress of mathematical thought; that the very fact that it is absurd that a proposition should be Caesar or Pompey proves that there will be no inconvenience, not in calling propositions what you mean by Caesar and Pompey, which, as you say, nobody could mean to do, but in generalizing the conception of Caesar, so as to make it include those propositions which are destined to triumph over the others. To protest against this, is virtually to protest against generalization; and to protest against generalization is to protest against thought; and to protest against thought is a pretty kind of logic. But still the unmathematical mind will ask, why not, however, adopt the v' and f'; for he cannot conquer his shrinking from any generalization that can be evaded. It is the spirit of conservatism, the shrinking from the outré, which is commendable in its proper place; only it is unmathematical: instead of shrinking from generalizations, the part of the mathematician is to go for them eagerly. However, it would not even answer the purpose to distinguish v' and f' from v and f,†P1 for the reason that there would be equal reason for distinguishing propositions about quantities being v or being f from propositions about quantities being v' or being f'; so that we should require a v'' and an f'', and so on, ad infinitum. Now this would hamper us, because we should find we had occasion to form many a proposition about two propositions, as to whether one of the two was v'' or f'', for example, and at the same time whether

the other were, say, vIV or fIV, etc. We should, therefore, require still other v's and f's all to no mathematical purpose whatsoever; but, on the contrary, interfering fatally with a very different diversification of v's and f's which, we shall find, really will be needed. Peirce: CP 4.264 Cross-Ref:†† 264. If we assign the values v and f to propositions, we must either say that x_ has the same value as x, in which case ~x will have the contrary value, and x x ξ ξ _ ετχ., ωηιλε ∼(ξ ∼ξ), ∼(∼ξ ∼∼ξ), σο τηατ, ξ ∼∼ξ ∼∼∼∼ξ ετχ., ∼ξ ∼∼ξ ετχ., ορ ελσε ωε μυστ σαψ τηατ ∼ξ ηασ τηε σαμε ϖαλυε ασ ξ, ιν τηε ωηιχη χασε, ξ_ ωιλλ ηαϖε τηε χοντραρψ ϖαλυε, σο τηατ ωε σηαλλ ηαϖε ξ ∼ξ ∼∼ξ ετχ. Βυτ ξ ξ ξ ετχ., ξ ξ ετχ. ανδ ∼(ξ ξ_), ∼(ξ ξ _), ετχ. Α χηοιχε ηασ το βε μαδε; ανδ τηερε ισ νο ρεασον φορ ονε χηοιχε ρατηερ τηαν τηε οτηερ, εξχεπτ τηατ Ι ηαϖε σελεχτεδ τηε λεττερσ ϖ ανδ φ, ανδ τηε οτηερ σιγνσ, σο ασ το μακε τηε φορμερ χηοιχε αχχορδ ωιτη υσυαλ χονϖεντιονσ αβουτ σιγνσ. Peirce: CP 4.264 Cross-Ref:†† Adopting that former convention, we shall make the value of x_ the same as that of x. Where it becomes necessary, as it sometimes will, to distinguish them, we may either use the bar, or vinculum, below the line, or we may make use of the admirable invention of Albert Girard, who, in 1629, introduced the practice of enclosing an expression in parentheses to show that it was to be understood as signifying a quantity.†1 For example, x y†2 signifies that x is f and y is f. Then (x y) z, or (x y) ζ, 3 ωιλλ σιγνιφψ τηατ ζ ισ φ, βυτ τηατ τηε στατεμεντ τηατ ξ ανδ ψ αρε βοτη φ ισ ιτσελφ φ, τηατ ισ, ισ φαλσε. Ηενχε, τηε ϖαλυε οφ ξ ξ ισ τηε σαμε ασ τηατ οφ ∼ξ; ανδ τηε ϖαλυε οφ (ξ ξ) ξ 4 ισ φ, βεχαυσε ιτ ισ νεχεσσαριλψ φαλσε; ωηιλε τηε ϖαλυε οφ (ξ ψ) (ξ ψ)_ 5 ισ ονλψ φ ιν χασε ξ ψ ισ ϖ; ανδ ((ξ ξ) ξ) (ξ (ξ ξ)_) 6 ισ νεχεσσαριλψ τρυε, σο τηατ ιτσ ϖαλυε ισ ϖ. Peirce: CP 4.264 Cross-Ref:†† With these two signs, the vinculum (with its equivalents, parentheses, brackets, braces, etc.) and the sign , which I will call the ampheck (from {amphékés}, cutting both ways), all assertions as to the values of quantities can be expressed.

Thus, x is (x

x)

∼ξ ισ ξ

ξ

(ξ

ξ)_

ξ:∴/:∼ξ ισ ((ξ ξ)_ ξ•∼ξ ισ (ξ ξ) ∼(ξ∼ξ ψ∼ψ) ισ {(ξ ψ)_)}

ξ) (ξ (ξ ξ)_) ξ

ψ)

ξ∼ξ ψ∼ψ ισ (ξ ψ)

((ξ ((ξ

ψ) ψ)

(ξ (ξ

ψ)_)} {((ξ ψ) ψ)_)

(ξ

ψ)_) (ξ

ξ ψ ισ (ξ (ψ ψ)_) ((ξ ξ) ∼(ξ ψ) ισ (ξ

ψ)

ξ∴/ψ ισ (ξ ψ) ∼ξ∴/∼ψ ισ ((ξ

ξ)

∼ξ∴/ψ ισ ((ξ

ψ)

ξ•ψ ισ (ξ ξ) ∼ξ•ψ ισ ξ { ∼ξ•∼ψ ισ ξ

ξ)

(ξ

ψ)_

(ψ ψ)_)

ξ)

{

∼ψ ισ ψ ψ

ξ)

(ψ ψ)_) [ορ ξ∼

ψ]

ψ)_ ορ ξ (ψ ψ)_ } 1

(ψ ψ)_)

ψ ισ (ψ ψ)

((ξ

ψ) (ψ (ξ ψ)_) [ορ (ψ (ξ ξ)_) (ψ (ξ ξ)_)]

ψ

{

(ψ ψ)_)

(ψ ψ)_

(ξ

{ ξ∴/∼ψ ισ (ξ { ξ•∼ψ ισ (ξ

((ξ

ψ)

(ξ

ψ

(ψ ψ)_) } }

(ψ ψ)_

}

}

Peirce: CP 4.265 Cross-Ref:†† 265. It is equally possible to express all propositions concerning more than two quantities. Thus, the one between three noticed above †2 is {x [(y z) ((ψ ψ) (ζ ζ)_)]} {(ξ ξ) [(ψ (ζ ζ)_) ((ψ ψ) ζ)]}. Τηατ ωε χαν εθυαλλψ εξπρεσσ εϖερψ προποσιτιον βψ μεανσ οφ τηε ϖινχυλυμ [ανδ] ονε ∼ ισ συφφιχιεντλψ σηοων βψ τηε φαχτ τηατ ξ ψ χαν βε σο εξπρεσσεδ. 3 Ιτ ισ ((ξ(∼

)ξ)_(∼

)(ψ(∼

)ψ)_)(∼

)((ξ(∼

)ξ)_(∼

)(ψ(∼

)ψ)_) . . .

Peirce: CP 4.266 Cross-Ref:†† 266. In order that a sign, say O, should be associative, it is requisite either that whatever quantity x may be, x O x = x, or else, that whatever quantities x and y may be, x O y = y O x and either x = y, or x O x = x, or y O y = y. This may be otherwise stated as follows: Peirce: CP 4.266 Cross-Ref:†† First, Suppose vOv = v and fOf = f. Then I will show that the operation is associative. For if not, it would be possible to give such values to p, q, r, that ~{p O(q Or)} (pOq)Or. But of these three values, p, q, r, some two must be equal. But all three cannot be equal, since then, because of vOv = v and fOf = f, the inequality would not hold. Suppose then first that p r, ~p q. If then pOq qOp, substituting p for r, pO(qOp) pO(pOq) (pOq)Op (pOq)Or, contrary to the hypothesis. Suppose, secondly, then, that q r, ~q p. Then, substituting q for r, pO(qOq) pOq; and since this is unequal to (pOq)Oq, it follows that ~(p pOq). But in that case, there being only two different values possible, pOq q, and pO(qOq) pOq q while (pOq)Oq qOq q, contrary to the hypothesis. The third supposition, that p q would evidently lead to an

absurdity analogous to the last; so that in no way can the associativeness fail in this case. Peirce: CP 4.266 Cross-Ref:†† Second, Suppose vOv = f and fOf = v. Then I will show that the operation is not associative. For on the one hand, (vOv)Of fOf v; while, on the other hand, whether vOf = v, so that vO(vOf) vOv f, or whether vOf = f, so that vO(vOf) vOf f, in either case, the associative rule is broken. Peirce: CP 4.266 Cross-Ref:†† Third, Suppose vOv = fOf and vOf = fOv. Then I will show that the operation is associative. For otherwise it would be possible to give such values to p, q, r, that ~{pO(qOr)} (pOq)Or. But since vOf = fOv, it follows that the second side of the inequality would be equal to rO(qOp) so that the inequality requires ~p r. But then also ~(qOp) qOr and consequently, the two assumed equations are inconsistent with the inequality, and the operation must be associative. Peirce: CP 4.266 Cross-Ref:†† Fourth, Suppose vOv fOf, but ~(vOf) fOv. Then I will show that the operation is not associative. For either (vOv)Ov fOv, while vO(vOv) vOf or (fOf)Of vOf, while fO(fOf) fOv; and in either case since ~(vOf) fOv, the rule of association is violated. The four propositions thus proved, when taken together, are equivalent to the proposition [at the top of page 217]. Of these four, the first shows that \/, ·, x,†1 y†1 are associative; the third that :\/:~, , ~ , ~(:\/:~), are so. The second shows that (~ ), ~x,†1 ~y,†1 , are non-associative; the fourth that -(0/1). This can always be done if a little ingenuity varies the process a little. Thus, which is the greater 487/830 or 301/513?

Peirce: CP 4.340 Cross-Ref:†† Proceed thus

[Click here to view] Peirce: CP 4.340 Cross-Ref:†† This series everywhere increases in the same direction. Hence 487/830 > 301/513. Fractions can be added on the same principle.

Peirce: CP 4.341 Cross-Ref:†† X ANALYSIS OF SOME DEMONSTRATIONS CONCERNING DEFINITE POSITIVE INTEGERS†1P

341. Let the Universe of 1. c. italics be the aggregate of all definite Integers not negative. Let the Universe of Greek Minuscules be the aggregate of possible characters of such Integers. Let q[αu] mean, as in 3.398, that the Integer, u, has the character, α. Hypotheses (1) π[α]Σ[β]π[u] (q[αu] [q[βu])·(~q[αu] ~q[βu]) i.e., every character has a negative. (2)π[α]π[β]Σ[{g}]π[u](~q[αu] [~q[βu] q[{g}u])·(q[αu]·q[βu] ~q[{g}u]) i.e., of every two possibilities there is a compounded possibility. Instead of introducing an unanalyzed relation of 'as small as,' let us, at first, conceive a character of characters, consisting in each of certain characters belonging to every integer lower than an integer to which it belongs. (3) π[u]Σ[α]q[αu]·s[α],†2 every Integer has a character common to all lower Integers. (A formal proposition.) (4) π[a]Σ[u]~s[α] ~q[αu] i.e., there is no highest Integer. (5) Σ[u]π[α]π[v] q[αu] ~s[α] ~q[αv] i.e., there is a lowest Integer (i.e., Zero).

(6) π[α]π[β]π[u]π[v] ~s[α] ~s[β] ~q[αu]q[βu] q[αv]~q[βv] i.e., an Integer having any s-character that another has not has every one that other has. (7) π[u]π[v]Σ[α]π[β] s[α]·(q[αu] q[αv])·(~q[αu] ~q[αv]) q[βu]·q[βv] ~q[βu]·~q[βv] i.e., unless one of any two integers has an s-character which the other has not, they are alike in all characters, and therefore, being definite, are identical. (8) π[u]Σ[α]Σ[v]π[w]π[β]π[{g}] q[αu]·s[α]·~q[αv]·(~s[β] ~q[βu] q[βw] ~s[{g}] ~q[{g}w] q[{g}v]) i.e., every integer has another next higher. This is a consequence of the following with (4): (9) π[α]π[u]Σ[v]π[w]π[β] ~q[αu] q[αv]·(~q[αw] ~s[β] ~q[βw] q[βv]) i.e., every class of integers has a lowest member. Peirce: CP 4.341 Cross-Ref:†† For formula (10) see below [343]. Addition Peirce: CP 4.341 Cross-Ref:†† Let (i+j)[k] mean that the integer k can result from adding the integer i to the integer j. This can be negatived by an obelus over it like any other expression. Peirce: CP 4.341 Cross-Ref:†† Addition is definable by the following six formulæ: (11) π[i]π[j]Σ[k](i+j)[k] (12) π[k]π[j]Σ[i]Σ[α] s[α]·q[αj]·~q[αk] (i+j)[k] (13) π[i]π[k]Σ[j]Σ[α] s[α]·q[αi]·~q[αk] (i+j)[k]. (14) π[i]π[j]π[k]π[u]π[v]π[w]Σ[α]π[β]~(i+j)[k] ~(u+v)[w] s[α]·(q[αu]·~q[αi] q[αv]·~q[αj])

~s[β] ~q[βw] q[βk]

(15) π[i]π[j]π[k]π[u]π[v]π[w]Σ[α]π[β]~(i+j)[k] ~(u+v)[w] s[α]·(q[αv]·~q[αj] q[αw]·~q[αk])

~s[β] ~q[βu] q[βi]

(16) π[i]π[j]π[k]π[u]π[v]π[w]Σ[α]π[β]~(i+j)[k] ~(u+v)[w] s[α]·(q[αu]·~q[αi] q[αw]·~q[αk])

~s[β] ~q[βv] q[βj]

Peirce: CP 4.342 Cross-Ref:†† 342. It would be illuminating to exhibit the above fifteen propositions scribed in existential graphs;†1 but it would be aside from my present purpose. I proceed to indicate sketchily in what manner the leading theorems concerning the addition of positive integers can be deduced from the fifteen propositions by means of the rules given in 3.396. (Though those rules might now be amended much, so as to render them more efficient.) If (14) be iterated, it becomes π[i]π[j]π[k]π[u]π[v]π[w]Σ[α]π[i']π[j']π[k']π[u']π[v']π[w']Σ[α']π[β] π[β']{~(i+j)[k] ~(u+v)[w] s[α]·(q[αu]·~q[αi] q[αv]·~q[αj]) ~s[β] ~q[βw] q[βk]}·{~(i'+j')[k'] ~(u'+v')[w'] s[α']·(q[α'u']·~q[α'i'] q[α'v']·~q[α'j']) ~s[β']

~q[β'w'] q[β'k']. Peirce: CP 4.342 Cross-Ref:†† Next (I go into detail with this first example farther than I shall with others), we may, by the fifth rule, identify u, i', and u', with i; v, j', and v' with j; k' with w; w' with k; and β' with β (for though the rule as given in the memoir is the right one, theoretically, yet in practice the operation of this and part of the sixth can generally be reduced with convenience to the identification of the index of any π with any index to the left of it in the quantifier). We, at the same time, apply Rule 6 somewhat, remembering that q[αi]·~q[αi] 0 etc. and applying the principle (A B)·(A C) = A B·C, and then applying Rule 7 we get π[i]π[j]π[k]π[w]π[β]~(i+j)[k] ~(i+j)[w] ~s[β] (~qβw q[βk])·(~q[βk] q[βw]) or π[i]π[j]π[k]π[w]π[β]~(i+j)[k] ~(i+j)[w] ~s[β] ~qβw·~q[βk]) q[βw]·q[βk]). Peirce: CP 4.342 Cross-Ref:†† Let us now compound this with (7) in which, to avoid confusion, we may write m for u, n for v, {y} for α, and φ for β. We thus get π[m]π[n]Σ[{y}]π[φ]π[i]π[j]π[k]π[w]π[β] {~(i+j)[k] ~(i+j)[w] ~s[β] ~qβw·~q[βk]) q[βw]·q[βk])} {s[{y}]·(q[{y}m] q[{y}n])·(~q[{y}m] ~q[{y}n]) q[φm]·q[φn] ~q[φm]·~q[φn]}. Now identifying β with {y}, w with m, k with u, the formula with an obvious reduction of the Boolian, becomes (17) π[m]π[n]π[φ]π[i]π[j] ~(i+j)[n] ~(i+j)[m] q[φm]·q[φn] ~q[φm]·~q[φn] i.e., if (i+j)[m] and (i+j)[n], then m = n; or the sum of two definite positive integers has but a single value. Peirce: CP 4.342 Cross-Ref:†† Without writing down the formulae, a little close attention will enable one to convince himself that (15) and (16), treated almost exactly as (14) has been above, show that if (18) (i+j)[k] and (u+j)[k] then u = i and that (19) if (i+j)[k] and (i+v)[k] then j = v. Peirce: CP 4.343 Cross-Ref:†† 343. Abbreviations. Having thus illustrated how the notation works, it will be well to introduce some abbreviations. First, although obviously indefinite individuals may be alike in respect to every character, yet different in their (real or pretended) brute existence, such as the different parts of space and the different vertices of the regular dodecahedron of pure mathematics, still since the Universe of l. c. italics is confined to definite integers, we may, by introducing [ij] to mean that i and j are the same individual, write the following principle: (10) π[u]π[v]Σ[α]s[α]·(q[αu] q[αv])·(~q[αu] ~q[αv])Ψ [υϖ]. Peirce: CP 4.343 Cross-Ref:†† Of course, the negative of [ij] will be

[ij].

Peirce: CP 4.344 Cross-Ref:†† 344. One may entertain the theory that all vagueness is due to a defect of cogitation or cognition. It is a natural kind of nominalism the justice of which it would be remote from the purpose of this analysis to consider. The vagueness of characters is of different kinds. The quality of redness and the quality of blueness differ without differing in any essential character which one has but the other lacks. The otherness of them is as irrational as the qualities themselves, if not more so. It appears to consist in a mutual war between them, in our taste. But the characters of integers are not of this irrational kind. In another regard, however, they are vague. Thus we say that the two characters of 4, of being the sum of 2 and 2, and of being the product of ~ and 2, are different characters, so that we cannot, in imitation of (10), write π[α]π[α]Σ[n](q[αn] q[βn])·(~q[αn] ~q[βn])Ψ [αβ]

Peirce: CP 4.344 Cross-Ref:†† This is [because] we do not think out the meaning of 2+2 and 2X2 to the very bottom. In this respect, the objects we denote by Greek minuscules are not generally definite. Peirce: CP 4.345 Cross-Ref:†† 345. The character, , which I introduced in 1882, when I was teaching logic in the Johns Hopkins University, was in my mind one of a class of notations which I left unmentioned in order that some one of my pupils might have the pleasure of finding it out for himself; but as nobody has, so far as I have noticed, in the three-fourths of a generation that has elapsed, I will give some illustrations of the class: [ij] means j is a member of the singlet, i. 2[ij] means j is a member of the doublet, i, or unordered pair, or couple. 3[ij] means j is a member of the triplet, i, or unordered trio, or leash. 4[ij] means j is a member of the quadruplet, i, or unordered collection of 4. 9[ij] means j is a member of the nonuplet, i, or unordered collection of 9. x[ij] means j is a member of the decuplet, i, or unordered collection of 10. Ordered collections I call, medads (0), monads, dyads, triads, etc. Indeterminate as to being ordered are binion (or pair), trine, quaternion, quine, senion, septene, octone, novene, dene (or denion), etc. Peirce: CP 4.345 Cross-Ref:†† By an ordered collection, I mean one of which each member has a peculiar relation to the whole; as for example, if one is definitely the first, another definitely the second, a definite one the third, etc., or if there is any other formal relation by which each is different from all the others. There are also diduct collections which are formally divided into subcollections and it may be in more than one way, whether inadequately, adequately, or superfluously. By adequately, I mean just sufficiently to make the collection an ordered one.

Peirce: CP 4.345 Cross-Ref:†† With this notation (7) can be expressed as follows, using Hebrew letters to denote definite collections: π[ℵ]Σ[i]Σ[j]Σ[α] 2[ℵi]·2[ℵj]·s[α]·q[αi]·~q[αj] Peirce: CP 4.345 Cross-Ref:†† The utility of the symbols 1, 2, 3, etc. is increased by employing them as follows: [i], [ij], [ijk], etc. means that the indices denote the same existing individual. 2[ℵi], 2[ℵij], 2[ℵijk], etc. mean that the individuals denoted by the indices belong to the doublet ℵ. 2[ijk], 2[ijkl], etc. mean that all the individuals denoted by the indices are members of one doublet. (2· )[ij], (2· )[ijk], etc., mean that the individuals denoted by the indices belong to one doublet but are not all one individual. 3[ℵi], 3[ℵij], 3[ℵijk], 3[ℵijkl], mean that i, j, etc. all belong to the triplet ℵ. 3[ijkl], 3[ijklm], etc. mean that i, j, etc. all belong to one triplet. (3· )[ijkl], means that i, j, k, l all belong to one triplet but are not all identical. (3·~2)[ijkl] means that i, j, k, l are three different existing individuals. (32)[ijklmnpq] (where note the absence of a dot -- not 3·2, but 32) means that the individuals indicated are all members of a triplet of doublets. (3 2)[ijklmno] means that every individual denoted by an index is either a member of a triplet or of a doublet. Peirce: CP 4.345 Cross-Ref:†† I would use a special form of parenthesis (I will not recommend any particular form as more appropriate than another) which I would use in the following way: π[i][ Ψ ][ι]; μεανσ ανψ οβϕεχτ ωηιχη ισ συν ισ, ασ συχη, τηε μεμβερ οφ α σινγλετ, ι.ε., π[i]π[j] [i] [j]Ψ [ιϕ]. Peirce: CP 4.345 Cross-Ref:†† If + means is a satellite of Jupiter, then π[i][~ + 5][i]; means that whatever is a satellite of Jupiter is, as such, a member of a quintuplet, i.e., Σ[i]Σ[j]Σ[k]Σ[l]Σ[m]π[n] +[i]· +[j]· +[k]· +[l]· +[m]· [ij]· [ik]· [il]· [im]· [jk]· [jl]· [jm]· [kl]· [km]· [lm]·(~ +[n]Ψ [ιν]Ψ [ϕν]Ψ [κν]Ψ [λν]Ψ [μν]). Peirce: CP 4.345 Cross-Ref:†† The saving here is enormous. Peirce: CP 4.345 Cross-Ref:††

Intimately connected with these abbreviations are others, some of which I have mentioned elsewhere. The rules of their application would form an elaborate logical doctrine, which I have not time to develop, because I am working at more fundamental parts of logic. Whoever undertakes it in the light of what I have said here and elsewhere will have other symbols forced upon his attention. Peirce: CP 4.345 Cross-Ref:†† I pass to another and very simple abbreviation, which consists in using the symbol σ so that σ[ij] shall mean that j is at least as low an integer as i. That is, (20) σ[ij]=π[α]~s[α] ~q[αi]q[αj] ~σ[ij]=Σ[α]s[α]·q[αi]~q[αj] Peirce: CP 4.345 Cross-Ref:†† It immediately follows that (21)π[i]σ[ii] and (22)π[i]π[j]π[k] ~σ[ij] ~σ[jk]Ψσ[ik] Peirce: CP 4.345 Cross-Ref:†† From (20) and (10) it follows that (23) π[i]π[j]π[k] σ[ij]Ψσ[jk] ~σ[ik] Hypothesis (5) represents that there is an integer as low as any and by (10) this is lower than any other. We may therefore give it the proper name, o, which will possess the singularity of being definable. Thus (24) π[i] [oi] ~σ[oi] π[i]σ[io] a definition which is also singular in being in a single proposition. But this is owing to (10). Peirce: CP 4.345 Cross-Ref:†† The long formula (8) requires abbreviation; and we may write (25) H[uv] = Σ[α]π[β]π[{g}]π[w] s[α]·q[αu]·~q[αv]·(~s[β] ~q[βu] q[βw] ~s[{g}] q[βv] ~q[βw]) = π[w]~σ[uv]·(σ[uw]Ψσ[wv]) We may further take the index 1 as such a proper name that (26) H[01]. I will also write r[Hi] for Σ[j]r[j]·H[ij] and (H[i]+H[j])[k] for Σ[u]H[iu]·Σ[v]H[wv]·(u+v)[k] Peirce: CP 4.345 Cross-Ref:†† Formulæ (14)-(16) may be put in the form (14) π[i]π[j]π[k]π[u]π[v]π[w] ~σ[ui] ~σ[vj] ~(i+j)[k] ~(u+v)[w]Ψσ[wk] (15) π[i]π[j]π[k]π[u]π[v]π[w] ~σ[wk] ~σ[vj] ~(i+j)[k] ~(u+v)[w]Ψσ[ui] (16) π[i]π[j]π[k]π[u]π[v]π[w] ~σ[ui] ~σ[wk] ~(i+j)[k] ~(u+v)[w]Ψσ[vj]

Peirce: CP 4.345 Cross-Ref:†† Putting, in (14), o for i, j, and w, it becomes

π[k]π[u]π[v](o+o)[k] (u+v)[o] ~σ[uo] ~σ[vo]Ψσ[ok] Peirce: CP 4.345 Cross-Ref:†† Multiplying this by the third power of (24), i.e., (24)·(24)·(24), we get (27)(o+o)[o] (9) may be put in the form (9')π[α]π[u]Σ[v]π[w] ~q[αu] q[αv]·(~q[αw]Ψσ[wv]) Putting for q[α] the expression ~σ[u], this becomes (28)π[u]Σ[v]π[w] ~σ[uv]·(σ[uw]Ψσ[wv]) Peirce: CP 4.346 Cross-Ref:†† 346. I will now return to addition. I will remark, by the way (for I do not make this paper at all systematic), that Schröder's notation Σ and π u

{y}

and the like, which is his chief modification of my two logical algebras (which, by the way, can perfectly well be mixed), made long after my second intentional section of the paper, No. XIII in vol. 3, has several advantages over mine, both theoretical and practical, and ought to be employed freely. But it fails to do what my invention was made in order to do, namely, to enable us to perform the operation of hypostatic abstraction, and freely make use of entia rationis. But that is neither here nor there. Peirce: CP 4.346 Cross-Ref:†† I will start with (14) in its last form and will trace out the steps of the algebraical transformation in closer detail than I purpose generally to do in this paper. For the inference I am coming to employs the Rule of Diduction, or diversification, which I fully treated of in a paper I drew up in Grammercy Park in 1885†1, but the dignity of science does not permit me to go begging to have its results printed. This paper I am writing will probably never be seen by other eyes than those that see it written; but I record this for my own gratification. The rule is that after any quantifier of the Peircean (whether it be π or Σ) can be inserted a Σ with a new index, into which the preceding index can be transmuted in any of the places where it occurs, remaining untransmuted in the other places. Thus π[i]l[ii] everybody loves himself, can be changed to π[i]Σ[j]l[ij], everybody loves somebody. Identifying v with j, in (14) we get π[i]π[j]π[k]π[w]π[u]~(i+j)[k]Ψσ[wk] ~(u+j)[w] ~σ[ui] Let us insert the aggregant ~q[βu]: π[i]π[j]π[k]π[w]π[β]π[u]~(i+j)[k]Ψσ[wk] ~q[βu] ~(u+j)[w] ~σ[ui] The insertion of this aggregant authorizes the insertion of its negative as component of another aggregant. π[i]π[j]π[k]π[w]π[β]π[u]~(i+j)[k]Ψσ[wk] ~q[βu] ~(u+j)[w] q[βu] ~σ[ui].

Let the index u now be diduced, becoming x in the last aggregant: π[i]π[j]π[k]π[w]π[β]π[u]Σ[x]~(i+j)[k]Ψσ[wk] ~q[βu] ~(u+j)[w] q[βx] ~σ[xi]. Let ~q[αw] be inserted as an aggregant: π[i]π[j]π[k]π[α]π[w]π[β]π[u]Σ[x]~(i+j)[k] ~q[αw]Ψσ[wk] ~q[βu] ~(u+j)[w] q[βx] ~σ[xi]. The insertion of this aggregant authorizes the insertion of its negative as a component of another aggregant. π[i]π[j]π[k]π[α]π[w]π[β]π[u]Σ[x]~(i+j)[k] ~q[αw]Ψσ[wk] q[αw]·{~q[βu] ~(u+j)[w]} q[βx] ~σ[xi]. We now diduce w, transmuting it in one term into z, and thus obtain finally, (29) π[i]π[j]π[k]π[α]π[w]Σ[z]π[β]π[u]Σ[x]~(i+j)[k] ~q[αw]Ψσ[wk] q[αz]·{~q[βu] ~(u+j)[z]} q[βx] ~σ[xi]. Peirce: CP 4.346 Cross-Ref:†† Here we have a nice little theoremidion, obvious though not self-evident. Namely, if any three positive integers, i, j, k, are such that k can result from adding i to j, then, selecting any class of integers we please, and speaking of the character of being an integer of this class as "the character α" either all integers of this class are as large as or larger than any integer k that can result from adding i to j, or else (if that is not the case) there is an integer of this class, z, if we take any second class of integers whatever (inclusion in which shall be called the character β) no integer u of this second class can on being added to j give the integer z, unless there be an integer x of the second class which is smaller than i. The form of statement is too strictly logical and formal for an ordinary mind readily to grasp it; but let us dilute it with a little verbiage, as follows. Suppose k is a positive integer which can result as the sum of j, as augend, and i, as addend. We select a first class of positive integers, say for example the cubes above 0 and 1. Now it may be that k does not exceed any of these. As to that case we say nothing. But should there be one or more of the first class that exceed k, then it may be that one of them is such that it cannot result from adding any positive integer to j as augend, because it may be less than j. It would have been better if, instead of writing Σ[z]π[β]π[u] in the Peircean, I had written π[β]π[u]Σ[z] for it is always allowable to carry Σ's to the right. Then the second class being selected first, it might happen that there was an integer of the first class that could not result from adding any integer of the second class to j. . . .

Peirce: CP 4.347 Cross-Ref:†† BOOK II EXISTENTIAL GRAPHSP

MY CHEF D'OEUVRE

CHAPTER 1 EULER'S DIAGRAMSE

§1. LOGICAL DIAGRAM †1

347. A diagram composed of dots, lines, etc., in which logical relations are signified by such spatial relations that the necessary consequences of these logical relations are at the same time signified, or can, at least, be made evident by transforming the diagram in certain ways which conventional 'rules' permit. Peirce: CP 4.348 Cross-Ref:†† 348. In order to form a system of graphs which shall represent ordinary syllogisms, it is only necessary to find spatial relations analogous to the relations expressed by the copula of inclusion and its negative and to the relation of negation. Now all the formal properties of the copula of inclusion are involved in the principle of identity and the dictum de omni. That is, if r is the relation of the subject of a universal affirmative to its predicate, then, whatever terms X, Y, Z may be, Every X is r to an X; and if every X is r to a Y, and every Y is r to a Z, every X is r to a Z. Peirce: CP 4.348 Cross-Ref:†† Now, it is easily proved by the logic of relatives, that to say that a relation r is subject to these two rules, implies neither more nor less than to say that there is a relation l, such that, whatever individuals A and B may be, If nothing is in the relation l to A without being also in the same relation l to B, then A is in the relation r to B; and conversely, that, If A is r to B, there is nothing that is l to A except what is l to B. Peirce: CP 4.349 Cross-Ref:†† 349. Consequently, in order to construct such a system of graphs, we must find some spatial relation by which it shall appear plain to the eye whether or not there is anything that is in that relation to one thing without being in that relation to the other. The popular Euler's diagrams fulfill one-half of this condition well by representing A as an oval inside the oval B. Then, l is the relation of being included within; and it is plain that nothing can be inside of A without being inside B. The relation of the copula is thus represented by the spatial relation of 'enclosing only what is enclosed by'. In order to represent the negation of the copula of inclusion (which, unlike that copula, asserts the existence of its subject), a dot may be drawn to represent some existing individual. In this case the subject and predicate ovals must be drawn to intersect each other, in order to avoid asserting too much. If an oval already exists cutting the space in which the dot is to be placed, the latter should be

put on the line of that oval, to show that it is doubtful on which side it belongs; or, if an oval is to be drawn through the space where a dot is, it should be drawn through the dot; and it should further be remembered that if two dots lie on the boundaries of one compartment, there is nothing to prevent their being identical. The relation of negation here appears as 'entirely outside of'. For a later practical improvement see Venn, Symbolic Logic, chapter xi.

Peirce: CP 4.350 Cross-Ref:†† §2. OF EULER'S DIAGRAMS †1

350. In the second volume of the great Leonard Euler's Lettres à une Princesse d'Allegmagne, which appeared in 1772 (four years after the first volume), the nature of the syllogism is illustrated by means of circles, in substantially the following manner. Let the syllogism whose cogency is to be exhibited be the following: All men are passionate, All saints are men; Therefore, All saints are passionate. Imagine the entire collection of saints and nothing else to be enclosed in the imaginary circle, S, of Fig. 1; imagine the entire collection of men and nothing else to be enclosed in the imaginary circle, M -- which will therefore enclose whatever is enclosed in the circle S, since all saints are men. Imagine the entire collection of passionate beings and nothing else to be enclosed in the circle, P, which will thus enclose whatever is enclosed in the circle M, since all men are passionate. We see, then, that whatever is enclosed in the circle, S, is enclosed in the circle, P; that is, that all saints are passionate.

Fig. 1

[Click here to view]

Fig. 2

[Click here to view] It will be remarked that the way in which any facts of enclosure relating to the circle M can inform us about any relation of enclosure between two other circles, is by the facts about M being that one of the other circles is on the inside of M and the other on the outside of it; so that, so far as this mode of representation exhibits the true nature of the syllogism, there ought to be just two kinds of syllogisms, one corresponding to Fig. 1, and the other to Fig. 2, which latter figure illustrates the following syllogism: No man is perfect, But any saint is a man; Hence, no saint is perfect.

Peirce: CP 4.351 Cross-Ref:†† 351. . . . We may now define the system of Euler's Diagrams, as he left it, in the following rules: Peirce: CP 4.351 Cross-Ref:†† First, every area of the diagram represents the entire collection, or aggregate, of possibilities of a certain description. Peirce: CP 4.351 Cross-Ref:†† Second, if a circle is drawn within an area, the part of the area within the circle represents the entire aggregate of those possibilities represented by the area to which a certain description applies, while the area outside the circle represents the entire aggregate of those possibilities represented by the area to which that description does not apply; and in general the area common to two areas represents the entire aggregate of possibilities which are at once represented by each of those two areas. Peirce: CP 4.352 Cross-Ref:†† 352. From these two principles it follows that to draw two circles or other areas, A and B, so that they have no common area, is to represent that there is no possibility which is at once of the description represented by A and of the description represented by B; and this is the only way in which a Euler's diagram can represent a state of things to be a fact. It is essential that this should be understood. Thus, in

Fig. 3 [Click here to view], let the entire area of the sheet represent all men now living. Let the circle G enclose all Greeks, and the circle C all courageous men. Then the four parts into which the two circles divide the whole sheet represent respectively whatever, i, Courageous Greeks; ii, Greeks not Courageous; iii, Courageous men not Greeks; iv, Men, neither Courageous nor Greeks, there may be among men now living. These are represented merely as possible classes, without any assertion. Fig. 4

[Click here to view] represents that all men consist of whatever Greeks not Courageous, Courageous Men not Greeks and Men neither Greeks nor Courageous there may be, and thus asserts that no Greek is Courageous; while Fig. 5

[Click here to view] represents that men consist of whatever Courageous Greeks, Courageous Men not Greeks, and Men neither Courageous nor Greek there may be, and thus asserts that whatever Greek men there may be now living are courageous. Peirce: CP 4.353 Cross-Ref:†† 353. The history of this System of Graphs has been discussed by Mr. Venn;†1 and though he does not consider every historical question upon which one might desire to be informed, nothing additional will here be brought forward. The results are briefly these: Eight years before Euler's publication appeared, the Neues Organon of John Henry Lambert (Alsatian by birth, French by descent, but German by residence and by the honor and support that country rendered him) in which †1 the author made the same use of the stretches of parallel lines essentially as Euler did of the areas of circles, with an additional feature of dotted lines and extensions of lines. Lambert, however, does not seem to aim at any mathematical accuracy of thought in using his lines. He certainly does not attain it; nor could he do so, as long as he failed to perceive that the only purpose such diagrams could subserve is that of representing the necessity with which the conclusion follows from the premisses of a necessary reasoning, and that that necessity is not a compulsion in thinking (although there is

such a compulsion) but is a relation between the facts represented in the premisses and the facts represented in the conclusion. The failure to comprehend the true nature of logical necessity and the confounding of it with a psychological compulsion is common to German logicians generally,†2 excepting only the Herbartians. Thus, he represents 'Some A is B' by this diagram.

[Click here to view]

B

-------

A....... Fig. 6 Thus, a distinction is represented between 'Some A is B,' and 'Some B is A,' although they express the same fact. No doubt, there is a way of regarding such a fact, or supposed fact, as that 'Some Germans are given to subjective ways of thinking' which renders that a more natural mode of expression than 'Some men given to subjective ways of thinking [are] Germans.' But it is one thing to admit that this is so and quite another to admit that the sentence "expresses" that way of thinking, rather than the fact itself. The sentence is an assertion; and an assertion is of a fact and not of a way of thinking the fact. When a writer makes an assertion, his principal purpose is to induce the reader to believe in the reality of the fact asserted. He has the subsidiary design of causing the reader to follow along his line of thinking. . . . Throughout Lambert's whole treatment of syllogistic, the way of thinking is made the principal thing. Under these circumstances, it was impossible for him to have a clear conception of the proper nature of a system of syllogistic graphs. Peirce: CP 4.353 Cross-Ref:†† My reason for insisting at such length upon this point is that it is a passage of Lambert's Architektonik†1 which is the principal authority for one of the main points of the current account of the history of the Eulerian diagrams. The usual assertion is that the voluminous pedagogist Christian Weise, the author of two works on logic (Doctrina Logica, 1690, and Nucleus Logicæ, 1691), who died 1708 October 21, made use of the system of diagrams in question. Nobody has ever examined Weise's own editions to see whether they bear out the assertion.†2 But it is said that one Johann Christian Lange in a book by him (von ihm verfassten) and entitled Nucleus Logicæ Weisianæ, published in 1712, tells how Weise so taught logic. Nobody, however, except Hamilton,†3 claims to have seen even this book. They appeal to a vague account of its contents in Lambert's Architektonik. Now this book of Lambert's

preceded Euler's publication by a year; and in view of Lambert's crude notions of what such diagrams ought to be, and in view of his not apparently being greatly struck by what would, to his mathematical mind in its benighted condition concerning syllogistic, have been a great light, the passage of Lambert is rather against the claims made for Weise than in their favor. It is curious that even Ueberweg †4 talks of Lange as the author of the publication of 1712. But to anybody familiar with such literature the title proclaims it to be a work by Weise probably with a running commentary or copious notes by Lange. The passage in Lambert's Architektonik was first brought to light by Drobisch in 1851 in the second edition of his Neue Darstellung der Logik.†5 But Hamilton in his fourteenth lecture on logic, publicly delivered in 1837-8 and regularly afterwards till his death, says †6 "I find it [i.e., the mode of sensualizing by circles the abstractions of logic] in the Nucleus Logicæ Weisianæ, which appeared in 1712; but this was a posthumous publication, and the author, Christian Weise, who was Rector of Zittan, died in 1708." Hamilton was mistaken in supposing that the book had not appeared before; for it was published originally in 1691. What Hamilton here attributes to Weise falls very far short indeed of the system of Eulerian diagrams. It is true that Hamilton appears to confound the two; but no careful student of this strikingly unmathematical scholar will attribute any importance to such a unification. In this very same passage, he attributes to Alstedius, in 1614, the use of Lambert's linear graphs, which his editors are compelled to admit is a gross exaggeration.†1 How utterly unfounded it was is shown by Venn.†2 When we think of the great reputation of Weise in his own day, it is almost incredible that so striking an idea as that of Euler's diagrams should have been developed by so prominent a man without attracting universal attention. Until further evidence is adduced his claims to their authorship must be pronounced quite unsupported. But Friedrich Albert Lange in his remarkable Logische Studien (p. 10) says that substantially the Eulerian method is mentioned by the celebrated Juan Luis Vives †3 early in the sixteenth century, and in an offhand manner (die schlichte Art ihrer Einführung) that would seem to indicate that it was traditional in the schools. Venn †4 copies the passage and diagram, which shows the cardinal idea of the Eulerian diagrams, that the middle term is like a boundary separating the two regions in which the other two terms respectively lie -and this much probably was traditional -- but gives no hint of any development of this idea into a sort of calculus, such as Euler's system is. Of this, the principal achievement, Euler is the author. After Euler several attempts were made to improve the system; but all of them were blunders until Venn's publications in 1880. Venn made a distinct improvement, and I shall endeavor to contribute others; but before giving an account of them, it will be requisite to study critically Euler's original proposal. Peirce: CP 4.354 Cross-Ref:†† 354. What is it, then, that these diagrams are supposed to accomplish? Is it to prove the validity of the syllogistic formula? That sounds rather ridiculous -- as if anything could be more evident than a syllogism -- yet that is not far from the opinion of Friedrich Albert Lange, a thinker of no ordinary force. Suppose we ask ourselves why it is that, if a circle P wholly encloses a circle M which itself wholly encloses a circle S, the circle P necessarily wholly encloses the circle S. In order to express the answer, it will be well to avail ourselves of a phraseology proper to the logic of relatives. I use the words relation and relative in a somewhat narrow sense, which I begin by explaining. Take, then, any assertion †P1 whatever about a number of designate individuals. These individuals may be persons, material objects, actions, collections of things, possible courses of events, qualities, abstractions of any kind,

and, in short, of any one nature or of any several natures whatsoever; only each of them must be well-known and rated by a proper name, and each must belong to some universe, or total aggregate of things of the same wide class, and the assertion must be such that if any one of the individuals did not really occur in its universe, independently of whether you, I, or any collection of men or other cognoscitive beings should opine that it did or that it did not, then that assertion would be false. For example, if in the assertion that Mrs. Harris was unbeknown to Betsy Prig except by hearsay, "unbeknown" be understood in such a sense that the nonexistence of Mrs. Harris would render it true, then not only does this assertion not fulfill the condition, but -- still taking "unbeknown" in the same sense -- no more would the assertion that Sairey Gamp was unbeknown to Betsy Prig except by hearsay; while if "unbeknown" be taken in such a sense that the first assertion is rendered false by the non-existence of Mrs. Harris, then, although that assertion would not fulfill the condition because Mrs. Harris did not belong to the universe of characters in Martin Chuzzlewit, yet, taking the word in this sense, the assertion, "Sairey Gamp is unbeknown to Betsy Prig except by hearsay," will perfectly fulfill the condition; and neither its falsity nor the fictitiousness of the universe to which Sairey Gamp and Betsy Prig belong are any objections. . . . Peirce: CP 4.354 Cross-Ref:†† An assertion fulfilling the condition having been obtained, let a number of the proper designations of individual subjects be omitted, so that the assertion becomes a mere blank form for an assertion which can be reconverted into an assertion by filling all the blanks with proper names. I term such a blank form a rheme. If the number of blanks it contains is zero, it may nevertheless be regarded as a rheme, and under this aspect, I term it a medad. A medad is, therefore, merely an assertion regarded in a certain way, namely as subject to the inquiry, How many blanks has it? If the number of blanks is one, I term the rheme a monad. If the number of blanks exceeds one, I term it a Relative Rheme. If the number of blanks is two, I term the rheme a Dyad, or Dyadic Relative. If the number of blanks exceeds two, I term it a Polyad, or Plural Relative, etc. A Relation is a substance whose being and identity precisely consist in this; its being, in the possibility of a fact which could be precisely asserted by filling the blanks of a corresponding relative rheme with proper names; its identity, in its being in all cases so expressible by the same relative rheme.†1 It must be confessed that it would have been better if a modifying adjective had been attached to the words relative and relation to form the technical terms to designate what have just been defined as a relative rheme and a relation. But now that these terms have been established by me, my convictions of the ethics of terminology †2 forbid me to attempt to alter the meanings attached to them. I use the word "signify" in such a sense that I say that a relative rheme signifies its corresponding relation. In the technical language of the logic of relatives, letters of the alphabet are employed as pronouns to denote relatives, just as, in ordinary and especially in legal language, they are often used as relative pronouns. The ancient grammarians defined a pronoun as a word used to replace a noun, a most preposterous attempt at analysis. It would have been far nearer the truth to describe a common noun as a word used in place of a pronoun.†1 In the middle ages, Duns Scotus and others brought a correcter definition into vogue; but the humanists of the reformation stickled for the ancient definition and that of the scholastics was quite forgotten . . . . A relative pronoun designates a subject by indicating, through its position and agreement, a noun that designates that subject. This nearly corresponds to the use of letters in the Catechism, "What is your Name? Answer N., or M." and the priest in dipping the child in the water so

"discretely and warily," is represented as saying "N. I baptize thee in the name" etc. The point is that in neither case is it meant that the letter is pronounced, but this letter designates the person through indicating by its position that it is to be replaced by the Christian name. . . . So in logic, Barbara is described as the syllogistic form Any M is P, Any S is M; .·.Any S is P. What is meant is that the letters S, M, and P, in this formula, may be replaced by any terms whatever; only each letter must everywhere in the formula be replaced by the same term. In the logic of relatives, the letters r, s, {r}, σ, etc., are frequently employed as substitutes for dyadic relatives, so that "A is r of B" and "B is r'd by A" stand for different expressions of the same fact, analogous to "A is lover of B" and "B is loved by A." Peirce: CP 4.355 Cross-Ref:†† 355. With this explanation of terms, we can intelligibly answer the question, "Why does a circle, P, that wholly encloses a circle M, itself wholly enclosing a circle, S, likewise necessarily enclose the circle, S?" Namely, understanding by this question, "What is the peculiarity of the relation of wholly enclosing which renders this necessary?" we answer, "It is because the relation of wholly enclosing is such that there is a dyadic relative, r, such that to say that any place, X, wholly encloses a place, Y, is equivalent to saying that X is at once r of Y and is r of everything that is r'd by Y." To show that this is the explanation, we must prove two propositions: firstly, that there is a dyadic relative, r, such that, on the one hand, if the place X wholly encloses the place Y, the place X is r of Y and is r of everything r'd by Y, while on the other hand, if X does not enclose Y, either X is not r of Y or else there is something r'd by Y that is not r'd by X; and secondly, that from that first proposition it necessarily results that if any circle, P, wholly encloses a circle, M, itself wholly enclosing a circle, S, then P wholly encloses the circle, S. Before proving the first of these propositions, it is to be remarked that whether we affirm that the place X wholly encloses the place Y, [or whether] we say that the place, X, does not wholly enclose the place, Y, we are to be understood as recognizing X and Y as definite places in space, so that if either of them is not of that nature, both the one assertion and the other are false. Now in order to prove the first proposition, it will suffice to make r signify the relation of not being quite at a distance from (with intervening place), so that the first clause of the first proposition will be that if the place, X, wholly encloses the place, Y, then X is not altogether at a distance from Y, nor is it altogether at a distance from any place from which Y is not altogether at a distance. That, if X wholly encloses Y it is not altogether at a distance from Y, is self-evident. Moreover, if we consider any place Z, from which Y is not altogether removed, there must be some point of Z from which Y is not altogether removed, and from this point, X will not be altogether removed. Hence it is evident that X is not altogether removed from any place from which Y is not altogether removed; and the first clause of the proposition is found to be true. The other clause of the proposition is that, if X does not wholly enclose Y, then either X is not r of Y or else there is something r'd by Y that is not r'd by X; that is, if X does not wholly enclose Y, either X is altogether remote from Y or else there is some place not altogether distant from Y from which X is entirely remote. This is plainly true, since if X does not wholly enclose Y, there is some point of Y which lies quite outside of X; and such a point will be a place from

which Y is not remote but from which X is remote. Thus the first proposition is true. It remains then to be shown that, from this peculiar form of the relation of total inclusion, it follows that a circle, P, wholly enclosing a circle, M, itself wholly enclosing a circle, S, likewise wholly encloses the circle, S. . . . Now it is clear that as long as P is r of whatever is r'd by M, if S is r'd by M, so long will P be r of S; while as long as P is r of whatever is r'd by M, and whatever is r'd by S is r'd by M, P is r of whatever is r'd by S. Thus if P is r of whatever is r'd by M and if both S and whatever is r'd by S are r'd by M, P is r both of S and of whatever is r'd by S; quod erat demonstrandum. Thus the reason, that the geometrically wholly enclosed by the wholly enclosed is itself wholly enclosed, is shown. But this is the very same reason substantially that Aristotle †1 gives for the validity of the syllogism in Barbara. Any M is P, Any S is M; .·.Any S is P. For Aristotle's doctrine is that this depends on the essential nature of being dictum de omni, or universally predicated. This essential nature he says is, that to say that X is predicated of the whole of Y, is to say that X is predicated of Y and of whatever Y is predicated of. Peirce: CP 4.355 Cross-Ref:†† That is, the relation of universal predication is also of the form, "At once r of and r of whatever is r'd by." He might have avoided the apparent circulus in definiendo by stating the matter thus: X is predicated of all Y if and only if X is not foreign to Y nor to any term to which Y is not foreign. Thus, as far as logical dependence goes, the validity of the syllogism and the property of the Eulerian diagram depend upon a common principle. They are analogous phenomena neither of which is, properly speaking, the cause or principle of the other. Lange †2 is of opinion that all reasoning proceeds by the observation of imaginary Euler's diagrams or of something closely similar; and I,†3 for my part, share his opinion so far as to admit that an imaginary observation is the most essential part of reasoning. But the psychological process is not the matter in question. This brings us back to the inquiry, What purpose are the diagrams fitted to subserve? They may help to analyze reasonings, and this either in a practical way by aiding a person in rendering his ideas clear, or theoretically. In either regard it is desirable that they should be adequate to represent the gist of every kind of deductive reasoning. Peirce: CP 4.356 Cross-Ref:†† 356. As Euler left the system, it had the following faults: Peirce: CP 4.356 Cross-Ref:†† First, two circles cannot be each inside the other; so that while, as Mrs. Franklin has shown (Johns Hopkins Studies in Logic, p. 64), there are fifteen or sixteen different ways in which two terms may be related in reference to the possibility or impossibility of their different combinations, Euler's original diagrams show but eight of these, as follows:

Figs. 7-14 [Click here to view]

The states of possibility not represented are as follows: Everything is either S or P; [S\/P] Everything is S; [S] No S is P, but everything but S is P; [S ~P] Everything is S and nothing is P; [S.~P] Everything is P; [P] Everything is both S and P; [S.P] Nothing is S but everything is P; [~S.P] The Universe is absurd and impossible; [P.~P] Peirce: CP 4.356 Cross-Ref:†† Second, in regard to every combination of terms (that is, in regard to each of the possible parts of the universe, when we are in complete ignorance), the system is limited to expressing its non-existence or to not expressing whether it exists or not. It cannot affirm the existence of any description of an object. But a categorical, though possibly partial, description of the universe in its relation to two terms can, in reference to each of the four possible parts into which those two terms can divide the universe of possibility, either affirm its existence, or deny its existence, or say nothing. Therefore, excluding the absurd assertion that nothing exists, there are 34-1, or eighty, possible categorical descriptions of the universe, of which this system can express but one tenth part. Peirce: CP 4.356 Cross-Ref:†† Third, the system affords no means of expressing a knowledge that one or another of several alternative states of things occurs. Of the sixteen possible dichotomic states-of things with reference to two terms, a state of knowledge may either exclude or admit each, though it cannot exclude all. There are therefore 216-1,

or 65535, possible states of dichotomous information about two terms of which the system permits the expression of only eight, or one out of every 8192. Peirce: CP 4.356 Cross-Ref:†† Fourth, the system affords no means of expressing any other than dichotomous, or qualitative, information. It cannot express enumerations, statistical facts, measurements, or probabilities. In short, it affords no room for the introduction of quantitative premisses into its reasonings. Peirce: CP 4.356 Cross-Ref:†† Fifth, the system affords no means of exhibiting reasoning, the gist of which is of a relational or abstractional kind. It does not extend to the logic of relatives. Peirce: CP 4.357 Cross-Ref:†† 357. Some of these imperfections are, however, easily removed. This first of them was done away with by an improvement

Figs. 15-18

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introduced by Mr. Venn in 1880. Namely, Mr. Venn in his Symbolic Logic (I use the first edition of 1881) recommends drawing the diagrams so as always to exhibit all the possible parts into which terms, to the number employed, would, in the absence of all information, divide the universe. That done, if information is received that certain of these parts do not exist, the corresponding regions of the diagrams are shaded. Thus the areas representing the terms may be arranged in one of the following ways according as they are one, two, three, or four in number. With more than four terms the system becomes cumbrous; yet, by having on hand lithographed blank forms showing the four-term figure on a large scale, all the compartments containing repetitions of one figure, whether that for one term, for two terms, for three or for four, and considering corresponding regions of all sixteen of the large compartments to represent together the extension of one term, it is possible without much inconvenience to increase the number of terms to eight. Beyond eight terms, the best way will simply be to make a list of the regions, numbered in the dichotomous system of arithmetical notation, one numerical place being appropriated to each term. Peirce: CP 4.357 Cross-Ref:†† Instead of shading excluded regions we may simply make them with the character 0, for zero. Peirce: CP 4.358 Cross-Ref:†† 358. The unmodified Eulerian system gives two syllogistic diagrams as shown

above, Figs. 1 and 2. These with the modification

Figs. 19-22

[Click here to view]†1†2

are shown in Figs. 19 and 20. The exclusions by different premisses are marked differently. Venn's modification furnishes two new syllogistic diagrams shown in Figs. 21 and 22. Peirce: CP 4.359 Cross-Ref:†† 359. The second imperfection of the system is also very readily remedied; and the remedy almost inevitably suggests a partial remedy for the third imperfection. Namely, why not draw the character X in any compartment in order to signify that something of the corresponding description occurs in the universe? We shall thus get these three forms of propositions:

Fig. 23 view] Some S is not P.

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Fig. 24 view] Some S is P.

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Fig. 25 view] There is something beside S and P.

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The precise denial of each of these is produced by substituting 0 for X. But when a third term is present some further rule has to be determined. How shall we mark the following diagram in order to express "Some S is not P "? The proposition

Fig. 26

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will here take the form, Either some S that is M but not P exists or some S that is neither M nor P exists. One suggestion would be that a cross be made on the circumference of M. But this would only provide for a special class of disjunctions. The question would then become, How shall we express, "Either something that is at once S and P exists or something that is neither S nor P exists"? Since we have drawn zeros at once in two compartments to signify the non-existence of either of two classes of objects, if we are to adhere to the principle that precise denial is produced by substituting crosses for zeros and conversely, it would follow that two crosses in two compartments would signify that something exists either of the one or of the other class. But this decision would render it impossible to give any systematic interpretation to a cross in one compartment and a zero in another. Suppose, then, that signs in different compartments, if disconnected, are to be taken conjunctively, and, if connected, disjunctively, or vice versa. Then precise denial will be effected by reversing the characters of the signs and of their relations as to connection or disconnection. There are perhaps no very compulsive reasons for adopting one interpretation of the connection of signs rather than the other. But it would seem strange if the insertion of a new and disconnected sign should cause a diagram to assert less; while the modification of an existing sign, by attaching to it a line of connection terminating in a new sign, might well enough diminish the assertion. It seems also quite natural that to mark the same compartment independently with contradictory signs, as in Fig. 27, should be absurd, while that if the two opposite signs are connected, as in Fig. 28, they should simply annul one another and be equivalent to no sign at all.

[Click here to view] Fig. 27 Fig. 28

Moreover, a cross on a boundary line may very naturally be understood to be equivalent to two connected crosses on the two sides of the boundary. Another consideration, perhaps more decisive, is that we shall necessarily regard the connected assertions as being put together directly, while the detached connexi of assertions are afterward compounded. It is therefore a question between using copulations of disjunctions [or] disjunctions of copulations. The former is the more convenient. . . . Peirce: CP 4.360 Cross-Ref:†† 360. Let this rule then be adopted: Connected assertions are made alternatively, but disconnected ones independently, i.e., copulatively. Peirce: CP 4.361 Cross-Ref:†† 361. As a consequence of this rule and of the introduction of the cross, the permissible transformations of diagrams, which transformations of course signify inferences, become so various that it is time to draw up a code of Rules for them. "Rules" is here used in the sense in which we speak of the "rules" of algebra; that is, as a permission under strictly defined conditions.†P1 Peirce: CP 4.362 Cross-Ref:†† 362. Rules of Transformation of Eulerian Diagrams Peirce: CP 4.362 Cross-Ref:†† Rule 1. Any entire sign of assertion (i.e., a cross, zero, or connected body of crosses and zeros) can be erased. Peirce: CP 4.362 Cross-Ref:†† Rule 2. Any sign of assertion can receive any accretion. Thus Fig. 29 may be transformed into Fig. 30.†1

Figs. 29-30

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Peirce: CP 4.362 Cross-Ref:†† Rule 3. Any assertion which could permissively be written, if there were no other assertion, can be written at any time, detachedly. Peirce: CP 4.362 Cross-Ref:†† Rule 4. In the same compartment repetitions of the same sign, whether mutually attached or detached, are equivalent to one writing of it. Two different signs in the same compartment, if attached to one another are equivalent to no sign at all, and may be erased or inserted. But if they are detached from one another they constitute an absurdity. All the foregoing supposes the signs to be unconnected with any in other compartments. If two contrary signs are written in the same compartment, the one being attached to certain others, P,

Figs. 31-34

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and the other to certain others, Q, it is permitted to attach P to Q and to erase the two contrary signs. Peirce: CP 4.362 Cross-Ref:†† Thus, Fig. 31 can be transformed into Fig. 32.†1 Peirce: CP 4.362 Cross-Ref:†† Rule 5. Any Area-boundary, representing a term, can be erased, provided that, if, in doing so, two compartments are thrown together containing independent zeros, those zeros be connected, while if there be a zero on one side of the boundary to be erased which is thrown into a compartment containing no independent zero, the zero and its whole connex be erased.

Peirce: CP 4.362 Cross-Ref:†† Thus, Fig. 33 can be transformed into Fig. 34.†2 Peirce: CP 4.362 Cross-Ref:†† Rule 6. Any new Term-boundary can be inserted; and if it cuts every compartment already present, any interpretation desired may be assigned to it. Only, where the new boundary passes through a compartment containing a cross, the new boundary must pass through the cross, or what is the same thing, a second cross connected with that already there must be drawn and the new boundary must pass between

Figs. 35-36

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them, regardless of what else is connected with the cross. If the new boundary passes through a compartment containing a zero, it will be permissible to insert a detached duplicate of the whole connex of that zero, so that one zero shall be on one side and the other on the other side of the new boundary. Peirce: CP 4.362 Cross-Ref:†† Thus, Fig. 35 can be converted into Fig. 36.†3 Peirce: CP 4.362 Cross-Ref:†† These six rules have been written down entirely without preconsideration; and it is probable that they might be simplified, and not unlikely that some have been overlooked. Peirce: CP 4.363 Cross-Ref:†† 363. As thus improved, Euler's diagrams are capable of giving an instructive development of the particular syllogism. The premisses of Darii are as follows:

Figs. 37-40

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Fig. 37: Any M is P. S being inserted this gives, by Rule 6, Fig. 39. Fig. 38: Some S is M, P being inserted, this becomes, by Rule 6, Fig. 40. Uniting Figs. 39 and 40 by Rule 3, we get Fig. 41, and by Rule 4, Fig. 42. Now erasing M by Rule 5, Figs. 41-43

[Click here to view] we get Fig. 43. Baroko, Bokardo, and Frisesomorum proceed in the same way. The premisses of the last are as follows:

Figs. 44-48

[Click here to view] Figs. 49-51

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Fig. 44: Some M is P, which, by Rule 6, gives Fig. 46. Fig. 45: No S is M, which, by Rule 6, may give Fig. 47. Combining these by Rule 3, Rule 4 gives Fig. 48 and Rule 5, Fig. 49. Let us now make the second premiss particular, as well as the first. We thus have Fig. 50 in place of Fig. 45; and on inserting P, we have Fig. 51 in place of Fig. 47. Uniting Figs. 46 and 51 we get Fig. 52. We now introduce two new and undescribed terms, as in Fig. 53, and on erasing M, we get Fig. 54 of which the interpretation is "Some S is

Figs. 52-54

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not some P." The objection may be raised that this method of dealing with the spurious †1 syllogism does not seem to follow from general principles, as a matter of course. In view of that objection we may put a single cross on the boundary instead of two connected crosses. The reasoning then proceeds, by uniting Figs. 44 and 50, as shown in Figs. 55 and 56.

[Click here to view] A portion of the boundary of M is retained in Fig. 56 to show that on whichever sides of the boundaries the two crosses may belong, they can in no case fall within the same region. Let it be noted, by the way, as a suggestive circumstance, that the portion of the boundary of M now remaining is simply a sign of negation. Peirce: CP 4.364 Cross-Ref:†† 364. This proposition "Some S is not some P" is called by Mr. B. I. Gilman, in

a paper which constitutes a distinct step in logical research, but which is buried in the Johns Hopkins Bulletins,†2 a proposition "particular in the second degree." An ordinary particular proposition asserts the existence of at least one individual of a given description. A proposition particular in the second degree asserts the existence of at least two individuals. It is an inference from two particular propositions each of which affirms the existence of one of the two individuals. We should therefore expect that, from a particular proposition of the second degree combined with one of the first degree, the inference should affirm the existence of three objects. Let us try the

experiment. Fig. 57 here to view] shows that the conclusions from the two premisses

[Click

Some S is not some M, and Some P is not M, is "Some S is other than something other than some P." But the S and the P in question are represented by the two lower crosses in the figure; and since these border upon the same compartment they may refer to the same individual. But if in addition to two ordinary particular premisses we take a universal premiss we can get a conclusion affirming the existence of three individuals. Take for instance the premisses Some S is not M Some M is P No N is P Some M is N These premisses are combined in Fig. 58

[Click here to view]; and it will be seen that the three connexes of crosses must be all different individuals; so that the conclusion is "Some S is other than and other than something other than some P." This line of study is far from being a trivial matter, however it may appear to superficial thinkers. But it does not enter into the purpose of the present paper to pursue it further. Peirce: CP 4.365 Cross-Ref:†† 365. In remedying the second imperfection we have gone far to remove the third and have even done something toward a treatment of the fourth. Let us consider a moment how far it can now be said that the method is inadequate to dealing with disjunctions. If by a disjunctive proposition we mean the sort of propositions usually given in the books as examples of this form, there never was any difficulty at all in dealing with them by Euler's diagrams in their original form. But such a proposition as "Every A is either B or C" which merely declares the non-existence of an A that is at once not B and not C, is not properly a disjunctive proposition. It is only disjunctions of conjunctions that cause some inconvenience; such as "Either some A is B while everything is either A or B, or else All A is B while some B is not A." Even here there is no serious difficulty. Fig. 59

[Click here to view] expresses this proposition. It is merely that there is a greater complexity in the expression than is essential to the meaning. There is, however, a very easy and very useful way of avoiding this. It is to draw an Euler's Diagram of Euler's Diagrams each surrounded by a circle to represent its Universe of Hypothesis. There will be no need of connecting lines in the enclosing diagram, it being understood that its compartments contain the several possible cases. Thus, Fig. 60

[Click here to view] expresses the same proposition as Fig. 59. Peirce: CP 4.366 Cross-Ref:†† 366. Let us now consider the fourth imperfection. We are already in condition to express minimal multitudes. Thus Fig. 61

[Click here to view] expresses that there are at least four A's. The precise denial of a minimal proposition will be a maximal proposition; and consequently, Fig. 62

[Click here to view] must express that there are not as many as four A's. It is necessary here that the whole area

of A should be covered by the parts. Peirce: CP 4.366 Cross-Ref:†† This mode of expression becoming impracticable, except for very small numbers, it naturally occurs to us to write a number in a compartment to express the precise multitude of individuals it contains. By extending this to algebraic expressions, not merely ratios but all sorts of numerical relations can be expressed. Peirce: CP 4.367 Cross-Ref:†† 367. The fifth fault of the system is by far the worst; and if there is any cure for it, not the smallest hopeful indication of its possibility appears at present. Peirce: CP 4.368 Cross-Ref:†† 368. Let us now endeavor to seize upon the spirit and characteristic of this system of graphs, and to estimate its value. Its beauty -- a violent inappropriate word, yet apparently the best there is to express the satisfactoriness of it upon mere contemplation -- and its other merits, which are fairly considerable, spring from its being veridically iconic, naturally analogous to the thing represented, and not a creation of conventions. It represents logic because it is governed by the same law. It works the syllogism as the planet integrates the equation of Laplace, or as the motion of the air about a pendulum solves a mathematical problem in ideal hydrodynamics. Still more closely, it resembles the application of geometry to algebra. By this I mean what is commonly called the application of algebra to geometry, but surely quite preposterously and contrarily to the spirit of the study. I hope no set argument is needed to defend this statement. The habitual neglect by students of analytical geometry of the real properties of loci, of which very little is known, and their almost exclusive interest in the imaginary properties, which are non-geometrical, sufficiently show that it is geometry that is the means, algebra the end. Geometry is not a perfect fit to algebra, in some respects falling short, in others over-running; elliptic in the absence of the imaginary, hyperbolic in presenting a continuity to which analytic quantity can hardly be said to make any approach. Yet even its partial analogy has been so helpful to modern algebra (and it was not less so to the older doctrine) that the phrase "it has been the making of it" is not too strong. For no doubt it was geometry that suggested the importance of the linear transformation, that of invariance, and in short almost all the profounder conceptions. The analogy of the doctrine of the Eulerian diagrams to non-relative logic is proportionately fully as great; although, owing to the greater simplicity of the subject and to its having fewer characters in all, the absolute number and weight of the points of resemblance are necessarily less. Such mathematics, as there may be connected with non-relative logic, we should have a right to expect would be much illuminated by the Eulerian Diagrams. Only this [is] mathematics of the most rudimentary conceivable kind; and hardly stands in need of any particular illumination. The different branches of pure mathematics are distinguished by their different systems of quantity; that is, of systems of points, units, or elements. In algebra, these points are so distributed over a surface that, in whatever manner any one is related to a single other exclusively, in that same manner is this other related to a third, and so on, ad infinitum; and moreover this infinite series may tend toward a definite limit, which limit is, in every case, included in the system. This is the most highly organized system of quantity that mathematicians have ever succeeded in definitely conceiving. On the other hand, the very simplest and most rudimentary of all conceivable systems of quantity is that one which distinguishes only two values. This [is] the system of evaluation which ethics applies to actions in dividing them into the right and the wrong, and which

non-relative logic applies to assertions in dividing them into the true and the false. The mathematics of such a system -- dichotomous mathematics -- amounts to very little. Those who seek to make a calculus of the algebra of logic struggle vainly after mathematical interest by complicating their problems. They do not succeed: mere complication has not even a mathematical interest. Peirce: CP 4.369 Cross-Ref:†† 369. Dichotomous mathematics does not amount to much, but it does amount to something. For example, the subject of higher particular propositions, in consequence of not being perfectly familiar, will call for considerable reflection to understand in its entirety and in its connections. Complicated questions of non-relative deductive reasoning are rare, it is true; still, they do occur, and if they are garbed in strange disguises, will now and then make the quickest minds hesitate or blunder. Euler's diagrams are the best aids in such cases, being natural, little subject to mistake, and every way satisfactory. It is true that there is a certain difficulty in applying them to problems involving many terms; but it is an easy art to learn to break such problems up into manageable fragments. The improved Boolian algebra has some advantages for those who are expert in its use, and who do not allow their instrument to rust from want of use. But the diagrams are always ready . . . .†1 Peirce: CP 4.370 Cross-Ref:†† 370. Any broad mathematical hypothesis, like that of a system of values, will attract three classes of students by three different interests that attach to it. The first is the special interest in the circumstance that that hypothesis necessarily involves certain relations among the things supposed, over and above those that were supposed in the definition of it. This is the mathematical interest proper. The second is the methodeutic interest in the devices which have to be employed to bring those new relations to light. This is a matter of supreme interest to the mathematician and of considerable, though subordinate, interest to the logician. The third is the analytical interest in the essential elements of the hypothesis and of the deductive processes of the second study, in their intellectual pedigrees and in their conceptual affiliations with ideas met with elsewhere. This is the logical interest, par excellence. In the case of non-relative deductive logic, that is, the doctrine of the relations of truth and falsity between combinations of non-relative terms, the methodeutic interest is slight owing to the extreme simplicity of the methods. The logical interest, on the other hand, limited as the subject is when relative terms are excluded, is very considerable, not to say great. In the inquiries which it prompts, it is the simplest cases which will chiefly attract attention, and therefore the circumstance, that the system of Eulerian diagrams becomes too cumbrous and laborious in complicated problems, is no objection to it. While the student cannot be counselled to confine himself to any single method of representation, the system of Eulerian diagrams is probably the best of any single one for the purely non-relative analysis of thought. Thus, it at once directs attention to the circumstance that the syllogism may be considered as a special case of the inference from Fig. 63 to Fig. 64, where the blots may either be zero or crosses or one a zero and the other a cross. Another example of the analytical interest of the system lies in the higher particular propositions, where we see an evolution of the conception of multitude. Multitude, or maniness, is a property of collections. Now a collection is an ens rationis, or abstraction; and abstraction appears as the highest product of the development of the logic

[Click here to view]

[Click here to view] Fig. 63

Fig. 64

of relatives. The student is thus directed to the deeply interesting and important problem of just how it is that the conception of multitude merges in the Eulerian diagrams. Peirce: CP 4.371 Cross-Ref:†† 371. The value of the system is thus considerable. Its fatal defect seems to be that it has no vital power of growth beyond the point to which it has here been carried. But this seeming may perhaps only be the reflection of the present student's own stupidity.

Peirce: CP 4.372 Cross-Ref:†† CHAPTER 2

SYMBOLIC LOGIC†1

372. If symbolic logic be defined as logic -- for the present only deductive logic -- treated by means of a special system of symbols, either devised for the purpose or extended to logical from other uses, it will be convenient not to confine the symbols used to algebraic symbols, but to include some graphical symbols as well. Peirce: CP 4.373 Cross-Ref:†† 373. The first requisite to understanding this matter is to recognize the purpose of a system of logical symbols. That purpose and end is simply and solely the investigation of the theory of logic, and not at all the construction of a calculus to aid the drawing of inferences. These two purposes are incompatible, for the reason that the system devised for the investigation of logic should be as analytical as possible, breaking up inferences into the greatest possible number of steps, and exhibiting them under the most general categories possible; while a calculus would aim, on the contrary, to reduce the number of processes as much as possible, and to specialize the symbols so as to adapt them to special kinds of inference. It should be recognized as a defect of a system intended for logical study that it has two ways of expressing the same fact, or any superfluity of symbols, although it would not be a serious fault for a calculus to have two ways of expressing a fact. Peirce: CP 4.374 Cross-Ref:†† 374. There must be operations of transformation. In that way alone can the symbol be shown determining its interpretant. In order that these operations should be as analytically represented as possible, each elementary operation should be either an insertion or an omission. Operations of commutation, like xy.·.yx, may be dispensed with by not recognizing any order of arrangement as significant. Associative transformations, like (xy)z.·.x(yz), which is a species of commutation, will be dispensed with in the same way; that is, by recognizing an equiparant †1 as what it is, a symbol of an unordered set. Peirce: CP 4.375 Cross-Ref:†† 375. It will be necessary to recognize two different operations, because of the difference between the relation of a symbol to its object and to its interpretant. Illative transformation (the only transformation, relating solely to truth, that a system of symbols can undergo) is the passage from a symbol to an interpretant, generally a partial interpretant. But it is necessary that the interpretant shall be recognized without the actual transformation. Otherwise the symbol is imperfect. There must, therefore, be a sign to signify that an illative transformation would be possible. That is to say, we must not only be able to express "A therefore B," but "If A then B." The symbol must, besides, separately indicate its object. This object must be indicated by a sign, and the relation of this to the significant element of the symbol is that both are signs of the same object. This is an equiparant, or commutative relation. It is therefore necessary to have an operation combining two symbols as referring to the same object. This, like the other operation, must have its actual and its potential state. The former makes the symbol a proposition "A is B;" that is, "Something A stands for, B stands for." The latter expresses that such a proposition might be expressed, "This stands for something which A stands for and B stands for." These relations might be expressed in roundabout ways; but two operations would always be necessary. In Jevons's modification †2 of Boole's algebra the two operations are aggregation and composition. Then, using non-relative terms, "nothing" is defined as that term which aggregated with any term gives that term, while "what is" is that term which

compounded with any term gives that term. But here we are already using a third operation; that is, we are using the relation of equivalence; and this is a composite relation. And when we draw an inference, which we cannot avoid, since it is the end and aim of logic, we use still another. It is true that if our purpose were to make a calculus, the two operations, aggregation and composition, would go admirably together. Symmetry in a calculus is a great point, and always involves superfluity, as in homogeneous coördinates and in quaternions. Superfluities which bring symmetry are immense economies in a calculus. But for purposes of analysis they are great evils. Peirce: CP 4.376 Cross-Ref:†† 376. A proposition de inesse relates to a single state of the universe, like the present instant. Such a proposition is altogether true or altogether false. But it is a question whether it is not better to suppose a general universe, and to allow an ordinary proposition to mean that it is sometimes or possibly true. Writing down a proposition under certain circumstances asserts it. Let these circumstances be represented in our system of symbols by writing the proposition on a certain sheet. If, then, we write two propositions on this same sheet, we can hardly resist understanding that both are asserted. This, then, will be the mode of representing that there is something which the one and the other represent -- not necessarily the same quasi-instantaneous state of the universe, but the same universe. If writing A asserts that A may be true, and writing B that B may be true, then writing both together will assert that A may be true and that B may be true. Peirce: CP 4.377 Cross-Ref:†† 377. By a rule of a system of symbols is meant a permission under certain circumstances to make a certain transformation; and we are to recognize no transformations as elementary except writing down and erasing. From the conventions just adopted, it follows, as Rule 1, that anything written down may be erased, provided the erasure does not visibly affect what else there may be which is written along with it. Peirce: CP 4.378 Cross-Ref:†† 378. Let us suppose that two facts are so related that asserting the one gives us the right to assert the other, because if the former is true, the latter must be true. If A having been written, we can add B, we may then, by our first rule, erase A; and consequently A may be transformed into B by two steps. We shall need to express the fact that writing A gives us a right, under all circumstances, to add B. Since this is not a reciprocal relation, A and B must be written differently; and since neither is positively asserted, neither must be written so that the other could be erased without affecting it. We need some place on our sheet upon which we can write a proposition without asserting it. The present writer's habit is to cut it off from the main sheet by enclosing it within an oval line; but in order to facilitate the printing, we will here enclose it in square brackets. In order, then, to express "If A can under any circumstances whatever be true, B can under some circumstances be true," we must certainly enclose A in square brackets. But what are we to do with B? We are not to assert positively that B can be true; yet it is to be more than hypothetically set forth, as A is. It must certainly, in some fashion, be enclosed within the brackets; for were it detached from the brackets, the brackets with their enclosed A could, by Rule 1, be erased; while in fact the dependence upon A cannot be omitted without danger of falsity. It is to be remarked that, in case we can assert that "If A can be true, B can be true," then, a fortiori, we can assert that "If both A and C can be true, B can be true,"

no matter what proposition C may be. Consequently, we have, as Rule 2, that, within brackets already written, anything whatever can be inserted. But the fact that "If A can be true, B can be true" does not generally justify the assertion "If A can be true, both B and D are true"; yet our second rule would imply that, unless the B were cut off, in some way, from the main field within the brackets. We will therefore enclose B in parentheses, and express the fact that "If A can be true, B can be true" by [A(B)] or [(B)A] or [A/(B)], etc. The arrangement is without significance. The fact that "If A can be true, both B and D can be true," or [A(BD)], justifies the assertion that "If A is true B is true," or [A(B)]. Hence the permission of Rule 1 may be enlarged, and we may assert that anything unenclosed or enclosed both in brackets and parentheses can be erased if it is separate from everything else. Let us now ask what [A] means. Rule 2 gives it a meaning; for by this rule [A] implies [A(X)], whatever proposition X may be. That is to say, that [A] can be true implies that "If A can under any circumstances be true, then anything you like, X, may be true." But we may like to make X express an absurdity. This, then, is a reductio ad absurdum of A; so that [A] implies, for one thing, that A cannot under any circumstances be true. The question is, Does it express anything further? According to this, [A (B)] expresses that A(B)is impossible. But what is this? It is that A can be true while something expressed by (B) can be true. Now, what can it be that renders the fact that "If A can ever be true, B can sometimes be true" incompatible with A's being able to be true? Evidently the falsity of B under all circumstances. Thus, just as [A] implies that A can never be true, so (B) implies that B can never be true. But further, to say that [A(B)], or "If A is ever true, B is sometimes true," is to say no more than that it is impossible that A is ever true, B being never true. Hence, the square brackets and the parentheses precisely deny what they enclose. A logical principle can be deduced from this: namely, if [A] is true [A(X)] is true. That is, if A is never true, then we have a right to assert that "If A is ever true, X is sometimes true," no matter what proposition X may be. Square brackets and parentheses, then, have the same meaning. Braces may be used for the same purpose. Peirce: CP 4.379 Cross-Ref:†† 379. Moreover, since two negatives make an affirmative, we have, as Rule 3, that anything can have double enclosures added or taken away, provided there be nothing within one enclosure but outside the other. Thus, if B can be true, so that B is written, Rule 3 permits us to write [(B)], and then Rule 2 permits us to write [X(B)]. That is, if B is sometimes true, then "If X is ever true, B is sometimes true." Let us make the apodosis of a conditional proposition itself a conditional proposition. That is, in (C{D}) let us put for D the proposition [A(B)]. We thus have (C{[A(B)]}). But, by Rule 3, this is the same as (CA(B)). Peirce: CP 4.380 Cross-Ref:†† 380. All our transformations are analysed into insertions and omissions. That is, if from A follows B, we can transform A into A B and then omit the B. Now, by Rule 1, from A B follows A. Treating this in the same way, we first insert the conclusion and say that from A B follows A B A. We thus get as Rule 4 that any detached portion of a proposition can be iterated. Peirce: CP 4.381 Cross-Ref:†† 381. It is now time to reform Rule 2 so as to state in general terms the effect of enclosures upon permissions to transform. It is plain that if we have written [A(B)]C, we can write [A(BC)]C, although the latter gives us no right to the former. In place,

then, of Rule 2 we have: Peirce: CP 4.381 Cross-Ref:†† Rule 2 (amended). Whatever transformation can be performed on a whole proposition can be performed upon any detached part of it under additional enclosures even in number, and the reverse transformation can be performed under additional enclosures odd in number. Peirce: CP 4.381 Cross-Ref:†† But this rule does not permit every transformation which can be performed on a detached part of a proposition to be performed upon the same expression otherwise situated. Peirce: CP 4.382 Cross-Ref:†† 382. Rule 4 permits, by virtue of Rule 2 (amended), all iteration under additional enclosures and erasure of a term inside enclosures if it is iterated outside some of them. Peirce: CP 4.383 Cross-Ref:†† 383. We can now exhibit the modi tollens et ponens. Suppose, for example, we have these premisses: "If A is ever true, B is sometimes true," and "B is never true." Writing them, we have [A(B)](B). By Rule 4, from (B) we might proceed to (B)(B). Hence, by Rule 2 (amended), from [A(B)](B) we can proceed to [A](B), and by Rule 1 to [A]. That is, "A is never true." Suppose, on the other hand, our premisses are [A(B)] and A. As before, we get [(B)]A, and by Rule 3, B A, and by Rule 1, B. That is, from the premisses of the modus ponens we get the conclusion. Let us take as premisses "If A is ever true, B is sometimes true," and "If B is ever true, C is sometimes true." That is, (A{B})[B(C)]. Then iterating [B(C)] within two enclosures, we get (A(B[B(C)]})[B(C)], or, by Rule 1, (A{B[B(C)]}). But we have just seen that B[B(C)] can be transformed to C. Performing this under two enclosures, we get (A{C}), which is the conclusion, "If A is ever true, C is sometimes true." Let us now formally deduce the principle of contradiction [A(A)]. Start from any premiss X. By Rule 3 we can insert [(X)], so that we have X[(X)]. By insertion under odd enclosures we have X[A(X)]. By iteration under additional enclosures we get X[A(A X)]; by erasures under even enclosures [A(A)]. Peirce: CP 4.384 Cross-Ref:†† 384. In complicated cases the multitude of enclosures become unmanageable. But by using ruled paper and drawing lines for the enclosures, composed of vertical and horizontal lines, always writing what is more enclosed lower than what is less enclosed, and what is evenly enclosed, on the left-hand part of the sheet, and what is oddly enclosed, on the right-hand part, this difficulty is greatly reduced. The diagram on page 325 (Fig. 65

[Click here to view]) illustrates the general style of arrangement recommended. Peirce: CP 4.385 Cross-Ref:†† 385. It is now time to make an addition to our system of symbols. Namely, A B signifies that A is at some quasi-instant true, and that B is at some quasi-instant true. But we wish to be able to assert that A and B are true at the same quasi-instant. We should always study to make our representations iconoidal; and a very iconoidal way of representing that there is one quasi-instant at which both A and B are true will be to connect them with a heavy line drawn in any shape, thus:

A-B or

[Click here to view]

If this line be broken, thus A- -B, the identity ceases to be asserted. We have evidently: Peirce: CP 4.385 Cross-Ref:†† Rule 5. A line of identity may be broken where unenclosed. -A will mean "At some quasi-instant A is true." It is equivalent to A simply. But -(-A) will differ from (-A) or (A) in merely asserting that at some quasi-instant A is not true, instead of asserting, with the latter forms, that at no quasi-instant is A true. Our quasi-instants may be individual things. In that case -A will mean "Something is A"; -(-A), "Something is not A"; [-(-A)], "Everything is A"; (-A), "Nothing is A." So A-B will express "Some A is B"; (A-B), "No A is B"; A-(-B), "Some A is not B"; [A-(-B)],

"Whatever A there may be is B";

[Click here to view] "There is

something besides A and B";†1 [ "Everything is either A or B."

[Click here to view]],

Peirce: CP 4.386 Cross-Ref:†† 386. The rule of iteration must now be amended as follows: Peirce: CP 4.386 Cross-Ref:†† Rule 4 (amended). Anything can be iterated under the same enclosures or under additional ones, its identical connections remaining identical. Peirce: CP 4.386 Cross-Ref:††

Thus, [A-(-B)] can be transformed to [Click here to view]. By the same rule A--(--B), i.e., "Something is A and nothing is B," by

iteration of the line of identity, can be transformed to [Click here to view] i.e., "Some A is not coexistent with anything that is B," whence,

by Rules 5 and 2 (amended), it can be further transformed to [Click here to view] i.e., "Some A is not B." Peirce: CP 4.387 Cross-Ref:†† 387. But it must be most carefully observed that two unenclosed parts cannot be illatively united by a line of identity. The enclosure of such a line is that of its least enclosed part. We can now exhibit any ordinary syllogism. Thus, the premisses of Baroko, "Any M is P" and "Some S is not P,"

Fig 66 view]

may be written

[Click here to

[Click here to view] Then, as

just seen, we can write

view] Then, by iteration, here to view] Breaking the line under even enclosures, we get

[Click here to

[Click

[Click here to view] But we have already shown that [P(P)] can be written unenclosed. Hence it can be struck out under one enclosure; and the unenclosed (P) can be erased. Thus we get

[Click here to view] or "Some S is not M." The great number of steps into which syllogism is thus analysed shows the perfection of the method for purposes of analysis. Peirce: CP 4.388 Cross-Ref:†† 388. In taking account of relations, it is necessary to distinguish between the different sides of the letters. Thus let l be taken in such a sense that X-l-Y means "X

loves Y." Then [Click here to view] will mean "Y loves X." Then, if m- means "Something is a man," and -w means "Something is a woman," m-l-w will mean "Some man loves some woman";

[Click here to view] will mean "Some man

loves all women"; "Every woman is loved by some man," etc.

[Click here to view] will mean

Peirce: CP 4.389 Cross-Ref:†† 389. Since enclosures signify negation, by enclosing a part of the line of

identity, the relation of otherness is represented. Thus, [Click here to view] will assert "Some A is not some B." Given the premisses "Some A is B" and "Some C is not B," they can be written

[Click here to view] By Rule 3, this can be written

[Click here to view]. By iteration, this gives

[Click here to view] The lines of identity are to be conceived as passing through the space between the braces outside of the brackets. By breaking the lines under even enclosures, we get

[Click here to view] As we have already seen, oddly enclosed [B(B)] can be erased. This, with erasure of the

detached (B), gives

under odd enclosures, we get "Some A is not some C."

[Click here to view] Joining the lines

[Click here to view] or

Peirce: CP 4.390 Cross-Ref:†† 390. For all considerable steps in ratiocination, the reasoner has to treat qualities, or collections, (they only differ grammatically), and especially relations, or systems, as objects of relation about which propositions are asserted and inferences drawn. It is, therefore, necessary to make a special study of the logical relatives "-- is a member of the collection --," and "-- is in the relation to --." The key to all that amounts to much in symbolical logic lies in the symbolization of these relations. But we cannot enter into this extensive subject in this article. Peirce: CP 4.391 Cross-Ref:†† 391. The system, of which the slightest possible sketch has been given, is not so iconoidal as the so-called Euler's diagrams; but it is by far the best general system which has yet been devised. The present writer has had it under examination for five years with continually increasing satisfaction. However, it is proper to notice some other systems that are now in use. Two systems which are merely extensions of Boole's algebra of logic may be mentioned. One of these is called by no more proper designation than the "general algebra of logic."†1 The other is called "Peirce's algebra of dyadic relatives."†1 In the former there are two operations -- aggregation, which Jevons †2 (to whom its use in algebra is due) signifies by a sign of division turned on its side, thus ·|·. (I prefer to join the two dots, in order to avoid mistaking the single character for three); and composition, which is best signified by a somewhat heavy dot, ·. Peirce: CP 4.391 Cross-Ref:†† Thus, if A and B are propositions, A·|·B is the proposition which is true if A is

true, is true if B is true, but is such that if A is false and B is false, it is false. A·B is the proposition which is true if A is true and B is true, but is false if A is false and false if B is false. Considered from an algebraical point of view, which is the point of view of this system, these expressions A·|·B and A·B are mean functions; for a mean function is defined as such a symmetrical function of several variables, that when the variables have the same value, it takes that same value. It is, therefore, wrong to consider them as addition and multiplication, unless it be that truth and falsity, the two possible states of a proposition, are considered as logarithmic infinity and zero. It is therefore well to let o represent a false proposition and ∞ (meaning logarithmic infinity, so that + ∞ and - ∞ are different) a true proposition. A heavy line, called an "obelus," over an expression negatives it. Peirce: CP 4.391 Cross-Ref:†† The letters i, j, k, etc., written below the line after letters signifying predicates, denote individuals, or supposed individuals, of which the predicates are true. Thus, l[ij] may mean that i loves j. To the left of the expression a series of letters π and Σ are written, each with a special one of the individuals i, j, k attached to it in order to show in what order these individuals are to be selected, and how. Σ[i] will mean that i is to be a suitably chosen individual, π[j] that j is any individual, no matter what. Thus, Σ[i]π[j]l[ji] means that there is an individual i such that every individual j loves i; and π[j]Σ[i]l[ji]

will mean that taking any individual j, no matter what, there is some individual i, whom j loves. This is the whole of this system, which has considerable power. This use of Σ and π was probably first introduced by O. C. Mitchell in his epoch-making paper in Studies in Logic,†1 by members of the Johns Hopkins University. Peirce: CP 4.392 Cross-Ref:†† 392. In Peirce's algebra of dyadic relatives the signs of aggregation and composition are used; but it is not usual to attach indices. In place of them two relative operations are used. Let l be "lover of," s "servant of." Then ls, called the relative product of s by l, denotes "lover of some servant of"; and l†s, called the relative sum of l to s, denotes "lover of whatever there may be besides servants of." In ms. the tail of the cross will naturally be curved. The sign is used to mean "numerically identical with," and to mean "other than." Schröder, who has written an admirable treatise on this system (though his characters are very objectionable, and should not be used †2), has considerably increased its power by various devices, and especially by writing, for example, π/u before an expression containing u to signify that u may be any relative whatever, or Σ/u to signify that it is a possible relative. In this way he introduces an abstraction or term of second intention. Peirce: CP 4.393 Cross-Ref:†† 393. Peano has made considerable use of a system of logical symbolization of his own. Mrs. Ladd-Franklin †3 advocates eight copula-signs to begin with, in order to exhibit the equal claim to consideration of the eight propositional forms. Of these she chooses "No a is b" and "Some a is b" (a(~\/)b and a\/b) as most desirable for the

elements of an algorithmic scheme; they are both symmetrical and natural. She thinks that a symbolic logic which takes "All a is b" (Boole, Schröder) as its basis is cumbrous; for every statement of a theorem, there is a corresponding statement necessary in terms of its contrapositive. This, she says, is the source of the parallel columns of theorems in Schröder's Logik; a single set of theorems is all-sufficient if a symmetrical pair of copulas is chosen. Some logicians (as C. S. P.) think the objections to Mrs. Ladd-Franklin's system outweigh its advantages. Other systems, as that of Wundt,†4 show a complete misunderstanding of the problem.

Peirce: CP 4.394 Cross-Ref:†† CHAPTER 3

EXISTENTIAL GRAPHS†1

A. THE CONVENTIONS

§1. ALPHA PART

394. Convention No. Zero. Any feature of these diagrams that is not expressly or by previous conventions of languages required by the conventions to have a given character may be varied at will. This "convention" is numbered zero, because it is understood in all agreements. Peirce: CP 4.395 Cross-Ref:†† 395. Convention No. I. These Conventions are supposed to be mutual understandings between two persons: a Graphist, who expresses propositions according to the system of expression called that of Existential Graphs, and an Interpreter, who interprets those propositions and accepts them without dispute. Peirce: CP 4.395 Cross-Ref:†† A graph is the propositional expression in the System of Existential Graphs of any possible state of the universe. It is a Symbol,†2 and, as such, general, and is accordingly to be distinguished from a graph-replica.†P1 A graph remains such though not actually asserted. An expression, according to the conventions of this system, of an impossible state of things (conflicting with what is taken for granted at the outset or has been asserted by the graphist) is not a graph, but is termed The pseudograph, all such expressions being equivalent in their absurdity. Peirce: CP 4.396 Cross-Ref:†† 396. It is agreed that a certain sheet, or blackboard, shall, under the name of The Sheet of Assertion, be considered as representing the universe of discourse, and as asserting whatever is taken for granted between the graphist and the interpreter to be true of that universe. The sheet of assertion is, therefore, a graph. Certain parts of the sheet, which may be severed from the rest, will not be regarded as any part of it.

Peirce: CP 4.397 Cross-Ref:†† 397. The graphist may place replicas of graphs upon the sheet of assertion; but this act, called scribing a graph on the sheet of assertion, shall be understood to constitute the assertion of the truth of the graph scribed. (Since by 395 the conventions are only "supposed to be" agreed to, the assertions are mere pretence in studying logic. Still they may be regarded as actual assertions concerning a fictitious universe.) "Assertion" is not defined; but it is supposed to be permitted to scribe some graphs and not others. Peirce: CP 4.397 Cross-Ref:†† Corollary. Not only is the sheet itself a graph, but so likewise is the sheet together with the graph scribed upon it. But if the sheet be blank, this blank, whose existence consists in the absence of any scribed graph, is itself a graph. Peirce: CP 4.398 Cross-Ref:†† 398. Convention No. II. A graph-replica on the sheet of assertion having no scribed connection with any other graph-replica that may be scribed on the sheet shall, as long as it is on the sheet of assertion in any way, make the same assertion, regardless of what other replicas may be upon the sheet. Peirce: CP 4.398 Cross-Ref:†† The graph which consists of all the graphs on the sheet of assertion, or which consists of all that are on any one area severed from the sheet, shall be termed the entire graph of the sheet of assertion or of that area, as the case may be. Any part of the entire graph which is itself a graph shall be termed a partial graph of the sheet or of the area on which it is. Peirce: CP 4.398 Cross-Ref:†† Corollaries. Two graphs scribed on the sheet are, both of them, asserted, and any entire graph implies the truth of all its partial graphs. Every blank part of the sheet is a partial graph. Peirce: CP 4.399 Cross-Ref:†† 399. Convention No. III. By a Cut shall be understood to mean a self-returning linear separation (naturally represented by a fine-drawn or peculiarly colored line) which severs all that it encloses from the sheet of assertion on which it stands itself, or from any other area on which it stands itself. The whole space within the cut (but not comprising the cut itself) shall be termed the area of the cut. Though the area of the cut is no part of the sheet of assertion, yet the cut together with its area and all that is on it, conceived as so severed from the sheet, shall, under the name of the enclosure of the cut, be considered as on the sheet of assertion or as on such other area as the cut may stand upon. Two cuts cannot intersect one another, but a cut may exist on any area whatever. Any graph which is unenclosed or is enclosed within an even number of cuts shall be said to be evenly enclosed; and any graph which is within an odd number of cuts shall be said to be oddly enclosed. A cut is not a graph; but an enclosure is a graph. The sheet or other area on which a cut stands shall be called the place of the cut. Peirce: CP 4.400 Cross-Ref:†† 400. A pair of cuts, one within the other but not within any other cut that that other is not within, shall be called a scroll. The outer cut of the pair shall be called the outloop, the inner cut the inloop, of the scroll. The area of the inloop shall be termed

the inner close of the scroll; the area of the outloop, excluding the enclosure of the inloop (and not merely its area), shall be termed the outer close of the scroll. Peirce: CP 4.401 Cross-Ref:†† 401. The enclosure of a scroll (that is, the enclosure of the outer cut of the pair) shall be understood to be a graph having such a meaning that if it were to stand on the sheet of assertion, it would assert de inesse that if the entire graph in its outer close is true, then the entire graph in its inner close is true. No graph can be scribed across a cut, in any way; although an enclosure is a graph. (A conditional proposition de inesse considers only the existing state of things, and is, therefore, false only in case the consequent is false while the antecedent is true. If the antecedent is false, or if the consequent is true, the conditional de inesse is true.) Peirce: CP 4.402 Cross-Ref:†† 402. The filling up of any entire area with whatever writing material (ink, chalk, etc.) may be used shall be termed obliterating that area, and shall be understood to be an expression of the pseudograph on that area. Peirce: CP 4.402 Cross-Ref:†† Corollary. Since an obliterated area may be made indefinitely small, a single cut will have the effect of denying the entire graph in its area. For to say that if a given proposition is true, everything is true, is equivalent to denying that proposition.

Peirce: CP 4.403 Cross-Ref:†† §2. BETA PART

403. Convention No. IV. The expression of a rheme in the system of existential graphs, as simple, that is without any expression, according to these conventions, of the analysis of its signification, and such as to occupy a superficial portion of the sheet or of any area shall be termed a spot. The word "spot" is to be used in the sense of a replica; and when it is desired to speak of the symbol of which it is the replica, this shall be termed a spot-graph. On the periphery of every spot, a certain place shall be appropriated to each blank of the rheme; and such a place shall be called a hook of the spot. No spot can be scribed except wholly in some area. Peirce: CP 4.404 Cross-Ref:†† 404. A heavy dot scribed at the hook of a spot shall be understood as filling the corresponding blank of the rheme of the spot with an indefinite sign of an individual, so that when there is a dot attached to every hook, the result shall be a proposition which is particular in respect to every subject. Peirce: CP 4.405 Cross-Ref:†† 405. Convention No. V. Every heavily marked point, whether isolated, the extremity of a heavy line, or at a furcation of a heavy line, shall denote a single individual, without in itself indicating what individual it is. Peirce: CP 4.406 Cross-Ref:†† 406. A heavily marked line without any sort of interruption (though its extremity may coincide with a point otherwise marked) shall, under the name of a

line of identity, be a graph, subject to all the conventions relating to graphs, and asserting precisely the identity of the individuals denoted by its extremities. Peirce: CP 4.406 Cross-Ref:†† Corollaries. It follows that no line of identity can cross a cut. Peirce: CP 4.406 Cross-Ref:†† Also, a point upon which three lines of identity abut is a graph expressing the relation of teridentity. Peirce: CP 4.407 Cross-Ref:†† 407. A heavily marked point may be on a cut; and such a point shall be interpreted as lying in the place of the cut and at the same time as denoting an individual identical with the individual denoted by the extremity of a line of identity on the area of the cut and abutting upon the marked point on the cut. Thus, in Fig. 67,

[Click here to view] if we refer to the individual denoted by the point where the two lines meet on the cut, as X, the assertion is, "Some individual, X, of the universe is a man, and nothing is at once mortal and identical with X"; i.e., some man is not mortal. So in Fig. 68

[Click here to view], if X and Y are the individuals denoted by the points on the [inner] cut, the interpretation is, Peirce: CP 4.407 Cross-Ref:†† "If X is the sun and Y is the sun, X and Y are identical." Peirce: CP 4.407 Cross-Ref:†† A collection composed of any line of identity together with all others that are connected with it directly or through still others is termed a ligature. Thus ligatures often cross cuts, and, in that case, are not graphs. Peirce: CP 4.408 Cross-Ref:†† 408. Convention No. VI. A symbol for a single individual, which individual is more than once referred to, but is not identified as the object of a proper name, shall be termed a Selective. The capital letters may be used as selectives, and may be made to abut upon the hooks of spots. Any ligature may be replaced by replicas of one selective placed at every hook and also in the outermost area that it enters. In the interpretation, it is necessary to refer to the outermost replica of each selective first, and generally to proceed in the interpretation from the outside to the inside of all cuts.

Peirce: CP 4.409 Cross-Ref:†† §3. GAMMA PART

409. Convention No. VII. The following spot-symbols shall be used, as if they were ordinary spot-symbols, except for special rules applicable to them: (Selectives are placed against the hooks in order to render the meanings of the new spot-symbols clearer). A[q], A is a monadic character; A[r], A is a dyadic relation;

A[s], A is a triadic relation; X /0\, X is a proposition or fact; X /1\ Y, Y possesses the character X; X /2\ Y/Z, Y stands in the dyadic relation X to Z; X Y//3\/W /Z, Y stands in the triadic relation X to Z for W. Peirce: CP 4.410 Cross-Ref:†† 410. Convention No. VIII. A cut with many little interruptions †1 aggregating about half its length shall cause its enclosure to be a graph, expressing that the entire graph on its area is logically contingent (non-necessary). Peirce: CP 4.411 Cross-Ref:†† 411. Convention No. IX. By a rim shall be understood an oval line making it, with its contents, the expression either of a rheme or a proper name of an ens rationis. Such a rim may be drawn as a line of peculiar texture, or a gummed label with a colored border may be attached to the sheet. A dotted rim containing a graph, some part of which is itself enclosed by a similar inner dotted oval and with heavy dotted lines proceeding from marked points of this graph to hooks on the rim, shall be a spot expressing that the individuals denoted by lines of identity attached to the hooks (or the single such individual) have the character, constituted by the truth of the graph, to be possessed by the individuals denoted by those points of it to which the heavy dotted lines are attached, in so far as they are connected with the partial graph within the inner oval. Peirce: CP 4.412 Cross-Ref:†† 412. A rim represented by a wavy line containing a graph, of which some marked points are connected by wavy lines with hooks on the rim, shall be a spot expressing that the individuals denoted by lines of identity abutting on these hooks form a collection of sets, of which collection each set has its members characterized in the manner in which those individuals must be which are denoted by the points of attachment of the interior graph, when that graph is true. Peirce: CP 4.413 Cross-Ref:†† 413. A rim shown as a saw line denotes an individual collection of individual single objects or sets of objects, the members of the collection being all those in existence, which are such individuals as the truth of the graph within makes those to be that are denoted by points of attachment of that graph to saw lines passing to hooks of the rim.

Peirce: CP 4.414 Cross-Ref:†† B. RULES OF TRANSFORMATION

Pure Mathematical Definition of Existential Graphs, Regardless of Their Interpretation

§1. ALPHA PART

414. 1. The System of Existential Graphs is a certain class of diagrams upon which it is permitted to operate certain transformations. 2. There is required a certain surface upon which it is practicable to scribe the diagrams and from which they can be erased in whole or in part. 3. The whole of this surface except certain parts which may be severed from it by "cuts" is termed the sheet of assertion. 4. A graph is a legisign (i.e., a sign which is of the nature of a general type) which is one of a certain class of signs used in this system. A graph-replica is any individual instance of a graph. The sheet of assertion itself is a graph-replica; and so is any part of it, being called the blank. Other graph-replicas can be scribed on the sheet of assertion, and when this is done the graphs of which those graph-replicas are instances is said to be "scribed on the sheet of assertion"; and when a graph-replica is erased, the graph is said to be erased. Two graphs scribed on the sheet of assertion constitute one graph of which they are said to be partial graphs. All that is at any time scribed on the sheet of assertion is called the entire scribed graph. 5. A cut is a self-returning finely drawn line. A cut is not a graph-replica. A cut drawn upon the sheet of assertion severs the surface it encloses, called the area of the cut, from the sheet of assertion; so that the area of a cut is no part of the sheet of assertion. A cut drawn upon the sheet of assertion together with its area and whatever is scribed upon that area constitutes a graph-replica scribed upon the sheet of assertion, and is called the enclosure of the cut. Whatever graph might, if permitted, be scribed upon the sheet of assertion might (if permitted) be scribed upon the area of any cut. Two graphs scribed at once on such area constitute a graph, as they would on the sheet of assertion. A cut can (if permitted) be drawn upon the area of any cut, and will sever the surface which it encloses from the area of the cut, while the enclosure of such inner cut will be a graph-replica scribed on the area of the outer cut. The sheet of assertion is also an area. Any blank part of any area is a graph-replica. Two cuts one of which has the enclosure of the other on its area and has nothing else there constitute a double cut. 6. No graph or cut can be placed partly on one area and partly on another.†1 7. No transformation of any graph-replica is permitted unless it is justified by the following code of Permissions.

Peirce: CP 4.415 Cross-Ref:†† Code of Permissions

415. Permission No. 1. In each special problem such graphs may be scribed on the sheet of assertion as the conditions of the special problem may warrant. Peirce: CP 4.415 Cross-Ref:†† Permission No. 2. Any graph on the sheet of assertion may be erased, except

an enclosure with its area entirely blank. Peirce: CP 4.415 Cross-Ref:†† Permission No. 3. Whatever graph it is permitted to scribe on the sheet of assertion, it is permitted to scribe on any unoccupied part of the sheet of assertion, regardless of what is already on the sheet of assertion. Peirce: CP 4.415 Cross-Ref:†† Permission No. 4. Any graph which is scribed on the inner area of a double cut on the sheet of assertion may be scribed on the sheet of assertion. Peirce: CP 4.415 Cross-Ref:†† Permission No. 5. A double cut may be drawn on the sheet of assertion; and any graph that is scribed on the sheet of assertion may be scribed on the inner area of any double cut on the sheet of assertion. Peirce: CP 4.415 Cross-Ref:†† Permission No. 6. The reverse of any transformation that would be permissible on the sheet of assertion is permissible on the area of any cut that is upon the sheet of assertion. Peirce: CP 4.415 Cross-Ref:†† Permission No. 7. Whenever we are permitted to scribe any graph we like upon the sheet of assertion, we are authorized to declare that the conditions of the special problem are absurd. Peirce: CP 4.416 Cross-Ref:†† §2. BETA PART

416. 8. The beta part adds to the alpha part certain signs to which new permissions are attached, while retaining all the alpha signs with the permissions attaching to them. 9. The line of identity is a Graph any replica of which, also called a line of identity, is a heavy line with two ends and without other topical singularity (such as a point of branching or a node), not in contact with any other sign except at its extremities. Otherwise, its shape and length are matters of indifference. All lines of identity are replicas of the same graph. 10. A spot is a graph any replica of which occupies a simple bounded portion of a surface, which portion has qualities distinguishing it from the replica of any other spot; and upon the boundary of the surface occupied by the spot are certain points, called the hooks of the spot, to each of which, if permitted, one extremity of one line of identity can be attached. Two lines of identity cannot be attached to the same hook; nor can both ends of the same line. 11. Any indefinitely small dot may be a spot replica called a spot of teridentity, and three lines of identity may be attached to such a spot. Two lines of identity, one outside a cut and the other on the area of the same cut, may have each an extremity at the same point on the cut. The totality of all the lines of identity that join one another is termed a ligature. A ligature is not generally a graph, since it may be part in one area and part in another. It is said to lie within any cut which it is wholly within.

Peirce: CP 4.417 Cross-Ref:†† 417. 12. The following are the additional permissions attaching to the beta part. Code of Permissions -- Continued Permission No. 8. All the above permissions apply to all spots and to the line of identity, as Graphs; and Permission No. 2 is to be understood as permitting the erasure of any portion of a line of identity on the sheet of assertion, so as to break it into two. Permission No. 3 is to be understood as permitting the extension of a line of identity on the sheet of assertion to any unoccupied part of the sheet of assertion. Permission No. 3 must not be understood [as stating that] that because it is permitted to scribe a graph without certain ligatures therefore it is permissible to scribe it with them, or the reverse. Peirce: CP 4.417 Cross-Ref:†† Permission No. 9. It is permitted to scribe an unattached line of identity on the sheet of assertion, and to join such unattached lines in any number by spots of teridentity. This is to be understood as permitting a line of identity, whether within or without a cut, to be extended to the cut, although such extremity is to be understood to be on both sides of the cut. But this does not permit a line of identity within a cut that is on the sheet of assertion to be retracted from the cut, in case it extends to the cut. Peirce: CP 4.417 Cross-Ref:†† Permission No. 10. If two spots are within a cut (whether on its area or not), and are not joined by any ligature within that cut, then a ligature joining them outside the cut is of no effect and may be made or broken. But this does not apply if the spots are joined by other hooks within the cut.†1 Peirce: CP 4.417 Cross-Ref:†† Permission No. 11. Permissions Nos. 4 and 5 do not cease to apply because of ligatures passing from without the outer of two cuts to within the inner one, so long as there is nothing else in the annular area.†2

Peirce: CP 4.418 Cross-Ref:†† CHAPTER 4

ON EXISTENTIAL GRAPHS, EULER'S DIAGRAMS, AND LOGICAL ALGEBRA†1P

§INTRODUCTION

418. A diagram is a representamen †2 which is predominantly an icon of

relations and is aided to be so by conventions. Indices are also more or less used. It should be carried out upon a perfectly consistent system of representation, founded upon a simple and easily intelligible basic idea. Peirce: CP 4.419 Cross-Ref:†† 419. A graph is a superficial diagram composed of the sheet upon which it is written or drawn, of spots or their equivalents, of lines of connection, and (if need be) of enclosures. The type, which it is supposed more or less to resemble, is the structural formula of the chemist. Peirce: CP 4.420 Cross-Ref:†† 420. A logical graph is a graph representing logical relations iconically, so as to be an aid to logical analysis. Peirce: CP 4.421 Cross-Ref:†† 421. An existential graph is a logical graph governed by a. system of representation founded upon the idea that the sheet upon which it is written, as well as every portion of that sheet, represents one recognized universe, real or fictive, and that every graph drawn on that sheet, and not cut off from the main body of it by an enclosure, represents some fact existing in that universe, and represents it independently of the representation of another such fact by any other graph written upon another part of the sheet, these graphs, however, forming one composite graph. Peirce: CP 4.422 Cross-Ref:†† 422. No other system of existential graphs than that herein set forth having hitherto been proposed, this one will need, for the present, no more distinctive designation. Should such designation hereafter become desirable, I desire that this system should be called the Existential System of 1897, in which year I wrote an account of it and offered it for publication to the Editor of The Monist, who declined it on the ground that it might later be improved upon. No changes have been found desirable since that date, although it has been under continual examination; but the exposition has been rendered more formal. Peirce: CP 4.423 Cross-Ref:†† 423. The following exposition of this system will be arranged as follows: Peirce: CP 4.423 Cross-Ref:†† Part I will explain the expression of ordinary forms of language in graphs and the interpretation of the latter into the former in three sections, as follows: Peirce: CP 4.423 Cross-Ref:†† A will state all the fundamental conventions of the system, separating those which are essentially different, showing the need which each is designed to meet together with the reasons for meeting it by the particular convention chosen, so far as these can be given at this stage of the development. A complete discussion will be given in an Appendix †1 to this part. To aid the understanding of all this, various logical analyses will be interspersed where they become pertinent. Peirce: CP 4.423 Cross-Ref:†† B will enunciate other rules of interpretation whose validity will be demonstrated from the fundamental conventions as premisses. This section will also introduce certain modifications of some of the signs established in A, the modified signs being convenient, although good reasons forbid their being considered fundamental.

Peirce: CP 4.423 Cross-Ref:†† C will redescribe the system in a compact form, which, on account of its uniting into one many rules that had, in the first instance; to be considered separately, is more easily grasped and retained in the mind. Peirce: CP 4.423 Cross-Ref:†† Part II will develop formal "rules," or permissions, by which one graph may be transformed into another without danger of passing from truth to falsity and without recurring to any interpretation of the graphs; such transformations being of the nature of immediate inferences. The part will be divided into sections corresponding to those of Part I. Peirce: CP 4.423 Cross-Ref:†† A will prove the basic rules of transformation directly from the fundamental conventions of A of Part I. Peirce: CP 4.423 Cross-Ref:†† B will deduce further rules of transformation from those of A, without further recourse to the principles of transformation. Peirce: CP 4.423 Cross-Ref:†† C will restate the rules in more compact form. Peirce: CP 4.423 Cross-Ref:†† Part III will show how the system may be made useful.†1

Peirce: CP 4.424 Cross-Ref:†† PART I. PRINCIPLES OF INTERPRETATIONP

A. Fundamental ConventionsP

§1. OF CONVENTIONS NOS. 1 AND 2†1P

424. In order to understand why this system of expression has the construction it has, it is indispensable to grasp the precise purpose of it, and not to confuse this with four other purposes, to wit: Peirce: CP 4.424 Cross-Ref:†† First, although the study of it and practice with it will be highly useful in helping to train the mind to accurate thinking, still that consideration has not had any influence in determining the characters of the signs employed; and an exposition of it, which should have that aim, ought to be based upon psychological researches of which it is impossible here to take account. Peirce: CP 4.424 Cross-Ref:†† Second, this system is not intended to serve as a universal language for mathematicians or other reasoners, like that of Peano.

Peirce: CP 4.424 Cross-Ref:†† Third, this system is not intended as a calculus, or apparatus by which conclusions can be reached and problems solved with greater facility than by more familiar systems of expression. Although some writers †2 have studied the logical algebras invented by me with that end apparently in view, in my own opinion their structure, as well as that of the present system, is quite antagonistic to much utility of that sort. The principal desideratum in a calculus is that it should be able to pass with security at one bound over a series of difficult inferential steps. What these abbreviated inferences may best be, will depend upon the special nature of the subject under discussion. But in my algebras and graphs, far from anything of that sort being attempted, the whole effort has been to dissect the operations of inference into as many distinct steps as possible. Peirce: CP 4.424 Cross-Ref:†† Fourth, although there is a certain fascination about these graphs, and the way they work is pretty enough, yet the system is not intended for a plaything, as logical algebra has sometimes been made, but has a very serious purpose which I proceed to explain. Peirce: CP 4.425 Cross-Ref:†† 425. Admirable as the work of research of the special sciences -- physical and psychical -- is, as a whole, the reasoning [employed in them] is of an elementary kind except when it is mathematical, and it is not infrequently loose. The philosophical sciences are greatly inferior to the special sciences in their reasoning. Mathematicians alone reason with great subtlety and great precision. But hitherto nobody has succeeded in giving a thoroughly satisfactory logical analysis of the reasoning of mathematics. That is to say, although every step of the reasoning is evidently such that the collective premisses cannot be true and yet the conclusion false, and although for each such step, A, we are able to draw up a self-evident general rule that from a premiss of such and such a form such and such a form of conclusion will necessarily follow, this rule covering the particular inferential step, A, yet nobody has drawn up a complete list of such rules covering all mathematical inferences. It is true that mathematics has its calculus which solves problems by rules which are fully proved; but, in the first place, for some branches of the calculus those proofs have not been reduced to self-evident rules, and in the second place, it is only routine work which can be done by simply following the rules of the calculus, and every considerable step in mathematics is performed in other ways. Peirce: CP 4.426 Cross-Ref:†† 426. If we consult the ordinary treatises on logic for an account of necessary reasoning, all the help that they afford is the rules of syllogism. They pretend that ordinary syllogism explains the reasoning of mathematics; and books have professed to exhibit considerable parts of the reasoning of the first book of Euclid's Elements stated in the form of syllogisms. But if this statement is examined, it will be found that it represents transformations of statements to be made that are not reduced to strict syllogistic form; and on examination it will be found that it is precisely in these transformations that the whole gist of the reasoning lies. The nearest approach to a logical analysis of mathematical reasoning that has ever been made was Schröder's statement, with improvements, in a logical algebra of my invention, of Dedekind's reasoning (itself in a sort of logical form) concerning the foundations of arithmetic.†1 But though this relates only to an exceptionally simple kind of mathematics, my opinion -- quite against my natural leanings toward my own creation -- is that the soul

of the reasoning has even here not been caught in the logical net. Peirce: CP 4.427 Cross-Ref:†† 427. No other book has, during the nineteenth century, been deeply studied by so large a proportion of the strong intellects of the civilized world as Kant's Critic of the Pure Reason; and the reason has undoubtedly been that they have all been greatly struck by Kant's logical power. Yet Kant, for all this unquestionable power, had paid so little attention to logic that he makes it manifest that he supposed that ordinary syllogism explains mathematical reasoning, and indeed [in] the simplest mood of syllogism, Barbara. Now, at the very utmost, from n propositions only 1/4n2 conclusions can be drawn by Barbara. In the thirteen books of Euclid's Elements there [are] 14 premisses (5 postulates and 9 axioms) excluding the definitions, which are merely verbal. Therefore, even if these premisses were related to one another in the most favorable way, which is far from being the case, there could only be 49 conclusions from them. But Euclid draws over ten times that number (465 propositions, 27 corollaries, and 17 lemmas) besides which his editors have inserted hundreds of corollaries. There are 48 propositions in the first book. Moreover, in Barbara or any sorites, or complexus of such syllogisms, to introduce the same premiss twice is idle. But throughout mathematics the same premisses are used over and over again. Moreover a person of fairly good mind and some logical training will instantly see the syllogistic conclusions from any number of premisses. But this is far from being true of mathematical inferences. Peirce: CP 4.428 Cross-Ref:†† 428. There is reason to believe that a thorough understanding of the nature of mathematical reasoning would lead to great improvements in mathematics. For when a new discovery is made in mathematics, the demonstration first found is almost always replaced later by another much simpler. Now it may be expected that, if the reasoning were thoroughly understood, the unnecessary complications of the first proof would be eliminable at once. Indeed, one might expect that the shortest route would be taken at the outset. Then again, consider the state of topical geometry, or geometrical topics, otherwise called topology. Here is a branch of geometry which not only leaves out of consideration the proportions of the different dimensions of figures and the magnitudes of angles (as does also graphics, or projective geometry -perspective, etc.) but also leaves out of account the straightness or mode of curvature of lines and the flatness or mode of bending of surfaces, and confines itself entirely to the connexions of the parts of figures (distinguishing, for example, a ring from a ball). Ordinary metric geometry equally depends on the connections of parts; but it depends on much besides. It, therefore, is a far more complicated subject, and can hardly fail to be of its own nature much the more difficult. And yet geometrical topics stands idle with problems to all appearance very simple staring it unsolved in the face, merely because mathematicians have not found out how to reason about it. Now a thorough understanding of mathematical reasoning must be a long stride toward enabling us to find a method of reasoning about this subject as well, very likely, as about other subjects that are not even recognized to be mathematical. Peirce: CP 4.429 Cross-Ref:†† 429. This, then, is the purpose for which my logical algebras were designed but which, in my opinion, they do not sufficiently fulfill. The present system of existential graphs is far more perfect in that respect, and has already taught me much about mathematical reasoning. Whether or not it will explain all mathematical inferences is not yet known.

Peirce: CP 4.429 Cross-Ref:†† Our purpose, then, is to study the workings of necessary inference. What we want, in order to do this, is a method of representing diagrammatically any possible set of premisses, this diagram to be such that we can observe the transformation of these premisses into the conclusion by a series of steps each of the utmost possible simplicity. Peirce: CP 4.430 Cross-Ref:†† 430. What we have to do, therefore, is to form a perfectly consistent method of expressing any assertion diagrammatically. The diagram must then evidently be something that we can see and contemplate. Now what we see appears spread out as upon a sheet. Consequently our diagram must be drawn upon a sheet. We must appropriate a sheet to the purpose, and the diagram drawn or written on the sheet is to express an assertion. We can, then, approximately call this sheet our sheet of assertion. The entire graph, or all that is drawn on the sheet, is to express a proposition, which the act of writing is to assert. Peirce: CP 4.431 Cross-Ref:†† 431. But what are our assertions to be about? The answer must be that they are to be about an arbitrarily hypothetical universe, a creation of a mind. For it is necessary reasoning alone that we intend to study; and the necessity of such reasoning consists in this, that not only does the conclusion happen to be true of a pre-determinate universe, but will be true, so long as the premisses are true, howsoever the universe may subsequently turn out to be determined. Thus, conformity to an existing, that is, entirely determinate, universe does not make necessity, which consists in what always will be, that is, what is determinately true of a universe not yet entirely determinate. Physical necessity consists in the fact that whatever may happen will conform to a law of nature; and logical necessity, which is what we have here to deal with, consists of something being determinately true of a universe not entirely determinate as to what is true, and thus not existent. In order to fix our ideas, we may imagine that there are two persons, one of whom, called the grapheus, creates the universe by the continuous development of his idea of it, every interval of time during the process adding some fact to the universe, that is, affording justification for some assertion, although, the process being continuous, these facts are not distinct from one another in their mode of being, as the propositions, which state some of them, are. As fast as this process in the mind of the grapheus takes place, that which is thought acquires being, that is, perfect definiteness, in the sense that the effect of what, is thought in any lapse of time, however short, is definitive and irrevocable; but it is not until the whole operation of creation is complete that the universe acquires existence, that is, entire determinateness, in the sense that nothing remains undecided. The other of the two persons concerned, called the graphist, is occupied during the process of creation in making successive modifications (i.e., not by a continuous process, since each modification, unless it be final, has another that follows next after it), of the entire graph. Remembering that the entire graph is whatever is, at any time, expressed in this system on the sheet of assertion, we may note that before anything has been drawn on the sheet, the blank is, by that definition, a graph. It may be considered as the expression of whatever must be well-understood between the graphist and the interpreter of the graph before the latter can understand what to expect of the graph. There must be an interpreter, since the graph, like every sign founded on convention, only has the sort of being that it has if it is interpreted; for a conventional sign is neither a mass of ink on a piece of paper or any other

individual existence, nor is it an image present to consciousness, but is a special habit or rule of interpretation and consists precisely in the fact that certain sorts of ink spots -- which I call its replicas -- will have certain effects on the conduct, mental and bodily, of the interpreter. So, then, the blank of the blank sheet may be considered as expressing that the universe, in process of creation by the grapheus, is perfectly definite and entirely determinate, etc. Hence, even the first writing of a graph on the sheet is a modification of the graph already written. The business of the graphist is supposed to come to an end before the work of creation is accomplished. He is supposed to be a mind-reader to such an extent that he knows some (perhaps all) the creative work of the grapheus so far as it has gone, but not what is to come. What he intends the graph to express concerns the universe as it will be when it comes to exist. If he risks an assertion for which he has no warrant in what the grapheus has yet thought, it may or may not prove true. Peirce: CP 4.432 Cross-Ref:†† 432. The above considerations constitute a sufficient reason for adopting the following convention, which is hereby adopted: Peirce: CP 4.432 Cross-Ref:†† Convention No. 1. A certain sheet, called the sheet of assertion, is appropriated to the drawing upon it of such graphs that whatever may be at any time drawn upon it, called the entire graph, shall be regarded as expressing an assertion by an imaginary person, called the graphist, concerning a universe, perfectly definite and entirely determinate, but the arbitrary creation of an imaginary mind, called the grapheus. Peirce: CP 4.433 Cross-Ref:†† 433. The convention which has next to be considered is the most arbitrary of all. It is, nevertheless, founded on two good reasons. A diagram ought to be as iconic as possible; that is, it should represent relations by visible relations analogous to them. Now suppose the graphist finds himself authorized to write each of two entire graphs. Say, for example, that he can draw: The pulp of some oranges is red; and that he is equally authorized to draw: To express oneself naturally is the last perfection of a writer's art. Each proposition is true independently of the other, and either may therefore be expressed on the sheet of assertion. If both are written on different parts of the sheet of assertion, the independent presence on the sheet of the two expressions is analogous to the independent truth of the two propositions that they would, when written separately, assert. It would, therefore, be a highly iconic mode of representation to understand, The pulp of some oranges is red. To express oneself naturally is the last perfection of a writer's art. where both are written on different parts of the sheet, as the assertion of both

propositions. Peirce: CP 4.434 Cross-Ref:†† 434. It is a subsidiary recommendation of a mode of diagrammatization, but one which ought to be accorded some weight, that it is one that the nature and habits of our minds will cause us at once to understand, without our being put to the trouble of remembering a rule that has no relation to our natural and habitual ways of expression. Certainly, no convention of representation could possess this merit in a higher degree than the plan of writing both of two assertions in order to express the truth of both. It is so very natural, that all who have ever used letters or almost any method of graphic communication have resorted to it. It seems almost unavoidable, although in my first invented system of graphs, which I call entitative graphs,†1 propositions written on the sheet together were not understood to be independently asserted but to be alternatively asserted. The consequence was that a blank sheet instead of expressing only what was taken for granted had to be interpreted as an absurdity. One system seems to be about as good as the other, except that unnaturalness and aniconicity haunt every part of the system of entitative graphs, which is a curious example of how late a development simplicity is. These two reasons will suffice to make every reader very willing to accede to the following convention, which is hereby adopted. Peirce: CP 4.434 Cross-Ref:†† Convention No. 2. Graphs on different parts of the sheet, called partial graphs, shall independently assert what they would severally assert, were each the entire graph.

Peirce: CP 4.435 Cross-Ref:†† §2. OF CONVENTION NO. 3P

435. If a system of expression is to be adequate to the analysis of all necessary consequences,†P1 it is requisite that it should be able to express that an expressed consequent, C, follows necessarily from an expressed antecedent, A. The conventions hitherto adopted do not enable us to express this. In order to form a new and reasonable convention for this purpose we must get a perfectly distinct idea of what it means to say that a consequent follows from an antecedent. It means that in adding to an assertion of the antecedent an assertion of the consequent we shall be proceeding upon a general principle whose application will never convert a true assertion into a false one. This, of course, means that so it will be in the universe of which alone we are speaking. But when we talk logic -- and people occasionally insert logical remarks into ordinary discourse -- our universe is that universe which embraces all others, namely The Truth, so that, in such a case, we mean that in no universe whatever will the addition of the assertion of the consequent to the assertion of the antecedent be a conversion of a true proposition into a false one. But before we can express any proposition referring to a general principle, or, as we say, to a "range of possibility," we must first find means to express the simplest kind of conditional proposition, the conditional de inesse, in which "If A is true, C is true" means only that, principle or no principle, the addition to an assertion of A of an assertion of C will not be a conversion of a true assertion into a false one. That is, it asserts that the graph of Fig. 69, anywhere on the sheet of assertion, might be transformed into the graph of Fig. 70

without passing from truth to falsity. a Fig. 69

ac Fig. 70

This conditional de inesse has to be expressed as a graph in such a way as distinctly to express in our system both a and c, and to exhibit their relation to one another. To assert the graph thus expressing the conditional de inesse, it must be drawn upon the sheet of assertion, and in this graph the expressions of a and of c must appear; and yet neither a nor c must be drawn upon the sheet of assertion. How is this to be managed? Let us draw a closed line which we may call a sep (sæpes, a fence), which shall cut off its contents from the sheet of assertion. Let this sep together with all that is within it, considered as a whole, be called an enclosure, this close, being written on the sheet of assertion, shall assert the conditional de inesse; but that which it encloses, considered separately from the sep, shall not be considered as on the sheet of assertion. Then, obviously, the antecedent and consequent must be in separate compartments of the close. In order to make the representation of the relation between them iconic, we must ask ourselves what spatial relation is analogous to their relation. Now if it be true that "If a is true, b is true" and "If b is true, c is true," then it is true that "If a is true, c is true." This is analogous to the geometrical relation of inclusion. So naturally striking is the analogy as to be (I believe) used in all languages to express the logical relation; and even the modern mind, so dull about metaphors, employs this one frequently. It is reasonable, therefore, that one of the two compartments should be placed within the other. But which shall be made the inner one? Shall we express the conditional de inesse by Fig. 71 or by Fig. 72? In order to decide which is the more appropriate mode of representation, one should observe that the consequent of a conditional proposition asserts what is true, not throughout the whole universe of possibilities considered, but in a subordinate universe marked off by the antecedent. This is not a fanciful notion, but a truth. Now in Fig. 72, the consequent appears in a special part of the sheet representing the universe, the space between the two lines containing the definition of the sub-universe.

Figs. 71-72

[Click here to view]†1

There is no such expressiveness in Fig. 71 -- or, if there be, it is only of a superficial and fanciful sort. Moreover, the necessity of using two kinds of enclosing lines -- a

necessity which, we shall find, does not exist in Fig. 72 -- is a defect of Fig. 71; and when we come to consider the question of convenience, the superiority of Fig. 72 will appear still more strongly. This, then, will be the method for us to adopt. Peirce: CP 4.436 Cross-Ref:†† 436. The two seps of Fig. 72, taken together, form a curve which I shall call a scroll. The node is of no particular significance. The scroll may equally well be drawn

as in Fig. 73. [Click here to view] The only essential feature is that there should be two seps, of which the inner, however drawn, may be called the inloop. The node merely serves to aid the mind in the interpretation, and will be used only when it can have this effect. The two compartments will be called the inner, or second, close, and the outer close, the latter excluding the former. The outer close considered as containing the inloop will be called the close. Peirce: CP 4.437 Cross-Ref:†† 437. Convention No. 3. An enclosure shall be a graph consisting of a scroll with its contents. Peirce: CP 4.437 Cross-Ref:†† The scroll shall be a real curve of two closed branches, the one within the other, called seps, and the inner specifically called the loop; and these branches may or may not be joined at a node. Peirce: CP 4.437 Cross-Ref:†† The contents of the scroll shall consist of whatever is in the area enclosed by the outer sep, this area being called the close and consisting of the inner, or second, close, which is the area enclosed by the loop, and the outer, or first close, which is the area outside the loop but inside the outer sep. Peirce: CP 4.437 Cross-Ref:†† When an enclosure is written on the sheet of assertion, although it is asserted as a whole, its contents shall be cut off from the sheet, and shall not be asserted in the assertion of the whole. But the enclosure shall assert de inesse that if every graph in the outer close be true, then every graph in the inner close is true.

Peirce: CP 4.438 Cross-Ref:†† §3. OF CONVENTIONS NOS. 4 TO 9†1P

438. Let a heavy dot or dash be used in place of a noun which has been erased from a proposition. A blank form of proposition produced by such erasures as can be filled, each with a proper name, to make a proposition again, is called a rhema, or, relatively to the proposition of which it is conceived to be a part, the predicate of that proposition. The following are examples of rhemata:

-- is good every man is the son of --- loves -God gives -- to -Every proposition has one predicate and one only. But what that predicate is considered to be depends upon how we choose to analyze it. Thus, the proposition God gives some good to every man may be considered as having for its predicate either of the following rhemata: -- gives -- to --- gives some good to --- gives -- to every man God gives -- to -God gives some good to -God gives -- to every man -- gives some good to every man God gives some good to every man.

In the last case the entire proposition is considered as predicate. A rhema which has one blank is called a monad; a rhema of two blanks, a dyad; a rhema of three blanks, a triad; etc. A rhema with no blank is called a medad, and is a complete proposition. A rhema of more than two blanks is a polyad. A rhema of more than one blank is a relative. Every proposition has an ultimate predicate, produced by putting a blank in every place where a blank can be placed, without substituting for some word its definition. Were this done we should call it a different proposition, as a matter of nomenclature. If on the other hand, we transmute the proposition without making any difference as to what it leaves unanalyzed, we say the expression only is different, as, if we say,

Some good is bestowed by God on every man. Each part of a proposition which might be replaced by a proper name, and still leave the proposition a proposition is a subject of the proposition.†P1 It is, however, the rhema which we have just now to attend to. Peirce: CP 4.439 Cross-Ref:†† 439. A rhema is, of course, not a proposition. Supposing, however, that it be written on the sheet of assertion, so that we have to adopt a meaning for it as a proposition, what can it most reasonably be taken to mean? Take, for example, Fig. 74. Shall this, since it represents the universe, be taken to mean that "Something in the universe is beautiful," or that "Anything in the universe is beautiful," or that "The universe, as a whole, is beautiful"? The last interpretation may be rejected at once for the reason that we are generally unable to assert anything of the universe not reducible to one of the other forms except what is well-understood between graphist and interpreter. We have, therefore, to choose between interpreting Fig. 74 to mean "Something is beautiful"

-----is beautiful

[Click here to view] Fig. 74

Fig. 75

and to mean "Anything is beautiful." Each asserts the rhema of an individual; but the former leaves that individual to be designated by the grapheus, while the latter allows the rhema [interpreter q to fill the blank with any proper name he likes. If Fig. 74 be taken to mean "Something is beautiful," then Fig. 75 will mean "Everything is beautiful"; while if Fig. 74 be taken to mean "Everything is beautiful," then Fig. 75 will mean "Something is beautiful." In either case, therefore, both propositions will be expressible, and the main question is, which gives the most appropriate expressions? The question of convenience is subordinate, as a general rule; but in this case the difference is so vast in this respect as to give this consideration more than its usual importance. Peirce: CP 4.440 Cross-Ref:†† 440. In order to decide the question of appropriateness, we must ask which form of proposition, the universal or the particular, "Whatever salamander there may be lives in fire," or "Some existing salamander lives in fire," is more of the nature of a conditional proposition; for plainly, these two propositions differ in form from "Everything is beautiful" and "Something is beautiful" respectively, only in their being limited to a subsidiary universe of salamanders. Now to say "Any salamander lives in fire" is merely to say "If anything, X, is a salamander, X lives in fire." It

differs from a conditional, if at all, only in the identification of X which it involves. On the other hand, there is nothing at all conditional in saying "There is a salamander, and it lives in fire." Peirce: CP 4.440 Cross-Ref:†† Thus the interpretation of Fig. 74 to mean "Something is beautiful" is decidedly the more appropriate; and since reasonable arrangements generally prove to be the most convenient in the end, we shall not be surprised when we come to find, as we shall, the same interpretation to be incomparably the superior in that respect also. Peirce: CP 4.441 Cross-Ref:†† 441. Convention No. 4. In this system, the unanalyzed expression of a rhema shall be called a spot. A distinct place on its periphery shall be appropriated to each blank, which place shall be called a hook. A spot with a dot at each hook shall be a graph expressing the proposition which results from filling every blank of the rhema with a separate sign of an indesignate individual existing in the universe and belonging to some determinate category, usually that of "things." Peirce: CP 4.442 Cross-Ref:†† 442. In many reasonings it becomes necessary to write a copulative proposition in which two members relate to the same individual so as to distinguish these members. Thus we have to write such a proposition as, A is greater than something that is greater than B, so as to exhibit the two partial graphs of Fig. 76. A is greater than --- is greater than B Fig. 76 The proposition we wish to express adds to those of Fig. 76 the assertion of the identity of the two "somethings." But this addition cannot be effected as in Fig. 77. A is greater than--- is greater than B -- is greater than -Fig. 77 For the "somethings," being indesignate, cannot be described in general terms. It is necessary that the signs of them should be connected in fact. No way of doing this can be more perfectly iconic than that exemplified in Fig. 78.

Fig. 78: [Click here to view]

Peirce: CP 4.442 Cross-Ref:†† Any sign of such identification of individuals may be called a connexus, and the particular sign here used, which we shall do well to adopt, may be called a line of identity. Peirce: CP 4.443 Cross-Ref:†† 443. Convention No. 5. Two coincident points, not more, shall denote the same individual. Peirce: CP 4.444 Cross-Ref:†† 444. Convention No. 6. A heavy line, called a line of identity, shall be a graph asserting the numerical identity of the individuals denoted by its two extremities. Peirce: CP 4.445 Cross-Ref:†† 445. The next convention to be laid down is so perfectly natural that the reader may well have a difficulty in perceiving that a separate convention is required for it. Namely, we may make a line of identity branch to express the identity of three

individuals. Thus, Fig. 79 [Click here to view] will express that some black bird is thievish. No doubt, it would have been easy to draw up Convention No. 4 in such a form as to cover this procedure. But it is not our object in this section to find ingenious modes of statement which, being borne in mind, may serve as rules for as many different acts as possible. On the contrary, what we are here concerned to do is to distinguish all proceedings that are essentially different. Now it is plain that no number of mere bi-terminal bonds, each terminal occupying a spot's hook, can ever assert the identity of three things, although when we once have a three-way branch, any higher number of terminals can be produced from it, as in Fig. 80

[Click here to view]. Peirce: CP 4.446 Cross-Ref:†† 446. We ought to, and must, then, make a distinct convention to cover this procedure, as follows: Peirce: CP 4.446 Cross-Ref:†† Convention No. 7. A branching line of identity shall express a triad rhema signifying the identity of the three individuals, whose designations are represented as filling the blanks of the rhema by coincidence with the three terminals of the line. Peirce: CP 4.447 Cross-Ref:†† 447. Remark how peculiar a sign the line of identity is. A sign, or, to use a more general and more definite term, a representamen, is of one or other of three kinds:†1 it is either an icon, an index, or a symbol. An icon is a representamen of what it represents and for the mind that interprets it as such, by virtue of its being an immediate image, that is to say by virtue of characters which belong to it in itself as a sensible object, and which it would possess just the same were there no object in nature that it resembled, and though it never were interpreted as a sign. It is of the nature of an appearance, and as such, strictly speaking, exists only in consciousness, although for convenience in ordinary parlance and when extreme precision is not called for, we extend the term icon to the outward objects which excite in consciousness the image itself. A geometrical diagram is a good example of an icon. A pure icon can convey no positive or factual information; for it affords no assurance that there is any such thing in nature. But it is of the utmost value for enabling its interpreter to study what would be the character of such an object in case any such did exist. Geometry sufficiently illustrates that. Of a completely opposite nature is the kind of representamen termed an index. This is a real thing or fact which is a sign of its object by virtue of being connected with it as a matter of fact and by also forcibly intruding upon the mind, quite regardless of its being interpreted as a sign. It may simply serve to identify its object and assure us of its existence and presence. But very often the nature of the factual connexion of the index with its object is such as to excite in consciousness an image of some features of the object, and in that way affords evidence from which positive assurance as to truth of fact may be drawn. A photograph, for example, not only excites an image, has an appearance, but, owing to its optical connexion with the object, is evidence that that appearance corresponds to a reality. A symbol is a representamen whose special significance or fitness to represent just what it does represent lies in nothing but the very fact of there being a habit, disposition, or other effective general rule that it will be so interpreted. Take,

for example, the word "man." These three letters are not in the least like a man; nor is the sound with which they are associated. Neither is the word existentially connected with any man as an index. It cannot be so, since the word is not an existence at all. The word does not consist of three films of ink. If the word "man" occurs hundreds of times in a book of which myriads of copies are printed, all those millions of triplets of patches of ink are embodiments of one and the same word. I call each of those embodiments a replica of the symbol. This shows that the word is not a thing. What is its nature? It consists in the really working general rule that three such patches seen by a person who knows English will effect his conduct and thoughts according to a rule. Thus the mode of being of the symbol is different from that of the icon and from that of the index. An icon has such being as belongs to past experience. It exists only as an image in the mind. An index has the being of present experience. The being of a symbol consists in the real fact that something surely will be experienced if certain conditions be satisfied. Namely, it will influence the thought and conduct of its interpreter. Every word is a symbol. Every sentence is a symbol. Every book is a symbol. Every representamen depending upon conventions is a symbol. Just as a photograph is an index having an icon incorporated into it, that is, excited in the mind by its force, so a symbol may have an icon or an index incorporated into it, that is, the active law that it is may require its interpretation to involve the calling up of an image, or a composite photograph of many images of past experiences, as ordinary common nouns and verbs do; or it may require its interpretation to refer to the actual surrounding circumstances of the occasion of its embodiment, like such words as that, this, I, you, which, here, now, yonder, etc. Or it may be pure symbol, neither iconic nor indicative, like the words and, or, of, etc. Peirce: CP 4.448 Cross-Ref:†† 448. The value of an icon consists in its exhibiting the features of a state of things regarded as if it were purely imaginary. The value of an index is that it assures us of positive fact. The value of a symbol is that it serves to make thought and conduct rational and enables us to predict the future. It is frequently desirable that a representamen should exercise one of those three functions to the exclusion of the other two, or two of them to the exclusion of the third; but the most perfect of signs are those in which the iconic, indicative, and symbolic characters are blended as equally as possible. Of this sort of signs the line of identity is an interesting example. As a conventional sign, it is a symbol; and the symbolic character, when present in a sign, is of its nature predominant over the others. The line of identity is not, however, arbitrarily conventional nor purely conventional. Consider any portion of it taken arbitrarily (with certain possible exceptions shortly to be considered) and it is an ordinary graph for which Fig. 81 might perfectly well be substituted. But when we consider the --is identical with-Fig. 81 connexion of this portion with a next adjacent portion, although the two together make up the same graph, yet the identification of the something, to which the hook of the one refers, with the something, to which the hook of the other refers, is beyond the power of any graph to effect, since a graph, as a symbol, is of the nature of a law, and is therefore general, while here there must be an identification of individuals. This identification is effected not by the pure symbol, but by its replica which is a thing. The termination of one portion and the beginning of the next portion denote the same individual by virtue of a factual connexion, and that the closest possible; for both are

points, and they are one and the same point. In this respect, therefore, the line of identity is of the nature of an index. To be sure, this does not affect the ordinary parts of a line of identity, but so soon as it is even conceived, [it is conceived] as composed of two portions, and it is only the factual junction of the replicas of these portions that makes them refer to the same individual. The line of identity is, moreover, in the highest degree iconic. For it appears as nothing but a continuum of dots, and the fact of the identity of a thing, seen under two aspects, consists merely in the continuity of being in passing from one apparition to another. Thus uniting, as the line of identity does, the natures of symbol, index, and icon, it is fitted for playing an extraordinary part in this system of representation. Peirce: CP 4.449 Cross-Ref:†† 449. There is no difficulty in interpreting the line of identity until it crosses a sep. To interpret it in that case, two new conventions will be required. Peirce: CP 4.449 Cross-Ref:†† How shall we express the proposition "Every salamander lives in fire," or "If it be true that something is a salamander then it will always be true that that something lives in fire"? If we omit the assertion of the identity of the somethings, the expression is obviously given in Fig. 82.

[Click here to view] To that, we wish to add the expression of individual identity. We ought to use our line of identity for that. Then, we must draw Fig. 83.

[Click here to view] It would be unreasonable, after having adopted the line of identity as our instrument for the expression of individual identity, to hesitate to employ it in this case. Yet to regularize such a mode of expression two new conventions are required. For, in the first place, we have not hitherto had any such sign as a line of identity crossing a sep. This part of the line of identity is not a graph; for a graph must be either outside or inside of each sep.†1 In order, therefore, to legitimate our interpretation of Fig. 83, we must agree that a line of identity crossing a sep simply asserts the identity of the individual denoted by its outer part and the individual denoted by its inner part. But this agreement does not of itself necessitate our interpretation of Fig. 83; since this might be understood to mean, "There is something which, if it be a salamander, lives in fire," instead of meaning, "If there be anything that is a salamander, it lives in fire." But although the last interpretation but one would involve itself in no positive contradiction, it would annul the convention that a line of identity crossing a sep still asserts the identity of its extremities -- not, indeed, by conflict with that convention, but by rendering it nugatory. What does it mean to assert de inesse that there is something, which if it be a salamander, lives in fire? It asserts, no doubt, that there is something. Now suppose that anything lives in fire. Then of that it will be true de inesse that if it be a salamander, it lives in fire; so that the proposition will then be true. Suppose that there is anything that is not a salamander. Then, of that it will be true de inesse that if it be a salamander, it lives in fire; and again the proposition will be true. It is only false in case whatever there may be is a salamander while nothing lives in fire. Consequently, Fig. 83 would be precisely equivalent to Fig. 84

[Click here to view], and there would be no need of any line of identity's crossing a sep. It would then be impossible to express a universal categorical analytically except by resorting to an unanalytic expression of such a proposition or something substantially equivalent to that.†P1 Peirce: CP 4.449 Cross-Ref:†† Two conventions, then, are necessary. In stating them, it will be well to avoid the idea of a graph's being cut through by a sep, and confine ourselves to the effects of joining dots on the sep to dots outside and inside of it. Peirce: CP 4.450 Cross-Ref:†† 450. Convention No. 8. Points on a sep shall be considered to lie outside the close of the sep so that the junction of such a point with any other point outside the sep by a line of identity shall be interpreted as it would be if the point on the sep were outside and away from the sep. Peirce: CP 4.451 Cross-Ref:†† 451. Convention No. 9. The junction by a line of identity of a point on a sep to a point within the close of the sep shall assert of such individual as is denoted by the point on the sep, according to the position of that point by Convention No. 8, a hypothetical conditional identity, according to the conventions applicable to graphs situated as is the portion of that line that is in the close of the sep. Peirce: CP 4.452 Cross-Ref:†† 452. It will be well to illustrate these conventions by some examples. Fig. 85 asserts that if it be true that something is good, then this assertion is false. That is, the assertion is that nothing is good. But in Fig. 86, the terminal of the line of identity on the outer sep asserts that something, X, exists, and it is only of this existing individual, X, that it is asserted that if that is good the assertion is false. It therefore means

Figs. 85-86

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"Something is not good." On Fig. 87 and Fig. 88 the points on the seps are marked with letters, for convenience of reference. Fig. 87 asserts that something, A, is a woman; and that if there is an individual, X, that is a catholic, and an individual, Y, that is identical with A, then X adores Y; that is, some woman is adored by all catholics, if there are any. Fig. 88 asserts that if there be an individual, X, and if X is a catholic, then X adores somebody that is a woman. That is, whatever

Figs. 87-88

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catholic there may be adores some woman or other. This does not positively assert that any woman exists, but only that if there is a catholic, then there is a woman whom he adores. Peirce: CP 4.453 Cross-Ref:†† 453. A triad rhema gives twenty-six affirmative forms of simple general propositions, as follows: Nos. Fig. 89. --blames_| to-- Somebody blames somebody to somebody

1

Fig. 90. [Click here to view] Everybody blames everybody to everybody

1

Fig. 91. [Click here to view] Somebody blames everybody to everybody

3 such

Fig. 92. [Click here to view] Everybody blames everybody to somebody or other

3 such

Fig. 93.

[Click here to view] Somebody blames somebody to everybody

3 such

Fig. 94.

[Click here to view] Everybody blames somebody to somebody

3 such

Fig. 95. [Click here to view] Somebody blames everybody to somebody or other

6 such

Fig. 96. [Click here to view] Everybody to somebody or other blames all

6 such ----------

Total

26

For a tetrad there are 150 such forms; for a pentad 1082; for a hexad 9366; etc.

Peirce: CP 4.454 Cross-Ref:†† B. Derived Principles of InterpretationP

§1. OF THE PSEUDOGRAPH AND CONNECTED SIGNSP

454. It is, as will soon appear, sometimes desirable to express a proposition either absurd, contrary to the understanding between the graphist and the interpreter, or at any rate well-known to be false. From any such proposition, as antecedent, any proposition whatever follows as a consequent de inesse. Hence, every such proposition may be regarded as implying that everything is true; and consequently all such propositions are equivalent. The expression of such a proposition may very well fill the entire close in which it is, since nothing can be added to what it already implies. Hence we may adopt the following secondary convention. Peirce: CP 4.454 Cross-Ref:†† Convention No. 10. The pseudograph, or expression in this system of a proposition implying that every proposition is true, may be drawn as a black spot entirely filling the close in which it is. Peirce: CP 4.455 Cross-Ref:†† 455. Since the size of signs has no significance, the blackened close may be drawn invisibly small. Thus Fig. 97 [may be scribed] as in Fig. 98, or even as in Fig. 99, Fig. 100, or lastly as in Fig. 101.†1

Figs. 97-101

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Peirce: CP 4.456 Cross-Ref:†† 456. Interpretational Corollary 1. A scroll with its contents having the pseudograph in the inner close is equivalent to the precise denial of the contents of the outer close. Peirce: CP 4.456 Cross-Ref:†† For the assertion, as in Fig. 97, that de inesse if a is true everything is true, is equivalent to the assertion that a is not true, since if the conditional proposition de inesse be true a cannot be true, and if a is not true the conditional proposition de inesse, having a for its antecedent, is true. Hence the one is always true or false with the other, and they are equivalent. Peirce: CP 4.456 Cross-Ref:†† This corollary affords additional justification for writing Fig. 97 as in Fig. 101, since the effect of the loop enclosing the pseudograph is to make a precise denial of the absurd proposition; and to deny the absurd is equivalent to asserting nothing. Peirce: CP 4.457 Cross-Ref:†† 457. Interpretational Corollary 2. A disjunctive proposition may be

expressed by placing its members in as many inloops of one sep. But this will not exclude the simultaneous truth of several members or of all. Peirce: CP 4.457 Cross-Ref:††

Thus, Fig. 102 [Click here to view] will express that either a or b or c or d or e is true. For it will deny the simultaneous denial of all. Peirce: CP 4.458 Cross-Ref:†† 458. Interpretational Corollary 3. A graph may be interpreted by copulations and disjunctions. Namely, if a graph within an odd number of seps be said to be oddly enclosed, and a graph within no sep or an even number of seps be said to be evenly enclosed, then spots in the same compartment are copulated when evenly enclosed, and disjunctively combined when oddly enclosed; and any line of identity whose outermost part is evenly enclosed refers to something, and any one whose outermost part is oddly enclosed refers to anything there may be. And the interpretation must begin outside of all seps and proceed inward. And spots evenly enclosed are to be taken affirmatively; those oddly enclosed negatively. Peirce: CP 4.458 Cross-Ref:†† For example, Fig. 83 may be read, Anything whatever is either not a salamander or lives in fire. Fig. 87 may be read, Something, A, is a woman, and whatever X may be, either X is not a catholic or X adores A. Fig. 88 may be read, Whatever X may be, either X is not a catholic or there is something Y, such that X adores Y and Y is a woman. Fig. 96 may be read, Whatever A may be, there is something C, such that whatever B may be, A blames B to C. Fig. 103

[Click here to view] may be read, Whatever X and Y may be, either X is not a saint or Y is not a saint or X loves Y; that is, Every saint there may be loves every saint. So Fig. 104

[Click here to view] may be read, Whatever X and Y may be, either X is not best or Y is not best or X is identical with Y; that is, there are not two bests. Fig. 105

[Click here to view] may be read, Whatever X and Y may be, either X does not love Y or Y does not love X; that is, no two love each other. Fig. 106

[Click here to view] may be read, Whatever X and Y may be either X does not love Y or there is something L and X is not L but Y loves L; that is, nobody loves anybody who does not love somebody else. Peirce: CP 4.459 Cross-Ref:†† 459. Interpretational Corollary 4. A sep which is vacant, except for a line of identity traversing it, expresses with its contents the non-identity of the extremities of that line.

Peirce: CP 4.460 Cross-Ref:†† §2. SELECTIVES AND PROPER NAMESP

460. It is sometimes impossible upon an ordinary surface to draw a graph so that lines of identity will not cross one another. If, for example, we express that x is a

value that can result from raising z to the power whose exponent is y, by means of Fig. 107, and express that u is a value that can result from multiplying w by v, by Fig. 108, then in order to express that

Fig. 107:

Fig. 108:

[Click here to view]

[Click here to view]

whatever values x, y, and z may be, there is a value resulting from raising x to a power whose exponent is a value of the product of z by y which same value is also one of the values resulting from raising to the power z a value resulting from raising x to the power y (this being one of the propositions expressed by the equation xyz = (xy)z) we may draw Fig. 109

[Click here to

view]; but there is an unavoidable intersection of two lines of identity. In such a case, and indeed in any case in which the lines of identity become too intricate to be perspicuous, it is advantageous to replace some of them by signs of a sort that in this system are called selectives. A selective is very much of the same nature as a proper name; for it denotes an individual and its outermost occurrence denotes a wholly indesignate individual of a certain category (generally a thing) existing in the universe, just as a proper name, on the first occasion of hearing it, conveys no more. But, just as on any subsequent hearing of a proper name, the hearer identifies it with that individual concerning which he has some information, so all occurrences of the selective other than the outermost must be understood to denote that identical individual. If, however, the outermost occurrence of any given selective is oddly enclosed, then, on that first occurrence the selective will refer to any individual whom the interpreter may choose, and in all other occurrences to the same individual. If there be no one outermost occurrence, then any one of those that are outermost may be considered as the outermost. The later capital letters are used for selectives. For example, Fig. 109 is otherwise expressed in Figs. 110 and 111.

Figs. 110-111

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Fig. 111 may be read, "Either no value is designated as U, or no value is designated as V, or no value is designated as W, or else a value designated as Y results from raising W to the V power, and a value designated as Z results from multiplying U by V, and a value designated as X results from raising Y to the U power, while this same value X results from raising W to the Z power." Peirce: CP 4.461 Cross-Ref:†† 461. Convention No. 11. The capital letters of the alphabet shall be used to denote single individuals of a well-understood category, the individual existing in the universe, the early letters preferably as proper names of well-known individuals, the later letters, called selectives, each on its first occurrence, as the name of an individual (that is, an object existing in the universe in a well-understood category; that is, having such a mode of being as to be determinate in reference to every character as wholly possessing it or else wholly wanting it), but an individual that is indesignate (that is, which the interpreter receives no warrant for identifying); while in every occurrence after the first, it shall denote that same individual. Of two occurrences of the same selective, either one may be interpreted as the earlier, if

and only if, enclosed by no sep that does not enclose the other. A selective at its first occurrence shall be asserted in the mode proper to the compartment in which it occurs. If it be on that occurrence evenly enclosed, it is only affirmed to exist under the same conditions under which any graph in the same close is asserted; and it is then asserted, under those conditions, to be the subject filling the rhema-blank corresponding to any hook against which it may be placed. If, however, at its first occurrence, it be oddly enclosed, then, in the disjunctive mode of interpretation, it will be denied, subject to the conditions proper to the close in which it occurs, so that its existence being disjunctively denied, a non-existence will be affirmed, and as a subject, it will be universal (that is, freed from the condition of wholly possessing or wholly wanting each character) and at the same time designate (that is, the interpreter will be warranted in identifying it with whatever the context may allow), and it will be, subject to the conditions of the close, disjunctively denied to be the subject filling the rhema-blank of the hook against which it may be placed. In all subsequent occurrences it shall denote the individual with which the interpreter may, on its first occurrence, have identified it, and otherwise will be interpreted as on its first occurrence. Peirce: CP 4.461 Cross-Ref:†† Resort must be had to the examples to trace out the sense of this long abstract statement; and the line of identity will aid in explaining the equivalent selectives. Fig. 112 may be read X is good Fig. 112

[Click here to view] Fig. 113

there exists something that may be called X and it is good. Fig. 113, the precise denial of Fig. 112, may be read "Either there is not anything to be called X or whatever there may be is not good," or "Anything you may choose to call X is not good," or "all things are non-good." "Anything" is not an individual subject, since the two propositions, "Anything is good" and "Anything is bad," do not exhaust the

possibilities. Both may be false. Peirce: CP 4.462 Cross-Ref:†† 462. Convention No. 12. The use of selectives may be avoided, where it is desired to do so, by drawing parallels on both sides of the lines of identity where they appear to cross.†1

Peirce: CP 4.463 Cross-Ref:†† §3. OF ABSTRACTION AND ENTIA RATIONIS †2P

463. The term abstraction bears two utterly different meanings in philosophy. In one sense it is applied to a psychological act by which, for example, on seeing a theatre, one is led to call up images of other theatres which blend into a sort of composite in which the special features of each are obliterated. Such obliteration is called precisive abstraction. We shall have nothing to do with abstraction in that sense. But when that fabled old doctor, being asked why opium put people to sleep, answered that it was because opium has a dormative virtue, he performed this act of immediate inference: Opium causes people to sleep; Hence, Opium possesses a power of causing sleep. The peculiarity of such inference is that the conclusion relates to something -- in this case, a power -- that the premiss says nothing about; and yet the conclusion is necessary. Abstraction, in the sense in which it will here be used, is a necessary inference whose conclusion refers to a subject not referred to by the premiss; or it may be used to denote the characteristic of such inference. But how can it be that a conclusion should necessarily follow from a premiss which does not assert the existence of that whose existence is affirmed by it, the conclusion itself? The reply must be that the new individual spoken of is an ens rationis; that is, its being consists in some other fact. Whether or not an ens rationis can exist or be real, is a question not to be answered until existence and reality have been very distinctly defined. But it may be noticed at once, that to deny every mode of being to anything whose being consists in some other fact would be to deny every mode of being to tables and chairs, since the being of a table depends on the being of the atoms of which it is composed, and not vice versa. Peirce: CP 4.464 Cross-Ref:†† 464. Every symbol is an ens rationis, because it consists in a habit, in a regularity; now every regularity consists in the future conditional occurrence of facts not themselves that regularity. Many important truths are expressed by propositions which relate directly to symbols or to ideal objects of symbols, not to realities. If we say that two walls collide, we express a real relation between them, meaning by a real relation one which involves the existence of its correlates. If we say that a ball is red, we express a positive quality of feeling really connected with the ball. But if we say that the ball is not blue, we simply express -- as far as the direct expression goes -- a relation of inapplicability between the predicate blue, and the ball or the sign of it. So it is with every negation. Now it has already been shown that every universal proposition involves a negation, at least when it is expressed as an existential graph.

On the other hand, almost every graph expressing a proposition not universal has a line of identity. But identity, though expressed by the line as a dyadic relation, is not a relation between two things, but between two representamens of the same thing. Peirce: CP 4.465 Cross-Ref:†† 465. Every rhema whose blanks may be filled by signs of ordinary individuals, but which signifies only what is true of symbols of those individuals, without any reference to qualities of sense, is termed a rhema of second intention. For second intention is thought about thought as symbol. Second intentions and certain entia rationis demand the special attention of the logician. Avicenna defined logic as the science of second intentions, and was followed in this view by some of the most acute logicians, such as Raymund Lully, Duns Scotus, Walter Burleigh, and Armandus de Bello Visu; while the celebrated Durandus à Sancto Porciano, followed by Gratiadeus Esculanus, made it relate exclusively to entia rationis, and quite rightly. Peirce: CP 4.466 Cross-Ref:†† 466. Interpretational Corollary 5. A blank, considered as a medad, expresses what is well-understood between graphist and interpreter to be true; considered as a monad, it expresses "--exists" or "--is true"; considered as a dyad, it expresses "--coexists with--" or "and." Peirce: CP 4.467 Cross-Ref:†† 467. Interpretational Corollary 6. An empty sep with its surrounding blank, as in Fig. 114, is the pseudograph. Whether it be taken as medad, monad, or dyad, for which purpose it will be written as in Figs. 115, 116, it is the denial of the blank.

Figs. 114-116

[Click here to view]

Peirce: CP 4.468 Cross-Ref:†† 468. Interpretational Corollary 7. A line of identity traversing a sep will

signify non-identity. Thus Fig. 117 to view] will express that there are at least two men.

[Click here

Peirce: CP 4.469 Cross-Ref:†† 469. Interpretational Corollary 8. A branching of a line of identity enclosed

in a sep, as in Fig. 118 express that three individuals are not all identical.

[Click here to view], will

Peirce: CP 4.469 Cross-Ref:†† We now come to another kind of graphs which may go under the general head of second intentional graphs.†1 Peirce: CP 4.470 Cross-Ref:†† 470. Convention No. 13. The letters, {r}0 {r}1, {r}2, {r}3, etc., each with a number of hooks greater by one than the subscript number, may be taken as rhemata, signifying that the individuals joined to the hooks, other than the one vertically above the {r}, taken in their order clockwise, are capable of being asserted of the rhema indicated by the line of identity joined vertically to the {r}. Peirce: CP 4.470 Cross-Ref:†† Thus, Fig. 119

[Click here to view] expresses that there is a relation in which every man stands to some woman to whom no other man stands in the same relation; that is, there is a woman corresponding to every man or, in other words, there are at least as many women as men. The dotted lines, between which, in Fig. 119, the line of identity denoting the ens rationis is placed, are by no means necessary. Peirce: CP 4.471 Cross-Ref:†† 471. Convention No. 14. The line of identity representing an ens rationis may be placed between two rows of dots, or it may be drawn in ink of another colour, and any graph, which is to be spoken of as a thing, may be enclosed in a dotted oval with a dotted line attached to it. Other entia rationis may be treated in the same way, the patterns of the dotting being varied for those of different category. Peirce: CP 4.471 Cross-Ref:†† The graph of Fig. 120

[Click here to view] is an example. It may be read, as follows: "Euclid †2 enunciates it as a postulate that if two straight lines are cut by a third straight line so that those angles the two make with the third, these angles lying between the first two lines ({tas entos gönias}) and on the same side of the third, are less than two right angles, then that those two lines shall meet on that same side; and in this enunciation, by a side, {meré} of the third line must be understood part of a plane that contains that third line, which part is bounded by that line and by the infinitely distant parts of the plane." . . .

Peirce: CP 4.472 Cross-Ref:†† C. RecapitulationP

472. The principles of interpretation may now be restated more concisely and more comprehensibly. In this resume, it will be assumed that selectives, which should be regarded as a mere abbreviating device, and which constitute a serious exception to the general idea of the system, are not used. A person, learning to use the system and not yet thoroughly expert in it, might be led to doubt whether every proposition is capable of being expressed without selectives. For a line of identity cannot identify two individuals within enclosures outside of one another without passing out of both enclosures, while a selective is not subject to that restriction. It can be shown, however, that this restriction is of no importance nor even helps to render thought clear. Suppose then that two designations of individuals are to be identified, each being within a separate nest of seps, and the two nests being within a common nest of outer seps. The question is whether this identification can always be properly effected by a line of identity that passes out of the two separate nests of seps, and if desired, still farther out. The answer is plain enough when we consider that, having to say something of individuals, some to be named by the grapheus, others by the graphist, we can perfectly well postpone what we have to say until all these individuals are indicated; that is to say, the order in which they are to be specified by one and the other party. But if this be done, these individuals will first appear, even if selectives are used, in one nest of seps entirely outside of all the spots; and then these selectives can be replaced by lines of identity. Peirce: CP 4.473 Cross-Ref:†† 473. The respect in which selectives violate the general idea of the system is this; the outermost occurrence of each selective has a different significative force from every other occurrence -- a grave fault, if it be avoidable, in any system of regular and exact representation. The consequence is that the meaning of a partial graph containing a selective depends upon whether or not there be another part, which may be written on a remote part of the sheet in which the same selective occurs farther out. But the idea of this system is that assertions written upon different parts of the sheet should be independent of one another, if, and only if, they have no common part. When lines of identity are used to the exclusion of selectives, no such inconvenience can occur, because each line of one partial graph will retain precisely the same significative force, no matter what part outside of it be removed (though if a line be broken, the identity of the individuals denoted by its two parts will no longer be affirmed); and even if everything outside a sep be removed (the sep being unbreakable by any removal of a partial graph, or part which written alone would express a proposition) still there remains a point on the sep which retains the same force as if the line had been broken quite outside and away from the sep. Peirce: CP 4.474 Cross-Ref:†† 474. Rejecting the selectives, then, the principles of interpretation reduce themselves to simple form, as follows: 1. The writing of a proposition on the sheet of assertion unenclosed is to be understood as asserting that proposition; and that, independently of any other proposition on the sheet, except so far as the two may have some part or point in common. 2. A "spot," or unanalyzed expression of a rhema, upon this system, has upon its periphery a place called a "hook" appropriated to every blank of the rhema; and whenever it is written a heavily marked point occupies each hook. Now every heavily marked point, whether isolated or forming a part of a heavy line, denotes an

indesignate individual, and being unenclosed affirms the existence of some such individual; and if it occupy a hook of a spot it is the corresponding subject of the rhema signified by the spot. A heavy line is to be understood as asserting, when unenclosed, that all its points denote the same individual, so that any portion of it may be regarded as a spot. 3. A sep, or lightly drawn oval, when unenclosed is with its contents (the whole being called an enclosure) a graph, entire or partial, which precisely denies the proposition which the entire graph within it would, if unenclosed, affirm. Since, therefore, an entire graph, by the above principles, copulatively asserts all the partial graphs of which it is composed, and takes every indesignate individual, denoted by a heavily marked point that may be a part of it, in the sense of "something," it follows that an unenclosed enclosure disjunctively denies all the partial graphs which compose the contents of its sep, and takes every heavily marked point included therein in the sense of "anything" whatever. Consequently, if an enclosure is oddly enclosed, its evenly enclosed contents are copulatively affirmed; while if it be evenly enclosed, its oddly enclosed contents are disjunctively denied. 4. A heavily marked point upon a sep, or line of enclosure, is to be regarded as no more enclosed than any point just outside of and away from the sep, and is to be interpreted accordingly. But the effect of joining a heavily marked point within a sep to such a point upon the sep itself by means of a heavy line is to limit the disjunctive denial of existence (which is the effect of the sep upon the point within it) to the individual denoted by the point upon the sep. No heavy line is to be regarded as cutting a sep; nor can any graph be partly within a sep and partly outside of it; although the entire enclosure (which is not inside the sep) may be part of a graph outside of the sep.†1 5. A dotted oval is sometimes used to show that that which is within it is to be regarded as an ens rationis.

Peirce: CP 4.475 Cross-Ref:†† PART II. THE PRINCIPLES OF ILLATIVE TRANSFORMATIONP

A. Basic PrinciplesP

§1. SOME AND ANY

475. The first part of this tract was a grammar of this language of graphs. But one has not mastered a language as long as one has to think about it in another language. One must learn to think in it about facts. The present part is designed to show how to reason in this language without translating it into another, the language of our ordinary thought. This reasoning, however, depends on certain first principles, for the justification of which we have to make a last appeal to instinctive thought. Peirce: CP 4.476 Cross-Ref:†† 476. The purpose of reasoning is to proceed from the recognition of the truth

we already know to the knowledge of novel truth. This we may do by instinct or by a habit of which we are hardly conscious. But the operation is not worthy to be called reasoning unless it be deliberate, critical, self-controlled. In such genuine reasoning we are always conscious of proceeding according to a general rule which we approve. It may not be precisely formulated, but still we do think that all reasoning of that perhaps rather vaguely characterized kind will be safe. This is a doctrine of logic. We never can really reason without entertaining a logical theory. That is called our logica utens.†2 Peirce: CP 4.477 Cross-Ref:†† 477. The purpose of logic is attained by any single passage from a premiss to a conclusion, as long as it does not at once happen that the premiss is true while the conclusion is false. But reasoning proceeds upon a rule, and an inference is not necessary, unless the rule be such that in every case the fact stated in the premiss and the fact stated in the conclusion are so related that either the premiss will be false or the conclusion will be true. (Or both, of course. "Either A or B" does not properly exclude "both A and B.") Even then, the reasoning may not be logical, because the rule may involve matter of fact, so that the reasoner cannot have sufficient ground to be absolutely certain that it will not sometimes fail. The inference is only logical if the reasoner can be mathematically certain of the excellence of his rule of reasoning; and in the case of necessary reasoning he must be mathematically certain that in every state of things whatsoever, whether now or a million years hence, whether here or in the farthest fixed star, such a premiss and such a conclusion will never be, the former true and the latter false. It would be far beyond the scope of this tract to enter upon any thorough discussion of how this can be. Yet there are some questions which concern us here -- as, for example, how far the system of rules of this section is eternal verity, and how far it merely characterizes the special language of existential graphs -- and yet trench closely upon the deeper philosophy of logic; so that a few remarks meant to illuminate those pertinent questions and to show how they are connected with the philosophy of logic seem to be quite in order. Peirce: CP 4.478 Cross-Ref:†† 478. Mathematical certainty is not absolute certainty. For the greatest mathematicians sometimes blunder, and therefore it is possible -- barely possible -that all have blundered every time they added two and two. Bearing in mind that fact, and bearing in mind the fact that mathematics deals with imaginary states of things upon which experiments can be enormously multiplied at very small cost, we see that it is not impossible that inductive processes should afford the basis of mathematical certainty; and any mathematician can find much in the history of his own thought, and in the public history of mathematics to show that, as a matter of fact, inductive reasoning is considerably employed in making sure of the first mathematical premisses. Still, a doubt will arise as to whether this is anything more than a psychological need, whether the reasoning really rests upon induction at all. A geometer, for example, may ask himself whether two straight lines can enclose an area of their plane. When this question is first put, it is put in reference to a concrete image of a plane; and, at first, some experiments will be tried in the imagination. Some minds will be satisfied with that degree of certainty: more critical intellects will not. They will reflect that a closed area is an area shut off from other parts of the plane by a boundary all round it. Such a thinker will no longer think of a closed area by a composite photograph of triangles, quadrilaterals, circles, etc. He will think of a predictive rule -- a thought of what experience one would intend to produce who should intend to establish a closed area.

Peirce: CP 4.479 Cross-Ref:†† 479. That step of thought, which consists in interpreting an image by a symbol, is one of which logic neither need nor can give any account, since it is subconscious, uncontrollable, and not subject to criticism. Whatever account there is to be given of it is the psychologist's affair. But it is evident that the image must be connected in some way with a symbol if any proposition is to be true of it. The very truth of things must be in some measure representative. Peirce: CP 4.480 Cross-Ref:†† 480. If we admit that propositions express the very reality, it is not surprising that the study of the nature of propositions should enable us to pass from the knowledge of one fact to the knowledge of another.†P1 Peirce: CP 4.481 Cross-Ref:†† 481. We frame a system of expressing propositions -- a written language -having a syntax to which there are absolutely no exceptions. We then satisfy ourselves that whenever a proposition having a certain syntactical form is true, another proposition definitely related to it -- so that the relation can be defined in terms of the appearance of the two propositions on paper -- will necessarily also be true. We draw up our code of basic rules of such illative transformations, none of these rules being a necessary consequence of others. We then proceed to express in our language the premisses of long and difficult mathematical demonstrations and try whether our rules will bring out their conclusions. If, in any case, not, and yet the demonstration appears sound, we have a lesson in logic to learn. Some basic rule has been omitted, or else our system of expression is insufficient. But after our system and its rules are perfected, we shall find that such analyses of demonstrations teach us much about those reasonings. They will show that certain hypotheses are superfluous, that others have been virtually taken for granted without being expressly. laid down; and they will show that special branches of mathematics are characterized by appropriate modes of reasoning, the knowledge of which will be useful in advancing them. We may now lay all that aside, and begin again, constructing an entirely different system of expression, developing it from an entirely different initial idea, and having perfected it, as we perfected the former system, we shall analyze the same mathematical demonstrations. The results of the two methods will agree as to what is and what is not a necessary consequence. But a consequence that either method will represent as an immediate application of a basic rule, and therefore as simple, the other will be pretty sure to analyze into a series of steps. If it be not so, in regard to some inference the one method will be merely a disguise of the other. To say that one thing is simpler than another is an incomplete proposition, like saying that one ball is to the right of another. It is necessary to specify what point of view is assumed, in order to render the sentence true or false. Peirce: CP 4.482 Cross-Ref:†† 482. This remark has its application to the business now in hand, which is to translate the effect of each simple illative transformation of an existential graph into the language of ordinary thought and thus show that it represents a necessary consequence. For it will be found that it is not the operations which are simplest in this system that are simplest from the point of view of ordinary thought; so that it will be found that the simplest way to establish by ordinary thought the correctness of our basic rules will be to begin by proving the legitimacy of certain operations that are less simple from the point of view of the existential graphs.

Peirce: CP 4.483 Cross-Ref:†† 483. The first proposition for assent to which I shall appeal to ordinary reason is this; when a proposition contains a number of anys and somes, or their equivalents, it is a delicate matter to alter the form of statement while preserving the exact meaning. Every some, as we have seen,†1 means that under stated conditions, an individual could be specified of which that which is predicated of the some is [true], while every any means that what is predicated is true of no matter what [specified] individual; and the specifications of individuals must be made in a certain order, or the meaning of the proposition will be changed. Consider, for example, the following proposition: "A certain bookseller only quotes a line of poetry in case it was written by some blind authoress, and he either is trying to sell any books she may have written to the person to whom he quotes the line or else intends to reprint some book of hers." Here the existence of a bookseller is categorically affirmed; but the existence of a blind authoress is only affirmed conditionally on that bookseller's quoting a line of poetry. As for any book by her, none such is positively said to exist, unless the bookseller is not endeavoring to sell all the books there may be by her to the person to whom he quotes the line. Peirce: CP 4.484 Cross-Ref:†† 484. Now the point to which I demand the assent of reason is that all those individuals, whose selection is so referred to, might be named to begin with, thus: "There is a certain individual, A, and no matter what Z and Y may be, an individual, B, can be found such that whatever X may be, there is something C, and A is a bookseller and if he quotes Z to Y, and if Z is a line of poetry and Y is a person, then B is a blind poetess who has written Z, and either X is not a book published by B or A tries to sell X to Y or else C is a book published by B and A intends to reprint C." This is the precise equivalent of the original proposition, and any proposition involving somes and anys, or their equivalents, might equally be expressed by first thus defining exactly what these somes and anys mean, and then going on to predicate concerning them whatever is to be predicated. This is so evident that any proof of it would only confuse the mind; and anybody who could follow the proof will easily see how the proof could be constructed. But after the somes and anys have thus been replaced by letters, denoting each one individual, the subsequent statement concerns merely a set of designate individuals.

Peirce: CP 4.485 Cross-Ref:†† §2. RULES FOR DINECTED GRAPHS

485. In order, then, to make evident to ordinary reason what are the simple illative transformations of graphs, I propose to imagine the lines of identity to be all replaced by selectives, whose first occurrences are entirely outside the substance of the graph in a nest of seps, where each selective occurs once only and with nothing but existence predicated of it (affirmatively or negatively according as it is evenly or oddly enclosed). I will then show that upon such a graph certain transformations are permissible, and then will suppose the selectives to be replaced by lines of identity again. We shall thus have established the permissibility of certain transformations without the intervention of selectives. Peirce: CP 4.486 Cross-Ref:††

486. There will therefore be two branches to our inquiry. First, what transformations may be made in the inner part of the graph where all the selectives have proper names, and secondly what transformations may be made in the outer part where each selective occurs but once. It will be found that the second inquiry almost answers itself after the first has been investigated, and further, that the first class of transformations are precisely the same as if all the first occurrences of selectives were erased and the others were regarded as proper names. We therefore begin by inquiring what transformations are permissible in a graph which has no connexi at all, neither lines of identity nor selectives. Peirce: CP 4.487 Cross-Ref:†† 487. First of all, let us inquire what are those modes of illative transformation by each of which any graph whatever, standing alone on the sheet of assertion, may be transformed, and, at the same time, what are those modes of illative transformation from each of which any graph whatever, standing alone on the sheet of assertion, might result. Let us confine ourselves, in the first instance, to transformations not only involving no connexi, but also involving no entia rationis nor seps. Let us suppose a graph, say that of Fig. 121, a Fig. 121 to be alone upon the sheet of assertion. In what ways can it be illatively transformed without using connexi nor seps nor other entia rationis? In the first place, it may be erased; for the result of erasure, asserting nothing at all, can assert nothing false. In the second place, it can be iterated, as in Fig. 122; aa Fig. 122 for the result of the iteration asserts nothing not asserted already. In the third place, any graph, well-understood (before the original graph was drawn) to be true, can be inserted, as in The Fig. 123. The universe is here a Fig. 123 Evidently, these are the only modes of transformation that conform to the assumed conditions. Next, let us inquire in what manner any graph, say that of Fig. 124, z Fig. 124 can result. It cannot, unless of a special nature, result from insertion, since the blank is true and the graph may be false; but it can result by any omission, say of y from the graph of Fig. 125, y z

Fig. 125 whether y be true or false, or whatever its relation to z, since the result asserts nothing not asserted in the graph from which it results. Peirce: CP 4.488 Cross-Ref:†† 488. We may now employ the following: Peirce: CP 4.488 Cross-Ref:†† Conditional Principle No. 1. If any graph, a, were it written alone on the sheet of assertion, would be illatively transformable into another graph, z, then if the former graph, a, is a partial graph of an entire graph involving no connexus or sep, and written on the sheet of assertion, a may still be illatively transformed in the same way. Peirce: CP 4.488 Cross-Ref:†† For let a be a partial graph of which the other part is m, in Fig. 126. a m Fig. 126

z m Fig. 127

Then, both a and m will be asserted. But since a would be illatively transformable into z if it were the entire graph, it follows that if a is true z is true. Hence, the result of the transformation asserts only m which is already asserted, and z which is true if a, which is already asserted, is true. Peirce: CP 4.489 Cross-Ref:†† 489. By means of this principle we can evidently deduce the following: Peirce: CP 4.489 Cross-Ref:†† Categorical Basic Rules for the Illative Transformation of Graphs dinectively built up from partial graphs not separated by seps. 1. Any partial graph may be erased. 2. Any partial graph may be iterated. 3. Any graph well-understood to be true may be inserted. Peirce: CP 4.489 Cross-Ref:†† It is furthermore clear that no transformation of such graphs is logical, that is, results from the mere form of the graph, that is not justified by these rules. For a transformation not justified by these rules must insert something not in the premiss and not well-understood to be true. But under those circumstances, it may be false, as far as appears from the form. Peirce: CP 4.490 Cross-Ref:†† 490. Let us now consider graphs having no connexi or entia rationis other than seps. Here we shall have the following Peirce: CP 4.490 Cross-Ref:†† Conditional Principle No. 2. If a graph, a, were it written alone on the sheet of assertion, would be illatively transformable into a sep containing nothing but a graph, z, then in case nothing is on the sheet of assertion except this latter graph, z, this will be illatively transformable into a sep containing nothing but a.

Peirce: CP 4.490 Cross-Ref:†† For to say that Fig. 123 [?121] is illatively transformable into Fig. 128, is to say that if a is true, then if z were true, anything you like would be true; while to say that Fig. 124 is illatively transformable into Fig. 129 is to say that if z is true, then if a were true, anything you like would be true. But each of these amounts to saying that if a and z were both true anything you like would be true. Therefore, if either [transformation] is true so is the other.

Figs. 128-129 here to view]

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Peirce: CP 4.491 Cross-Ref:†† 491. Conditional Principle No. 3. If a sep containing nothing but a graph, a, would, were it written alone on the sheet of assertion, be illatively transformable into a graph, z, then if a sep, containing nothing but the latter graph, z, were written alone on the sheet of assertion, [this would] be illatively transformable into the graph, a. Peirce: CP 4.491 Cross-Ref:†† For to say that Fig. 129 is illatively transformable into Fig. 124 is to say that by virtue of the forms of a and z, if a is false, z is true; in other words, by virtue of their forms, either a or z is true. But this is precisely the meaning of saying that Fig. 128 is illatively transformable into Fig. 123 [?121]. Peirce: CP 4.492 Cross-Ref:†† 492. By means of these principles we can deduce the following: Peirce: CP 4.492 Cross-Ref:†† Basic Categorical Rules for the Illative Transformation of Graphs dinectively built up from Partial Graphs and from Graphs separated by seps. Peirce: CP 4.492 Cross-Ref:†† Rule 1. Within an even finite number (including none) of seps, any graph may be erased; within an odd number any graph may be inserted. Peirce: CP 4.492 Cross-Ref:†† Rule 2. Any graph may be iterated within the same or additional seps, or if iterated, a replica may be erased, if the erasure leaves another outside the same or additional seps.

Peirce: CP 4.492 Cross-Ref:†† Rule 3. Any graph well-understood to be true (and therefore an enclosure having a pseudograph within an odd number of its seps) may be inserted outside all seps. Peirce: CP 4.492 Cross-Ref:†† Rule 4. Two seps, the one enclosing the other but nothing outside that other, can be removed. Peirce: CP 4.493 Cross-Ref:†† 493. These rules have now to be demonstrated. The former set of rules, already demonstrated, apply to every graph on the sheet of assertion composed of dinected partial graphs not enclosed; for the reasoning of the demonstrations so apply. It is now necessary to demonstrate, from Conditional Principle No. 2, the following Principle of Contraposition: If any graph, say that of Fig. 123 [?121], is illatively transformable into another graph, say that of Fig. 124, then an enclosure consisting of a sep containing nothing but the latter graph, as in Fig. 130, is illatively transformable into

Figs. 130-134

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an enclosure consisting of a sep containing nothing but the first graph, as in Fig. 131. In order to prove this principle, we must first prove that any graph on the sheet of assertion is illatively transformable by having two seps drawn round it, the one containing nothing but the other with its contents. For let z be the original graph. Then, it has to be shown that Fig. 124 is transformable into Fig. 132. Now Fig. 130 on the sheet of assertion is illatively transformable into itself since any graph is illatively transformable into any graph that by virtue of its form cannot be false unless the original graph be false, and Fig. 130 cannot be false unless Fig. 130 is false. But from this it follows, by Conditional Principle No. 2, that Fig. 124 is illatively transformable into Fig. 132. Q. E. D. The principle of contraposition, which can now be proved without further difficulty, is that if any graph, a, (Fig. 123[?121], is illatively transformable into any graph, z, (Fig. 124) then an enclosure (Fig. 130) consisting of a sep enclosing nothing but the latter graph, z, is transformable into an enclosure (Fig. 131) consisting of a sep containing nothing but the first graph, a. If a is transformable into z, then, by the rule just proved, it is transformable into Fig. 132, consisting of z doubly enclosed with nothing between the seps. But if Fig. 123 [?121] is illatively transformable into Fig. 132, then, by Conditional Principle No. 2, Fig. 130 is illatively transformable into Fig. 131, Q. E. D. Peirce: CP 4.494 Cross-Ref:†† 494. Supposing, now, that Rule 1 holds good for any insertion or omission within not more than any finite number, N, of seps, it will also hold good for every

insertion or omission within not more than N+1 seps. For in any graph on the sheet of insertions of which a partial graph is an enclosure consisting of a sep containing only a graph, z, involving a nest of N seps, let the partial graph outside this enclosure be m, so that Fig. 133 is the entire graph. Then application of the rule within the N+1 seps will transform z into another graph, say a, so that Fig. 134 will be the result. Then a, were it written on the sheet of assertion unenclosed and alone, would be illatively transformable into z, since the rule is supposed to be valid for an insertion or omission within N seps. Hence, by the principle of contraposition, Fig. 130 will be transformable into Fig. 131, and by Conditional Principle No. 1, Fig. 133 will be transformable into Fig. 134. It is therefore proved that if Rule 1 is valid within any number of seps up to any finite number, it is valid for the next larger whole number of seps. But by Rule 1 of the former set of rules, it is valid for N = 0, and hence it follows that it is valid within seps whose number can be reached from 0 by successive additions of unity; that is, for any finite number. Rule 1 is, therefore, valid as stated. It will be remarked that the partial graphs may have any multitude whatsoever; but the seps of a nest are restricted to a finite multitude, so far as this rule is concerned. A graph with an endless nest of seps is essentially of doubtful meaning, except in special cases. Thus Fig. 135

[Click here to view], supposed to continue the alternation endlessly, evidently merely asserts the truth of a.†1 But if instead of ba, b were everywhere to stand alone, the graph would certainly assert either a or b to be true and would certainly be true if a were true, but whether it would be true or false in case b were true and not a is essentially doubtful. Peirce: CP 4.495 Cross-Ref:†† 495. Rule 2 is so obviously demonstrable in the same way that it will be sufficient to remark that unenclosed iterations of unenclosed graphs are justified by Rule 2 of the former set of rules. Then, since Fig. 136 is illatively transformable into

Figs. 136-141

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Fig. 137, it follows from the principle of contraposition that Fig. 138 is illatively transformable into Fig. 139. Or we may reason that to say that Fig. 137 follows from Fig. 136 is to say that, am being true, an follows from n; while to say that Fig. 139 follows from Fig. 138, is to say that, am being true, as before, if from an anything you like follows, then from n anything you like follows. In the same way Fig. 140 is transformable into Fig. 141. Peirce: CP 4.496 Cross-Ref:†† 496. The transformations the reverse of these, that is of Fig. 137 into Fig. 136, of Fig. 139 into Fig. 138, and of Fig. 141 into Fig. 140 are permitted by Rule 1. Then by the same Fermatian reasoning by which Rule 1 was demonstrated, we easily show that a graph can anywhere be illatively inserted or omitted, if there is another occurrence of the same graph in the same compartment or farther out by one sep. For if Fig. 138 is transformable into Fig. 139, then by the principle of contraposition, Fig. 142 is transformable into Fig. 143, and by Conditional Principle No. 1, Fig. 144 is transformable in Fig. 145. Having thus proved that iterations and deiterations are always permissible in the same compartment as the leading replica or in a compartment within one additional sep, we have no difficulty in extending this to any finite interval.

Figs. 142-146

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Thus, Fig. 146 is transformable into Fig. 147, this into Fig. 148, this successively into Figs. 149 to 153. Thus, the second rule is fully demonstrable. Peirce: CP 4.496 Cross-Ref:†† Rule 3 is self-evident. Peirce: CP 4.497 Cross-Ref:†† 497. We have thus far had no occasion to appeal to Conditional Principle No. 3; but it is indispensable for the proof of Rule 4. We have to show that if any graph, which [we] may denote by z is surrounded by two seps with nothing

Figs. 147-150

[Click here to view] Figs. 151-153

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between as in Fig. 132, then the two seps may be illatively removed as in Fig. 124. Now if the graph, z, occurred within one sep, as in Fig. 130, this, as we have seen, would be transformed into itself. Hence, by Conditional Principle No. 3, Fig. 132, can be illatively transformed into Fig. 124. Q. E. D. Peirce: CP 4.498 Cross-Ref:†† 498. The list of rules given for dinected graphs is complete. This is susceptible of proof; but the proof belongs in the next section of this chapter, where I may perhaps insert it. It is not interesting.

Peirce: CP 4.499 Cross-Ref:†† B. Rules for Lines of Identity

499. We now pass to the consideration of graphs connected by lines of identity. A small addition to our nomenclature is required here. Namely, we have seen that a line of identity is a partial graph; and as a graph it cannot cross a sep. Let us, then, call a series of lines of identity abutting upon one another at seps, a ligature; and we may extend the meaning of the word so that even a single line of identity shall be called a ligature. A ligature composed of more than one line of identity may be distinguished as a compound ligature. A compound ligature is not a graph, because by a graph we mean something which, written or drawn alone on the sheet of assertion, would, according to this system, assert something. Now a compound ligature could not be written alone on the sheet of assertion, since it is only by means of the intercepting sep, which is no part of it, that it is rendered compound. The different spots, as well as the different hooks, upon which a ligature abuts, may be said to be ligated by that ligature; and two replicas of the same graph are said to have the same ligations only when all the corresponding hooks of the two are ligated to one another. When a ligature cuts a sep, the part of the ligature outside the sep may be

said to be extended to the point of intersection on the sep, while the part of the ligature inside may be said to be joined to that point. Peirce: CP 4.500 Cross-Ref:†† 500. It has already †1 been pointed out that the mass of ink on the sheet by means of which a graph is said to be "scribed" is not, strictly speaking, a symbol, but only a replica of a symbol of the nature of an index. Let it not be forgotten that the significative value of a symbol consists in a regularity of association, so that the identity of the symbol lies in this regularity, while the significative force of an index consists in an existential fact which connects it with its object, so that the identity of the index consists in an existential fact or thing. When symbols, such as words, are used to construct an assertion, this assertion relates to something real. It must not only profess to do so, but must really do so; otherwise, it could not be true; and still less, false. Let a witness take oath, with every legal formality, that John Doe has committed murder, and still he has made no assertion unless the name John Doe denotes some existing person. But in order that the name should do this, something more than an association of ideas is requisite. For the person is not a conception but an existent thing. The name, or rather, occurrences of the name, must be existentially connected with the existent person. Therefore, no assertion can be constructed out of pure symbols alone. Indeed, the pure symbols are immutable, and it is not them that are joined together by the syntax of the sentence, but occurrences of them -- replicas of them. My aim is to use the term "graph" for a graph-symbol, although I dare say I sometimes lapse into using it for a graph-replica. To say that a graph is scribed is accurate, because "to scribe" means to make a graphical replica of. By "a line of identity," on the other hand, it is more convenient to mean a replica of the linear graph of identity. For here the indexical character is more positive; and besides, one seldom has occasion to speak of the graph. But the only difference between a line of identity and an ordinary dyadic spot is that the latter has its hooks marked at points that are deemed appropriate without our being under any factual compulsion to mark them at all, while a simple line such as is naturally employed for a line of identity must, from the nature of things, have extremities which are at once parts of it and of whatever it abuts upon. This difference does not prevent the rules of the last list from holding good of such lines. The only occasion for any additional rule is to meet that situation, in which no other graph-replica than a line of identity can ever be placed, that of having a hook upon a sep. Peirce: CP 4.501 Cross-Ref:†† 501. As to this, it is to be remarked that an enclosure -- that is, a sep with its contents -- is a graph; and those points on its periphery, that are marked by the abuttal upon them of lines of identity, are simply the hooks of the graph. But the sep is outside its own close. Therefore an unmarked point upon it is just like any other vacant place outside the sep. But if a line inside the sep is prolonged to the sep, at the instant of arriving at the sep, its extremity suddenly becomes identified -- as a matter of fact, and there as a matter of signification -- with a point outside the sep; and thus the prolongation suddenly assumes an entirely different character from an ordinary, insignificant prolongation. This gives us the following: Peirce: CP 4.501 Cross-Ref:†† Conditional Principle No. 4. Only the connexions and continuity of lines of identity are significant, not their shape or size. The connexion or disconnexion of a line of identity outside a sep with a marked or an unmarked point on the sep follows the same rules as its connexion or disconnexion with any other marked or unmarked

point outside the sep, but the junction or disjunction of a line of identity inside the sep with a point upon the sep always follows the same rules as its connexion or disconnexion with a marked point inside the sep. Peirce: CP 4.501 Cross-Ref:†† In consequence of this principle, although the categorical rules hitherto given remain unchanged in their application to lines of identity, yet they require some modifications in their application to ligatures. Peirce: CP 4.502 Cross-Ref:†† 502. In order to see that the principle is correct, first consider Fig. 154. Now the rule of erasure of an unenclosed graph certainly allows the transformation of this into Fig. 155, which must therefore be interpreted to mean "Something is not ugly," and must not be confounded with Fig. 156, "Nothing is ugly." But Fig. 156 is transformable into Fig. 157; that is, the line of identity with a loose end can be carried to any vacant place within the sep. If, therefore, Fig. 155 were to be treated as if the end of the line were loose, it could be illatively transformed into Fig. 156. But the line can no more be separated from the point of the sep than it could from any marked point within the sep -- any more, for example, than

Figs. 154-157

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Fig. 158, "Nothing good is ugly" could be transformed into Fig. 159, "Either nothing is ugly or nothing is good." So Fig. 160 can, by the rule of insertion within odd seps, be transformed to Fig. 161, and must be interpreted, like that, "Everything acts on everything," and not, as in Fig. 162, "Everything acts on something or other." But if the vacant point on the sep could be treated like an ordinary point, Fig. 162 could be illatively transformed into Fig. 160, which the interpretation forbids. Although in this argument special graphs

Figs. 158-162

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are used, it is evident that the argument would be just the same whatever others were used, and the proof is just as conclusive as if we had talked of "any graph whatever, x," etc., as well as being clearer. The principle of contraposition renders

Figs. 163-177

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it evident that the same thing would hold for any finite nests of seps. Peirce: CP 4.503 Cross-Ref:†† 503. On the other hand, it is easy to show that the illative connexion or disconnexion of a line exterior to the sep with a point on the sep follows precisely the same rules as if the point were outside of and away from the sep. Peirce: CP 4.503 Cross-Ref:†† Figs. 163-177 furnish grounds for the demonstration of this. Fig. 163 asserts that there is an old king whom every wise person that knows him respects. The connexion of "is old" with "is king" can be illatively severed by the rule of erasure, as

in Fig. 164; so that the old person shall not be asserted to be identical with the king whom all wise people that know him respect; and once severed the connexion cannot be illatively restored. So it is precisely if the line of identity outside the outer sep is cut at the sep, as in Fig. 165, which asserts that somebody is respected by whatever wise person there may be that knows him, and asserts that there is an old king, but fails to assert that the old king is that respected person. Here, as before, the line can be illatively severed but cannot be illatively restored. It is evident that this is not because of the special significance of the "spots" or unanalyzed rhemata, but that it would be the same in all cases in which a line of identity should terminate at a point on a sep where a line inside that sep should also terminate. Fig. 166 shows both lines broken, so that this might equally and for the same reason result from the illative transformation of Fig. 164 or of Fig. 165. The lines, being broken as in Fig. 166, can be distorted in any way and their extremities can be carried to any otherwise vacant places outside the outer sep, and afterwards can be brought back to their present places. In this respect, a vacant point on a sep is just like any other vacant point outside the close of the sep. If the line of identity attached to "is old" be carried to the sep, as in Fig. 167, certainly no addition is thereby made to the assertion. Once the ligature is carried as far as the sep, the rule of insertion within an odd number of seps permits it to be carried still further, as is done in Fig. 167, with the ligature attached to "is a king." This whole graph may be interpreted, "Something is old and something is a king." But this last does not exist unless something is respected by whatever that is wise there may be that knows it. The graph of Fig. 167 can be illatively retransformed into Fig. 166, by first severing the ligature attached to "is a king" outside the sep by the rule of erasure, when the part of the ligature inside may be erased by the rule of deiteration, and finally the part outside the close of the sep may be erased by the rule of erasure. On the other hand the ligatures attached in Fig. 167 to "is old" and "is a king" might, after Fig. 167 had been converted in Fig. 168, be illatively joined inside the sep by the rule of insertion, as in Fig. 169, which asserts that there is something old and there is a king; and if there is an old king something is respected by whatever wise thing there may be that knows it. This is not illatively retransformable in Fig. 168. It thus abundantly shows that an unenclosed line can be extended to a point on an unenclosed sep under the same conditions as to any other unenclosed point. For there is evidently nothing peculiar about the characters of being old and of being a king which render them different in this respect from graphs in general. Let us now see how it is in regard to singly enclosed lines in their relations to points on seps in the same close. If in Fig. 163 we sever the ligature denoting the object accusative of "respects," just outside the inner sep, as in Fig. 170, the interpretation becomes, "There is an old king, and whoever that is wise there may be who knows him, respects everybody." This is illatively transformable into Fig. 163 by the rule of insertion under odd enclosures, just as if the marked point on the sep were a hook of any spot. We may, of course, by the rule of erasure within even seps, cut away the ligature from the sep internally, getting Fig. 171, "There is an old king, whom anybody that knows respects somebody or other." The point on the sep being now unmarked, it makes no difference whether the outside ligature is extended to it, as in Fig. 172, or not. It is the same if the ligature denoting the subject nominative of "respects" be broken outside the inner sep, as in Fig. 174. Whether this be done, or whether the line of identity joining "is wise" to "knows" be cut, as in Fig. 173, in either case we get a graph illatively transformable into Fig. 163, but not derivable from Fig. 163 by any illative transformation. If, however, the line of identity within the inner sep be retracted from the sep, as in Figs. 175 and 176, it makes no difference whether the line outside the sep be extended to the unmarked point on the sep or not. One cannot even say that

one form of interpretation better fits the one figure and another the other: they are absolutely equivalent. Thus, the unmarked point on the oddly enclosed sep is just like any other unmarked point exterior to the close of the sep as far as its relations with exterior lines of identity are concerned. Peirce: CP 4.504 Cross-Ref:†† 504. The principle of contraposition extends this Conditional Principle No. 4 to all seps, within any finite number of seps. Peirce: CP 4.504 Cross-Ref:†† By means of this principle the rules of illative transformation hitherto given will easily be extended so as to apply to graphs with ligatures attached to them, and the one rule which it is necessary to add to the list will also be readily deduced. In the following statement, each rule will first be enunciated in an exact and compendious form and then, if necessary, two remarks will be added, under the headings of "Note A" and "Note B." Note A will state more explicitly how the rule applies to a line of identity; while Note B will call attention to a transformation which might, without particular care, be supposed to be permitted by the rule but which is really not permitted.

Peirce: CP 4.505 Cross-Ref:†† C. Basic Categorical Rules for the Illative Transformation of All GraphsP 505. Rule 1. Called The rule of Erasure and of Insertion. In even seps, any graph-replica can be erased; in odd seps any graph-replica can be inserted. Peirce: CP 4.505 Cross-Ref:†† Note A. By even seps is meant any finite even number of seps, including none; by odd seps is meant any odd number of seps. Peirce: CP 4.505 Cross-Ref:†† This rule permits any ligature, where evenly enclosed, to be severed, and any two ligatures, oddly enclosed in the same seps, to be joined. It permits a branch with a loose end to be added to or retracted from any line of identity. Peirce: CP 4.505 Cross-Ref:†† It permits any ligature, where evenly enclosed, to be severed from the inside of the sep immediately enclosing that evenly enclosed portion of it, and to be extended to a vacant point of any sep in the same enclosure. It permits any ligature to be joined to the inside of the sep immediately enclosing that oddly enclosed portion of it, and to be retracted from the outside of any sep in the same enclosure on which the ligature has an extremity. Peirce: CP 4.505 Cross-Ref:†† Note B. In the erasure of a graph by this rule, all its ligatures must be cut. The rule does not permit a sep to be so inserted as to intersect any ligature, nor does it permit any erasure to accompany an insertion. Peirce: CP 4.505 Cross-Ref:†† It does not permit the insertion of a sep within even seps.

Peirce: CP 4.506 Cross-Ref:†† 506. Rule 2. Called The Rule of Iteration and Deiteration. Anywhere within all the seps that enclose a replica of a graph, that graph may be iterated with identical ligations, or being iterated, may be deiterated. Peirce: CP 4.506 Cross-Ref:†† Note A. The operation of iteration consists in the insertion of a new replica of a graph of which there is already a replica, the new replica having each hook ligated to every hook of a graph-replica to which the corresponding hook of the old replica is ligated, and the right to iterate includes the right to draw a new branch to each ligature of the original replica inwards to the new replica. The operation of deiteration consists in erasing a replica which might have illatively resulted from an operation of iteration, and of retracting outwards the ligatures left loose by such erasure until they are within the same seps as the corresponding ligature of the replica of which the erased replica might have been the iteration. Peirce: CP 4.506 Cross-Ref:†† The rule permits any loose end of a ligature to be extended inwards through a sep or seps or to be retracted outwards through a sep or seps. It permits any cyclical part of a ligature to be cut at its innermost part, or a cycle to be formed by joining, by inward extensions, the two loose ends that are the innermost parts of a ligature. Peirce: CP 4.506 Cross-Ref:†† If any hook of the original replica of the iterated graph is ligated to no other hook of any graph-replica, the same should be the case with the new replica. Peirce: CP 4.506 Cross-Ref:†† Note B. This rule does not confer a right to ligate any hook to another nor to deligate any hook from another unless the same hooks, or corresponding hooks of other replicas of the same graphs (these replicas being outside every sep that the hooks ligated or deligated are outside), be ligated otherwise, and outside of every sep that the new ligations or deligations are outside of. Peirce: CP 4.506 Cross-Ref:†† This rule does not confer the right to extend any ligature outwardly from within any sep, nor to retract any ligature inwardly from without any sep. Peirce: CP 4.507 Cross-Ref:†† 507. Rule 3. Called The Rule of Assertion. Any graph well-understood to be true may be scribed unenclosed. Peirce: CP 4.507 Cross-Ref:†† Note A. This rule is to be understood as permitting the explicit assertion of three classes of propositions; first, those that are involved in the conventions of this system of existential graphs; secondly, any propositions known to be true but which may not have been thought of as pertinent when the graph was first scribed or as pertinent in the way in which it is now seen to be pertinent (that is to say, premisses may be added if they are acknowledged to be true); thirdly, any propositions which the scription of the graph renders true or shows to be true. Thus, having graphically asserted that it snows, we may insert a graph asserting "that it snows is asserted" or "it is possible to assert that it snows without asserting that it is winter." Peirce: CP 4.508 Cross-Ref:††

508. Rule 4. Called The Rule of Biclosure. Two seps, one within the other, with nothing between them whose significance is affected by seps, may be withdrawn from about the graph they doubly enclose. Peirce: CP 4.508 Cross-Ref:†† Note A. The significance of a ligature is not affected by a sep except at its outermost part, or if it passes through the close of the sep; and therefore ligatures passing from outside the outer sep to inside the inner one will not prevent the withdrawal of the double sep; and such ligatures will remain unaffected by the withdrawal. Peirce: CP 4.508 Cross-Ref:†† Note B. A ligature passing twice through the outer sep without passing through the inner one, or passing from within the inner one into the intermediate space and stopping there, will be equivalent to a graph and will preclude the withdrawal. Peirce: CP 4.509 Cross-Ref:†† 509. Rule 5. Called The Rule of Deformation. All parts of the graph may be deformed in any way, the connexions of parts remaining unaltered; and the extension of a line of identity outside a sep to an otherwise vacant point on that sep is not to be considered to be a connexion.

Peirce: CP 4.510 Cross-Ref:†† CHAPTER 5

THE GAMMA PART OF EXISTENTIAL GRAPHS†1

510. The alpha part of graphs . . . is able to represent no reasonings except those which turn upon the logical relations of general terms. Peirce: CP 4.511 Cross-Ref:†† 511. The beta part . . . is able to handle with facility and dispatch reasonings of a very intricate kind, and propositions which ordinary language can only express by means of long and confusing circumlocutions. A person who has learned to think in beta graphs has ideas of the utmost clearness and precision which it is practically impossible to communicate to the mind of a person who has not that advantage. Its reasonings generally turn upon the properties of the relations of individual objects to one another. Peirce: CP 4.511 Cross-Ref:†† But it is able to do nothing at all with many ideas which we are all perfectly familiar with. Generally speaking it is unable to reason about abstractions. It cannot reason for example about qualities nor about relations as subjects to be reasoned about. It cannot reason about ideas. It is to supply that defect that the gamma part of the subject has been invented. But this gamma part is still in its infancy. It will be many years before my successors will be able to bring it to the perfection to which the

alpha and beta parts have been brought. For logical investigation is very slow, involving as it does the taking up of a confused mass of ordinary ideas, embracing we know not what and going through with a great quantity of analyses and generalizations and experiments before one can so much as get a new branch fairly inaugurated. . . . Peirce: CP 4.512 Cross-Ref:†† 512. The gamma part of graphs, in its present condition, is characterized by a great wealth of new signs; but it has no sign of an essentially different kind from those of the alpha and beta part. The alpha part has three distinct kinds of signs, the graphs, the sheet of assertion, and the cuts. The beta part adds two quite different kinds of signs, spots, or lexeis, and ligatures with selectives. It is true that a line of identity is a graph; but the terminal of such a line, especially a terminal on a cut where two lines of identity have a common point, is radically different. So far, all the gamma signs that have presented themselves, are of those same kinds. If anybody in my lifetime shall discover any radically disparate kind of sign, peculiar to the gamma part of the system, I shall hail him as a new Columbus. He must be a mind of vast power. But in the gamma part of the subject all the old kinds of signs take new forms. . . . Thus in place of a sheet of assertion, we have a book of separate sheets, tacked together at points, if not otherwise connected. For our alpha sheet, as a whole, represents simply a universe of existent individuals, and the different parts of the sheet represent facts or true assertions made concerning that universe. At the cuts we pass into other areas, areas of conceived propositions which are not realized. In these areas there may be cuts where we pass into worlds which, in the imaginary worlds of the outer cuts, are themselves represented to be imaginary and false, but which may, for all that, be true, and therefore continuous with the sheet of assertion itself, although this is uncertain. You may regard the ordinary blank sheet of assertion as a film upon which there is, as it were, an undeveloped photograph of the facts in the universe. I do not mean a literal picture, because its elements are propositions, and the meaning of a proposition is abstract and altogether of a different nature from a picture. But I ask you to imagine all the true propositions to have been formulated; and since facts blend into one another, it can only be in a continuum that we can conceive this to be done. This continuum must clearly have more dimensions than a surface or even than a solid; and we will suppose it to be plastic, so that it can be deformed in all sorts of ways without the continuity and connection of parts being ever ruptured. Of this continuum the blank sheet of assertion may be imagined to be a photograph. When we find out that a proposition is true, we can place it wherever we please on the sheet, because we can imagine the original continuum, which is plastic, to be so deformed as to bring any number of propositions to any places on the sheet we may choose. Peirce: CP 4.513 Cross-Ref:†† 513. So far I have called the sheet a photograph, so as not to overwhelm you with all the difficulties of the conception at once. But let us rather call it a map -- a map of such a photograph if you like. A map of the simplest kind represents all the points of one surface by corresponding points of another surface in such a manner as to preserve the continuity unbroken, however great may be the distortion. A Mercator's chart, however, represents all the surface of the earth by a strip, infinitely long, both north and south poles being at infinite distances, so that places near the poles are magnified so as to be many times larger than the real surfaces of the earth that they represent, while in longitude the whole equator measures only two or three feet; and you might continue the chart so as to represent the earth over and over again

in as many such strips as you pleased. Other kinds of map, such as my Quincuncial Projection which is drawn in the fourth volume of the American Journal of Mathematics,†1 show the whole earth over and over again in checkers, and there is no arrangement you can think of in which the different representations of the same place might not appear on a perfectly correct map. This accounts for our being able to scribe the same graph as many times as we please on any vacant places we like. Now each of the areas of any cut corresponds exactly to some locus of the sheet of assertion where there is mapped, though undeveloped, the real state of things which the graph of that area denies. In fact it is represented by that line of the sheet of assertion which the cut itself marks. Peirce: CP 4.514 Cross-Ref:†† 514. By taking time enough I could develop this idea much further, and render it clearer; but it would not be worth while, for I only mention it to prepare you for the idea of quite different kinds of sheets in the gamma part of the system. These sheets represent altogether different universes with which our discourse has to do. In the Johns Hopkins Studies in Logic†2 -- I printed a note of several pages on the universe of qualities -- marks, as I then called them. But I failed to see that I was then wandering quite beyond the bounds of the logic of relations proper. For the relations of which the so-called "logic of relatives" treats are existential relations, which the nonexistence of either relate or correlate reduces to nullity. Now, qualities are not, properly speaking, individuals. All the qualities you actually have ever thought of might, no doubt, be counted, since you have only been alive for a certain number of hundredths of seconds, and it requires more than a hundredth of a second actually to have any thought. But all the qualities, any one of which you readily can think of, are certainly innumerable; and all that might be thought of exceed, I am convinced, all multitude whatsoever. For they are mere logical possibilities, and possibilities are general, and no multitude can exhaust the narrowest kind of a general. Nevertheless, within limitations, which include most ordinary purposes, qualities may be treated as individuals. At any rate, however, they form an entirely different universe of existence. It is a universe of logical possibility. As we have seen, although the universe of existential fact can only be conceived as mapped upon a surface by each point of the surface representing a vast expanse of fact, yet we can conceive the facts [as] sufficiently separated upon the map for all our purposes; and in the same sense the entire universe of logical possibilities might be conceived to be mapped upon a surface. Nevertheless, in order to represent to our minds the relation between the universe of possibilities and the universe of actual existent facts, if we are going to think of the latter as a surface, we must think of the former as three-dimensional space in which any surface would represent all the facts that might exist in one existential universe. In endeavoring to begin the construction of the gamma part of the system of existential graphs, what I had to do was to select, from the enormous mass of ideas thus suggested, a small number convenient to work with. It did not seem to be convenient to use more than one actual sheet at one time; but it seemed that various different kinds of cuts would be wanted. Peirce: CP 4.515 Cross-Ref:†† 515. I will begin with one of the gamma cuts. I call it the broken cut. I scribe it thus

Fig. 178

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This does not assert that it does not rain. It only asserts that the alpha and beta rules do not compel me to admit that it rains, or what comes to the same thing, a person altogether ignorant, except that he was well versed in logic so far as it embodied in the alpha and beta parts of existential graphs, would not know that it rained.†1 Peirce: CP 4.516 Cross-Ref:†† 516. The rules of this cut are very similar to those of the alpha cut. Peirce: CP 4.516 Cross-Ref:†† Rule 1. In a broken cut already on the sheet of assertion any graph may be inserted. Peirce: CP 4.516 Cross-Ref:†† Rule 2. An evenly enclosed alpha cut may be half erased so as to convert it into a broken cut, and an oddly enclosed broken cut may be filled up to make an alpha cut. Whether the enclosures are by alpha or broken cuts is indifferent. Consequently

Fig. 179

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will mean that the graph g is beta-necessarily true.†2 By Rule 2, this is converted into

Fig. 180

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which is equivalent to g Fig. 181

the simple assertion of g. By the same rule Fig. 180 is transformable into

Fig. 182

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which means that the beta rules do not make g false.†1 That is g is beta-possible.†2 Peirce: CP 4.516 Cross-Ref:††

So if we start from

Fig. 183

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which denies the last figure and thus asserts that it is beta-impossible that g should be true,†3 Rule 2 gives

Fig. 184

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185

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the simple denial of g.†4 And from this we get again

Fig. 186

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Peirce: CP 4.517 Cross-Ref:†† 517. It must be remembered that possibility and necessity are relative to the state of information. Peirce: CP 4.517 Cross-Ref:†† Of a certain graph g let us suppose that I am in such a state of information that it may be true and may be false; that is I can scribe on the sheet of assertion Figs. 182 and 186. Now I learn that it is true. This gives me a right to scribe on the sheet Figs. 182, 186 and 181. But now relative to this new state of information, Fig. 186 ceases to be true; and therefore relatively to the new state of information we can scribe Fig. 179.†6 Peirce: CP 4.518 Cross-Ref:†† 518. You thus perceive that we should fall into inextricable confusion in dealing with the broken cut if we did not attach to it a sign to distinguish the particular state of information to which it refers. And a similar sign has then to be attached to the simple g, which refers to the state of information at the time of learning that graph to be true. I use for this purpose cross marks below, thus:

Fig. 187

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These selectives are very peculiar in that they refer to states of information as if they were individual objects. They have, besides, the additional peculiarity of having a definite order of succession, and we have the rule that from Fig. 188 we can infer Fig. 189.†1

Figs. 188-189

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These signs are of great use in cleaning up the confused doctrine of modal propositions as well as the subject of logical breadth and depth. Peirce: CP 4.519 Cross-Ref:†† 519. There is not much utility in a double broken cut. Yet it may be worth notice that Fig. 181 and

Fig. 190

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can neither of them be inferred from the other. The outer of the two broken cuts is not only relative to a state of information but to a state of reflection. The graph [190] asserts that it is possible that the truth of the graph g is necessary. It is only because I have not sufficiently reflected upon the subject that I can have any doubt of whether it is so or not. Peirce: CP 4.520 Cross-Ref:†† 520. It becomes evident, in this way, that a modal proposition is a simple assertion, not about the universe of things, but about the universe of facts that one is in a state of information sufficient to know. [Fig. 186] without any selective, merely asserts that there is a possible state of information in which the knower is not in a condition to know that the graph g is true, while Fig. 179 asserts that there is no such possible state of information. Suppose, however, we wish to assert that there is a conceivable state of information of which it would not be true that, in that state, the knower would not be in condition to know that g is true. We shall naturally express

this by Fig. 191. that there is a conceivable state

[Click here to view] But this is to say

of information in which the knower would know Fig. 191 that g is true. [This is expressed by] Fig. 188. Peirce: CP 4.521 Cross-Ref:†† 521. Now suppose we wish to assert that there is a conceivable state of information in which the knower would know g to be true and yet would not know another graph h to be true. We shall naturally express this by Fig. 192.

Fig. 192 view]

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Here we have a new kind of ligature, which will follow all the rules of ligatures. We have here a most important addition to the system of graphs. There will be some peculiar and interesting little rules, owing to the fact that what one knows, one has the means of knowing that one knows -- which is sometimes incorrectly stated in the form that whatever one knows, one knows that one knows, which is manifestly false. For if it were the same to say "A whale is not a fish" and "I know that a whale is not a fish," the precise denials of the two would be the same. Yet one is "A whale is a fish" and the other is "I do not know that a whale is not a fish." Peirce: CP 4.522 Cross-Ref:†† 522. The truth is that it is necessary to have a graph to signify that one state of

information follows after another. If we scribe, [Click here to view] to express that the state of information B follows after the state of information A, we shall have

Fig. 193 here to view]

[Click

Peirce: CP 4.523 Cross-Ref:†† 523. It is clear, however, that the matter must not be allowed to rest here. For it would be a strangely, and almost an ironically, imperfect kind of logic which should recognize only ignorance and should ignore error. Yet in order to recognize error in our system of graphs, we shall be obliged still further to introduce the idea of time, which will bring still greater difficulties. Time has usually been considered by logicians to be what is called "extra-logical" matter. I have never shared this opinion.†1 But I have thought that logic had not reached that state of development at which the introduction of temporal modifications of its forms would not result in great confusion; and I am much of that way of thinking yet. The idea of time really is involved in the very idea of an argument. But the gravest complications of logic would be involved, [if we took] account of time [so as] to distinguish between what one knows and what one has sufficient reason to be entirely confident of. The only difference, that there seems to be room for between these two, is that what one knows, one always will have reason to be confident of, while what one now has ample reason to be entirely confident of, one may conceivably in the future, in consequence of a new light, find reason to doubt and ultimately to deny. Whether it is really possible for this to occur, whether we can be said truly to have sufficient reason for entire confidence unless it is manifestly impossible that we should have any such new light in the future, is not the question. Be that as it may, it still remains conceivable that there should be that difference, and therefore there is a difference in the meanings of the two phrases. I confess that my studies heretofore have [not] progressed so far that I am able to say precisely what modification of our logical forms will be required when we come to take account, as some day we must, of all the effects of the possibilities of error, as we can now take account, in the doctrine of modals, of the possibilities of ignorance. Nor do I believe that the time has yet come when it would be profitable to introduce such complications. But I can see that, when that time does come, our logical forms will become very much more metamorphosed, by introducing that consideration, than they are in modal logic, where we take account of the possibility of ignorance as compared with the simple logic of propositions de inesse (as non-modal propositions, in which the ideas of possibility and necessity are not introduced, are called) . . . Peirce: CP 4.524 Cross-Ref:†† 524. I introduce certain spots which I term Potentials. They are shown on this diagram:

The Potentials A-p

means A is a primary individual

A-q

means A is a monadic character or "quality"

A-r

means A is a dyadic relation

A-s

means A is a legisign

A - /0\

means A is a graph

A - /1\ - B means B possesses the quality A

A - /2\ / B means B is in the relation A to C \C

/B / A - /3\ -- C means B is in the triadic relation A to C for D. \ \D

See image:

[Click here to view] Peirce: CP 4.525 Cross-Ref:†† 525. It is obvious that the lines of identity on the left-hand side of the potentials are quite peculiar, since the characters they denote are not, properly speaking, individuals. For that reason and others, to the left of the potentials I use

selectives not ligatures. Peirce: CP 4.526 Cross-Ref:†† 526. As an example of the use of the potentials, we may take this graph, which expresses a theorem of great importance: The proposition is that for every quality Q whatsoever, there is a dyadic relation, R, such that, taking any two different individuals both possessing this quality, Q, either the first stands in the relation R to some thing to which the second does not stand in that relation, while there is nothing to which the second stands in that relation without the first standing in the same relation to it; or else it is just the other way, namely that the second stands in the relation, R, to which the first does not stand in that relation, while there is nothing to which the first stands in that relation, R, without the second also standing in the same relation to it. The proof of this, which is a little too intricate to be followed in an oral statement

Fig. 194 [Click here to view]

(although in another lecture †1 I shall substantially prove it) depends upon the fact that a relation is in itself a mere logical possibility. Peirce: CP 4.527 Cross-Ref:†† 527. I will now pass to another quite indispensable department of the gamma graphs. Namely, it is necessary that we should be able to reason in graphs about graphs. The reason is that a reasoning about graphs will necessarily consist in showing that something is true of every possible graph of a certain general description. But we cannot scribe every possible graph of any general description, and therefore if we are to reason in graphs we must have a graph which is a general description of the kind of graph to which the reasoning is to relate. Peirce: CP 4.528 Cross-Ref:††

528. For the alpha graphs, it is easy to see what is wanted. Let [Click here to view], the old Greek form of the letter A, denote the sheet of assertion.

Let ~{g} be "is a graph." Let Y [Click here to view]X mean that X is scribed or placed on Y. Let W - k - Z mean that Z is the area of

the cut W. Let U [Click here to view] mean that U is a graph, precisely expressing V. It is necessary to place V in the saw-rim, as I call the line about it, because in thus speaking of a sign materialiter, as they said in the middle ages, we require that it should have a hook that it has not got. For example

Fig. 195

[Click here to view]

asserts, of course, that if it hails, it is cold de inesse. Now a graph asserting that this graph is scribed on the sheet of assertion, will be

Fig. 196

[Click here to view]

This graph only asserts what the other does assert. It does not say what the other does not assert. But there would be no difficulty in expressing that. We have only to place

instead of

[Click here to view], wherever it occurs,

[Click here to view] Peirce: CP 4.529 Cross-Ref:†† 529. We come now to the graphical expressions of beta graphs. Here we require the following symbols, Gamma Expressions of Beta Graphs

[Click here to view]means Y is a ligature whose outermost part is on X.

[Click here to view]means g is expressed by a monad spot on X whose hook is joined to the ligature Y on X.

[Click here to view]means g is expressed by a dyad graph on X whose first and second hooks respectively are joined on X to the ligatures Y and Z.

[Click here to view]means g is expressed by a triad graph on X whose first, second, and third hooks are joined on X to the ligatures Y, Z, W, respectively.

[Click here to view]means g is expressed by a tetrad spot on X whose first to fourth hooks are joined to Y, Z, U, V, respectively.†1

Peirce: CP 4.530 Cross-Ref:†† CHAPTER 6

PROLEGOMENA TO AN APOLOGY OR PRAGMATICISM†1

§1. SIGNSE†2

530. Come on, my Reader, and let us construct a diagram to illustrate the general course of thought; I mean a System of diagrammatization by means of which any course of thought can be represented with exactitude. Peirce: CP 4.530 Cross-Ref:†† "But why do that, when the thought itself is present to us?" Such, substantially, has been the interrogative objection raised by more than one or two superior intelligences, among whom I single out an eminent and glorious General. Peirce: CP 4.530 Cross-Ref:†† Recluse that I am, I was not ready with the counter-question, which should have run, "General, you make use of maps during a campaign, I believe. But why should you do so, when the country they represent is right there?" Thereupon, had he replied that he found details in the maps that were so far from being "right there," that they were within the enemy's lines, I ought to have pressed the question, "Am I right, then, in understanding that, if you were thoroughly and perfectly familiar with the country, as, for example, if it lay just about the scenes of your childhood, no map of it would then be of the smallest use to you in laying out your detailed plans?" To that he could only have rejoined, "No, I do not say that, since I might probably desire the maps to stick pins into, so as to mark each anticipated day's change in the situations of the two armies." To that again, my sur-rejoinder should have been, "Well, General,

that precisely corresponds to the advantages of a diagram of the course of a discussion. Indeed, just there, where you have so clearly pointed it out, lies the advantage of diagrams in general. Namely, if I may try to state the matter after you, one can make exact experiments upon uniform diagrams; and when one does so, one must keep a bright lookout for unintended and unexpected changes thereby brought about in the relations of different significant parts of the diagram to one another. Such operations upon diagrams, whether external or imaginary, take the place of the experiments upon real things that one performs in chemical and physical research. Chemists have ere now, I need not say, described experimentation as the putting of questions to Nature. Just so, experiments upon diagrams are questions put to the Nature of the relations concerned." The General would here, may be, have suggested (if I may emulate illustrious warriors in reviewing my encounters in afterthought), that there is a good deal of difference between experiments like the chemist's, which are trials made upon the very substance whose behavior is in question, and experiments made upon diagrams, these latter having no physical connection with the things they represent. The proper response to that, and the only proper one, making a point that a novice in logic would be apt to miss, would be this: "You are entirely right in saying that the chemist experiments upon the very object of investigation, albeit, after the experiment is made, the particular sample he operated upon could very well be thrown away, as having no further interest. For it was not the particular sample that the chemist was investigating; it was the molecular structure. Now he was long ago in possession of overwhelming proof that all samples of the same molecular structure react chemically in exactly the same way; so that one sample is all one with another. But the object of the chemist's research, that upon which he experiments, and to which the question he puts to Nature relates, is the Molecular Structure, which in all his samples has as complete an identity as it is in the nature of Molecular Structure ever to possess. Accordingly, he does, as you say, experiment upon the Very Object under investigation. But if you stop a moment to consider it, you will acknowledge, I think, that you slipped in implying that it is otherwise with experiments made upon diagrams. For what is there the Object of Investigation? It is the form of a relation. Now this Form of Relation is the very form of the relation between the two corresponding parts of the diagram. For example, let f[1] and f[2] be the two distances of the two foci of a lens from the lens. Then, 1/f[1] + 1/f[2] = 1/f[o] Peirce: CP 4.530 Cross-Ref:†† This equation is a diagram of the form of the relation between the two focal distances and the principal focal distance; and the conventions of algebra (and all diagrams, nay all pictures, depend upon conventions) in conjunction with the writing of the equation, establish a relation between the very letters f[1], f[2], f[o] regardless of their significance, the form of which relation is the Very Same as the form of the relation between the three focal distances that these letters denote. This is a truth quite beyond dispute. Thus, this algebraic Diagram presents to our observation the very, identical object of mathematical research, that is, the Form of the harmonic mean, which the equation aids one to study. (But do not let me be understood as saying that a Form possesses, itself, Identity in the strict sense; that is, what the logicians, translating {arithmö}, call 'numerical identity.')" Peirce: CP 4.531 Cross-Ref:†† 531. Not only is it true that by experimentation upon some diagram an experimental proof can be obtained of every necessary conclusion from any given

Copulate of Premisses, but, what is more, no "necessary" conclusion is any more apodictic than inductive reasoning becomes from the moment when experimentation can be multiplied ad libitum at no more cost than a summons before the imagination. I might furnish a regular proof of this, and am dissuaded from doing so now and here only by the exigency of space, the ineluctable length of the requisite explanations, and particularly by the present disposition of logicians to accept as sufficient F. A. Lange's persuasive and brilliant, albeit defective and in parts even erroneous, apology for it.†1 Under these circumstances, I will content myself with a rapid sketch of my proof. First, an analysis of the essence of a sign, (stretching that word to its widest limits, as anything which, being determined by an object, determines an interpretation to determination, through it, by the same object), leads to a proof that every sign is determined by its object, either first, by partaking in the characters of the object, when I call the sign an Icon; secondly, by being really and in its individual existence connected with the individual object, when I call the sign an Index; thirdly, by more or less approximate certainty that it will be interpreted as denoting the object, in consequence of a habit (which term I use as including a natural disposition), when I call the sign a Symbol.†P1 I next examine into the different efficiencies and inefficiencies of these three kinds of signs in aiding the ascertainment of truth. A Symbol incorporates a habit, and is indispensable to the application of any intellectual habit, at least. Moreover, Symbols afford the means of thinking about thoughts in ways in which we could not otherwise think of them. They enable us, for example, to create Abstractions, without which we should lack a great engine of discovery. These enable us to count; they teach us that collections are individuals (individual = individual object), and in many respects they are the very warp of reason. But since symbols rest exclusively on habits already definitely formed but not furnishing any observation even of themselves, and since knowledge is habit, they do not enable us to add to our knowledge even so much as a necessary consequent, unless by means of a definite preformed habit. Indices, on the other hand, furnish positive assurance of the reality and the nearness of their Objects. But with the assurance there goes no insight into the nature of those Objects. The same Perceptible may, however, function doubly as a Sign. That footprint that Robinson Crusoe found in the sand, and which has been stamped in the granite of fame, was an Index to him that some creature was on his island, and at the same time, as a Symbol, called up the idea of a man. Each Icon partakes of some more or less overt character of its Object. They, one and all, partake of the most overt character of all lies and deceptions -- their Overtness. Yet they have more to do with the living character of truth than have either Symbols or Indices. The Icon does not stand unequivocally for this or that existing thing, as the Index does. Its Object may be a pure fiction, as to its existence. Much less is its Object necessarily a thing of a sort habitually met with. But there is one assurance that the Icon does afford in the highest degree. Namely, that which is displayed before the mind's gaze -- the Form of the Icon, which is also its object -must be logically possible. This division of Signs is only one of ten different divisions of Signs which I have found it necessary more especially to study.†1 I do not say that they are all satisfactorily definite in my mind. They seem to be all trichotomies, which form an attribute to the essentially triadic nature of a Sign. I mean because three things are concerned in the functioning of a Sign; the Sign itself, its Object, and its Interpretant. I cannot discuss all these divisions in this article; and it can well be believed that the whole nature of reasoning cannot be fully exposed from the consideration of one point of view among ten. That which we can learn from this division is of what sort a Sign must be to represent the sort of Object that reasoning is concerned with. Now reasoning has to make its conclusion manifest. Therefore, it

must be chiefly concerned with forms, which are the chief objects of rational insight. Accordingly, Icons are specially requisite for reasoning. A Diagram is mainly an Icon, and an Icon of intelligible relations. It is true that what must be is not to be learned by simple inspection of anything. But when we talk of deductive reasoning being necessary, we do not mean, of course, that it is infallible. But precisely what we do mean is that the conclusion follows from the form of the relations set forth in the premiss. Now since a diagram, though it will ordinarily have Symbolide Features, as well as features approaching the nature of Indices, is nevertheless in the main an Icon of the forms of relations in the constitution of its Object, the appropriateness of it for the representation of necessary inference is easily seen.

Peirce: CP 4.532 Cross-Ref:†† §2. COLLECTIONSE

532. But since you may, perhaps, be puzzled to understand how an Icon can exhibit a necessity -- a Must-be -- I will here give, as an example of its doing so, my proof †2 that the single members of no collection or plural, are as many as are the collections it includes, each reckoned as a single object, or, in other words, that there can be no relation in which every collection composed of members of a given collection should (taken collectively as a single object) stand to some member of the latter collection to which no other such included collection so stands. This is another expression of the following proposition, namely: that, taking any collection or plural, whatsoever, be it finite or infinite, and calling this the given collection; and considering all the collections, or plurals, each of which is composed of some of the individual members of the given collection (but including along with these Nothing which is to be here regarded as a collection having no members at all; and also including the single members of the given collection, conceived as so many collections each of a single member), and calling these the involved collections; the proposition is that there is no possible relation in which each involved collection (considered as a single object), stands to a member of the given collection, without any other of the involved collections standing in the same relation to that same member of the given collection. This purely symbolic statement can be rendered much more perspicuous by the introduction of Indices, as follows. The proposition is that no matter what collection C may be, and no matter what relation R may be, there must be some collection, c', composed exclusively of members of C, which does not stand in the relation R to any member, k, of C, unless some other collection, c'', likewise composed of members of C, stands in the same relation R to the same k. The theorem is important in the doctrine of multitude, since it is the same as to say that any collection, no matter how great, is less multitudinous than the collection of possible collections composed exclusively of members of it; although formerly this was assumed to be false of some infinite collections. The demonstration begins by insisting that, if the proposition be false, there must be some definite relation of which it is false. Assume, then, that the letter R is an index of any one such relation you please. Next divide the members of C into four classes as follows: Peirce: CP 4.532 Cross-Ref:†† Class I is to consist of all those members of C (if there be any such) to each of which no collection of members of C stands in the relation R.

Peirce: CP 4.532 Cross-Ref:†† Class II is to consist of all those members of C to each of which one and only one collection of members of C stands in the relation R; and this class has two subclasses, as follows: Sub-Class 1 is to consist of whatever members of Class II there may be, each of which is contained in that one collection of members of C that is in the relation R to it. Sub-Class 2 is to consist of whatever members of Class II there may be, none of which is contained in that one collection of members of C that is in the relation R to it. Peirce: CP 4.532 Cross-Ref:†† Class III is to consist of all those members of C, if there be any such, to each of which more than one collection of members of C are in the relation R. Peirce: CP 4.532 Cross-Ref:†† This division is complete; but everybody would consider the easy diagrammatical proof that it is so as needless to the point of nonsense, implicitly relying on a Symbol in his memory which assures him that every Division of such construction is complete. Peirce: CP 4.532 Cross-Ref:†† I ought already to have mentioned that, throughout the enunciation and demonstration of the proposition to be proved, the term "collection included in the given collection" is to be taken in a peculiar sense to be presently defined. It follows that there is one "possible collection" that is included in every other, that is, which excludes whatever any other excludes. Namely, this is the "possible collection" which includes only the Sphinxes, which is the same that includes only the Basilisks, and is identical with the "possible collection" of all the Centaurs, the unique and ubiquitous collection called "Nothing," which has no member at all. If you object to this use of the term "collection," you will please substitute for it, throughout the enunciation and the demonstration, any other designation of the same object. I prefix the adjective "possible," though I must confess it does not express my meaning, merely to indicate that I extend the term "collection" to Nothing, which, of course, has no existence. Were the suggested objection to be persisted in by those soi-disant reasoners who refuse to think at all about the object of this or that description, on the ground that it is "inconceivable," I should not stop to ask them how they could say that, when that involves thinking of it in the very same breath, but should simply say that for them it would be necessary to except collections consisting of single individuals. Some of these mighty intellects refuse to allow the use of any name to denote single individuals, and also plural collections along with them; and for them the proposition ceases to be true of pairs. If they would not allow pairs to be denoted by any term that included all higher collections, the proposition would cease to be true of triplets and so on. In short, by restricting the meaning of "possible collection," the proposition may be rendered false of small collections. No general formal restriction can render it false of greater collections. Peirce: CP 4.532 Cross-Ref:†† I shall now assume that you will permit me to use the term "possible collection" according to the following definition. A "possible collection" is an ens

rationis of such a nature that the definite plural of any noun, or possible noun of definite signification, (as "the A's," "the B's," etc.) denotes one, and only one, "possible collection" in any one perfectly definite state of the universe; and there is a certain relation between some "possible collections," expressed by saying that one "possible collection" includes another (or the same) "possible collection," and if, and only if, of two nouns one is universally and affirmatively predicable of the other in any one perfectly definite state of the universe, then the "possible collection" denoted by the definite plural of the former includes whatever "possible collection" is included by the "possible collection" denoted by the definite plural of the latter, and of any two different "possible collections," one or other must include something not included by the other. Peirce: CP 4.532 Cross-Ref:†† A diagram of the definition of "possible collection" being compared with a diagram embracing whatever members of subclasses 1 and 2 that it may, excluding all the rest, will now assure us that any such aggregate is a possible collection of members of the class C, no matter what individuals of Classes I and III be included or excluded in the aggregate along with those members of Class II, if any there be in the aggregate. Peirce: CP 4.532 Cross-Ref:†† We shall select, then, a single possible collection of members of C to which we give the proper name, c, and this possible collection shall be one which contains no individual of Subclass 1, but contains whatever individual there may be of Subclass 2. We then ask whether or not it is true that c stands in the relation R to a member of C to which no other possible collection of members of C stands in the same relation; or, to put this question into a more convenient shape, we ask, Is there any member of the Class C to which c and no other possible collection of members of C stands in the relation R? If there be such a member or members of C, let us give one of them the proper name T. Then T must belong to one of our four divisions of this class. That is, either T belongs to Class I (but that cannot be, since by the definition of Class I, to no member of this class is any possible collection of members of C in the relation R); or T belongs to Subclass 1 (but that cannot be, since by the definition of that subclass, every member of it is a member of the only possible collection of members of C that is R to it, which possible collection cannot be c, because c is only known to us by a description which forbids its containing any member of Subclass 1. Now it is c, and c only, that is in the relation R to T); or T belongs to Subclass 2 (but that cannot be, since by the definition of that subclass, no member of it is a member of the only possible collection of members of C that is R to it, which possible collection cannot be c, because the description by which alone c can be recognized makes it contain every member of Subclass 2. Now it is c only that is in the relation R to T); or T belongs to Class III (but this cannot be, since to every member of that class, by the definition of it, more than one collection of members of C stand in the relation R, while to T only one collection, namely, c, stands in that relation). Peirce: CP 4.532 Cross-Ref:††

Thus, T belongs to none of the classes of members of C, and consequently is not a member of C. Consequently, there is no such member of C; that is, no member of C to which c, and no other possible collection of members of C, stands in the relation R. But c is the proper name we were at liberty to give to whatever possible collection of members of C we pleased. Hence, there is no possible collection of members of C that stands in the relation R to a member of the class C to which no other possible collection of members of C stands in this relation R. But R is the name of any relation we please, and C is any class we please. It is, therefore, proved that no matter what class be chosen, or what relation be chosen, there will be some possible collection of members of that class (in the sense in which Nothing is such a collection) which does not stand in that relation to any member of that class to which no other such possible collection stands in the same relation.

Peirce: CP 4.533 Cross-Ref:†† §3. GRAPHS AND SIGNSE

533. When I was a boy, my logical bent caused me to take pleasure in tracing out upon a map of an imaginary labyrinth one path after another in hopes of finding my way to a central compartment. The operation we have just gone through is essentially of the same sort, and if we are to recognize the one as essentially performed by experimentation upon a diagram, so must we recognize that the other is performed. The demonstration just traced out brings home to us very strongly, also, the convenience of so constructing our diagram as to afford a clear view of the mode of connection of its parts, and of its composition at each stage of our operations upon it. Such convenience is obtained in the diagrams of algebra. In logic, however, the desirability of convenience in threading our way through complications is much less than in mathematics, while there is another desideratum which the mathematician as such does not feel. The mathematician wants to reach the conclusion, and his interest in the process is merely as a means to reach similar conclusions. The logician does not care what the result may be; his desire is to understand the nature of the process by which it is reached. The mathematician seeks the speediest and most abridged of secure methods; the logician wishes to make each smallest step of the process stand out distinctly, so that its nature may be understood. He wants his diagram to be, above all, as analytical as possible. Peirce: CP 4.534 Cross-Ref:†† 534. In view of this, I beg leave, Reader, as an Introduction to my defence of pragmatism, to bring before you a very simple system of diagrammatization of propositions which I term the System of Existential Graphs. For, by means of this, I shall be able almost immediately to deduce some important truths of logic, little understood hitherto, and closely connected with the truth of pragmaticism;†P1 while discussions of other points of logical doctrine, which concern pragmaticism but are not directly settled by this system, are nevertheless much facilitated by reference to it. Peirce: CP 4.535 Cross-Ref:†† 535. By a graph (a word overworked of late years), I, for my part, following my friends Clifford †1 and Sylvester,†2 the introducers of the term, understand in general a diagram composed principally of spots and of lines connecting certain of the spots. But I trust it will be pardoned to me that, when I am discussing Existential

Graphs, without having the least business with other Graphs, I often omit the differentiating adjective and refer to an Existential Graph as a Graph simply. But you will ask, and I am plainly bound to say, precisely what kind of a Sign an Existential Graph, or as I abbreviate that phrase here, a Graph is. In order to answer this I must make reference to two different ways of dividing all Signs. It is no slight task, when one sets out from none too clear a notion of what a Sign is -- and you will, I am sure, Reader, have noticed that my definition of a Sign is not convincingly distinct -- to establish a single vividly distinct division of all Signs. The one division which I have already given has cost more labor than I should care to confess. But I certainly could not tell you what sort of a Sign an Existential Graph is, without reference to two other divisions of Signs. It is true that one of these involves none but the most superficial considerations, while the other, though a hundredfold more difficult, resting as it must for a clear comprehension of it upon the profoundest secrets of the structure of Signs, yet happens to be extremely familiar to every student of logic. But I must remember, Reader, that your conceptions may penetrate far deeper than mine; and it is to be devoutly hoped they may. Consequently, I ought to give such hints as I conveniently can, of my notions of the structure of Signs, even if they are not strictly needed to express my notions of Existential Graphs. Peirce: CP 4.536 Cross-Ref:†† 536. I have already noted that a Sign has an Object and an Interpretant, the latter being that which the Sign produces in the Quasi-mind that is the Interpreter by determining the latter to a feeling, to an exertion, or to a Sign, which determination is the Interpretant. But it remains to point out that there are usually two Objects, and more than two Interpretants. Namely, we have to distinguish the Immediate Object, which is the Object as the Sign itself represents it, and whose Being is thus dependent upon the Representation of it in the Sign, from the Dynamical Object, which is the Reality which by some means contrives to determine the Sign to its Representation. In regard to the Interpretant we have equally to distinguish, in the first place, the Immediate Interpretant, which is the interpretant as it is revealed in the right understanding of the Sign itself, and is ordinarily called the meaning of the sign; while in the second place, we have to take note of the Dynamical Interpretant which is the actual effect which the Sign, as a Sign, really determines. Finally there is what I provisionally term the Final Interpretant, which refers to the manner in which the Sign tends to represent itself to be related to its Object. I confess that my own conception of this third interpretant is not yet quite free from mist.†1 Of the ten divisions of signs which have seemed to me to call for my special study, six turn on the characters of an Interpretant and three on the characters of the Object.†1 Thus the division into Icons, Indices, and Symbols depends upon the different possible relations of a Sign to its Dynamical Object.†2 Only one division is concerned with the nature of the Sign itself, and this I now proceed to state. Peirce: CP 4.537 Cross-Ref:†† 537. A common mode of estimating the amount of matter in a MS. or printed book is to count the number of words.†P1 There will ordinarily be about twenty the's on a page, and of course they count as twenty words. In another sense of the word "word," however, there is but one word "the" in the English language; and it is impossible that this word should lie visibly on a page or be heard in any voice, for the reason that it is not a Single thing or Single event. It does not exist; it only determines things that do exist. Such a definitely significant Form, I propose to term a Type.†3 A Single event which happens once and whose identity is limited to that one happening or a Single object or thing which is in some single place at any one instant of time,

such event or thing being significant only as occurring just when and where it does, such as this or that word on a single line of a single page of a single copy of a book, I will venture to call a Token.†3 An indefinite significant character such as a tone of voice can neither be called a Type nor a Token. I propose to call such a Sign a Tone;†3 In order that a Type may be used, it has to be embodied in a Token which shall be a sign of the Type, and thereby of the object the Type signifies. I propose to call such a Token of a Type an Instance of the Type. Thus, there may be twenty Instances of the Type "the" on a page. The term (Existential) Graph will be taken in the sense of a Type; and the act of embodying it in a Graph-Instance will be termed scribing the Graph (not the Instance), whether the Instance be written, drawn, or incised. A mere blank place is a Graph-Instance, and the Blank per se is a Graph; but I shall ask you to assume that it has the peculiarity that it cannot be abolished from any Area on which it is scribed, as long as that Area exists. Peirce: CP 4.538 Cross-Ref:†† 538. A familiar logical triplet is Term, Proposition, Argument.†1 In order to make this a division of all signs, the first two members have to be much widened. By a Seme,†2 I shall mean anything which serves for any purpose as a substitute for an object of which it is, in some sense, a representative or Sign. The logical Term, which is a class-name, is a Seme. Thus, the term "The mortality of man" is a Seme. By a Pheme†3 I mean a Sign which is equivalent to a grammatical sentence, whether it be Interrogative, Imperative, or Assertory. In any case, such a Sign is intended to have some sort of compulsive effect on the Interpreter of it. As the third member of the triplet, I sometimes use the word Delome (pronounce deeloam, from {délöma}), though Argument would answer well enough. It is a Sign which has the Form of tending to act upon the Interpreter through his own self-control, representing a process of change in thoughts or signs, as if to induce this change in the Interpreter. Peirce: CP 4.538 Cross-Ref:†† A Graph is a Pheme, and in my use hitherto, at least, a Proposition. An Argument is represented by a series of Graphs.

Peirce: CP 4.539 Cross-Ref:†† §4. UNIVERSES AND PREDICAMENTSE

539. The Immediate Object of all knowledge and all thought is, in the last analysis, the Percept. This doctrine in no wise conflicts with Pragmaticism, which holds that the Immediate Interpretant of all thought proper is Conduct. Nothing is more indispensable to a sound epistemology than a crystal-clear discrimination between the Object and the Interpretant of knowledge; very much as nothing is more indispensable to sound notions of geography than a crystal-clear discrimination between north latitude and south latitude; and the one discrimination is not more rudimentary than the other. That we are conscious of our Percepts is a theory that seems to me to be beyond dispute; but it is not a fact of Immediate Perception. A fact of Immediate Perception is not a Percept, nor any part of a Percept; a Percept is a Seme, while a fact of Immediate Perception or rather the Perceptual Judgment of which such fact is the Immediate Interpretant, is a Pheme that is the direct Dynamical Interpretant of the Percept, and of which the Percept is the Dynamical Object, and is with some considerable difficulty (as the history of psychology shows), distinguished

from the Immediate Object, though the distinction is highly significant.†1 But not to interrupt our train of thought, let us go on to note that while the Immediate Object of a Percept is excessively vague, yet natural thought makes up for that lack (as it almost amounts to), as follows. A late Dynamical Interpretant of the whole complex of Percepts is the Seme of a Perceptual Universe that is represented in instinctive thought as determining the original Immediate Object of every Percept.†2 Of course, I must be understood as talking not psychology, but the logic of mental operations. Subsequent Interpretants furnish new Semes of Universes resulting from various adjunctions to the Perceptual Universe. They are, however, all of them, Interpretants of Percepts. Peirce: CP 4.539 Cross-Ref:†† Finally, and in particular, we get a Seme of that highest of all Universes which is regarded as the Object of every true Proposition, and which, if we name it [at] all, we call by the somewhat misleading title of "The Truth." Peirce: CP 4.540 Cross-Ref:†† 540. That said, let us go back and ask this question: How is it that the Percept, which is a Seme, has for its direct Dynamical Interpretant the Perceptual Judgment, which is a Pheme? For that is not the usual way with Semes, certainly. All the examples that happen to occur to me at this moment of such action of Semes are instances of Percepts, though doubtless there are others. Since not all Percepts act with equal energy in this way, the instances may be none the less instructive for being Percepts. However, Reader, I beg you will think this matter out for yourself, and then you can see -- I wish I could -- whether your independently formed opinion does not fall in with mine. My opinion is that a pure perceptual Icon -- and many really great psychologists have evidently thought that Perception is a passing of images before the mind's eye, much as if one were walking through a picture gallery -- could not have a Pheme for its direct Dynamical Interpretant. I desire, for more than one reason, to tell you why I think so, although that you should today appreciate my reasons seems to be out of the question. Still, I wish you to understand me so far as to know that, mistaken though I be, I am not so sunk in intellectual night as to be dealing lightly with philosophic Truth when I aver that weighty reasons have moved me to the adoption of my opinion; and I am also anxious that it should be understood that those reasons have not been psychological at all, but are purely logical. My reason, then, briefly stated and abridged, is that it would be illogical for a pure Icon to have a Pheme for its Interpretant, and I hold it to be impossible for thought not subject to self-control, as a Perceptual Judgment manifestly is not, to be illogical. I dare say this reason may excite your derision or disgust, or both; and if it does, I think none the worse of your intelligence. You probably opine, in the first place, that there is no meaning in saying that thought which draws no Conclusion is illogical, and that, at any rate, there is no standard by which I can judge whether such thought is logical or not; and in the second place, you probably think that, if self-control has any essential and important relation to logic, which I guess you either deny or strongly doubt, it can only be that it is that which makes thought logical, or else which establishes the distinction between the logical and the illogical, and that in any event it has to be such as it is, and would be logical, or illogical, or both, or neither, whatever course it should take. But though an Interpretant is not necessarily a Conclusion, yet a Conclusion is necessarily an Interpretant. So that if an Interpretant is not subject to the rules of Conclusions there is nothing monstrous in my thinking it is subject to some generalization of such rules. For any evolution of thought, whether it leads to a Conclusion or not, there is a certain normal course, which is to be determined by considerations not in the least

psychological, and which I wish to expound in my next article;†1 and while I entirely agree, in opposition to distinguished logicians, that normality can be no criterion for what I call rationalistic reasoning, such as alone is admissible in science, yet it is precisely the criterion of instinctive or common-sense reasoning, which, within its own field, is much more trustworthy than rationalistic reasoning. In my opinion, it is self-control which makes any other than the normal course of thought possible, just as nothing else makes any other than the normal course of action possible; and just as it is precisely that that gives room for an ought-to-be of conduct, I mean Morality, so it equally gives room for an ought-to-be of thought, which is Right Reason; and where there is no self-control, nothing but the normal is possible. If your reflections have led you to a different conclusion from mine, I can still hope that when you come to read my next article, in which I shall endeavor to show what the forms of thought are, in general and in some detail, you may yet find that I have not missed the truth. Peirce: CP 4.541 Cross-Ref:†† 541. But supposing that I am right, as I probably shall be in the opinions of some readers, how then is the Perceptual Judgment to be explained? In reply, I note that a Percept cannot be dismissed at will, even from memory. Much less can a person prevent himself from perceiving that which, as we say, stares him in the face. Moreover, the evidence is overwhelming that the perceiver is aware of this compulsion upon him; and if I cannot say for certain how this knowledge comes to him, it is not that I cannot conceive how it could come to him, but that, there being several ways in which this might happen, it is difficult to say which of those ways actually is followed. But that discussion belongs to psychology; and I will not enter upon it. Suffice it to say that the perceiver is aware of being compelled to perceive what he perceives. Now existence means precisely the exercise of compulsion. Consequently, whatever feature of the percept is brought into relief by some association and thus attains a logical position like that of the observational premiss of an explaining Abduction,†P1 the attribution of Existence to it in the Perceptual Judgment is virtually and in an extended sense, a logical Abductive Inference nearly approximating to necessary inference. But my next paper will throw a flood of light upon the logical affiliation of the Proposition, and the Pheme generally, to coercion. Peirce: CP 4.542 Cross-Ref:†† 542. That conception of Aristotle which is embodied for us in the cognate origin of the terms actuality and activity is one of the most deeply illuminating products of Greek thinking. Activity implies a generalization of effort; and effort is a two-sided idea, effort and resistance being inseparable, and therefore the idea of Actuality has also a dyadic form. Peirce: CP 4.543 Cross-Ref:†† 543. No cognition and no Sign is absolutely precise, not even a Percept; and indefiniteness is of two kinds, indefiniteness as to what is the Object of the Sign, and indefiniteness as to its Interpretant, or indefiniteness in Breadth and in Depth.†1 Indefiniteness in Breadth may be either Implicit or Explicit. What this means is best conveyed in an example. The word donation is indefinite as to who makes the gift, what he gives, and to whom he gives it. But it calls no attention, itself, to this indefiniteness. The word gives refers to the same sort of fact, but its meaning is such that that meaning is felt to be incomplete unless those items are, at least formally, specified; as they are in "Somebody gives something to some person (real or artificial)." An ordinary Proposition †2 ingeniously contrives to convey novel information through Signs whose significance depends entirely on the interpreter's

familiarity with them; and this it does by means of a "Predicate," i.e., a term explicitly indefinite in breadth, and defining its breadth by means of "Subjects," or terms whose breadths are somewhat definite, but whose informative depth (i.e., all the depth except an essential superficies) is indefinite, while conversely the depth of the Subjects is in a measure defined by the Predicate. A Predicate is either non-relative, or a monad, that is, is explicitly indefinite in one extensive respect, as is "black"; or it is a dyadic relative, or dyad, such as "kills," or it is a polyadic relative, such as "gives." These things must be diagrammatized in our system. Peirce: CP 4.543 Cross-Ref:†† Something more needs to be added under the same head. You will observe that under the term "Subject" I include, not only the subject nominative, but also what the grammarians call the direct and the indirect object, together, in some cases, with nouns governed by prepositions. Yet there is a sense in which we can continue to say that a Proposition has but one Subject, for example, in the proposition, "Napoleon ceded Louisiana to the United States," we may regard as the Subject the ordered triplet, "Napoleon -- Louisiana -- the United States," and as the Predicate, "has for its first member, the agent, or party of the first part, for its second member the object, and for its third member the party of the second part of one and the same act of cession." The view that there are three subjects is, however, preferable for most purposes, in view of its being so much more analytical, as will soon appear. Peirce: CP 4.544 Cross-Ref:†† 544. All general, or definable, Words, whether in the sense of Types or of Tokens, are certainly Symbols. That is to say, they denote the objects that they do by virtue only of there being a habit that associates their signification with them. As to Proper Names, there might perhaps be a difference of opinion, especially if the Tokens are meant. But they should probably be regarded as Indices, since the actual connection (as we listen to talk), of Instances of the same typical words with the same Objects, alone causes them to be interpreted as denoting those Objects. Excepting, if necessary, propositions in which all the subjects are such signs as these, no proposition can be expressed without the use of Indices.†P1 If, for example, a man remarks, "Why, it is raining!" it is only by some such circumstances as that he is now standing here looking out at a window as he speaks, which would serve as an Index (not, however, as a Symbol) that he is speaking of this place at this time, whereby we can be assured that he cannot be speaking of the weather on the satellite of Procyon, fifty centuries ago. Nor are Symbols and Indices together generally enough. The arrangement of the words in the sentence, for instance, must serve as Icons, in order that the sentence may be understood. The chief need for the Icons is in order to show the Forms of the synthesis of the elements of thought. For in precision of speech, Icons can represent nothing but Forms and Feelings. That is why Diagrams are indispensable in all Mathematics, from Vulgar Arithmetic up, and in Logic are almost so. For Reasoning, nay, Logic generally, hinges entirely on Forms. You, Reader, will not need to be told that a regularly stated Syllogism is a Diagram; and if you take at random a half dozen out of the hundred odd logicians who plume themselves upon not belonging to the sect of Formal Logic, and if from this latter sect you take another half dozen at random, you will find that in proportion as the former avoid diagrams, they utilize the syntactical Form of their sentences. No pure Icons represent anything but Forms; no pure Forms are represented by anything but Icons. As for Indices, their utility especially shines where other Signs fail. Extreme precision being desired in the description of a red color, should I call it vermillion, I may be criticized on the ground that vermillion differently prepared has quite different hues, and thus I may be driven

to the use of the color-wheel, when I shall have to Indicate four disks individually, or I may say in what proportions light of a given wave-length is to be mixed with white light to produce the color I mean. The wave-length being stated in fractions of a micron, or millionth of a meter, is referred through an Index to two lines on an individual bar in the Pavillon de Breteuil, at a given temperature and under a pressure measured against gravity at a certain station and (strictly) at a given date, while the mixture with white, after white has been fixed by an Index of an individual light, will require at least one new Index. But of superior importance in Logic is the use of Indices to denote Categories and Universes,†P1 which are classes that, being enormously large, very promiscuous, and known but in small part, cannot be satisfactorily defined, and therefore can only be denoted by Indices. Such, to give but a single instance, is the collection of all things in the Physical Universe. If anybody, your little son for example, who is such an assiduous researcher, always asking, What is the Truth ({Ti estin alétheia}); but like "jesting Pilate," will not always stay for an answer, should ask you what the Universe of things physical is, you may, if convenient, take him to the Rigi-Kulm, and about sunset, point out all that is to be seen of Mountains, Forests, Lakes, Castles, Towns, and then, as the stars come out, all there is to be seen in the heavens, and all that though not seen, is reasonably conjectured to be there; and then tell him, "Imagine that what is to be seen in a city back yard to grow to all you can see here, and then let this grow in the same proportion as many times as there are trees in sight from here, and what you would finally have would be harder to find in the Universe than the finest needle in America's yearly crop of hay." But such methods are perfectly futile: Universes cannot be described. Peirce: CP 4.545 Cross-Ref:†† 545. Oh, I overhear what you are saying, O Reader: that a Universe and a Category are not at all the same thing; a Universe being a receptacle or class of Subjects, and a Category being a mode of Predication, or class of Predicates. I never said they were the same thing; but whether you describe the two correctly is a question for careful study. Peirce: CP 4.546 Cross-Ref:†† 546. Let us begin with the question of Universes. It is rather a question of an advisable point of view than of the truth of a doctrine. A logical universe is, no doubt, a collection of logical subjects, but not necessarily of meta-physical Subjects, or "substances"; for it may be composed of characters, of elementary facts, etc. See my definition in Baldwin's Dictionary.†1 Let us first try whether we may not assume that there is but one kind of Subjects which are either existing things or else quite fictitious. Let it be asserted that there is some married woman who will commit suicide in case her husband fails in business. Surely that is a very different proposition from the assertion that some married woman will commit suicide if all married men fail in business. Yet if nothing is real but existing things, then, since in the former proposition nothing whatever is said as to what the lady will or will not do if her husband does not fail in business, and since of a given married couple this can only be false if the fact is contrary to the assertion, it follows it can only be false if the husband does fail in business and if the wife then fails to commit suicide. But the proposition only says that there is some married couple of which the wife is of that temper. Consequently, there are only two ways in which the proposition can be false, namely, first, by there not being any married couple, and secondly, by every married man failing in business while no married woman commits suicide. Consequently, all that is required to make the proposition true is that there should either be some

married man who does not fail in business, or else some married woman who commits suicide. That is, the proposition amounts merely to asserting that there is a married woman who will commit suicide if every married man fails in business. The equivalence of these two propositions is the absurd result of admitting no reality but existence. If, however, we suppose that to say that a woman will suicide if her husband fails, means that every possible course of events would either be one in which the husband would not fail or one in which the wife would commit suicide, then, to make that false it will not be requisite for the husband actually to fail, but it will suffice that there are possible circumstances under which he would fail, while yet his wife would not commit suicide. Now you will observe that there is a great difference between the two following propositions:

First, There is some one married woman who under all possible conditions would commit suicide or else her husband would not have failed.

Second, Under all possible circumstances there is some married woman or other who would commit suicide, or else her husband would not have failed.

Peirce: CP 4.546 Cross-Ref:†† The former of these is what is really meant by saying that there is some married woman who would commit suicide if her husband were to fail, while the latter is what the denial of any possible circumstances except those that really take place logically leads to [our] interpreting (or virtually interpreting), the Proposition as asserting. Peirce: CP 4.547 Cross-Ref:†† 547. In other places,†1 I have given many other reasons for my firm belief that there are real possibilities. I also think, however, that, in addition to actuality and possibility, a third mode of reality must be recognized in that which, as the gipsy fortune-tellers express it, is "sure to come true," or, as we may say, is destined,†P1 although I do not mean to assert that this is affirmation rather than the negation of this Mode of Reality. I do not see by what confusion of thought anybody can persuade himself that he does not believe that tomorrow is destined to come. The point is that it is today really true that to-morrow the sun will rise; or that, even if it does not, the clocks or something, will go on. For if it be not real it can only be fiction: a Proposition is either True or False. But we are too apt to confound destiny with the impossibility of the opposite. I see no impossibility in the sudden stoppage of everything. In order to show the difference, I remind you that "impossibility" is that which, for example, describes the mode of falsity of the idea that there should be a collection of objects so multitudinous that there would not be characters enough in the universe of characters to distinguish all those things from one another. Is there anything of that sort about the stoppage of all motion? There is, perhaps, a law of nature against it; but that is all. However, I will postpone the consideration of that point. Let us, at least, provide for such a mode of being in our system of diagrammatization, since it may turn out to be needed and, as I think, surely will. Peirce: CP 4.548 Cross-Ref:†† 548. I will proceed to explain why, although I am not prepared to deny that

every proposition can be represented, and that I must say, for the most part very conveniently, under your view that the Universes are receptacles of the Subjects alone, I, nevertheless, cannot deem that mode of analyzing propositions to be satisfactory. Peirce: CP 4.548 Cross-Ref:†† And to begin with, I trust you will all agree with me that no analysis, whether in logic, in chemistry, or in any other science, is satisfactory, unless it be thorough, that is, unless it separates the compound into components each entirely homogeneous in itself, and therefore free from the smallest admixture of any of the others. It follows that in the Proposition, "Some Jew is shrewd," the Predicate is "Jew-that-is-shrewd," and the Subject is Something, while in the proposition "Every Christian is meek," the Predicate is "Either not Christian or else meek," while the Subject is Anything; unless, indeed, we find reason to prefer to say that this Proposition means, "It is false to say that a person is Christian of whom it is false to say that he is meek." In this last mode of analysis, when a Singular Subject is not in question (which case will be examined later), the only Subject is Something. Either of these two modes of analysis [differentiates] quite [clearly] the Subject from any Predicative ingredients; and at first sight, either seems quite favorable to the view that it is only the Subjects which belong to the Universes. Let us, however, consider the following two forms of propositions:

A †1 Any adept alchemist could produce a philosopher's stone of some kind or other, B There is one kind of philosopher's stone that any adept alchemist could produce. Peirce: CP 4.548 Cross-Ref:†† We can express these in the principle that the Universes are receptacles of Subjects as follows: A1 The Interpreter having selected any individual he likes, and called it A, an object B can be found, such that, Either A would not be an adept alchemist, or B would be a philosopher's stone of some kind, and A could produce B. B1 Something, B, might be found, such that, no matter what the Interpreter might select and call A, B would be a philosopher's stone of some kind, while either A would not be an adept alchemist, or else A could produce B.

Peirce: CP 4.548 Cross-Ref:†† In these forms there are two Universes, the one of individuals selected at pleasure by the interpreter of the proposition, the other of suitable objects. Peirce: CP 4.548 Cross-Ref:†† I will now express the same two propositions on the principle that each Universe consists, not of Subjects, but the one of True assertions, the other of False, but each to the effect that there is something of a given description. 1. This is false: That something, P, is an adept alchemist, and that this is false, that while something, S, is a philosopher's stone of some kind, P could produce S. 2. This is true: That something, S, is a philosopher's stone of some kind; and this is

false, that something, P, is an adept alchemist while this is false, that P could produce S. Peirce: CP 4.548 Cross-Ref:†† Here, the whole proposition is mostly made up of the truth or falsity of assertions that a thing of this or that description exists, the only conjunction being "and." That this method is highly analytic is manifest. Now since our whole intention is to produce a method for the perfect analysis of propositions, the superiority of this method over the other for our purpose is undeniable. Moreover, in order to illustrate how that other might lead to false logic, I will tack the predicate of B1, in its objectionable form, upon the subject of A1 in the same form, and vice versa. I shall thus obtain two propositions which that method represents as being as simple as are Nos. 1 and 2. Peirce: CP 4.548 Cross-Ref:†† We shall see whether they are so. Here they are:†1 3. The Interpreter having designated any object to be called A, an object B may be found such that B is a philosopher's stone of some kind, while either A is not an adept alchemist or else A could produce B. 4. Something, B, may be found, such that, no matter what the interpreter may select, and call A, Either A would not be an adept alchemist, or B would be a philosopher's stone of some kind, and A could produce B. Peirce: CP 4.548 Cross-Ref:†† Proposition 3 may be expressed in ordinary language thus: There is a kind of philosopher's stone, and if there be any adept alchemist, he could produce a philosopher's stone of some kind. That is, No. 3 differs from A, A1 and 1 only in adding that there is a kind of philosopher's stone. It differs from B, B1 and 2 in not saying that any two adepts could produce the same kind of stone (nor that any adept could produce any existing kind); while B, B1 and 2 assert that some kind is both existent and could be made by every adept. Peirce: CP 4.548 Cross-Ref:†† Proposition 4, in ordinary language, is: If there be (or were) an adept alchemist, there is (or would be) a kind of philosopher's stone that any adept could produce. This asserts the substance of B, B1 and 2, but only conditionally upon the existence of an adept; but it asserts, what A, A1 and 1 do not, that all adepts could produce some one kind of stone, and this is precisely the difference between No. 4 and A1. Peirce: CP 4.548 Cross-Ref:†† To me it seems plain that the propositions 3 and 4 are both less simple than No. 1 and less simple than No. 2, each adding some thing to one of the pair first given and asserting the other conditionally. Yet the method of treating the Universes as receptacles for the metaphysical Subjects only, involves as a consequence the representation of 3 and 4 as quite on a par with 1 and 2. Peirce: CP 4.548 Cross-Ref:†† It remains to show that the other method does not carry this error with it. [If] it is the states of things affirmed or denied that are contained in the universes, then the

propositions [3 and 4] become as follows: 5. This is true: that there is a philosopher's stone of some kind, S, and that it is false that there is an adept, A, and that it is false that A could produce a philosopher's stone of some kind, S'. (Where it is neither asserted nor denied that S and S' are the same, thus distinguishing this from 2.) 6. This is false: That there is an adept, A, and that this is false: That there is a stone of a kind, S, and this is false: That there is an adept, A', and that this is false: That A' could produce a stone of the kind S. (Where again it is neither asserted nor denied that A and A' are identical, but the point is that this proposition holds even if they are not identical, thus distinguishing this from 1.) Peirce: CP 4.548 Cross-Ref:†† These forms exhibit the greater complexity of Propositions 3 and 4, by showing that they really relate to three individuals each; that is to say, 3 to two possible different kinds of stone, as well as to an adept; and 4 to two possible different adepts, and to a kind of stone. Indeed, the two forms 3 and 4†1 are absolutely identical in meaning with the following different forms on the same theory. Now it is, to say the least, a serious fault in a method of analysis that it can yield two analyses so different of one and the same compound. 7. An object, B, can be found, such that whatever object the interpreter may select and call A, an object, B', can thereupon be found such that B is an existing kind of philosopher's stone, and either A would not be an adept or else B' is a kind of philosopher's stone such as A could produce. 8. Whatever individual the Interpreter may choose to call A, an object, B, may be found, such that whatever individual the Interpreter may choose to call A', Either A is not an adept or B is an existing kind of philosopher's stone, and either A' is not an adept or else A' could produce a stone of the kind B. Peirce: CP 4.548 Cross-Ref:†† But while my forms are perfectly analytic, the need of diagrams to exhibit their meaning to the eye (better than merely giving a separate line to every proposition said to be false) is painfully obtrusive.†P1 Peirce: CP 4.549 Cross-Ref:†† 549. I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates. That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think through, into being subjects thought of. We thus think of the thought-sign itself, making it the object of another thought-sign. Thereupon, we can repeat the operation of hypostatic abstraction, and from these second intentions derive third intentions. Does this series proceed endlessly? I think not. What then are the characters of its different members? My thoughts on this subject are not yet harvested. I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being:†1 Actuality, Possibility, Destiny (or Freedom from Destiny). On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being. Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into Realms for the different Predicaments.

Peirce: CP 4.550 Cross-Ref:†† 550. All the various meanings of the word "Mind," Logical, Metaphysical, and Psychological, are apt to be confounded more or less, partly because considerable logical acumen is required to distinguish some of them, and because of the lack of any machinery to support the thought in doing so, partly because they are so many, and partly because (owing to these causes), they are all called by one word, "Mind." In one of the narrowest and most concrete of its logical meanings, a Mind is that Seme of The Truth, whose determinations become Immediate Interpretants of all other Signs whose Dynamical Interpretants are dynamically connected.†2 In our Diagram the same thing which represents The Truth must be regarded as in another way representing the Mind, and indeed, as being the Quasi-mind of all the Signs represented on the Diagram. For any set of Signs which are so connected that a complex of two of them can have one interpretant, must be Determinations of one Sign which is a Quasi-mind. Peirce: CP 4.551 Cross-Ref:†† 551. Thought is not necessarily connected with a brain. It appears in the work of bees, of crystals, and throughout the purely physical world; and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there. Consistently adhere to that unwarrantable denial, and you will be driven to some form of idealistic nominalism akin to Fichte's. Not only is thought in the organic world, but it develops there. But as there cannot be a General without Instances embodying it, so there cannot be thought without Signs. We must here give "Sign" a very wide sense, no doubt, but not too wide a sense to come within our definition. Admitting that connected Signs must have a Quasi-mind, it may further be declared that there can be no isolated sign. Moreover, signs require at least two Quasi-minds; a Quasi-utterer and a Quasi-interpreter; and although these two are at one (i.e., are one mind) in the sign itself, they must nevertheless be distinct. In the Sign they are, so to say, welded. Accordingly, it is not merely a fact of human Psychology, but a necessity of Logic, that every logical evolution of thought should be dialogic. You may say that all this is loose talk; and I admit that, as it stands, it has a large infusion of arbitrariness. It might be filled out with argument so as to remove the greater part of this fault; but in the first place, such an expansion would require a volume -- and an uninviting one; and in the second place, what I have been saying is only to be applied to a slight determination of our system of diagrammatization, which it will only slightly affect; so that, should it be incorrect, the utmost certain effect will be a danger that our system may not represent every variety of non-human thought.

Peirce: CP 4.552 Cross-Ref:†† §5. TINCTURED EXISTENTIAL GRAPHSE

552. There now seems to remain no reason why we should not proceed forthwith to formulate and agree upon

THE CONVENTIONS

DETERMINING THE FORMS AND INTERPRETATIONS OF EXISTENTIAL GRAPHS Convention the First: Of the Agency of the Scripture. We are to imagine that two parties †P1 collaborate in composing a Pheme, and in operating upon this so as to develop a Delome. (Provision shall be made in these Conventions for expressing every kind of Pheme as a Graph;†P2 and it is certain that the Method could be applied to aid the development and analysis of any kind of purposive thought. But hitherto no Graphs have been studied but such as are Propositions; so that, in the resulting uncertainty as to what modifications of the Conventions might be required for other applications, they have mostly been here stated as if they were only applicable to the expression of Phemes and the working out of necessary conclusions.) Peirce: CP 4.552 Cross-Ref:†† The two collaborating parties shall be called the Graphist and the Interpreter. The Graphist shall responsibly scribe each original Graph and each addition to it, with the proper indications of the Modality to be attached to it, the relative Quality†P1 of its position, and every particular of its dependence on and connections with other graphs. The Interpreter is to make such erasures and insertions of the Graph delivered to him by the Graphist as may accord with the "General Permissions" deducible from the Conventions and with his own purposes. Peirce: CP 4.553 Cross-Ref:†† 553. Convention the Second; Of the Matter of the Scripture, and the Modality†P1 of the Phemes expressed. The matter which the Graph-instances are to determine, and which thereby becomes the Quasi-mind in which the Graphist and Interpreter are at one, being a Seme of The Truth,†P2 that is, of the widest Universe of Reality, and at the same time, a Pheme of all that is tacitly taken for granted between the Graphist and Interpreter, from the outset of their discussion, shall be a sheet, called the Phemic Sheet, upon which signs can be scribed and from which any that are already scribed in any manner (even though they be incised) can be erased. But certain parts of other sheets not having the significance of the Phemic

THE TINCTURES OF COLOR Fig. 197

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sheet, but on which Graphs can be scribed and erased, shall be sometimes inserted in the Phemic sheet and exposed to view, as the Third Convention shall show. Every part of the exposed surface shall be tinctured in one or another of twelve tinctures. These are divided into three classes of four tinctures each, the class-characters being called Modes of Tincture, or severally, Color, Fur, and Metal. The tinctures of Color are Azure, Gules, Vert, and Purpure. Those of Fur are Sable, Ermine, Vair, and Potent. Those of Metal are Argent, Or, Fer, and Plomb. The Tinctures will in practice be represented as in Fig. 197.†P1 The whole of any continuous part of the exposed surface in one tincture shall be termed a Province. The border of the sheet has one tincture all round; and we may imagine that it was chosen from among twelve, in agreement between the Graphist and the Interpreter at the outset. The province of the border may be called the March. Provinces adjacent to the March are to be regarded as overlying it; Provinces adjacent to those Provinces, but not to the March, are to be regarded as overlying the provinces adjacent to the March, and so on. We are to imagine that the Graphist always finds provinces where he needs them. Peirce: CP 4.554 Cross-Ref:†† 554. When any representation of a state of things consisting in the applicability of a given description to an individual or limited set of individuals otherwise indesignate is scribed, the Mode of Tincture of the province on which it is scribed shows whether the Mode of Being which is to be affirmatively or negatively attributed to the state of things described is to be that of Possibility, when Color will

be used; or that of Intention, indicated by Fur; or that of Actuality shown by Metal. Special understandings may determine special tinctures to refer to special varieties of the three genera of Modality. Finally, the Mode of Tincture of the March may determine whether the Entire Graph is to be understood as Interrogative, Imperative, or Indicative. Peirce: CP 4.555 Cross-Ref:†† 555. Convention the Third: Of Areas enclosed within, but severed from, the Phemic Sheet. The Phemic Sheet is to be imagined as lying on the smoother of the two surfaces or sides of a Leaf, this side being called the recto, and to consist of so much of this side as is continuous with the March. Other parts of the recto may be exposed to view. Every Graph-instance on the Phemic Sheet is posited unconditionally (unless, according to an agreement between Graphist and Interpreter, the Tincture of its own Province or of the March should indicate a condition) and every Graph-instance on the recto is posited affirmatively and, in so far as it is indeterminate, indefinitely. Peirce: CP 4.556 Cross-Ref:†† 556. Should the Graphist desire to negative a Graph, he must scribe it on the verso, and then, before delivery to the Interpreter, must make an incision, called a Cut, through the Sheet all the way round the Graph-instance to be denied, and must then turn over the excised piece, so as to expose its rougher surface carrying the negatived Graph-instance. This reversal of the piece is to be conceived to be an inseparable part of the operation of making a Cut.†P1 But if the Graph to be negatived includes a Cut, the twice negatived Graph within that Cut must be scribed on the recto, and so forth. The part of the exposed surface that is continuous with the part just outside the Cut is called the Place of the Cut. A Cut is neither a Graph nor a Graph-instance; but the Cut, together with all that it encloses, exposed is termed an Enclosure, and is conceived to be an Instance of a Graph scribed on the Place of the Cut, which is also termed the Place of the Enclosure. The surface within the Cut, continuous with the parts just within it, is termed the Area of the Cut and of the Enclosure; and the part of the recto continuous with the March (i.e., the Phemic Sheet), is likewise termed an Area, namely the Area of the Border. The Copulate of all that is scribed on any one Area, including the Graphs of which the Enclosures whose Place is this Area are Instances, is called the Entire Graph of that Area; and any part of the Entire Graph, whether graphically connected with or disconnected from the other parts, provided it might be the Entire Graph of the Sheet, is termed a Partial Graph of the Area. Peirce: CP 4.557 Cross-Ref:†† 557. There may be any number of Cuts, one within another, the Area of one being the Place of the next, and since the Area of each is on the side of the leaf opposite to its Place, it follows that recto Areas may be exposed which are not parts of the Phemic Sheet. Every Graph-instance on a recto Area is affirmatively posited, but is posited conditionally upon whatever may be signified by the Graph on the Place of the Cut of which this Area is the Area. (It follows that Graphs on Areas of different Enclosures on a verso Place are only alternately affirmed, and that while only the Entire Graph of the Area of an Enclosure on a recto Place is denied, but not its different Partial Graphs, except alternatively, the Entire Graphs of Areas of different Enclosures on one recto Place are copulatively denied.) Peirce: CP 4.558 Cross-Ref:††

558. Every Graph-instance must lie upon one Area,†P1 although an Enclosure may be a part of it. Graph-instances on different Areas are not to be considered as, nor by any permissible latitude of speech to be called, Parts of one Graph-instance, nor Instances of Parts of one Graph; for it is only Graph-instances on one Area that are called Parts of one Graph-instance, and that only of a Graph-instance on that same Area; for though the Entire Graph on the Area of an enclosure is termed the Graph of the Enclosure, it is no Part of the Enclosure and is connected with it only through a denial. Peirce: CP 4.559 Cross-Ref:†† 559. Convention the Fourth: Concerning Signs of Individuals and of Individual Identity. A single dot, not too minute, or single congeries of contiguous pretty large dots, whether in the form of a line or surface, when placed on any exposed Area, will refer to a single member of the Universe to which the Tincture of that Area refers, but will not thereby be made to refer determinately to any one. But do not forget that separate dots, or separate aggregates of dots, will not necessarily denote different Objects. Peirce: CP 4.560 Cross-Ref:†† 560. By a rheme, or predicate, will here be meant a blank form of proposition which might have resulted by striking out certain parts of a proposition, and leaving a blank in the place of each, the parts stricken out being such that if each blank were filled with a proper name, a proposition (however nonsensical) would thereby be recomposed. An ordinary predicate of which no analysis is intended to be represented will usually be written in abbreviated form, but having a particular point on the periphery of the written form appropriated to each of the blanks that might be filled with a proper name. Such written form with the appropriated points shall be termed a Spot; and each appropriated point of its periphery shall be called a Peg of the Spot. If a heavy dot is placed at each Peg, the Spot will become a Graph expressing a proposition in which every blank is filled by a word (or concept) denoting an indefinite individual object, "something." Peirce: CP 4.561 Cross-Ref:†† 561. A heavy line shall be considered as a continuum of contiguous dots; and since contiguous dots denote a single individual, such a line without any point of branching will signify the identity of the individuals denoted by its extremities, and the type of such unbranching line shall be the Graph of Identity, any instance of which (on one area, as every Graph-instance must be) shall be called a Line of Identity. The type of a three-way point of such a line (Fig. 198) shall be the Graph of Teridentity; and it shall be considered as composed of three contiguous Pegs of a Spot of Identity. An extremity

Figs. 198-199

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of a Line of Identity not abutting upon another such Line in another area shall be called a Loose End. A heavy line, whether confined to one area or not (and therefore not generally being a Graph-instance) of which two extremities abut upon pegs of spots shall be called a Ligature. Two lines cannot abut upon the same peg other than a point of teridentity. (The purpose of this rule is to force the recognition of the demonstrable logical truth that the concept of teridentity is not mere identity. It is identity and identity, but this "and" is a distinct concept, and is precisely that of teridentity.) A Ligature crossing a Cut is to be interpreted as unchanged in meaning by erasing the part that crosses to the Cut and attaching to the two Loose Ends so produced two Instances of a Proper Name nowhere else used; such a Proper name (for which a capital letter will serve) being termed a Selective.†P1 In the interpretation of Selectives it is often necessary to observe the rule which holds throughout the System, that the Interpretation of Existential Graphs must be endoporeutic, that is, the application of a Graph on the Area of a Cut will depend on the predetermination of the application of that which is on the Place of the Cut. Peirce: CP 4.561 Cross-Ref:†† In order to avoid the intersection of Lines of Identity, either a Selective may be employed, or a Bridge, which is imagined to be a bit of paper ribbon, but will in practice be pictured as in Fig. 199. Peirce: CP 4.562 Cross-Ref:†† 562. Convention the Fifth: Of the Connections of Graph-Instances. Two partial Graph-Instances are said to be individually and directly connected, if, and only if, in the Entire Graph, one individual is, either unconditionally or under some condition, and whether affirmatively or negatively, made a Subject of both. Two Graph-Instances connected by a ligature are explicitly and definitely individually and directly connected. Two Graph-Instances in the same Province are thereby explicitly, although indefinitely, individually and directly connected, since both, or one and the negative of the other, or the negative of both, are asserted to be true or false together, that is, under the same circumstances, although these circumstances are not formally defined, but are left to be interpreted according to the nature of the case. Two Graph-Instances not in the same Province, though on the same Mode of Tincture, are only in so far connected that both are in the same Universe. Two Graph-Instances in different Modes of Tincture are only in so far connected that both, or one and the negative of the other, or the negative of both, are posited as appertaining to the Truth. They cannot be said to have any individual and direct connection. Two Graph-Instances that are not individually connected within the innermost Cut which

contains them both cannot be so connected at all; and every ligature connecting them is meaningless and may be made or broken. Peirce: CP 4.563 Cross-Ref:†† 563. Relations which do not imply the occurrence in their several universes of all their correlates must not be expressed by Spots or single Graphs,†P1 but all such relations can be expressed in the System. Peirce: CP 4.564 Cross-Ref:†† 564. I will now proceed to give a few examples of Existential Graphs in order to illustrate the method of interpretation, and also the Permissions of Illative Transformation of them. Peirce: CP 4.564 Cross-Ref:†† If you carefully examine the above conventions, you will find that they are simply the development, and excepting in their insignificant details, the inevitable result of the development of the one convention that if any Graph, A, asserts one state of things to be real and if another graph, B, asserts the same of another state of things, then AB, which results from setting both A and B upon the sheet, shall assert that both states of things are real. This was not the case with my first system of Graphs, described in Vol. VII of The Monist,†1 which I now call Entitative Graphs. But I was forced to this principle by a series of considerations which ultimately arrayed themselves into an exact logical deduction of all the features of Existential Graphs which do not involve the Tinctures. I have no room for this here; but I state some of the points arrived at somewhat in the order in which they first presented themselves. Peirce: CP 4.564 Cross-Ref:†† In the first place, the most perfectly analytical system of representing propositions must enable us to separate illative transformations into indecomposable parts. Hence, an illative transformation from any proposition, A, to any other, B, must in such a system consist in first transforming A into AB, followed by the transformation of AB into B. For an omission and an insertion appear to be indecomposable transformations and the only indecomposable transformations. That is, if A can be transformed by insertion into AB, and AB by omission in B, the transformation of A into B can be decomposed into an insertion and an omission. Accordingly, since logic has primarily in view argument, and since the conclusiveness of an argument can never be weakened by adding to the premisses nor by subtracting from the conclusion, I thought I ought to take the general form of argument as the basal form of composition of signs in my diagrammatization; and this necessarily took the form of a "scroll," that is (see Figs. 200, 201, 202) a curved line without contrary flexure and returning

Figs. 200-202

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into itself after once crossing itself, and thus forming an outer and an inner "close." I shall call the outer boundary the Wall; and the inner, the Fence. In the outer I scribed the Antecedent, in the inner the Consequent, of a Conditional Proposition de inesse. The scroll was not taken for this purpose at hap-hazard, but was the result of experiments and reasonings by which I was brought to see that it afforded the most faithful Diagram of such a Proposition. This form once obtained, the logically inevitable development brought me speedily to the System of Existential Graphs. Namely, the idea of the scroll was that Fig. 200, for example, should assert that if A be true (under the actual circumstances), then C and D are both true. This justifies Fig. 201, that if both A and B are true, then both C and D are true, no matter what B may assert, any insertion being permitted in the outer close, and any omission from the inner close. By applying the former clause of this rule to Fig. 202, we see that this scroll with the outer close void, justifies the assertion that if no matter what be true, C is in any case true; so that the two walls of the scroll, when nothing is between them, fall together, collapse, disappear, and leave only the contents of the inner close standing, asserted, in the open field. Supposing, then, that the contents of the inner scroll had been CD, these would have been left standing, both asserted; and we thus return to the principle that writing assertions together on the open sheet asserts them all. Now, Reader, if you will just take pencil and paper and scribe the scroll expressing that if A be true, then it is true that if B be true C and D are true, and compare this with Fig. 201, which amounts to the same thing in meaning, you will see that scroll walls with a void between them collapse even when they belong to different scrolls; and you will

Figs. 203-204

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further see that a scroll is really nothing but one oval within another. Since a Conditional de inesse (unlike other conditionals) only asserts that either the antecedent is false or the consequent is true, it all but follows that if the latter alternative be suppressed by scribing nothing but the antecedent, which may be any proposition, in an oval, that antecedent is thereby denied.†P1 The use of a heavy line as a juncture signifying identity is inevitable; and since Fig. 203 must mean that if anything is a man, it is mortal, it will follow that Fig. 204 must mean "Something is a man."

Peirce: CP 4.565 Cross-Ref:†† 565. The first permission of illative transformation is now evident as follows: Peirce: CP 4.565 Cross-Ref:†† First Permission, called "The Rule of Deletion and Insertion." Any Graph-Instance can be deleted from any recto Area (including the severing of any Line of Identity), and any Graph-Instance can be inserted on any verso Area (including as a Graph-Instance the juncture of any two Lines of Identity or Points of Teridentity). Peirce: CP 4.566 Cross-Ref:†† 566. The justice of the following will be seen instantly by students of any form of Logical Algebra, and with very little difficulty by others: Peirce: CP 4.566 Cross-Ref:†† Second Permission, called "The Rule of Iteration and Deiteration." Any Graph scribed on any Area may be Iterated in or (if already Iterated) may be Deiterated by a deletion from that Area or from any other Area included within that. This involves the Permission to distort a line of Identity, at will. Peirce: CP 4.566 Cross-Ref:†† To iterate a Graph means to scribe it again, while joining by Ligatures every Peg of the new Instance to the corresponding Peg of the Original Instance. To deiterate a Graph is to erase a second Instance of it, of which each Peg is joined by a Ligature to a first Instance of it. One Area is said to be included within another if, and only if, it either is that Area or else is the Area of a Cut whose Place is an Area which, according to this definition, must be regarded as included within that other. By this Permission, Fig. 205 may be transformed into Fig. 206, and thence, by Permission No. 1, into Fig. 207.

Figs. 205-207

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Peirce: CP 4.567 Cross-Ref:†† 567. We now come to the Third Permission, which I shall state in a form which is valid, sufficient for its purpose, and convenient in practice, but which cannot be assumed as an undeduced Permission, for the reason that it allows us to regard the Inner close, after the Scroll is removed, as being a part of the Area on which the Scroll lies. Now this is not strictly either an Insertion or a Deletion; and a perfectly analytical System of Permissions should permit only the indecomposable operations of Insertion and Deletion of Graphs that are simple in expression. The more scientific way would be to substitute for the Second and Third Permissions the following Permission: Peirce: CP 4.567 Cross-Ref:†† If an Area, {G}, and an Area, , be related in any of these four ways, viz., (1) If {G} and are the same Area; (2) If is the Area of an Enclosure whose Place is {G}; (3) if is the Area of an Enclosure whose Place is the Area of a

second Enclosure whose Place is {G}; or (4) if is the Place of an Enclosure whose Area is vacant except that it is the Place of an Enclosure whose Area is {G}, and except that it may contain ligatures, identifying Pegs in with Pegs in {G}; then, if be a recto area, any simple Graph already scribed upon {G} may be iterated upon ; while if be a verso Area, any simple Graph already scribed upon {G} and iterated upon may be deiterated by being deleted or abolished from . Peirce: CP 4.567 Cross-Ref:†† These two Rules (of Deletion and Insertion, and of Iteration and Deiteration) are substantially all the undeduced Permissions needed; the others being either Consequences or Explanations of these. Only, in order that this may be true, it is necessary to assume that all indemonstrable implications of the Blank have from the beginning been scribed upon distant parts of the Phemic Sheet, upon any part of which they may, therefore, be iterated at will. I will give no list of these implications, since it could serve no other purpose than that of warning beginners that necessary propositions not included therein were deducible from the other permissions. I will simply notice two principles the neglect of which might lead to difficulties. One of these is that it is physically impossible to delete or otherwise get rid of a Blank in any Area that contains a Blank, whether alone or along with other Graph-Instances. We may, however, assume that there is one Graph, and only one, an Instance of which entirely fills up an Area, without any Blank. The other principle is that, since a Dot merely asserts that some individual object exists, and is thus one of the implications of the Blank, it may be inserted in any Area; and since the Dot will signify the same thing whatever its size, it may be regarded as an Enclosure whose Area is filled with an Instance of that sole Graph that excludes the Blank. The Dot, then, denies that Graph, which may, therefore, be understood as the absurd Graph, and its signification may be formulated as "Whatever you please is true." The absurd Graph may also take the form of an Enclosure with its Area entirely Blank, or enclosing only some Instance of a Graph implied in the Blank. These two principles will enable the Graphist to thread his way through some Transformations which might otherwise appear paradoxical and absurd. Peirce: CP 4.567 Cross-Ref:†† Third Permission, called "The Rule of the Double Cut." Two Cuts one within another, with nothing between them, unless it be Ligatures passing from outside the outer Cut to inside the inner one, may be made or abolished on any Area. Peirce: CP 4.568 Cross-Ref:†† 568. Let us now consider the Interpretation of such Ligatures. For that purpose, I first note that the Entire Graph of any recto Area is a wholly particular and affirmative Proposition or Copulation of such Propositions. By "wholly particular," I mean, having for every Subject an indesignate individual. The Entire Graph of any verso Area is a wholly universal negative proposition or a disjunction of such propositions. Peirce: CP 4.568 Cross-Ref:†† The first time one hears a Proper Name pronounced, it is but a name, predicated, as one usually gathers, of an existent, or at least historically existent, individual object, of which, or of whom, one almost always gathers some additional information. The next time one hears the name, it is by so much the more definite;

and almost every time one hears the name, one gains in familiarity with the object. A Selective is a Proper Name met with by the Interpreter for the first time. But it always occurs twice, and usually on different areas. Now the Interpretation, by Convention No. 3, is to be Endoporeutic, so that it is the outermost occurrence of the Name that is the earliest. Peirce: CP 4.569 Cross-Ref:†† 569. Let us now analyze the interpretation of a Ligature passing through a Cut. Take, for example, the Graph of Fig. 208.

[Click here to view] The partial Graph on the Place of the Cut asserts that there exists an individual denoted by the extremity of the line of identity on the Cut, which is a millionaire. Call that individual C. Then, since contiguous dots denote the same individual objects, the extremity of the line of identity on the Area of the cut is also C, and the Partial Graph on that Area, asserts that, let the Interpreter choose whatever individual he will, that individual is either not C, or else is not unfortunate. Thus, the Entire Graph asserts that there exists a millionaire who is not unfortunate. Furthermore, the Enclosure lying in the same Argent Province as the "millionaire," it is asserted that this individual's being a millionaire is connected with his not being unfortunate. This example shows that the Graphist is permitted to extend any Line of Identity on a recto Area so as to carry an end of it to any Cut in that area. Let us next interpret Fig. 209.

[Click here to view] It obviously asserts that there exists a Turk who is at once the husband of an Individual denoted by a point on the Cut, which individual we may name U, and is the husband of an Individual, whom we may name V, denoted by another point on the Cut. And the Graph on the Area of the cut, declares that whatever Fig. 209 Individual the Interpreter may select either is not, and cannot be, U, or is not and cannot be V. Thus, the Entire Graph asserts that there is an existent Turk who is husband of two existent persons; and the "husband," the "Turk" and the enclosure, all being in the same Argent province, although the Area of the Enclosure is on color, and thus denies the possibility of the identity of U and V, all four predications are true together, that is, are true under the same circumstances, which circumstances should be defined by a special convention when anything may turn upon what they are. For the sake of illustrating this, I shall now scribe Fig. 210

[Click here to view] all in one province. This may be read, "There is some married woman who will commit suicide in case her husband fails in business." This evidently goes far beyond saying that if every married man fails in business some married woman will commit suicide. Yet note that since the Graph is on Metal it asserts a conditional proposition de inesse and only means that there is a married woman whose husband does not fail or else she commits suicide. That, at least, is all it will seem to mean if we fail to take account of the fact, that being all in one Province, it is said that her suicide is connected with his failure. Neglecting that, the proposition only denies that every married man fails, while no married woman commits suicide.†1 The logical principle is that to say that there is some one individual of which one or other of two predicates is true is no more than to say that there either is some individual of which one is true or else there is some individual of which the other is true. Or, to state the matter as an illative permission of the System of Existential Graphs, Peirce: CP 4.569 Cross-Ref:†† Fourth Permission. If the smallest Cut which wholly contains a Ligature connecting two Graphs in different Provinces has its Area on the side of the Leaf opposite to that of the Area of the smallest Cut that contains those two Graphs, then such Ligature may be made or broken at pleasure, as far as these two Graphs are concerned.†2 Peirce: CP 4.570 Cross-Ref:†† 570. Another somewhat curious problem concerning ligatures is to say by what principle it is true, as it evidently is true, that the passage of ligatures from without the outer of two Cuts to within the inner of them will not prevent the two from collapsing in case there is no other Graph-Instance between them. A little study suffices to show that this may depend upon the ligatures' being replaceable by Selectives where they cross the Cuts, and that a Selective is always, at its first occurrence, a new predicate. For it is a principle of Logic that in introducing a new predicate one has a right to assert what one likes concerning it, without any restriction, as long as one implies no assertion concerning anything else. I will leave it to you, Reader, to find out how this principle accounts for the collapse of the two Cuts. Another solution of this problem, not depending on the superfluous device of Selectives, is afforded by the second enunciation of the Rule of Iteration and Deiteration; since this permits the Graph of the Inner Close to be at once iterated on the Phemic Sheet. One may choose between these two methods of solution. Peirce: CP 4.571 Cross-Ref:†† 571. The System of Existential Graphs which I have now sufficiently described -- or, at any rate, have described as well as I know how, leaving the further perfection of it to others -- greatly facilitates the solution of problems of Logic, as will be seen in the sequel, not by any mysterious properties, but simply by substituting for the symbols in which such problems present themselves, concrete visual figures concerning which we have merely to say whether or not they admit

Figs. 211-212

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certain describable relations of their parts. Diagrammatic reasoning is the only really fertile reasoning. If logicians would only embrace this method, we should no longer see attempts to base their science on the fragile foundations of metaphysics or a psychology not based on logical theory; and there would soon be such an advance in logic that every science would feel the benefit of it. Peirce: CP 4.571 Cross-Ref:†† This System may, of course, be applied to the analysis of reasonings. Thus, to separate the syllogistic illation, "Any man would be an animal, and any animal would be mortal; therefore, any man would be mortal," the Premisses are first scribed as in Fig. 211. Then by the rule of Iteration, a first illative transformation gives Fig. 212. Next, by the permission to erase from a recto Area, a second step gives Fig. 213. Then, by the permission to deform a line of Identity on a recto Area, a third step gives Fig. 214. Next, by the permission to insert in a verso Area, a fourth step gives Fig. 215. Next, by Deiteration, a fifth step gives Fig. 216. Next, by

Figs. 213-218

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the collapse of two Cuts, a sixth step gives Fig. 217; and finally, by omission from a recto Area, a seventh step gives the conclusion Fig. 218. The analysis might have been carried a little further, by means of the Rule of Iteration and Deiteration, so as to increase the number of distinct inferential steps to nine, showing how complex a process the drawing of a syllogistic conclusion really is. On the other hand, it need

scarcely be said that there are a number of deduced liberties of transformation, by which even much more complicated inferences than a syllogism can be performed at a stroke. For that sort of problem, however, which consists in drawing a conclusion or assuring oneself of its correctness, this System is not particularly adapted. Its true utility is in the assistance it renders -- the support to the mind, by furnishing concrete diagrams upon which to experiment -- in the solution of the most difficult problems of logical theory. Peirce: CP 4.572 Cross-Ref:†† 572. I mentioned on an early page of this paper that this System leads to a different conception of the Proposition and Argument from the traditional view that a Proposition is composed of Names, and that an Argument is composed of Propositions. It is a matter of insignificant detail whether the term Argument be taken in the sense of the Middle Term, in that of the Copulate of Premisses, in that of the setting forth of Premisses and Conclusion, or in that of the representation that the real facts which the premisses assert (together, it may be, with the mode in which those facts have come to light) logically signify the truth of the Conclusion. In any case, when an Argument is brought before us, there is brought to our notice (what appears so clearly in the Illative Transformations of Graphs) a process whereby the Premisses bring forth the Conclusion, not informing the Interpreter of its Truth, but appealing to him to assent thereto. This Process of Transformation, which is evidently the kernel of the matter, is no more built out of Propositions than a motion is built out of positions. The logical relation of the Conclusion to the Premisses might be asserted; but that would not be an Argument, which is essentially intended to be understood as representing what it represents only in virtue of the logical habit which would bring any logical Interpreter to assent to it. We may express this by saying that the Final (or quasi-intended) Interpretant of an Argument represents it as representing its Object after the manner of a Symbol. In an analogous way the relation of Predicate to Subject which is stated in a Proposition might be merely described in a Term. But the essence of the Proposition is that it intends, as it were, to be regarded as in an existential relation to its Object, as an Index is, so that its assertion shall be regarded as evidence of the fact. It appears to me that an assertion and a command do not differ essentially in the nature of their Final Interpretants as in their Immediate, and so far as they are effective, in their Dynamical Interpretants; but that is of secondary interest. The Name, or any Seme, is merely a substitute for its Object in one or another capacity in which respect it is all one with the Object. Its Final Interpretant thus represents it as representing its Object after the manner of an Icon, by mere agreement in idea. It thus appears that the difference between the Term, the Proposition, and the Argument, is by no means a difference of complexity, and does not so much consist in structure as in the services they are severally intended to perform. Peirce: CP 4.572 Cross-Ref:†† For that reason, the ways in which Terms and Arguments can be compounded cannot differ greatly from the ways in which Propositions can be compounded. A mystery, or paradox, has always overhung the question of the Composition of Concepts. Namely, if two concepts, A and B, are to be compounded, their composition would seem to be necessarily a third ingredient, Concept C, and the same difficulty will arise as to the Composition of A and C. But the Method of Existential Graphs solves this riddle instantly by showing that, as far as propositions go, and it must evidently be the same with Terms and Arguments, there is but one general way in which their Composition can possibly take place; namely, each component must be indeterminate in some respect or another; and in their composition each determines

the other. On the recto this is obvious: "Some man is rich" is composed of "Something is a man" and "something is rich," and the two somethings merely explain each other's vagueness in a measure. Two simultaneous independent assertions are still connected in the same manner; for each is in itself vague as to the Universe or the "Province" in which its truth lies, and the two somewhat define each other in this respect. The composition of a Conditional Proposition is to be explained in the same way. The Antecedent is a Sign which is Indefinite as to its Interpretant; the Consequent is a Sign which is Indefinite as to its Object. They supply each the other's lack. Of course, the explanation of the structure of the Conditional gives the explanation of negation; for the negative is simply that from whose Truth it would be true to say that anything you please would follow de inesse. Peirce: CP 4.572 Cross-Ref:†† In my next paper, the utility of this diagrammatization of thought in the discussion of the truth of Pragmaticism shall be made to appear.†1

Peirce: CP 4.573 Cross-Ref:†† CHAPTER 7

AN IMPROVEMENT ON THE GAMMA GRAPHS†1E

573. In working with Existential Graphs, we use, or at any rate imagine that we use, a sheet of paper of different tints on its two sides. Let us say that the side we call the recto is cream white while the verso is usually of somewhat bluish grey, but may be of yellow or of a rose tint or of green. The recto is appropriated to the representation of existential, or actual, facts, or what we choose to make believe are such. The verso is appropriated to the representation of possibilities of different kinds according to its tint, but usually to that of subjective possibilities, or subjectively possible truths. The special kind of possibility here called subjective is that which consists in ignorance. If we do not know that there are not inhabitants of Mars, it is subjectively possible that there are such beings. . . . Peirce: CP 4.574 Cross-Ref:†† 574. The verso is usually appropriated to imparting information about subjective possibilities or what may be true for aught we know. To scribe a graph is to impart an item of information; and this item of information does one of two things. It either adds to what we know to exist or it cuts off something from our list of subjective possibilities. Hence, it must be that a graph scribed on the verso is thereby denied. Peirce: CP 4.575 Cross-Ref:†† 575. Now the denial of a subjective possibility usually, if not always, involves the assertion of a truth of existence; and consequently what is put upon the verso must usually have a definite connection with a place on the recto. Peirce: CP 4.576 Cross-Ref:†† 576. In my former exposition of Existential Graphs, I said that there must be a

department of the System which I called the Gamma part into which I was as yet able to gain mere glimpses, sufficient only to show me its reality, and to rouse my intense curiosity, without giving me any real insight into it. The conception of the System which I have just set forth is a very recent discovery. I have not had time as yet to trace out all its consequences. But it is already plain that, in at least three places, it lifts the veil from the Gamma part of the system. Peirce: CP 4.577 Cross-Ref:†† 577. The new discovery which sheds such a light is simply that, as the main part of the sheet represents existence or actuality, so the area within a cut, that is, the verso of the sheet, represents a kind of possibility. Peirce: CP 4.578 Cross-Ref:†† 578. From thence I immediately infer several things that I did not understand before, as follows: Peirce: CP 4.578 Cross-Ref:†† First, the cut may be imagined to extend down to one or another depth into the paper, so that the overturning of the piece cut out may expose one stratum or another, these being distinguished by their tints; the different tints representing different kinds of possibility. Peirce: CP 4.578 Cross-Ref:†† This improvement gives substantially, as far as I can see, nearly the whole of that Gamma part which I have been endeavoring to discern. Peirce: CP 4.579 Cross-Ref:†† 579. Second, In a certain partly printed but unpublished "Syllabus of Logic," which contains the only formal or full description of Existential Graphs that I have ever undertaken to give, I laid it down, as a rule, that no graph could be partly in one area and partly in another;†1 and this I said simply because I could attach no interpretation to a graph which should cross a cut. As soon, however, as I discovered that the verso of the sheet represents a universe of possibility, I saw clearly that such a graph was not only interpretable, but that it fills the great lacuna in all my previous developments of the logic of relatives. For although I have always recognized that a possibility may be real, that it is sheer insanity to deny the reality of the possibility of my raising my arm, even if, when the time comes, I do not raise it; and although, in all my attempts to classify relations, I have invariably recognized, as one great class of relations, the class of references, as I have called them, where one correlate is an existent, and another is a mere possibility; yet whenever I have undertaken to develop the logic of relations, I have always left these references out of account, notwithstanding their manifest importance, simply because the algebras or other forms of diagrammatization which I employed did not seem to afford me any means of representing them.†1 I need hardly say that the moment I discovered in the verso of the sheet of Existential Graphs a representation of a universe of possibility, I perceived that a reference would be represented by a graph which should cross a cut, thus subduing a vast field of thought to the governance and control of exact logic. Peirce: CP 4.580 Cross-Ref:†† 580. Third, My previous account of Existential Graphs

Fig. 219 here to view]†2

[Click

was marred by a certain rule which, from the point of view from which I thought the system ought to be regarded, seemed quite out of place and inacceptable, and yet which I found myself unable to dispute.†3 I will just illustrate this matter by an example. Suppose we wish to assert that there is a man every dollar of whose indebtedness will be paid by some man

Figs. 220-221

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or other, perhaps one dollar being paid by one man and another by another man, or perhaps all paid by the same man. We do not wish to say how that will be. Here will be our graph, Fig. 219. But if we wish to assert that one man will pay the whole, without saying in what relation the payer stands to the debtor, here will be our graph, Fig. 220. Now suppose we wish to add that this man who will pay all those debts is the very same man who owes them. Then we insert two graphs of teridentity and a line of identity as in Fig. 221. The difference between the graph with and without this added line is obvious, and is perfectly represented in all my systems. But here it will be observed that the graph "owes" and the graph "pays" are not only united on the left by a line outside the smallest area that contains them both, but likewise on the right, by a line inside that smallest common area. Now let us consider a case in which this inner connection is lacking. Let us assert that there is a man A and a man B, who may or may not be the same man, and if A becomes bankrupt then B will suicide. Then, if we add that A and B are the same man, by drawing a line outside the smallest common area of the

Figs. 222-223

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graphs joined, which are here bankrupt and suicide, the strange rule to which I refer is that such outer line, because there is no connecting line within the smallest common area, is null and void, that is, it does not affect the interpretation in the least. . . . The proposition that there is a man who if he goes bankrupt will commit suicide is false only in case, taking any man you please, he will go bankrupt, and will not suicide. That is, it is falsified only if every man goes bankrupt without suiciding. But this is the same as the state of things under which the other proposition is false; namely, that every man goes broke while no man suicides. This reasoning is irrefragable as long as a mere possibility is treated as an absolute nullity. Some years ago,†1 however, when in consequence of an invitation to deliver a course of lectures in Harvard University upon Pragmatism, I was led to revise that doctrine, in which I had already found difficulties, I soon discovered, upon a critical analysis, that it was absolutely necessary to insist upon and bring to the front, the truth that a mere possibility may be quite real. That admitted, it can no longer be granted that every conditional proposition whose antecedent does not happen to be realized is true, and the whole reasoning just given breaks down.

Peirce: CP 4.581 Cross-Ref:†† 581. I often think that we logicians are the most obtuse of men, and the most devoid of common sense. As soon as I saw that this strange rule, so foreign to the general idea of the System of Existential Graphs, could by no means be deduced from the other rules nor from the general idea of the system, but has to be accepted, if at all, as an arbitrary first principle -- I ought to have asked myself, and should have asked myself if I had not been afflicted with the logician's bêtise, What compels the adoption of this rule? The answer to that must have been that the interpretation requires it; and the inference of common sense from that answer would have been that the interpretation was too narrow. Yet I did not think of that until my operose method like that of a hydrographic surveyor sounding out a harbour, suddenly brought me up to the important truth that the verso of the sheet of Existential Graphs represents a universe of possibilities. This, taken in connection with other premisses, led me back to the same conclusion to which my studies of Pragmatism had already brought me, the reality of some possibilities. This is a striking proof of the superiority of the System of Existential Graphs to either of my algebras of logic.†1 For in both of them the incongruity of this strange rule is completely hidden behind the superfluous machinery which is introduced in order to give an appearance of symmetry to logical law, and in order to facilitate the working of these algebras considered as reasoning machines. I cannot let this remark pass without protesting, however, that in the construction of no algebra was the idea of making a calculus which would turn out conclusions by a regular routine other than a very secondary purpose. . . .†2 Peirce: CP 4.582 Cross-Ref:†† 582. The sheet of the graphs in all its states collectively, together with the laws of its transformations, corresponds to and represents the Mind in its relation to its thoughts, considered as signs. That thoughts are signs has been more especially urged by nominalistic logicians; but the realists are, for the most part, content to let the proposition stand unchallenged, even when they have not decidedly affirmed its truth. The scribed graphs are determinations of the sheet, just as thoughts are determinations of the mind; and the mind itself is a comprehensive thought just as the sheet considered in all its actual transformation-states and transformations, taken collectively, is a graph-instance and taken in all its permissible transformations is a graph. Thus the system of existential graphs is a rough and generalized diagram of the Mind, and it gives a better idea of what the mind is, from the point of view of logic, than could be conveyed by any abstract account of it. Peirce: CP 4.583 Cross-Ref:†† 583. The System of Existential Graphs recognizes but one mode of combination of ideas, that by which two indefinite propositions define, or rather partially define, each other on the recto and by which two general propositions mutually limit each other upon the verso; or, in a unitary formula, by which two indeterminate propositions mutually determine each other in a measure. I say in a measure, for it is impossible that any sign whether mental or external should be perfectly determinate. If it were possible such sign must remain absolutely unconnected with any other. It would quite obviously be such a sign of its entire universe, as Leibniz and others have described the omniscience of God to be, an intuitive representation amounting to an indecomposable feeling of the whole in all its details, from which those details would not be separable. For no reasoning, and consequently no abstraction, could connect itself with such a sign. This consideration, which is obviously correct, is a strong argument to show that what the system of

existential graphs represents to be true of propositions and which must be true of them, since every proposition can be analytically expressed in existential graphs, equally holds good of concepts that are not propositional; and this argument is supported by the evident truth that no sign of a thing or kind of thing -- the ideas of signs to which concepts belong -- can arise except in a proposition; and no logical operation upon a proposition can result in anything but a proposition; so that non-propositional signs can only exist as constituents of propositions. But it is not true, as ordinarily represented, that a proposition can be built up of non-propositional signs. The truth is that concepts are nothing but indefinite problematic judgments. The concept of man necessarily involves the thought of the possible being of a man; and thus it is precisely the judgment, "There may be a man." Since no perfectly determinate proposition is possible, there is one more reform that needs to be made in the system of existential graphs. Namely, the line of identity must be totally abolished, or rather must be understood quite differently. We must hereafter understand it to be potentially the graph of teridentity by which means there always will virtually be at least one loose end in every graph. In fact, it will not be truly a graph of teridentity but a graph of indefinitely multiple identity. Peirce: CP 4.584 Cross-Ref:†† 584. We here reach a point at which novel considerations about the constitution of knowledge and therefore of the constitution of nature burst in upon the mind with cataclysmal multitude and resistlessness. It is that synthesis of tychism and of pragmatism for which I long ago proposed the name, Synechism,†1 to which one thus returns; but this time with stronger reasons than ever before. But I cannot, consistently with my own convictions, ask the Academy to listen to a discourse upon Metaphysics.

Peirce: CP 4.585 Cross-Ref:†† BOOK III

THE AMAZING MAZES

CHAPTER 1

THE FIRST CURIOSITY†1E

"Mazes intricate. Eccentric, interwov'd, yet regular Then most, when most irregular they seem."

Milton's Description of the Mystical Angelic Dance.

§1. STATEMENT OF THE FIRST CURIOSITYE

585. About 1860 I cooked up a mélange of effects of most of the elementary principles of cyclic arithmetic; and ever since, at the end of some evening's card-play, I have occasionally exhibited it in the form of a "trick" (though there is really no trick about the phenomenon) with the uniform result of interesting and surprising all the company, albeit their mathematical powers have ranged from a bare sufficiency for an altruistic tolerance of cards up to those of some of the mightiest mathematicians of the age, who assuredly with a little reflection could have unraveled the marvel. Peirce: CP 4.586 Cross-Ref:†† 586. The following shall describe what I do; but you, Reader, must do it too, if you are to appreciate the curiosity of the effect. So be good enough as to take two packets of playing-cards, the one consisting of a complete red suit and the other of a black suit without the king, the cards of each being arranged in regular order in the packet, so that the face-value of every card is equal to its ordinal number in the packet. Peirce: CP 4.586 Cross-Ref:†† N.B. Throughout all my descriptions of manipulations of cards, it is to be understood, once for all, that the observance of the following STANDING RULES is taken for granted in all cases where the contrary is not expressly directed: Firstly, that a pack or packet of cards held in the hand is, unless otherwise directed, to be held with backs up (though not, of course, while they are in process of arrangement or rearrangement), while a pile of cards FORMED on the table (in contra-distinction to a pile placed, ready formed, on the table, as well as to rows of single cards spread upon the table) is always to be formed with the faces displayed, and left so until they are gathered up. Secondly, that, whether a packet in the hand or a pile on the table be referred to, by the "ordinal, or serial, number" of a single card or of a larger division of the whole is meant its number, counting in the order of succession in the packet or pile, from the card or other part at the BACK of the packet or at the BOTTOM of the pile as "Number 1," to the card or other part at the FACE of the packet or the TOP of the pile; the ordinal or serial number of this last being equal to the cardinal number of cards (or larger divisions COUNTED) in the whole packet or pile; and the few exceptions to this rule will be noted as they occur; Thirdly, that by the "face-value" is meant the number of pips on a plain card, the ace counting as one; while, of the picture-cards, the knave, for which J will usually be written, will count as eleven, the queen, or Q, as twelve, and the king, K, either as thirteen or as the zero of the next suit; and Fourthly, that when a number of piles that have been formed upon the table by dealing out the cards, are to be gathered up, the uniform manner of doing so is to be as follows: The first pile to be taken (which pile this is to be will appear in due time) is to be grasped as a whole and placed (faces up) upon the pile that is to be taken next. Then those two piles are to be grasped as a whole, and placed (faces up) upon the pile that is next to be taken; and so on, until all the piles have been gathered up; when, in accordance with the first Standing Rule, the whole packet is to be turned back up. And note, by the way,

that in consequence of the manner in which the piles are gathered, each, after the first, being placed at the back of those already taken, while in observance of the second Standing Rule, we always count places in a packet from the back of it, it follows that the last pile taken will be the first in the regathered packet, while the first taken will become the last, and all the others in the same complementary way, the ordinal numbers of their gathering and those of their places in the regathered packet adding up to one more than the total number of piles. Peirce: CP 4.587 Cross-Ref:†† 587. Of course, while the red packet and the black packet are getting arranged so that the face-value of each card shall also be its ordinal, or serial, number in the packet, the cards must needs be held faces up. But as soon as they have been arranged, the packet of thirteen cards is to be laid on the table, back up. You then deal -- for, let me repeat it, Reader, by the inexorable laws of psychology, if you do not actually take cards (and the United States Playing-Card Company's "Fauntleroy" playing-cards are the most suitable, although any that run smoothly will do), and actually go through the processes, the whole description can mean nothing to you; -you deal, then, the twelve black cards, one by one, into two piles, the first card being turned to form the bottom of the first pile, the second that of the second pile (on the right hand of the first pile), the third card going on the first pile again, the fourth on the second, and every following card being placed immediately upon the card whose ordinal, or serial, number in the packet before the deal was two lower than the former's ordinal, or serial, number then was. The last card, however, is to be exceptionally treated. Instead of being placed on the top of the second pile according to the rule just given, it is to be placed on the table, face up, and apart from the other cards, to make the bottom card of an isolated pile, to be called the "discard pile"; while, in place of it, the first card of the pile of cards of the red suit, which card will, of course, be the ace, is to be placed face up on the top of the second of the two piles formed by the dealing, where that discarded card would naturally have gone. Now you gather up these two piles by grasping the first, or left-hand pile, placing it, face up, upon the second, or right-hand, pile, and taking up the two together; and you then at once turn the packet back up in compliance with the first standing rule. This whole operation of firstly, dealing out into two piles the packet that was at first entirely composed of black cards; but secondly, placing the last card, face up, on the discard pile, and thirdly, substituting for it the card then at the top of the pile of red cards, by placing this latter, face up, upon the top of the second pile of the deal, and then, fourthly, putting the left-hand, or first, pile of the deal, face up, upon the second, and having taken up the whole packet, turning it with its back up -- this whole quadripartite operation, I say, is to be performed, in all, twelve times in succession. My statement that in this operation the last card is treated exceptionally was quite correct, since its treatment made an exception to the rule of placing each card on the card that before the deal came two places in advance of it in the packet. Had I said it was treated irregularly, I should have written very carelessly, since it is just one of those cases in which a violation of a regularity of a low order establishes a regularity of a much higher order (if John Milton knew the meaning of the word "regular") -- a pronouncement which must be left for the issue of the performance to ratify; and you shall see, Reader, that the event will ratify it with striking emphasis. Already, we begin to see some regularity in the process, since each of the twelve cards placed on the discard-pile in the twelve performances of the quadripartite operation is seen to belong to the black suit; so that the packet held in the hand and dealt out, from being originally entirely black, has now become entirely red. Having placed the red king

upon the face of this packet, you now lay down the latter in order to have your hands free to manipulate the discard-pile. Holding this discard pile as the first standing rule directs, you take the cards singly from the top and range them, one by one, from left to right, in a row upon the table, with their backs up. The length of the table from left to right ought to be double that of the row; and this is one of the reasons for preferring cards of a small size. To guard against any mistake, you may take a peek at the seventh card, to make sure that it is the ace, as it should be. The row being formed, I remark to the company, as you should do in substance, that I reserve the right to move as many of these black cards as I please, at any and all times, from one end of the row to the other; but that beyond doing that, I renounce all right to disarrange those cards. Then, taking up the red cards, and holding the packet with its back up, I (and so must you) request any person to cut it. When he does so, you place the cards he leaves in your hand at the back of the partial packet he removes. This is my proceeding, and must be yours. You then ask some person to say into how many piles (less than thirteen) the red cards shall be dealt. When he has prescribed the number of piles, you are to hold the packet of red cards back up, and deal cards one by one from the back of it, placing each card on the table face up, and each to the right of the last card dealt. When you have dealt out enough to form the bottom cards of piles to the number commanded, you return to the extreme left-hand pile, which you are to imagine as lying next to, and to the right of, the extreme right-hand pile -- as in fact it would come next in clockwise order, if the row were bent down at the ends in the manner

shown in Fig. 224 [Click here to view], where the piles (here supposed to be eight in all) are numbered in the order in which their bottom cards are laid down. Indeed, when more than seven piles are ordered, it is not a bad plan actually to arrange them so. So, counting the piles round and round, whether you place them in a circle or not, you place each card on the pile that comes clockwise next after, or to the right of the pile upon which the card next before it was placed (regulating your imagination as above stated), and so you continue until you have dealt out the whole packet of thirteen cards. You now proceed to gather up the piles according to the Fourth Standing Rule. Peirce: CP 4.588 Cross-Ref:†† 588. That rule, however, does not determine the order of succession in which the piles are to be taken up. I will now give the rule for this. It applies to the dealing

of any prime number of cards, or of any number of cards one less than a prime number, into any number of piles less than that prime number. It happens that that form of statement of this rule which is decidedly the most convenient when the number of piles does not exceed seven, as well as when the whole number of cards differs by less than three from some multiple of the number of piles, becomes quite confusing in other cases. A slight modification of it which I will give as a second form of the rule, sometimes greatly mitigates the inconvenience; and it will be well to acquaint yourself with it. But for the most part, when the first form threatens to be confusing, it will be best to resort to that form of the rule which I describe as the third. Peirce: CP 4.588 Cross-Ref:†† For the purpose of this "first curiosity" (indeed, in every case where a prime number of real cards are dealt out), it matters not what pile you take up first. But in certain cases we shall have occasion to deal out into piles a number of cards, such as 52, which is one less than a prime number. In such case, it will be necessary to add an imaginary card to the pack, since a real card would interfere with certain operations. Now imaginary cards, if allowed to get mixed in with real ones, are liable to get lost. Consequently, in such cases, we have to keep the imaginary card constantly at the face of the pack by taking up first the pile on which it is imagined to fall, that is, the pile next to the right of the one on which the last real card falls. I now proceed to state, in its three forms, the rule for determining what pile is to be taken up next after any given pile that has just been taken. It is assumed that the whole pack of cards dealt consists of a prime number of cards; but, of these cards, the last may be an imaginary one, provided the pile on which it is imagined regularly to fall be taken up first. Peirce: CP 4.588 Cross-Ref:†† First Form of the Rule. Count from the place of the extreme right-hand pile, as zero, either way round, clockwise or counter-clockwise -- preferably in the shortest way -- to the place of the pile on which the last card, real or imaginary, fell. Then, counting the original places of piles, whether the piles themselves still remain in those places or have already been picked up, from the place of the pile last taken, in the same direction, up to the same number, you will reach the place of the next pile to be taken.

Fig. 225

[Click here to view]

Peirce: CP 4.588 Cross-Ref:†† Example. If 13 cards are dealt into five piles, the thirteenth card will fall on the second pile from the extreme right-hand pile going round counter-clockwise. Supposing, then, that the first pile taken is the right-handmost but one, they are all to be taken in the order marked in Fig. 225. Peirce: CP 4.588 Cross-Ref:†† Second Form. Proceed as in the first form of the rule until you have repassed the place of the first pile taken. You will then always find that the place of the last pile taken is nearer to that of some pile, P, previously taken, than it is to the place of that taken immediately before it. Then, the next pile to be taken will be in the same relation of places to the pile taken next after the pile P. Peirce: CP 4.588 Cross-Ref:†† Example. Let 13 cards be dealt into 9 piles. Then the last card will fall on the pile removed 4 places clockwise from the extreme right-hand pile. Then, when you have removed four piles according to the first form of the rule, you will at once perceive, as shown in Fig. 226 (where it is assumed that

Fig. 226 [Click here to view]

the extreme left-hand pile was the one to be taken up first), that for the rest of the regathering, you have simply to take the pile that stands immediately to the left of the place of the last previous removal but one. Peirce: CP 4.588 Cross-Ref:†† Third Form. In this form of the rule vacant places are not counted, but only the remaining piles, which is sometimes much less confusing. It is requisite, however, carefully to note the place of the pile first taken. You begin as in the first form of the rule; but every time you pass over the place whence the first pile was removed, you diminish the number of your count by one, beginning with the count then in progress; and you adhere to this number until you pass the same place again, and consequently again diminish the number of your count, which will thus ultimately be reduced to one, when you will take every pile you come to. Peirce: CP 4.588 Cross-Ref:†† Example. Let a pack of 52 cards be dealt into 22 piles. The first pile taken up must be the one upon which the imaginary fifty-third card falls. It is assumed that, before the deal the cards were arranged in suits in the order ♦ ♠ ♥ ♣ and in each suit in the order of their face-values. Then the different columns of Fig. 227 show the cards at the tops of the different piles while the different horizontal rows show what piles remain, just before you come to count the left-handmost of the remaining piles,

as your countings successively pass through the whole row of piles. The gap between the columns. just after the place where the imaginary card is supposed to have fallen, contains the direction thereafter to diminish by one the number of piles you count. Beneath the designations of the top cards are small type numbers which are the numbers in your different countings through the row of piles; and the last number in each count is followed by a note of admiration that is to be understood as a command to gather up that pile. Beneath it is a heavy faced number, which is the ordinal number of that removal.

Fig. 227

[Click here to view] Peirce: CP 4.589 Cross-Ref:†† 589. I hate to bore readers who are capable of exact thought with redundancies; but others often deploy such brilliant talents in not understanding the plainest statements that have no familiar jingle, that I must beg my more active-minded readers to have patience under the infliction while I exhibit in Fig. 228 the orders in which 5, 8, 9, 10, and 11 piles formed by dealing 13 cards are to be taken up. Peirce: CP 4.590 Cross-Ref:†† 590. When the red cards have thus been regathered, you again hold out the packet to somebody to cut, and again request somebody to say into how many piles

they shall be dealt "in order that the mixing may be as thorough as it may." You follow his directions, and regather the piles according to the same rule as before. If your company is not too intelligent, you might venture to ask somebody, before you regather the piles, to say what pile you shall take up first; but this will be presuming a good deal upon the stupidity of the company; for an inference might be drawn which would go far toward destroying the surprise of the result. Nothing absolutely prevents the cards from being cut and dealt any number of times.

Fig. 228 [Click here to view] Peirce: CP 4.591 Cross-Ref:†† 591. When the number of piles for the last dealing has been given out, you will have to ascertain what transposition of the black cards is required. There are three alternative ways of doing this, which I proceed to describe. The best way is to multiply together the numbers of piles of the different dealings of the red cards, subtracting from each product the highest multiple of 13, if there be any, that is less than that product. The result is the cyclical product. By "the different dealings," you here naturally understand those that have taken place since the last shifting of the black row. If a wrong shift has been made, the simplest way to correct it, after new cuttings and dealings, is to resort to a peep at the black ace, and to determining where it ought to be in the third way explained below. Peirce: CP 4.591 Cross-Ref:†† Thus, if the red cards have been dealt into 5 piles and into 3 piles, since 3 times 5 make 15, and 15-13 = 2, the cyclical product is 2. You now proceed to ascertain how many times 1 has to be cyclically doubled to make that cyclical product. But if 6 doublings do not give it -- which six doublings will give 1 doubling, twice 1 are 2, 2 doublings, twice 2 are 4,

3 doublings, twice 4 are 8, 4 doublings, twice 8 less 13 make 3, 5 doublings, twice 3 are 6, 6 doublings, twice 6 are Q,-Peirce: CP 4.591 Cross-Ref:†† I say if none of the first six doublings gives the cyclical product of the numbers of piles in the dealings, you resort to successive cyclical halvings of 1. The cyclical half of an even number is the simple half; but to get the cyclical half of an odd number, add 7 to half of one less than that number. Thus, The cyclical half of 1 is (0/2)+7 = 7; The cyclical half of 7 is (6/2)+7 = X; The cyclical half of X is 5; The cyclical half of 5 is (4/2)+7 = 9; The cyclical half of 9 is (8/2)+7 = J; The cyclical half of J is (X/2)+7 = Q. Peirce: CP 4.591 Cross-Ref:†† If the cyclical product of the numbers of piles in the dealings is one of the first six results of doubling one, you will have (when the time comes) to bring one card from the right-hand end of the row of black cards to the left-hand end for each such doubling. Thus, if the red cards have twice been deal into 4 piles, four cards must be brought from the right end to the left end of the row of black cards. For 4X4-13 = 3 and 1X24-13 = 3. But if that cyclical product is one of the first six results of successive cyclical halvings of one, one card must be carried from the left to the right end of the row of black cards for every halving. Thus, if the red cards have been dealt into 6 and into 8 piles, 4 black cards must be carried from the left-hand end of the row to the right-hand end of the row. 6X8-3X13 = 9 and it takes 4 cyclical halvings to give 9. If the product of the numbers of piles in the dealings is one more than a multiple of 13, the row of black cards is to remain unshifted. Peirce: CP 4.591 Cross-Ref:†† The second way of determining how the black cards are to be transposed is simply, during the last of the dealings, to note what card is laid upon the king. The face-value of this card is the ordinal, or serial place in the row, counting from the left-hand extremity of it, which the ace must be brought to occupy. Now if you remember, as you always ought to do, where the ace is in the row, you will know how many cards to carry from one end to the other so as to bring the ace into that place. But if in the last dealing the king happens to fall at the top of one of the piles, two lines of conduct are open to you. One would be, in regathering the piles, by a pretended awkwardness in taking up the pile that is to be taken next before the one that the king heads, at first to leave its bottom card on the table, so as to get a glimpse of it before you take it up, as you would regularly have done at first; and if the king should happen to be the last card dealt, the card at the back of the packet would be the

one for you to get sight of, by a similar imitation blunder. In either case, the card you so aim to get sight of would show the right place for the ace in the row. But if you doubt your ability to be gracefully awkward, it always remains open to you to ask to have the red packet cut again and a number of piles for a new deal to be ordered. Peirce: CP 4.591 Cross-Ref:†† The third way of determining the proper transposition of the black cards is a slight modification of the second. It consists in looking at the card whose back is against the face of the king, when you come to cut the red packet so as to bring the king to the face. (Any practical psychologist, such as a prestidigitator must be, can, with the utmost ease, look for the card he wants to see, and can inspect it without detection.) Peirce: CP 4.591 Cross-Ref:†† But whichever of these methods you employ, you should not touch the row of black cards until the red cards having been regathered after the last dealing, you have said something like this: "Now I think that all these dealings and cuttings and exchanges of the last cards have sufficiently mixed up the red cards to give a certain interest to the fact that I am going to show you; namely, that this row of black cards forms an index showing where any red card you would like to see is to be found in the red pack. But since there is no black king in the row, of course the place of the red king cannot be indicated; and for that reason, I shall just cut the pack of red cards so as to bring the king to the face of it, and so render any searching for that card needless." You then cut the red cards. That speech is quite important as restraining the minds of the company from reflecting upon the relation between the effect of your cutting and that of theirs. Without much pause you go on to say that you shall leave the row of black cards just as they are, simply putting so many of them from one end of the row to the other. You now ask some one, "Now, what red card would you like to find?" On his naming the face-value of a card, you begin at the left-hand end of the row of black cards and count them aloud and deliberately, pointing to each one as you count it, until you come to the ordinal number which equals the face-value of the red card called for; and in case that card is the knave or queen, you call "knave" instead of "eleven" on pointing at the eleventh card, and "queen" on pointing at the last card. When you come to call the number that equals that of the red card called for, you turn the card you are pointing at face up. Suppose it is the six, for example. Then you say, naming the card called for, that that card will be the sixth; or if the card turned up was the knave, you say that the card called for will be "in the knave-place," and so in other cases. You then take up the red packet, and counting them out, aloud and deliberately, from one hand to the other, and from the back toward the face of the packet, when you come to the number that equals the face-value of the black card turned, you turn over this card as soon as you have counted it, and lo! it will be the card called for. Peirce: CP 4.592 Cross-Ref:†† 592. The company never fail to desire to see the thing done again; and on their expressing this wish, after impressing on your memory the present place of the black ace, you have only to hold out the red cards to be cut again, and you again go through the rest of the performance, now abbreviating it by having the cards dealt only once. The third time you do it, since you will now have given them the enjoyment of their little astonishment, there will no longer be any reason for not asking somebody to say what pile you shall take up first, although that will soon lead to their seeing that all the cuttings are entirely nugatory. Still they will not thoroughly understand the phenomenon.

Peirce: CP 4.593 Cross-Ref:†† 593. If you wish for an explanation of it, the wish shows that you are not thoroughly grounded in cyclic arithmetic, and that you consequently still have before you the delight of assimilating the first three Abschnitte (for that matter the first hundred pages would suffice to reveal the foundations of the present mystery; but I confess I do not particularly admire the first Abschnitt) of Dedekind's lucid and elegant redaction of the unerring Lejeune-Dirichlet's "Vorlesungen über Zahlentheorie." But, perhaps, on another occasion †1 I will myself give a little essay on the subject, "adapted to the meanest capacity," as some of the books of my boyhood used, not too respectfully, to express it.

Peirce: CP 4.594 Cross-Ref:†† §2. EXPLANATION OF CURIOSITY THE FIRST †2

594. You remember that at the end of my description of the card "trick" that made my first curiosity, I half promised to give, some time, an explanation of its rationale. This half promise I proceed to half redeem. Peirce: CP 4.594 Cross-Ref:†† Suppose a prime number, P, of cards to be dealt into S (for strues) piles, where S]

[Click here to view] Peirce: CP 4.612 Fn P1 p 508 Cross-Ref:†† †P1 See Charmides, p. 160A, and the last chapter of the First Posterior Analytics [A:34]. Peirce: CP 4.613 Fn 1 p 509 †1 Cf. 2.267. Peirce: CP 4.613 Fn P1 p 510 Cross-Ref:†† †P1 {theörémation} is entered in L. & S. [Liddell & Scott, Greek-English Lexicon], with a reference to the Diatribes of Epictetus. Peirce: CP 4.614 Fn 1 p 510 †1 Cf. 239f. Peirce: CP 4.616 Fn 1 p 512 †1 But cf. the edition of 1884, p. 199. Peirce: CP 4.617 Fn 1 p 514 †1 Cf. 3.499. Peirce: CP 4.617 Fn 2 p 514

†2 Cf. 3.492ff. Peirce: CP 4.617 Fn 3 p 514 †3 See his Algebra u. Logik der Relative, passim. Peirce: CP 4.617 Fn 4 p 514 †4 E.g., by Russell in his Principles of Mathematics, p. 10. Peirce: CP 4.617 Fn 1 p 515 †1 See 565-69. Peirce: CP 4.617 Fn 2 p 515 †2 See 569 and 580. Peirce: CP 4.621 Fn 1 p 517 †1 See 560. Peirce: CP 4.624 Fn 1 p 520 †1 See 613. Peirce: CP 4.631 Fn P1 p 527 Cross-Ref:†† †P1 See Note at the end of the article [639ff]. Peirce: CP 4.633 Fn 1 p 530 †1 E.g., Schröder. Peirce: CP 4.633 Fn 2 p 530 †2 Cf. 332ff, 659ff, 673ff. Peirce: CP 4.633 Fn 3 p 530 †3 See 635. Peirce: CP 4.639 Fn 1 p 537 †1 This note was referred to in 631. Cf. also 121ff, 200ff, 219ff. Peirce: CP 4.639 Fn 1 p 539 †1 See e.g. 3.567f. Peirce: CP 4.640 Fn 1 p 540 †1 E.g., Russell, Principles of Mathematics, p. 437. Peirce: CP 4.641 Fn 2 p 540 †2 See e.g., 5.289. Peirce: CP 4.642 Fn 1 p 541 †1 That paper does not seem to have been written. Peirce: CP 4.643 Fn 1 p 543 †1 The Monist, pp. 36-45, vol. 19, January 1909, Peirce's last published paper. Peirce: CP 4.647 Fn 1 p 551 †1 From "Some Amazing Mazes, Fourth Curiosity," c. 1909. Neither the third nor the fourth papers of this series were previously published. The "Third Curiosity" contains little new. In the manuscript the present chapter follows shortly after 6.348. Peirce: CP 4.647 Fn 2 p 551 †2 Cf. 1.203ff.

Peirce: CP 4.648 Fn 3 p 551 †3 Cf. 3.66. Peirce: CP 4.648 Fn 1 p 552 †1 See e.g., Russell, Principles of Mathematics, p. 68. Peirce: CP 4.649 Fn 2 p 552 †2 Cf. 3.537n. Peirce: CP 4.651 Fn 1 p 553 †1 Paradoxien des Unendlichen, §22, Leipzig (1851). Peirce: CP 4.652 Fn 1 p 554 †1 See 3.546. Peirce: CP 4.652 Fn 2 p 554 †2 Cf. 321, 635, 3.232. Peirce: CP 4.652 Fn 3 p 554 †3 See 3.547f. Peirce: CP 4.654 Fn 1 p 555 †1 Cf. 113, 218, 674 and 3.550. Peirce: CP 4.655 Fn 1 p 556 †1 Cf. 3.288. Peirce: CP 4.656 Fn 2 p 556 †2 Leçons sur la théorie des Fonctions, Paris, 1898. Borel does not prove the point here at issue. Peirce: CP 4.657 Fn 3 p 556 †3 See Georg Cantor, Gesammelte Abhandlung, S. 282, Berlin (1932). Peirce: CP 4.657 Fn 4 p 556 †4 See 3.546f. Peirce: CP 4.658 Fn 1 p 557 †1 Liber Abaci (1202). Peirce: CP 4.658 Fn 2 p 557 †2 Ars Geometricæ, Leipzig (1867). Peirce: CP 4.658 Fn 3 p 557 †3 Arithmetica demonstrata (1496). Peirce: CP 4.658 Fn 4 p 557 †4 Tractatus de Arte Numerandi, Strasburg (1488). Peirce: CP 4.658 Fn 5 p 557 †5 Opus Majus, Part 4. Peirce: CP 4.658 Fn 6 p 557 †6 Regule Abaci, Bull. di Bibliographia, T. XIV. Peirce: CP 4.658 Fn 7 p 557

†7 Arithmetica speculativa, Paris (1495). Peirce: CP 4.658 Fn 8 p 557 †8 Opuscula, Strasburg (1490). Peirce: CP 4.658 Fn 9 p 557 †9 Algorisimus, Padua (1483). Peirce: CP 4.658 Fn 10 p 557 †10 By U. Wagner (1482). Peirce: CP 4.658 Fn 11 p 557 †11 Betrede und hubsche Rechnung, Pforzheim (1489). Peirce: CP 4.658 Fn 12 p 557 †12 Anonymous. Peirce: CP 4.658 Fn 13 p 557 †13 Libro de Abacho de Arithmetica, Venice (1484). Peirce: CP 4.658 Fn 14 p 557 †14 Suma, Venice (1494). Peirce: CP 4.658 Fn 15 p 557 †15 By Nicolas Chuquet (1484). Peirce: CP 4.658 Fn 1 p 558 †1 Protomathesis, Paris (1530). Peirce: CP 4.658 Fn 2 p 558 †2 Arithmetica Integra, Nürnberg (1544), Peirce: CP 4.658 Fn 3 p 558 †3 Published in 1522. Peirce: CP 4.658 Fn 4 p 558 †4 Published in 1543. Peirce: CP 4.658 Fn 5 p 558 †5 Arithmeticke, London (1592). Peirce: CP 4.658 Fn 6 p 558 †6 Exercises, London (1594). Peirce: CP 4.658 Fn 7 p 558 †7 The Arte of Vulgar Arithmeticke, London (1600). Peirce: CP 4.658 Fn 8 p 558 †8 Clavis Mathematicæ, London (1631). Peirce: CP 4.658 Fn 9 p 558 †9 Arithmetick, ed. by Hawkins (1678). Peirce: CP 4.658 Fn 10 p 558 †10 Elements of Arithmetic, Philadelphia (1844). Peirce: CP 4.658 Fn 11 p 558

†11 Elements of Arithmetic, Philadelphia (1851, 1855). Peirce: CP 4.660 Fn 1 p 560 †1 Was sind. u. was sollen die Zahlen, §73, §161. Peirce: CP 4.660 Fn 2 p 560 †2 Lehrbuch der Arithmetik u. Algebra, Leipzig (1873). Peirce: CP 4.660 Fn 3 p 560 †3 Op. cit., S. 284f. Peirce: CP 4.663 Fn 1 p 562 †1 See 107, 154, 3.242 and 3.331. Peirce: CP 4.664 Fn 2 p 562 †2 Cf. 3.253ff. Peirce: CP 4.666 Fn 1 p 564 †1 Cf. 190f., 3.262f., 3.562H. Peirce: CP 4.666 Fn P1 p 564 Cross-Ref:†† †P1 When I write a+b, I conceive a to be addend and b to be the augend, on the general principle of putting the operator before the operand, though addition is usually conceived to violate this rule. Peirce: CP 4.667 Fn 1 p 565 †1 Cf. 3.130, 3.327, 3.647. Peirce: CP 4.668 Fn 2 p 565 †2 Cf. 193ff., 3.263f., 3.5621. Peirce: CP 4.668 Fn 1 p 566 †1 x/y = x/y; y\x = y·x. Peirce: CP 4.668 Fn 2 p 566 †2 Hermann Grassmann, Die Ausdehnungslehre, S. 11 (1878). Peirce: CP 4.669 Fn P1 p 566 Cross-Ref:†† †P1 I remark that in my memoir of 1870 on "The Logic of Relatives" [3.53f.], although I insisted with emphasis on there generally being these two kinds of multiplication, I made no reference to Grassmann nor designated them as "internal" and "external" which I am all but absolutely sure that I should have done had I been acquainted with either of Grassmann's volumes. [But cf. 3.152, 3.242n.] So I infer that the too exclusive admiration of Hamilton in our household prevented my acquaintance with that great system. The matter interests me as showing that a person who was studying algebra purely from the point of view of logic was quite independently led to the recognition of the presence of the two kinds of multiplication in associative systems generally, in spite of an undisputed admiration for Hamilton. Peirce: CP 4.674 Fn 1 p 571 †1 See 3.548f. Peirce: CP 4.674 Fn 1 p 572 †1 See 113, 218, 3.550. Peirce: CP 4.675 Fn 2 p 572

†2 Op. cit., S. 168, S. 312. Peirce: CP 4.677 Fn 1 p 574 †1 Op. cit., S. 324f.

Peirce: CP 5 Title-Page COLLECTED PAPERS OF CHARLES SANDERS PEIRCE

EDITED BY CHARLES HARTSHORNE AND PAUL WEISS VOLUME V PRAGMATISM AND PRAGMATICISM CAMBRIDGE HARVARD UNIVERSITY PRESS 1934

Peirce: CP 5 Copyright Page COPYRIGHT, 1934 BY THE PRESIDENT AND FELLOWS OF HARVARD COLLEGE

Peirce: CP 5 Introduction p iii INTRODUCTION

Peirce's punctuation and spelling have, wherever possible, been retained. Titles supplied by the editors for papers previously published are marked with an E, while Peirce's titles for unpublished papers are marked with a P. Peirce's titles for previously published papers and the editors' titles for unpublished papers are not marked. Remarks and additions by the editors are enclosed in light-face square brackets. The editors' footnotes are indicated by various typographical signs, while Peirce's are indicated by numbers. Paragraphs are numbered consecutively throughout the volume. At the top of each page the numbers signify the volume and the first paragraph of that page. All references in the indices are to the numbers of the paragraphs.

Peirce: CP 5 Introduction p iii The department and the editors desire to express their gratitude to Dr. Henry S. Leonard and Miss Margaret Unangst for their assistance with the proofs, references and editorial footnotes.

HARVARD UNIVERSITY MARCH, 1934.

Peirce: CP 5 Editorial Note p v EDITORIAL NOTE

According to William James, the philosophical doctrine known as pragmatism was originated by Charles Sanders Peirce. The present volume contains practically everything of importance which Peirce is known to have written concerning his famous theory of "how to make ideas clear." There is, however, a short passage in a review, written in 1871, of Frazer's edition of the Works of Berkeley, quoted below for its historical interest:

A better rule [than Berkeley's] for avoiding the deceits of language is this: Do things fulfil the same function practically? Then let them be signified by the same word. Do they not? Then let them be distinguished. If I have learned a formula in gibberish which in any way jogs my memory so as to enable me in each single case to act as though I had a general idea, what possible utility is there in distinguishing between such a gibberish and formula and an idea? Why use the term a general idea in such a sense as to separate things which, for all experiential purposes, are the same? (North American Review, vol. 113, p. 469.)

Peirce: CP 5 Editorial Note p v About half of the present volume consists of previously unpublished papers. Their significance lies in the light which they throw upon certain obscure aspects of pragmatism. For anyone other than Peirce himself, the attempt to summarize the gist of his pragmatic point of view will inevitably be difficult and perhaps impossible. It is here attempted only provisionally and subject to the reader's own judgment. Peirce: CP 5 Editorial Note p v In the first place, we learn more specifically than hitherto what Peirce regarded as the alternative to a pragmatic theory of meaning. This alternative was the traditional philosophical view that the abstract explains the concrete, and that the most abstract ideas are ultimate and unanalyzable (see 177, 207, 289, 294, 500ff). Pragmatism or pragmaticism (Peirce's term to indicate his divergencies from other pragmatists) was thus Peirce's way of insisting that abstractions must give an account of themselves, and must do it in terms of concrete experience. He held this position as early as 1868 (cf. 289, 294f, 504n). Only simple qualities of sense or feeling, or blind

reactions between these, can be indefinable; concepts are relational and definable. Peirce: CP 5 Editorial Note p vi In the second place, it is impossible that abstract generalities should be defined, as older empiricisms affirmed, in terms of mere qualities of sensation or emotion. For these qualities are incommunicable (at least in the present state of science; cf. 506) and particular (cf. 299ff, 312); whereas intellectual meanings or concepts must be public and general (312, 467). Peirce: CP 5 Editorial Note p vi Thirdly, that which is most general and public is a habit of behavior (486) directed towards an end (135, 491). The element of generality, which is never absent from the given (181ff, 212, 299ff, 371n), reaches its maximum in purpose; that is, a value capable of being embodied in a wide variety of existents (3, 433). Peirce: CP 5 Editorial Note p vi Fourthly, logic is subsidiary to ethics and esthetics (108ff). The ultimate meaning of an intellectual conception is given by its conceivable bearings upon deliberate or self-controlled conduct; but conduct that is fully deliberate in this sense is ethical and the end which it realizes is esthetic (129ff, 533). Thus pragmatism does not subordinate contemplative values to those of expediency (3, 402n, 429). Peirce: CP 5 Editorial Note p vi Fifthly, pragmatism is conceived to be a method in logic rather than a principle of metaphysics. It provides a maxim which determines the admissibility of explanatory hypotheses (195ff). Peirce: CP 5 Editorial Note p vi Sixthly, it entails scholastic realism, which in its final pragmatic interpretation (503) means the ascription of purposive habits to nature (107, 603). Peirce: CP 5 Editorial Note p vi The first book of the volume contains the Pragmatism Lectures of 1903 which deal primarily with questions of phenomenology, epistemology and value. The second book contains previously published papers. The first three papers show the drift towards pragmatism which characterized Peirce's thought in 1868. Papers IV to VII contain the published accounts of pragmatism on the basis of which his theory has hitherto been judged. Except for the first five papers of the second book, all of the selections in the volume date from 1898 (in most cases from 1903-1907), and all the unpublished papers which compose the third book of the volume are from this later period. There are other topics more or less tenuously related to pragmatism which are discussed at some length in the volume, e.g., the doctrine of critical common-sensism (books II and III; 265 containing the earliest statement); logical theory (318-341; 574-604), and philosophical terminology (413, 610f); while in volume 2, book II, and volume 4, book II, there are papers dealing with other matters which were conceived by Peirce to be particularly germane to the topic of pragmatism. The preface to volume 4 may serve perhaps as a useful guide to relate the papers in volumes 2-4 with those in the present volume, which itself reveals its affiliations with volume 1.

BRYN MAWR COLLEGE UNIVERSITY OF CHICAGO

MARCH, 1934.

Peirce: CP 5 Contents p ix CONTENTS Paragraph Numbers Introduction Editorial Note Preface 1. A Definition of Pragmatic and Pragmatism

1

2. The Architectonic Construction of Pragmatism

5

3. Historical Affinities and Genesis 11

Peirce: CP 5 Contents Book 1 Lecture 1 p ix BOOK I. LECTURES ON PRAGMATISM

LECTURE I. PRAGMATISM: THE NORMATIVE SCIENCES 1. Two Statements of the Pragmatic Maxim 14 2. The Meaning of Probability

19

3. The Meaning of "Practical" Consequences 4. The Relations of the Normative Sciences 34

Peirce: CP 5 Contents Book 1 Lecture 2 p ix LECTURE II. THE UNIVERSAL CATEGORIES 1. Presentness 41 2. Struggle

45

3. Laws: Nominalism 59

Peirce: CP 5 Contents Book 1 Lecture 3 p ix LECTURE III. THE CATEGORIES CONTINUED 1. Degenerate Thirdness

66

2. The Seven Systems of Metaphysics

77

25

3. The Irreducibility of the Categories

82

Peirce: CP 5 Contents Book 1 Lecture 4 p ix LECTURE IV. THE REALITY OF THIRDNESS 1. Scholastic Realism 93 2. Thirdness and Generality 102 3. Normative Judgments

108

4. Perceptual Judgments

115

Peirce: CP 5 Contents Book 1 Lecture 5 p ix LECTURE V. THREE KINDS OF GOODNESS 1. The Divisions of Philosophy

120

2. Ethical and Esthetical Goodness

129

3. Logical Goodness 137

Peirce: CP 5 Contents Book 1 Lecture 6 p x LECTURE VI. THREE TYPES OF REASONING 1. Perceptual Judgments and Generality

151

2. The Plan and Steps of Reasoning 158 3. Inductive Reasoning

167

4. Instinct and Abduction

171

5. The Meaning of an Argument

175

Peirce: CP 5 Contents Book 1 Lecture 7 p x LECTURE VII. PRAGMATISM AND ABDUCTION 1. The Three Cotary Propositions

180

2. Abduction and Perceptual Judgments

182

3. Pragmatism -- the Logic of Abduction

195

4. The Two Functions of Pragmatism206

Peirce: CP 5 Contents Book 2 Paper 1 p x BOOK II. PUBLISHED PAPERS

Paper I. QUESTIONS CONCERNING CERTAIN FACULTIES CLAIMED FOR MAN Question 1. Whether by the simple contemplation of a cognition, independently of any previous knowledge and without reasoning from signs, we are enabled rightly to judge whether that cognition has been determined by a previous cognition or whether it refers immediately to its object 213 Question 2. Whether we have an intuitive self-consciousness

225

Question 3. Whether we have an intuitive power of distinguishing between the subjective elements of different kinds of cognitions 238 Question 4. Whether we have any power of introspection, or whether our whole knowledge of the internal world is derived from the observation of external facts 244 Question 5. Whether we can think without signs

250

Question 6. Whether a sign can have any meaning, if by its definition it is the sign of something absolutely incognizable 254 Question 7. Whether there is any cognition not determined by a previous cognition

Peirce: CP 5 Contents Book 2 Paper 2 p xi II. SOME CONSEQUENCES OF FOUR INCAPACITIES 1. The Spirit of Cartesianism 264 2. Mental Action

266

3. Thought-Signs

283

4. Man, a Sign 310

Peirce: CP 5 Contents Book 2 Paper 3 p xi III. GROUNDS OF VALIDITY OF THE LAWS OF LOGIC: FURTHER CONSEQUENCES OF FOUR INCAPACITIES 1. Objections to the Syllogism

318

2. The Three Kinds of Sophisms

333

3. The Social Theory of Logic

341

Peirce: CP 5 Contents Book 2 Paper 4 p xi IV. THE FIXATION OF BELIEF 1. Science and Logic 358 2. Guiding Principles 365

259

3. Doubt and Belief

370

4. The End of Inquiry 374 5. Methods of Fixing Belief 377

Peirce: CP 5 Contents Book 2 Paper 5 p xi V. HOW TO MAKE OUR IDEAS CLEAR 1. Clearness and Distinctness 388 2. The Pragmatic Maxim

394

3. Some Applications of the Pragmatic Maxim 4. Reality

403

405

Peirce: CP 5 Contents Book 2 Paper 6 p xi VI. WHAT PRAGMATISM IS 1. The Experimentalists' View of Assertion 411 2. Philosophical Nomenclature 3. Pragmaticism

413

414

4. Pragmaticism and Hegelian Absolute Idealism

436

Peirce: CP 5 Contents Book 2 Paper 7 p xi VII. ISSUES OF PRAGMATICISM 1. Six Characters of Critical Common-Sensism

438

2. Subjective and Objective Modality453

Peirce: CP 5 Contents Book 3 Chapter 1 p xii BOOK III. UNPUBLISHED PAPERS

Chapter 1. A SURVEY OF PRAGMATICISM 1. The Kernel of Pragmatism 464 2. The Valency of Concepts 469 3. Logical Interpretants

470

4. Other Views of Pragmatism

494

Peirce: CP 5 Contents Book 3 Chapter 2 p xii 2. PRAGMATICISM AND CRITICAL COMMON-SENSISM

497

Peirce: CP 5 Contents Book 3 Chapter 3 p xii 3. CONSEQUENCES OF CRITICAL COMMON-SENSISM 1. Individualism

502

2. Critical Philosophy and the Philosophy of Common-Sense 3. The Generality of the Possible 4. Valuation

526

533

Peirce: CP 5 Contents Book 3 Chapter 4 p xii 4. BELIEF AND JUDGMENT 1. Practical and Theoretical Beliefs 538 2. Judgment and Assertion

546

Peirce: CP 5 Contents Book 3 Chapter 5 p xii 5. TRUTH 1. Truth as Correspondence 549 2. Truth and Satisfaction

555

3. Definitions of Truth

565

Peirce: CP 5 Contents Book 3 Chapter 6 p xii 6. METHODS FOR ATTAINING TRUTH 1. The First Rule of Logic

574

2. On Selecting Hypotheses 590

Peirce: CP 5 Contents Appendix p xi APPENDIX 1. Knowledge 605 2. Representationism 607 3. Ultimate

608

4. Mr. Peterson's Proposed Discussion

610

505

Peirce: CP 5.1 Cross-Ref:†† PREFACE

§1. A DEFINITION OF PRAGMATIC AND PRAGMATISM †1E

1. Pragmatic anthropology, according to Kant,†2 is practical ethics. Peirce: CP 5.1 Cross-Ref:†† Pragmatic horizon is the adaptation of our general knowledge to influencing our morals. Peirce: CP 5.2 Cross-Ref:†† 2. The opinion that metaphysics is to be largely cleared up by the application of the following maxim for attaining clearness of apprehension: "Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object." Peirce: CP 5.2 Cross-Ref:†† [The doctrine that the whole "meaning" of a conception expresses itself in practical consequences, consequences either in the shape of conduct to be recommended, or in that of experiences to be expected, if the conception be true; which consequences would be different if it were untrue, and must be different from the consequences by which the meaning of other conceptions is in turn expressed. If a second conception should not appear to have other consequences, then it must really be only the first conception under a different name. In methodology it is certain that to trace and compare their respective consequences is an admirable way of establishing the differing meanings of different conceptions.]†3 Peirce: CP 5.3 Cross-Ref:†† 3. This maxim was first proposed by C.S. Peirce in the Popular Science Monthly for January, 1878 (xii. 287);†4 and he explained how it was to be applied to the doctrine of reality. The writer was led to the maxim by reflection upon Kant's Critic of the Pure Reason. Substantially the same way of dealing with ontology seems to have been practised by the Stoics. The writer subsequently saw that the principle might easily be misapplied, so as to sweep away the whole doctrine of incommensurables, and, in fact, the whole Weierstrassian way of regarding the calculus. In 1896 William James published his Will to Believe,†1 and later †2 his Philosophical Conceptions and Practical Results, which pushed this method to such extremes as must tend to give us pause. The doctrine appears to assume that the end of man is action -- a stoical axiom which, to the present writer at the age of sixty, does not recommend itself so forcibly as it did at thirty. If it be admitted, on the contrary, that action wants an end, and that that end must be something of a general description, then the spirit of the maxim itself, which is that we must look to the upshot of our concepts in order rightly to apprehend them, would direct us towards something different from practical facts, namely, to general ideas, as the true interpreters of our thought. Nevertheless, the maxim has approved itself to the writer, after many years

of trial, as of great utility in leading to a relatively high grade of clearness of thought. He would venture to suggest that it should always be put into practice with conscientious thoroughness, but that, when that has been done, and not before, a still higher grade of clearness of thought can be attained by remembering that the only ultimate good which the practical facts to which it directs attention can subserve is to further the development of concrete reasonableness †3; so that the meaning of the concept does not lie in any individual reactions at all, but in the manner in which those reactions contribute to that development. Indeed, in the article of 1878, above referred to, the writer practised better than he preached; for he applied the stoical maxim most unstoically, in such a sense as to insist upon the reality of the objects of general ideas in their generality.†4 Peirce: CP 5.4 Cross-Ref:†† 4. A widely current opinion during the last quarter of a century has been that reasonableness is not a good in itself, but only for the sake of something else. Whether it be so or not seems to be a synthetical question, not to be settled by an appeal to the principle of contradiction -- as if a reason for reasonableness were absurd. Almost everybody will now agree that the ultimate good lies in the evolutionary process in some way. If so, it is not in individual reactions in their segregation, but in something general or continuous. Synechism †1 is founded on the notion that the coalescence, the becoming continuous, the becoming governed by laws, the becoming instinct with general ideas, are but phases of one and the same process of the growth of reasonableness. This is first shown to be true with mathematical exactitude in the field of logic, and is thence inferred to hold good metaphysically. It is not opposed to pragmatism in the manner in which C.S. Peirce applied it, but includes that procedure as a step.

Peirce: CP 5.5 Cross-Ref:†† §2. THE ARCHITECTONIC CONSTRUCTION OF PRAGMATISM †2

5. . . . Pragmatism was not a theory which special circumstances had led its authors to entertain. It had been designed and constructed, to use the expression of Kant,†3 architectonically. Just as a civil engineer, before erecting a bridge, a ship, or a house, will think of the different properties of all materials, and will use no iron, stone, or cement, that has not been subjected to tests; and will put them together in ways minutely considered, so, in constructing the doctrine of pragmatism the properties of all indecomposable concepts †4 were examined and the ways in which they could be compounded. Then the purpose of the proposed doctrine having been analyzed, it was constructed out of the appropriate concepts so as to fulfill that purpose. In this way, the truth of it was proved.†5 There are subsidiary confirmations of its truth; but it is believed that there is no other independent way of strictly proving it. . . . Peirce: CP 5.6 Cross-Ref:†† 6. But first, what is its purpose? What is it expected to accomplish? It is expected to bring to an end those prolonged disputes of philosophers which no observations of facts could settle, and yet in which each side claims to prove that the other side is in the wrong. Pragmatism maintains that in those cases the disputants must be at cross-purposes. They either attach different meanings to words, or else one

side or the other (or both) uses a word without any definite meaning. What is wanted, therefore, is a method for ascertaining the real meaning of any concept, doctrine, proposition, word, or other sign. The object of a sign is one thing; its meaning is another. Its object is the thing or occasion, however indefinite, to which it is to be applied. Its meaning is the idea which it attaches to that object, whether by way of mere supposition, or as a command, or as an assertion. Peirce: CP 5.7 Cross-Ref:†† 7. Now every simple idea is composed of one of three classes; and a compound idea is in most cases predominantly of one of those classes. Namely, it may, in the first place, be a quality of feeling, which is positively such as it is, and is indescribable; which attaches to one object regardless of every other; and which is sui generis and incapable, in its own being, of comparison with any other feeling, because in comparisons it is representations of feelings and not the very feelings themselves that are compared.†1 Or, in the second place, the idea may be that of a single happening or fact, which is attached at once to two objects, as an experience, for example, is attached to the experiencer and to the object experienced.†2 Or, in the third place, it is the idea of a sign or communication conveyed by one person to another (or to himself at a later time) in regard to a certain object well known to both. . . .†3 Now the bottom meaning of a sign cannot be the idea of a sign, since that latter sign must itself have a meaning which would thereby become the meaning of the original sign. We may therefore conclude that the ultimate meaning of any sign consists either in an idea predominantly of feeling or in one predominantly of acting and being acted on.†4 For there ought to be no hesitation in assenting to the view that all those ideas which attach essentially to two objects take their rise from the experience of volition and from the experience of the perception of phenomena which resist direct efforts of the will to annul or modify them. Peirce: CP 5.8 Cross-Ref:†† 8. But pragmatism does not undertake to say in what the meanings of all signs consist, but merely to lay down a method of determining the meanings of intellectual concepts, that is, of those upon which reasonings may turn. Now all reasoning that is not utterly vague, all that ought to figure in a philosophical discussion involves, and turns upon, precise necessary reasoning. Such reasoning is included in the sphere of mathematics, as modern mathematicians conceive their science. "Mathematics," said Benjamin Peirce, as early as 1870, "is the science which draws necessary conclusions";†1 and subsequent writers have substantially accepted this definition, limiting it, perhaps, to precise conclusions. The reasoning of mathematics is now well understood.†2 It consists in forming an image of the conditions of the problem, associated with which are certain general permissions to modify the image, as well as certain general assumptions that certain things are impossible. Under the permissions, certain experiments are performed upon the image, and the assumed impossibilities involve their always resulting in the same general way. The superior certainty of the mathematician's results, as compared, for example, with those of the chemist, are due to two circumstances. First, the mathematician's experiments being conducted in the imagination upon objects of his own creation, cost next to nothing; while those of the chemist cost dear. Secondly, the assurance of the mathematician is due to his reasoning only concerning hypothetical conditions, so that his results have the generality of his conditions; while the chemist's experiments relating to what will happen as a matter of fact are always open to the doubt whether unknown conditions may not alter. Thus, the mathematician knows that a column of figures will add up the same, whether it be set down in black ink or in red; because he goes on the

assumption that the sum of any two numbers of which one is M and the other one more than N will be one more than the sum of M and N; and this assumption says nothing about the color of the ink. The chemist assumes that when he mixes two liquids in a test-tube, there will or will not be a precipitate whether the Dowager Empress of China happens to sneeze at the time, because his experience has always been that laboratory experiments are not affected by such distant conditions. Still, the solar system is moving through space at a great rate, and there is a bare possibility that it may just then have entered a region in which sneezing has very surprising force. Peirce: CP 5.9 Cross-Ref:†† 9. Such reasonings and all reasonings turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions. Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a "practical consideration." Hence is justified the maxim, belief in which constitutes pragmatism; namely, Peirce: CP 5.9 Cross-Ref:†† In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception; and the sum of these consequences will constitute the entire meaning of the conception. Peirce: CP 5.10 Cross-Ref:†† 10. Many plausible arguments in favor of this doctrine could easily be adduced; but the only way hitherto discovered of really proving its truth, without in any measure begging the question, is by following the thorny path that we have thus very roughly sketched.

Peirce: CP 5.11 Cross-Ref:†† §3. HISTORICAL AFFINITIES AND GENESIS †1

11. . . . Any philosophical doctrine that should be completely new could hardly fail to prove completely false; but the rivulets at the head of the river of pragmatism are easily traced back to almost any desired antiquity. Peirce: CP 5.11 Cross-Ref:†† Socrates bathed in these waters. Aristotle rejoices when he can find them. They run, where least one would suspect them, beneath the dry rubbish-heaps of Spinoza. Those clean definitions that strew the pages of the Essay concerning Humane Understanding (I refuse to reform the spelling), had been washed out in these same pure springs. It was this medium, and not tar-water, that gave health and strength to Berkeley's earlier works, his Theory of Vision and what remains of his Principles. From it the general views of Kant derive such clearness as they have. Auguste Comte made still more -- much more -- use of this element; as much as he saw his way to using. Unfortunately, however, both he and Kant, in their rather opposite ways, were in the habit of mingling these sparkling waters with a certain mental sedative to which many men are addicted -- and the burly business men very likely to their benefit, but which plays sad havoc with the philosophical constitution. I refer to the habit of cherishing contempt for the close study of logic.

Peirce: CP 5.12 Cross-Ref:†† 12. So much for the past. The ancestry of pragmatism is respectable enough; but the more conscious adoption of it as lanterna pedibus in the discussion of dark questions, and the elaboration of it into a method in aid of philosophic inquiry came, in the first instance, from the humblest souche imaginable. It was in the earliest seventies †1 that a knot of us young men in Old Cambridge, calling ourselves, half-ironically, half-defiantly, "The Metaphysical Club," -- for agnosticismwas then riding its high horse, and was frowning superbly upon all metaphysics -- used to meet, sometimes in my study, sometimes in that of William James. It may be that some of our old-time confederates would today not care to have such wild-oats-sowings made public, though there was nothing but boiled oats, milk, and sugar in the mess. Mr. Justice Holmes, however, will not, I believe, take it ill that we are proud to remember his membership; nor will Joseph Warner, Esq.†2 Nicholas St. John Green was one of the most interested fellows, a skillful lawyer and a learned one, a disciple of Jeremy Bentham. His extraordinary power of disrobing warm and breathing truth of the draperies of long worn formulas, was what attracted attention to him everywhere. In particular, he often urged the importance of applying Bain's †3 definition of belief, as "that upon which a man is prepared to act." From this definition, pragmatism is scarce more than a corollary; so that I am disposed to think of him as the grandfather of pragmatism. Chauncey Wright,†1 something of a philosophical celebrity in those days, was never absent from our meetings. I was about to call him our corypheus; but he will better be described as our boxing-master whom we -- I particularly -- used to face to be severely pummelled. He had abandoned a former attachment to Hamiltonianism to take up with the doctrines of Mill, to which and to its cognate agnosticism he was trying to weld the really incongruous ideas of Darwin. John Fiske and, more rarely, Francis Ellingwood Abbot, were sometimes present, lending their countenances to the spirit of our endeavours, while holding aloof from any assent to their success. Wright, James, and I were men of science, rather scrutinizing the doctrines of the metaphysicians on their scientific side than regarding them as very momentous spiritually. The type of our thought was decidedly British. I, alone of our number, had come upon the threshing-floor of philosophy through the doorway of Kant, and even my ideas were acquiring the English accent. Peirce: CP 5.13 Cross-Ref:†† 13. Our metaphysical proceedings had all been in winged words (and swift ones, at that, for the most part), until at length, lest the club should be dissolved, without leaving any material souvenir behind, I drew up a little paper expressing some of the opinions that I had been urging all along under the name of pragmatism. This paper was received with such unlooked-for kindness, that I was encouraged, some half dozen years later, on the invitation of the great publisher, Mr. W.H. Appleton, to insert it, somewhat expanded, in the Popular Science Monthly for November, 1877 and January, 1878, not with the warmest possible approval of the Spencerian editor, Dr. Edward Youmans. The same paper appeared the next year in a French redaction in the Revue Philosophique (Vol. VI, 1878, p. 553; Vol. VII, 1879, p. 39). In those medieval times, I dared not in type use an English word to express an idea unrelated to its received meaning. The authority of Mr. Principal Campbell †2 weighed too heavily upon my conscience. I had not yet come to perceive, what is so plain today, that if philosophy is ever to stand in the ranks of the sciences, literary elegance must be sacrificed -- like the soldier's old brilliant uniforms -- to the stern requirements of efficiency, and the philosophist must be encouraged -- yea, and required -- to coin new terms to express such new scientific concepts as he may

discover, just as his chemical and biological brethren are expected to do. Indeed, in those days, such brotherhood was scorned, alike on the one side and on the other -- a lamentable but not surprising state of scientific feeling. As late as 1893, when I might have procured the insertion of the word pragmatism in the Century Dictionary, it did not seem to me that its vogue was sufficient to warrant that step.†P1

Peirce: CP 5.14 Cross-Ref:†† BOOK I LECTURES ON PRAGMATISMP†1

PRAGMATISM AND PRAGMATICISM

LECTURE I PRAGMATISM: THE NORMATIVE SCIENCES

§1. TWO STATEMENTS OF THE PRAGMATIC MAXIM

14. A certain maxim of Logic which I have called Pragmatism has recommended itself to me for divers reasons and on sundry considerations. Having taken it as my guide in most of my thought, I find that as the years of my knowledge of it lengthen, my sense of the importance of it presses upon me more and more. If it is only true, it is certainly a wonderfully efficient instrument. It is not to philosophy only that it is applicable. I have found it of signal service in every branch of science that I have studied. My want of skill in practical affairs does not prevent me from perceiving the advantage of being well imbued with pragmatism in the conduct of life. Peirce: CP 5.15 Cross-Ref:†† 15. Yet I am free to confess that objections to this way of thinking have forced themselves upon me and have been found more formidable the further my plummet has been dropped into the abyss of philosophy, and the closer my questioning at each new attempt to fathom its depths. Peirce: CP 5.15 Cross-Ref:†† I propose, then, to submit to your judgment in half a dozen lectures an examination of the pros and cons of pragmatism by means of which I hope to show you the result of allowing to both pros and cons their full legitimate values. With more time I would gladly follow up the guiding thread so caught up and go on to ascertain what are the veritable conclusions, or at least the genera of veritable conclusions to which a carefully rectified pragmatism will truly lead. If you find what

I say acceptable, you will have learned something worth your while. If you can refute me, the gain will be chiefly on my side; but even in that I anticipate your acknowledging, when I take my leave of you, that the discussion has not been without profit; and in future years I am confident that you will recur to these thoughts and find that you have more to thank me for than you could understand at first. Peirce: CP 5.16 Cross-Ref:†† 16. I suppose I may take it for granted that you all know what pragmatism is. I have met with a number of definitions of it lately, against none of which I am much disposed to raise any violent protest. Yet to say exactly what pragmatism is describes pretty well what you and I have to puzzle out together. Peirce: CP 5.16 Cross-Ref:†† We must start with some rough approximation of it, and I am inclined to think that the shape in which I first stated [it] will be the most useful one to adopt as matter to work upon, chiefly because it is the form most personal to your lecturer, and [upon] which for that reason he can discourse most intelligently. Besides pragmatism and personality are more or less of the same kidney. Peirce: CP 5.17 Cross-Ref:†† 17. I sent forth my statement in January 1878; and for about twenty years never heard from it again. I let fly my dove; and that dove has never come back to me to this very day. But of late quite a brood of young ones have been fluttering about, from the feathers of which I might fancy that mine had found a brood. To speak plainly, a considerable number of philosophers have lately written as they might have written in case they had been reading either what I wrote but were ashamed to confess it, or had been reading something that some reader of mine had read. For they seem quite disposed to adopt my term pragmatism. I shouldn't wonder if they were ashamed of me. What could be more humiliating than to confess that one has learned anything of a logician? But for my part I am delighted to find myself sharing the opinions of so brilliant a company. The new pragmatists seem to be distinguished for their terse, vivid and concrete style of expression together with a certain buoyancy of tone as if they were conscious of carrying about them the master key to all the secrets of metaphysics. Peirce: CP 5.17 Cross-Ref:†† Every metaphysician is supposed to have some radical fault to find with every other, and I cannot find any direr fault to find with the new pragmatists than that they are lively. In order to be deep it is requisite to be dull. Peirce: CP 5.18 Cross-Ref:†† 18. On their side, one of the faults that I think they might find with me is that I make pragmatism to be a mere maxim of logic instead of a sublime principle of speculative philosophy. In order to be admitted to better philosophical standing I have endeavored to put pragmatism as I understand it into the same form of a philosophical theorem. I have not succeeded any better than this: Peirce: CP 5.18 Cross-Ref:†† Pragmatism is the principle that every theoretical judgment expressible in a sentence in the indicative mood is a confused form of thought whose only meaning, if it has any, lies in its tendency to enforce a corresponding practical maxim expressible as a conditional sentence having its apodosis in the imperative mood. Peirce: CP 5.18 Cross-Ref:††

But the Maxim of Pragmatism, as I originally stated it, Revue philosophique VII, is as follows: Peirce: CP 5.18 Cross-Ref:†† Considérer quels sont les effets pratiques que nous pensons pouvoir être produits par l'objet de notre conception. La conception de tous ces effets est la conception complète de l'objet. [p. 48.] Peirce: CP 5.18 Cross-Ref:†† Pour développer le sens d'une pensée, il faut donc simplement déterminer quelles habitudes elle produit, car le sens d'une chose consiste simplement dans les habitudes qu'elle implique. Le caractère d'une habitude dépend de la façon dont elle peut nous faire agir non pas seulement dans telle circonstance probable, mais dans toute circonstance possible, si improbable qu'elle puisse être. Ce qu'est une habitude dépend de ces deux points: quand et comment elle fait agir. Pour le premier point: quand? tout stimulant à l'action dérive d'une perception; pour le second point: comment? le but de toute action est d'amener au résultat sensible. Nous atteignons ainsi le tangible et le pratique comme base de toute différence de pensée, si subtile qu'elle puisse être. [p. 47.]

Peirce: CP 5.19 Cross-Ref:†† §2. THE MEANING OF PROBABILITY

19. The utility of the maxim, provided it is only true, appears in a sufficient light in the original article. I will here add a few examples which were not given in that paper. Peirce: CP 5.19 Cross-Ref:†† There are many problems connected with probabilities which are subject to doubt. One of them, for example, is this: Suppose an infinitely large company of infinitely rich men sit down to play against an infinitely rich bank at a game of chance, at which neither side has any advantage, each one betting a franc against a franc at each bet. Suppose that each player continues to play until he has netted a gain of one franc and then retires, surrendering his place to a new player. Peirce: CP 5.19 Cross-Ref:†† The chance that a player will ultimately net a gain of a franc may be calculated as follows: Peirce: CP 5.19 Cross-Ref:†† Let XL be a player's chance, if he were to continue playing indefinitely, of ever netting a gain of 1 franc. Peirce: CP 5.19 Cross-Ref:†† But after he has netted a gain of 1 franc, his chance of doing which is X[1], he is no richer than before, since he is infinitely rich. Consequently his chance of winning the second franc, after he has won the first, is the same as his chance of winning the first franc. That is, it is X[1] and his chance of winning both is X[2] = (X[1])2. And so in general, X[L] = (X[1])L. Peirce: CP 5.19 Cross-Ref:††

Now his chance of netting a gain of 1 franc, X[1], is the sum of the chances of the two ways in which it may come about; namely by first winning the first bet of which the chance is 1/2, and by first losing the first bet and then netting a gain of 2 francs of which the chance is 1/2 X[1]2. Therefore

X[1] = 1/2 + 1/2 X[1]2 or

X[1]2 - 2X[1] + 1 = 0

or

(X[1] - 1)2 = 0.

But if the square of a number is zero, the number itself is zero. Therefore

X[1] - 1 = 0 or

X[1] = 1.

Consequently, the books would say it was dead certain that any player will ultimately net his winning of a franc and retire. If so it must be certain that every player would win his franc and would retire. Peirce: CP 5.19 Cross-Ref:†† Consequently there would be a continual outflow of money from the bank. And yet, since the game is an even one, the banker would not net any loss. How is this paradox to be explained? Peirce: CP 5.20 Cross-Ref:†† 20. The theory of probabilities is full of paradoxes and puzzles. Let us, then, apply the maxim of pragmatism to the solution of them. Peirce: CP 5.20 Cross-Ref:†† In order to do this, we must ask What is meant by saying that the probability of an event has a certain value, p? According to the maxim of pragmatism, then, we must ask what practical difference it can make whether the value is p or something else. Then we must ask how are probabilities applied to practical affairs. The answer is that the great business of insurance depends upon it. Probability is used in insurance to determine how much must be paid on a certain risk to make it safe to pay a certain sum if the event insured against should occur. Then, we must ask how can it be safe to engage to pay a large sum if an uncertain event occurs. The answer is that the insurance company does a very large business and is able to ascertain pretty closely out of a thousand risks of a given description how many in any one year will be losses. The business problem is this. The number of policies of a certain description that can be sold in a year will depend on the price set upon them. Let p be that price, and let n be the number that can be sold at that price, so that the larger p is, the smaller n will be. Now n being a large number a certain proportion q of these policies, qn in all, will be losses during the year; and if I be the loss on each, qnl will be the total loss. Then what the insurance company has to do is to set p at such a figure that pn-qln or (p-ql)n shall reach its maximum possible value.

The solution of this equation is:

p = q l + ((δp/δn)(n))

where δp/δn is the amount by which the price would have to be lowered in order to sell one policy more. Of course if the price were raised instead of lowered just one policy fewer would be sold. For then by so lowering the profit from being

(p - ql)n

[it] would be changed to

(p - ql - δp/δn)(n + 1)

that is to

(p - ql)n + p - ql - δp/δn(n + 1)

and this being less than before ql + δp/δn(n + 1) > p

and by raising it, the change would be to

(p - ql + δp/δn)(n - 1)

that is to

(p - ql)(n - p + ql + δp/δn(n - 1)

and this being less than before

p > ql + δp/δn(n-1)

so since p is intermediate between

ql + (δp/δn)n + δp/δn

and

ql + (δp/δn)n - δp/δn

and δp/δn is very small, it must be close to the truth to write

p = ql + δp/δn(n).

Peirce: CP 5.21 Cross-Ref:†† 21. This is the problem of insurance. Now in order that probability may have any bearing on this problem, it is obvious that it must be of the nature of a real fact and not a mere state of mind. For facts only enter into the solution of the problem of insurance. And this fact must evidently be a fact of statistics. Peirce: CP 5.21 Cross-Ref:†† Without now going into certain reasons of detail that I should enter into if I were lecturing on probabilities, it must be that probability is a statistical ratio; and further, in order to satisfy still more special conditions, it is convenient, for the class of problems to which insurance belongs, to make it the statistical ratio of the number of experiential occurrences of a specific kind to the number of experiential occurrences of a generic kind, in the long run.†1 Peirce: CP 5.21 Cross-Ref:†† In order, then, that probability should mean anything, it will be requisite to specify to what species of event it refers and to what genus of event it refers. Peirce: CP 5.21 Cross-Ref:†† It also refers to a long run, that is, to an indefinitely long series of occurrences taken together in the order of their occurrence in possible experience. Peirce: CP 5.21 Cross-Ref:†† In this view of the matter, we note, to begin with, that a given species of event considered as belonging to a given genus of events does not necessarily have any definite probability. Because [it may be the case that] the probability is the ratio of one infinite multitude to another. Now infinity divided by infinity is altogether indeterminate, except in special cases.

Peirce: CP 5.22 Cross-Ref:†† 22. It is very easy to give examples of events that have no definite probability. If a person agrees to toss up a cent again and again forever and beginning as soon as the first head turns up whenever two heads are separated by any odd number of tails in the succession of throws, to pay 2 to that power in cents, provided that whenever the two successive heads are separated by any even number of throws he receives 2 to that power in cents,†P1 it is impossible to say what the probability will be that he comes out a winner. In half of the cases after the first head the next throw will be a head and he will receive (-2)0 = 1 cent. Which since it happens half the time will be in the long run a winning of 1/2 a cent per head thrown. Peirce: CP 5.22 Cross-Ref:†† But in half of the other half the cases, that is in 1/4 of all the cases, one tail will intervene and he will have to receive (-2)1 = -2 cents, i.e., he will have to pay 2 cents, which happening 1/4 of the time will make an average loss of 1/2 a cent per head thrown. Peirce: CP 5.22 Cross-Ref:†† But in half the remaining quarter of the cases, i.e., of all the cases, two tails will intervene and he will receive (-2)2 = 4 cents which happening one every eight times will be worth 1/2 a cent per head thrown and so on; so that his account in the long run will be 1/2-1/2+1/2-1/2+1/2-1/2+1/2-1/2 ad infinitum, the sum of which may be 1/2 or may be zero. Or rather it is quite indeterminate. Peirce: CP 5.22 Cross-Ref:†† If instead of being paid (-2)n when n is the number of intervening tails, he were paid (-2)n2 the result would be he would probably either win or lose enormously without there being any definite probability that it would be winning rather than losing. Peirce: CP 5.22 Cross-Ref:†† I think I may recommend this game with confidence to gamblers as being the most frightful ruin yet invented; and a little cheating would do everything in it. Peirce: CP 5.23 Cross-Ref:†† 23. Now let us revert to our original problem †1 and consider the state of things after every other bet. After the second, 1/4 of the players will have gained, gone out, and been replaced by players who have gained and gone out, so that a number of francs equal to half the number of seats will have been paid out by the bank, 1/4 of the players will have gained and gone out and been replaced by players who have lost, making the bank even; 1/4 of the players will have lost and then gained, making the bank and them even; 1/4 of the players will have lost twice, making a gain to the bank of half as many francs as there are seats at the table. The bank then will be where it was. Players to the number of three-quarters of the seats will have netted their franc each; but players to the number of a quarter of the seats will have lost two francs each and another equal number one franc each, just paying for the gains of those who have retired. That is the way it will happen every time. Peirce: CP 5.23 Cross-Ref:†† Just before the fifth bet of the players at the table, 3/8 will have lost nothing, 1/4 will have lost one franc, 1/4 two francs, 1/16 three francs and 1/16 four francs.

Thus some will always have lost a good deal. Those who sit at the table will among them always have paid just what those who have gone out have carried away. Peirce: CP 5.24 Cross-Ref:†† 24. But it will be asked: How then can it happen that all gain? I reply that I never said that all would gain, I only said that the probability was 1 that anyone would ultimately gain his franc. But does not probability 1 mean certainty? Not at all, it only means that the ratio of the number of those who ultimately gain to the total number is 1. Since the number of seats at the table is infinite the ratio of the number of those who never gain to the number of seats may be zero and yet they may be infinitely numerous. So that probabilities 1 and 0 are very far from corresponding to certainty pro and con.†2

Peirce: CP 5.25 Cross-Ref:†† §3. THE MEANING OF "PRACTICAL" CONSEQUENCES

25. If I were to go into practical matters, the advantage of pragmatism, of looking at the substantial practical issue, would be still more apparent. But here pragmatism is generally practised by successful men. In fact, the genus of efficient men [is] mainly distinguished from inefficient precisely by this. Peirce: CP 5.26 Cross-Ref:†† 26. There is no doubt, then, that pragmatism opens a very easy road to the solution of an immense variety of questions. But it does not at all follow from that, that it is true. On the contrary, one may very properly entertain a suspicion of any method which so resolves the most difficult questions into easy problems. No doubt Ockham's razor is logically sound. A hypothesis should be stripped of every feature which is in no wise called for to furnish an explanation of observed facts. Entia non sunt multiplicanda praeter necessitatem; only we may very well doubt whether a very simple hypothesis can contain every factor that is necessary. Certain it is that most hypotheses which at first seemed to unite great simplicity with entire sufficiency have had to be greatly complicated in the further progress of science. Peirce: CP 5.27 Cross-Ref:†† 27. What is the proof that the possible practical consequences of a concept constitute the sum total of the concept? The argument upon which I rested the maxim in my original paper †1 was that belief consists mainly in being deliberately prepared to adopt the formula believed in as the guide to action. If this be in truth the nature of belief, then undoubtedly the proposition believed in can itself be nothing but a maxim of conduct. That I believe is quite evident. Peirce: CP 5.28 Cross-Ref:†† 28. But how do we know that belief is nothing but the deliberate preparedness to act according to the formula believed? Peirce: CP 5.28 Cross-Ref:†† My original article carried this back to a psychological principle. The conception of truth, according to me, was developed out of an original impulse to act consistently, to have a definite intention. But in the first place, this was not very clearly made out, and in the second place, I do not think it satisfactory to reduce such

fundamental things to facts of psychology. For man could alter his nature, or his environment would alter it if he did not voluntarily do so, if the impulse were not what was advantageous or fitting. Why has evolution made man's mind to be so constructed? That is the question we must nowadays ask, and all attempts to ground the fundamentals of logic on psychology are seen to be essentially shallow. Peirce: CP 5.29 Cross-Ref:†† 29. The question of the nature of belief, or in other words the question of what the true logical analysis of the act of judgment is, is the question upon which logicians of late years have chiefly concentrated their energies. Is the pragmatistic answer satisfactory? Peirce: CP 5.29 Cross-Ref:†† Do we not all perceive that judgment is something closely allied to assertion?†1 That is the view that ordinary speech entertains. A man or woman will be heard to use the phrase, "I says to myself." That is, judgment is held to be either no more than an assertion to oneself or at any rate something very like that. Peirce: CP 5.30 Cross-Ref:†† 30. Now it is a fairly easy problem to analyze the nature of assertion.†2 To find an easily dissected example, we shall naturally take a case where the assertive element is magnified -- a very formal assertion, such as an affidavit. Here a man goes before a notary or magistrate and takes such action that if what he says is not true, evil consequences will be visited upon him, and this he does with a view to thus causing other men to be affected just as they would be if the proposition sworn to had presented itself to them as a perceptual fact. Peirce: CP 5.30 Cross-Ref:†† We thus see that the act of assertion is an act of a totally different nature from the act of apprehending the meaning of the proposition and we cannot expect that any analysis of what assertion is (or any analysis of what judgment or belief is, if that act is at all allied to assertion), should throw any light at all on the widely different question of what the apprehension of the meaning of a proposition is. Peirce: CP 5.31 Cross-Ref:†† 31. What is the difference between making an assertion and laying a wager? Both are acts whereby the agent deliberately subjects himself to evil consequences if a certain proposition is not true. Only when he offers to bet he hopes the other man will make himself responsible in the same way for the truth of the contrary proposition; while when he makes an assertion he always (or almost always) wishes the man to whom he makes it to be led to do what he does. Accordingly in our vernacular "I will bet" so and so, is the phrase expressive of a private opinion which one does not expect others to share, while "You bet" is a form of assertion intended to cause another to follow suit. Peirce: CP 5.32 Cross-Ref:†† 32. Such then seems at least in a preliminary glance at the matter to be a satisfactory account of assertion. Now let us pass to judgment and belief. There can, of course, be no question that a man will act in accordance with his belief so far as his belief has any practical consequences. The only doubt is whether this is all that belief is, whether belief is a mere nullity so far as it does not influence conduct. What possible effect upon conduct can it have, for example, to believe that the diagonal of a square is incommensurable with the side? Name a discrepancy e no matter how small,

and the diagonal differs from a rational quantity by much less than that. Professor Newcomb in his calculus and all mathematicians of his rather antiquated fashion think that they have proved two quantities to be equal when they have proved that they differ by less than any assignable quantity. I once tried hard to make Newcomb say whether the diagonal of the square differed from a rational fraction of the side or not; but he saw what I was driving at and would not answer. The proposition that the diagonal is incommensurable has stood in the textbooks from time immemorial without ever being assailed and I am sure that the most modern type of mathematician holds to it most decidedly. Yet it seems quite absurd to say that there is any objective practical difference between commensurable and incommensurable.†1 Peirce: CP 5.33 Cross-Ref:†† 33. Of course you can say if you like that the act of expressing a quantity as a rational fraction is a piece of conduct and that it is in itself a practical difference that one kind of quantity can be so expressed and the other not. But a thinker must be shallow indeed if he does not see that to admit a species of practicality that consists in one's conduct about words and modes of expression is at once to break down all the bars against the nonsense that pragmatism is designed to exclude. Peirce: CP 5.33 Cross-Ref:†† What the pragmatist has his pragmatism for is to be able to say: here is a definition and it does not differ at all from your confusedly apprehended conception because there is no practical difference. But what is to prevent his opponent from replying that there is a practical difference which consists in his recognizing one as his conception and not the other? That is, one is expressible in a way in which the other is not expressible. Peirce: CP 5.33 Cross-Ref:†† Pragmatism is completely volatilized if you admit that sort of practicality.

Peirce: CP 5.34 Cross-Ref:†† §4. THE RELATIONS OF THE NORMATIVE SCIENCES †1

34. It must be understood that all I am now attempting to show is that Pragmatism is apparently a matter of such great probable concern, and at the same time so much doubt hangs over its legitimacy, that it will be well worth our while to make a methodical, scientific, and thorough examination of the whole question, so as to make sure of our ground, and obtain some secure method for such a preliminary filtration of questions as pragmatism professes to furnish. Peirce: CP 5.34 Cross-Ref:†† Let us, then, enter upon this inquiry. But before doing so let us mark out the proposed course of it. That should always be done in such cases, even if circumstances subsequently require the plan to be modified, as they usually will. Peirce: CP 5.34 Cross-Ref:†† Although our inquiry is to be an inquiry into truth, whatever the truth may turn out to be, and therefore, of course, is not to be influenced by any liking for pragmatism or any pride in it as an American doctrine, yet still we do not come to this inquiry, any more than anybody comes to any inquiry, in that blank state that the

lawyers pretend to insist upon as desirable, though I give them credit for enough common-sense to know better. Peirce: CP 5.35 Cross-Ref:†† 35. We have some reason already to think there is some truth in pragmatism although we also have some reason to think that there is something wrong with it. For unless both branches of this statement were true we should do wrong to waste time and energy upon the inquiry we are undertaking. Peirce: CP 5.35 Cross-Ref:†† I will, therefore, presume that there is enough truth in it to render a preliminary glance at ethics desirable. For if, as pragmatism teaches us, what we think is to be interpreted in terms of what we are prepared to do, then surely logic, or the doctrine of what we ought to think, must be an application of the doctrine of what we deliberately choose to do, which is Ethics. Peirce: CP 5.36 Cross-Ref:†† 36. But we cannot get any clue to the secret of Ethics -- a most entrancing field of thought but soon broadcast with pitfalls -- until we have first made up our formula for what it is that we are prepared to admire. I do not care what doctrine of ethics be embraced, it will always be so. Suppose, for example, our maxim of ethics to be Pearson's †1 that all our action ought to be directed toward the perpetuation of the biological stock to which we belong. Then the question will arise, On what principle should it be deemed such a fine thing for this stock to survive -- or a fine thing at all? Is there nothing in the world or in posse that would be admirable per se except copulation and swarming? Is swarming a fine thing at all, apart from any results that it may lead to? The course of thought will follow a parallel line if we consider Marshall's ethical maxim: Act to restrain the impulses which demand immediate reaction, in order that the impulse-order determined by the existence of impulses of less strength, but of wider significance, may have full weight in the guidance of your life. Although I have not as clear an apprehension as I could wish of the philosophy of this very close, but too technical, thinker, yet I presume that he would not be among those who would object to making Ethics dependent upon Esthetics. Certainly, the maxim which I have just read to you from his latest book †2 supposes that it is a fine thing for an impulse to have its way, but yet not an equally fine thing for one impulse to have its way and for another impulse to have its way. There is a preference which depends upon the significance of impulses, whatever that may mean. It supposes that there is some ideal state of things which, regardless of how it should be brought about and independently of any ulterior reason whatsoever, is held to be good or fine. In short, ethics must rest upon a doctrine which, without at all considering what our conduct is to be, divides ideally possible states of things into two classes, those that would be admirable and those that would be unadmirable, and undertakes to define precisely what it is that constitutes the admirableness of an ideal. Its problem is to determine by analysis what it is that one ought deliberately to admire per se in itself regardless of what it may lead to and regardless of its bearings upon human conduct. I call that inquiry Esthetics, because it is generally said that the three normative sciences are logic, ethics, and esthetics, being the three doctrines that distinguish good and bad; Logic in regard to representations of truth, Ethics in regard to efforts of will, and Esthetics in objects considered simply in their presentation. Now that third Normative science can, I think, be no other than that which I have described. It is evidently the basic normative science upon which as a foundation, the doctrine of ethics must be reared to be surmounted in its turn by the doctrine of logic.

Peirce: CP 5.37 Cross-Ref:†† 37. But before we can attack any normative science, any science which proposes to separate the sheep from the goats, it is plain that there must be a preliminary inquiry which shall justify the attempt to establish such dualism. This must be a science that does not draw any distinction of good and bad in any sense whatever, but just contemplates phenomena as they are, simply opens its eyes and describes what it sees; not what it sees in the real as distinguished from figment -- not regarding any such dichotomy -- but simply describing the object, as a phenomenon, and stating what it finds in all phenomena alike. This is the science which Hegel made his starting-point, under the name of the Phänomenologie des Geistes -- although he considered it in a fatally narrow spirit, since he restricted himself to what actually forces itself on the mind and so colored his whole philosophy with the ignoration of the distinction of essence and existence and so gave it the nominalistic and I might say in a certain sense the pragmatoidal character in which the worst of the Hegelian errors have their origin. I will so far follow Hegel as to call this science Phenomenology although I will not restrict it to the observation and analysis of experience but extend it to describing all the features that are common to whatever is experienced or might conceivably be experienced or become an object of study in any way direct or indirect.†1 Peirce: CP 5.38 Cross-Ref:†† 38. Hegel was quite right in holding that it was the business of this science to bring out and make clear the Categories or fundamental modes. He was also right in holding that these Categories are of two kinds; the Universal Categories all of which apply to everything, and the series of categories consisting of phases of evolution. Peirce: CP 5.38 Cross-Ref:†† As to these latter, I am satisfied that Hegel has not approximated to any correct catalogue of them. It may be that here and there, in the long wanderings of his Encyclopædia he has been a little warmed by the truth. But in all its main features his catalogue is utterly wrong, according to me. I have made long and arduous studies of this matter, but I have not been able to draw up any catalogue that satisfies me. My studies,†2 if they are ever published, will I believe be found helpful to future students of this most difficult problem, but in these lectures I shall have little to say on that subject. The case is quite different with the three Universal Categories, which Hegel, by the way, does not look upon as Categories at all, or at least he does not call them so, but as three stages of thinking. In regard to these, it appears to me that Hegel is so nearly right that my own doctrine might very well be taken for a variety of Hegelianism, although in point of fact it was determined in my mind by considerations entirely foreign to Hegel, at a time when my attitude toward Hegelianism was one of contempt. There was no influence upon me from Hegel unless it was of so occult a kind as to entirely escape my ken; and if there was such an occult influence, it strikes me as about as good an argument for the essential truth of the doctrine, as is the coincidence that Hegel and I arrived in quite independent ways substantially to the same result. Peirce: CP 5.39 Cross-Ref:†† 39. This science of Phenomenology, then, must be taken as the basis upon which normative science is to be erected, and accordingly must claim our first attention. Peirce: CP 5.39 Cross-Ref:††

This science of Phenomenology is in my view the most primal of all the positive sciences. That is, it is not based, as to its principles, upon any other positive science. By a positive science I mean an inquiry which seeks for positive knowledge; that is, for such knowledge as may conveniently be expressed in a categorical proposition. Logic and the other normative sciences, although they ask, not what is but what ought to be, nevertheless are positive sciences since it is by asserting positive, categorical truth that they are able to show that what they call good really is so; and the right reason, right effort, and right being, of which they treat, derive that character from positive categorical fact. Peirce: CP 5.40 Cross-Ref:†† 40. Perhaps you will ask me whether it is possible to conceive of a science which should not aim to declare that something is positively or categorically true. I reply that it is not only possible to conceive of such a science, but that such science exists and flourishes, and Phenomenology, which does not depend upon any other positive science, nevertheless must, if it is to be properly grounded, be made to depend upon the Conditional or Hypothetical Science of Pure Mathematics, whose only aim is to discover not how things actually are, but how they might be supposed to be, if not in our universe, then in some other.†1 A Phenomenology which does not reckon with pure mathematics, a science hardly come to years of discretion when Hegel wrote, will be the same pitiful club-footed affair that Hegel produced.

Peirce: CP 5.41 Cross-Ref:†† LECTURE II †1 THE UNIVERSAL CATEGORIES

§1. PRESENTNESS †2

41. . . . Be it understood, then, that what we have to do, as students of phenomenology, is simply to open our mental eyes and look well at the phenomenon and say what are the characteristics that are never wanting in it, whether that phenomenon be something that outward experience forces upon our attention, or whether it be the wildest of dreams, or whether it be the most abstract and general of the conclusions of science. Peirce: CP 5.42 Cross-Ref:†† 42.†3 The faculties which we must endeavor to gather for this work are three. The first and foremost is that rare faculty, the faculty of seeing what stares one in the face, just as it presents itself, unreplaced by any interpretation, unsophisticated by any allowance for this or for that supposed modifying circumstance. This is the faculty of the artist who sees for example the apparent colors of nature as they appear. When the ground is covered by snow on which the sun shines brightly except where shadows fall, if you ask any ordinary man what its color appears to be, he will tell you white, pure white, whiter in the sunlight, a little greyish in the shadow. But that is not what is before his eyes that he is describing; it is his theory of what ought to be seen. The

artist will tell him that the shadows are not grey but a dull blue and that the snow in the sunshine is of a rich yellow. That artist's observational power is what is most wanted in the study of phenomenology. The second faculty we must strive to arm ourselves with is a resolute discrimination which fastens itself like a bulldog upon the particular feature that we are studying, follows it wherever it may lurk, and detects it beneath all its disguises. The third faculty we shall need is the generalizing power of the mathematician who produces the abstract formula that comprehends the very essence of the feature under examination purified from all admixture of extraneous and irrelevant accompaniments. Peirce: CP 5.43 Cross-Ref:†† 43. A very moderate exercise of this third faculty suffices to show us that the word Category bears substantially the same meaning with all philosophers. For Aristotle, for Kant, and for Hegel, a category is an element of phenomena of the first rank of generality. It naturally follows that the categories are few in number, just as the chemical elements are. The business of phenomenology is to draw up a catalogue of categories and prove its sufficiency and freedom from redundancies, to make out the characteristics of each category, and to show the relations of each to the others. I find that there are at least two distinct orders of categories, which I call the particular and the universal. The particular categories form a series, or set of series, only one of each series being present, or at least predominant, in any one phenomenon. The universal categories, on the other hand, belong to every phenomenon, one being perhaps more prominent in one aspect of that phenomenon than another but all of them belonging to every phenomenon. I am not very well satisfied with this description of the two orders of categories, but I am pretty well satisfied that there are two orders. I do not recognize them in Aristotle, unless the predicaments and the predicables are the two orders. But in Kant we have Unity, Plurality, and Totality not all present at once; Reality, Negation, and Limitation not all present at once; Inherence, Causation, and Reaction not all present at once; Possibility, Necessity, and Actuality not all present at once. On the other hand Kant's four greater categories, Quantity, Quality, Relation, and Modality, form what I should recognize as Kant's Universal Categories. In Hegel his long list which gives the divisions of his Encyclopædia are his Particular Categories. His three stages of thought, although he does not apply the word Category to them, are what I should call Hegel's Universal Categories. My intention this evening is to limit myself to the Universal, or Short List of Categories, and I may say, at once, that I consider Hegel's three stages as being, roughly speaking, the correct list of Universal Categories. . . . Peirce: CP 5.44 Cross-Ref:†† 44. When anything is present to the mind, what is the very first and simplest character to be noted in it, in every case, no matter how little elevated the object may be? Certainly, it is its presentness. So far Hegel is quite right. Immediacy is his word. To say, however, that presentness, presentness as it is present, present presentness, is abstract, is Pure Being, is a falsity so glaring, that one can only say that Hegel's theory that the abstract is more primitive than the concrete blinded his eyes to what stood before them. Go out under the blue dome of heaven and look at what is present as it appears to the artist's eye. The poetic mood approaches the state in which the present appears as it is present. Is poetry so abstract and colorless? The present is just what it is regardless of the absent, regardless of past and future. It is such as it is, utterly ignoring anything else. Consequently, it cannot be abstracted (which is what Hegel means by the abstract) for the abstracted is what the concrete, which gives it whatever being it has, makes it to be. The present, being such as it is while utterly

ignoring everything else, is positively such as it is. Imagine, if you please, a consciousness in which there is no comparison, no relation, no recognized multiplicity (since parts would be other than the whole), no change, no imagination of any modification of what is positively there, no reflexion -- nothing but a simple positive character. Such a consciousness might be just an odour, say a smell of attar; or it might be one infinite dead ache; it might be the hearing of a piercing eternal whistle. In short, any simple and positive quality of feeling would be something which our description fits that it is such as it is quite regardless of anything else. The quality of feeling is the true psychical representative of the first category of the immediate as it is in its immediacy, of the present in its direct positive presentness. Qualities of feeling show myriad-fold variety, far beyond what the psychologists admit. This variety however is in them only insofar as they are compared and gathered into collections. But as they are in their presentness, each is sole and unique; and all the others are absolute nothingness to it -- or rather much less than nothingness, for not even a recognition as absent things or as fictions is accorded to them. The first category, then, is Quality of Feeling, or whatever is such as it is positively and regardless of aught else.

Peirce: CP 5.45 Cross-Ref:†† §2. STRUGGLE †1

45. The next simplest feature that is common to all that comes before the mind, and consequently, the second category, is the element of Struggle. It is convenient enough, although by no means necessary, to study this, at first, in a psychological instance. Imagine yourself making a strong muscular effort, say that of pressing with all your might against a half-open door. Obviously, there is a sense of resistance. There could not be effort without an equal resistance any more than there could be a resistance without an equal effort that it resists. Action and reaction are equal. If you find that the door is pushed open in spite of you, you will say that it was the person on the other side that acted and you that resisted, while if you succeed in pushing the door to, you will say that it was you who acted and the other person that resisted. In general, we call the one that succeeds by means of his effort the agent and the one that fails the patient. But as far as the element of Struggle is concerned, there is no difference between being an agent and being a patient. It is the result that decides; but what it is that is deemed to be the result for the purpose of this distinction is a detail into which we need not enter. If while you are walking quietly along the sidewalk a man carrying a ladder suddenly pokes you violently with it in the back of the head and walks on without noticing what he has done, your impression probably will be that he struck you with great violence and that you made not the slightest resistance; although in fact you must have resisted with a force equal to that of the blow. Of course, it will be understood that I am not using force in the modern sense of a moving force but in the sense of Newton's actio†2; but I must warn you that I have not time to notice such trifles. In like manner, if in pitch darkness a tremendous flash of lightning suddenly comes, you are ready to admit having received a shock and being acted upon, but that you reacted you may be inclined to deny. You certainly did so, however, and are conscious of having done so. The sense of shock is as much a sense of resisting as of being acted upon. So it is when anything strikes the senses. The outward excitation succeeds in producing its effect on you, while you in turn produce no discernible effect on it; and therefore you call it the agent, and overlook

your own part in the reaction. On the other hand, in reading a geometrical demonstration, if you draw the figure in your imagination instead of on paper, it is so easy to add to your image whatever subsidiary line is wanted, that it seems to you that you have acted on the image without the image having offered any resistance. That it is not so, however, is easily shown. For unless that image had a certain power of persisting such as it is and resisting metamorphosis, and if you were not sensible of its strength of persistence, you never could be sure that the construction you are dealing with at one stage of the demonstration was the same that you had before your mind at an earlier stage. The main distinction between the Inner and the Outer Worlds is that inner objects promptly take any modifications we wish, while outer objects are hard facts that no man can make to be other than they are. Yet tremendous as this distinction is, it is after all only relative. Inner objects do offer a certain degree of resistance and outer objects are susceptible of being modified in some measure by sufficient exertion intelligently directed.†1 Peirce: CP 5.46 Cross-Ref:†† 46. Two very serious doubts arise concerning this category of struggle which I should be able completely to set to rest, I think, with only a little more time. But as it is, I can only suggest lines of reflexion which, if you perseveringly follow out, ought to bring you to the same result to which they have brought me. The first of these doubts is whether this element of struggle is anything more than a very special kind of phenomenon, and withal an anthropomorphic conception and therefore not scientifically true. Peirce: CP 5.46 Cross-Ref:†† The other doubt is whether the idea of Struggle is a simple and irresolvable element of the phenomenon; and in opposition to its being so, two contrary parties will enter into a sort of [alliance] without remarking how deeply they are at variance with one another. One of these parties will be composed of those philosophers who understand themselves as wishing to reduce everything in the phenomenon to qualities of feeling. They will appear in the arena of psychology and will declare that there is absolutely no such thing as a specific sense of effort. There is nothing, they will say, but feelings excited upon muscular contraction, feelings which they may or may not be disposed to say have their immediate excitations within the muscles. The other party will be composed of those philosophers who say that there can be only one absolute and only one irreducible element, and since Nous is such an element, Nous is really the only thoroughly clear idea there is. These philosophers will take a sort of pragmatistic stand. They will maintain that in saying that one thing acts upon another, absolutely the only thing that can be meant is that there is a law according to which under all circumstances of a certain general description certain phenomena will result; and therefore to speak of one thing as acting upon another hic et nunc regardless of uniformity, regardless of what will happen on all occasions, is simple nonsense. Peirce: CP 5.47 Cross-Ref:†† 47. I shall have to content myself with giving some hints as to how I would meet this second double-headed objection, leaving the first to your own reflexions. In the course of considering the second objection, the universality of the element of struggle will get brought to light without any special arguments to that end. But as to its being unscientific because anthropomorphic, that is an objection of a very shallow kind, that arises from prejudices based upon much too narrow considerations. "Anthropomorphic" is what pretty much all conceptions are at bottom; otherwise

other roots for the words in which to express them than the old Aryan roots would have to be found. And in regard to any preference for one kind of theory over another, it is well to remember that every single truth of science is due to the affinity of the human soul to the soul of the universe, imperfect as that affinity no doubt is.†1 To say, therefore, that a conception is one natural to man, which comes to just about the same thing as to say that it is anthropomorphic, is as high a recommendation as one could give to it in the eyes of an Exact Logician.†P1 Peirce: CP 5.48 Cross-Ref:†† 48. As for the double-headed objection, I will first glance at that branch of it that rests upon the idea that the conception of action involves the notion of law or uniformity so that to talk of a reaction regardless of anything but the two individual reacting objects is nonsense. As to that I should say that a law of nature left to itself would be quite analogous to a court without a sheriff. A court in that predicament might probably be able to induce some citizen to act as sheriff; but until it had so provided itself with an officer who, unlike itself, could not discourse authoritatively but who could put forth the strong arm, its law might be the perfection of human reason but would remain mere fireworks, brutum fulmen. Just so, let a law of nature -- say the law of gravitation -- remain a mere uniformity -- a mere formula establishing a relation between terms -- and what in the world should induce a stone, which is not a term nor a concept but just a plain thing, to act in conformity to that uniformity? All other stones may have done so, and this stone too on former occasions, and it would break the uniformity for it not to do so now. But what of that? There is no use talking reason to a stone. It is deaf and it has no reason. I should ask the objector whether he was a nominalist or a scholastic realist. If he is a nominalist, he holds that laws are mere generals, that is, formulae relating to mere terms; and ordinary good sense ought to force him to acknowledge that there are real connections between individual things regardless of mere formulae. Now any real connection whatsoever between individual things involves a reaction between them in the sense of this category. The objector may, however, take somewhat stronger ground by confessing himself to be a scholastic realist, holding that generals may be real. A law of nature, then, will be regarded by him as having a sort of esse in futuro. That is to say they will have a present reality which consists in the fact that events will happen according to the formulation of those laws. It would seem futile for me to attempt to reply that when, for example, I make a great effort to lift a heavy weight and perhaps am unable to stir it from the ground, there really is a struggle on this occasion regardless of what happens on other occasions; because the objector would simply admit that on such an occasion I have a quality of feeling which I call a feeling of effort, but he would urge that the only thing which makes this designation appropriate to the feeling is the regularity of connection between this feeling and certain motions of matter. Peirce: CP 5.49 Cross-Ref:†† 49. This is a position well enough taken to merit a very respectful reply. But before going into that reply, there is an observation which I should like to lay before the candid objector. Your argument against this category of Struggle is that a struggle regardless of law is not intelligible. Yet you have just admitted that my so-called sense of effort involves a peculiar quality of feeling. Now a quality of feeling is not intelligible, either. Nothing can be less so. One can feel it, but to comprehend it or express it in a general formula is out of the question. So it appears that unintelligibility does not suffice to destroy or refute a Category. Indeed, if you are to accept scholastic realism, you would seem to be almost bound to admit that Nous, or

intelligibility, is itself a category; and in that case far from non-intelligibility's refuting a category, intelligibility would do so -- that is, would prove that a conception could not be a category distinct from the category of Nous, or intelligibility. If it be objected that the unintelligibility of a Quality of Feeling is of a merely privative kind quite different from the aggressive and brutal anti-intelligibility of action regardless of law, the rejoinder will be that if intelligibility be a category, it is not surprising but rather inevitable that other categories should be in different relations to this one. Peirce: CP 5.50 Cross-Ref:†† 50. But without beating longer round the bush, let us come to close quarters. Experience is our only teacher. Far be it from me to enunciate any doctrine of a tabula rasa. For, as I said a few minutes ago, there manifestly is not one drop of principle in the whole vast reservoir of established scientific theory that has sprung from any other source than the power of the human mind to originate ideas that are true. But this power, for all it has accomplished, is so feeble that as ideas flow from their springs in the soul, the truths are almost drowned in a flood of false notions; and that which experience does is gradually, and by a sort of fractionation, to precipitate and filter off the false ideas, eliminating them and letting the truth pour on in its mighty current. Peirce: CP 5.51 Cross-Ref:†† 51. But precisely how does this action of experience take place? It takes place by a series of surprises. There is no need of going into details. At one time a ship is sailing along in the trades over a smooth sea, the navigator having no more positive expectation than that of the usual monotony of such a voyage, when suddenly she strikes upon a rock. The majority of discoveries, however, have been the result of experimentation. Now no man makes an experiment without being more or less inclined to think that an interesting result will ensue; for experiments are much too costly of physical and psychical energy to be undertaken at random and aimlessly. And naturally nothing can possibly be learned from an experiment that turns out just as was anticipated. It is by surprises that experience teaches all she deigns to teach us. Peirce: CP 5.51 Cross-Ref:†† In all the works on pedagogy that ever I read -- and they have been many, big, and heavy -- I don't remember that any one has advocated a system of teaching by practical jokes, mostly cruel. That, however, describes the method of our great teacher, Experience. She says,

Open your mouth and shut your eyes And I'll give you something to make you wise;

and thereupon she keeps her promise, and seems to take her pay in the fun of tormenting us. Peirce: CP 5.52 Cross-Ref:†† 52. The phenomenon of surprise in itself is highly instructive in reference to this category because of the emphasis it puts upon a mode of consciousness which can be detected in all perception, namely, a double consciousness at once of an ego and a

non-ego, directly acting upon each other.†1 Understand me well. My appeal is to observation -- observation that each of you must make for himself. Peirce: CP 5.53 Cross-Ref:†† 53. The question is what the phenomenon is. We make no vain pretense of going beneath phenomena. We merely ask, what is the content of the Percept? Everybody should be competent to answer that of himself. Examine the Percept in the particularly marked case in which it comes as a surprise. Your mind was filled [with] an imaginary object that was expected. At the moment when it was expected the vividness of the representation is exalted, and suddenly, when it should come, something quite different comes instead. I ask you whether at that instant of surprise there is not a double consciousness, on the one hand of an Ego, which is simply the expected idea suddenly broken off, on the other hand of the Non-Ego, which is the strange intruder, in his abrupt entrance. Peirce: CP 5.54 Cross-Ref:†† 54. The whole question is what the perceptual facts are, as given in direct perceptual judgments. By a perceptual judgment, I mean a judgment asserting in propositional form what a character of a percept directly present to the mind is.†2 The percept of course is not itself a judgment, nor can a judgment in any degree resemble a percept. It is as unlike it as the printed letters in a book, where a Madonna of Murillo is described, are unlike the picture itself. Peirce: CP 5.55 Cross-Ref:†† 55. You may adopt any theory that seems to you acceptable as to the psychological operations by which perceptual judgments are formed. For our present purpose it makes no difference what that theory is. All that I insist upon is that those operations, whatever they may be, are utterly beyond our control and will go on whether we are pleased with them or not. Now I say that taking the word "criticize" in the sense it bears in philosophy, that of apportioning praise and blame, it is perfectly idle to criticize anything over which you can exercise no sort of control. You may wisely criticize a reasoning, because the reasoner, in the light of your criticism, will certainly go over his reasoning again and correct it if your blame of it was just. But to pronounce an involuntary operation of the mind good or bad, has no more sense than to pronounce the proportion of weights in which hydrogen and chlorine combine, that of 1 to 35.11 to be good or bad. I said it was idle; but in point of fact "nonsensical" would have been an apter word. Peirce: CP 5.55 Cross-Ref:†† If, therefore, our careful direct interpretation of perception, and more emphatically of such perception as involves surprise, is that the perception represents two objects reacting upon one another, that is not only a decision from which there is no appeal, but it is downright nonsense to dispute the fact that in perception two objects really do so react upon one another. Peirce: CP 5.56 Cross-Ref:†† 56. That, of course, is the doctrine of Immediate Perception which is upheld by Reid, Kant, and all dualists who understand the true nature of dualism, and the denial of which led Cartesians to the utterly absurd theory of divine assistance upon which the preestablished harmony of Leibniz is but a slight improvement. Every philosopher who denies the doctrine of Immediate Perception -- including idealists of every stripe -- by that denial cuts off all possibility of ever cognizing a relation. Nor will he better his position by declaring that all relations are illusive appearances, since

it is not merely true knowledge of them that he has cut off, but every mode of cognitive representation of them. Peirce: CP 5.57 Cross-Ref:†† 57.†1 When a man is surprised he knows that he is surprised. Now comes a dilemma. Does he know he is surprised by direct perception or by inference? First try the hypothesis that it is by inference. This theory would be that a person (who must be supposed old enough to have acquired self-consciousness) on becoming conscious of that peculiar quality of feeling which unquestionably belongs to all surprise, is induced by some reason to attribute this feeling to himself. It is, however, a patent fact that we never, in the first instance, attribute a Quality of Feeling to ourselves. We first attribute it to a Non-Ego and only come to attribute it to ourselves when irrefragable reasons compel us to do so. Therefore, the theory would have to be that the man first pronounces the surprising object a wonder, and upon reflection convinces himself that it is only a wonder in the sense that he is surprised. That would have to be the theory. But it is in conflict with the facts which are that a man is more or less placidly expecting one result, and suddenly finds something in contrast to that forcing itself upon his recognition. A duality is thus forced upon him: on the one hand, his expectation which he had been attributing to Nature, but which he is now compelled to attribute to some mere inner world, and on the other hand, a strong new phenomenon which shoves that expectation into the background and occupies its place. The old expectation, which is what he was familiar with, is his inner world, or Ego. The new phenomenon, the stranger, is from the exterior world or Non-Ego. He does not conclude that he must be surprised because the object is so marvellous. But on the contrary, it is because of the duality presenting itself as such that he [is] led by generalization to a conception of a quality of marvellousness. Peirce: CP 5.58 Cross-Ref:†† 58. Try, then, the other alternative that it is by direct perception, that is, in a direct perceptual judgment, that a man knows that he is surprised. The perceptual judgment, however, certainly does not represent that it is he himself who has played a little trick upon himself. A man cannot startle himself by jumping up with an exclamation of Boo! Nor could the perceptual judgment have represented anything so out of nature. The perceptual judgment, then, can only be that it is the Non-Ego, something over against the Ego and bearing it down, is what has surprised him. But if that be so, this direct perception presents an Ego to which the smashed expectation belonged, and the Non-Ego, the sadder and wiser man, to which the new phenomenon belongs. . . .

Peirce: CP 5.59 Cross-Ref:†† §3. LAWS: NOMINALISM †1

59. Thus far, gentlemen, I have been insisting very strenuously upon what the most vulgar common sense has every disposition to assent to and only ingenious philosophers have been able to deceive themselves about. But now I come to a category which only a more refined form of common sense is prepared willingly to allow, the category which of the three is the chief burden of Hegel's song, a category toward which the studies of the new logico-mathematicians, Georg Cantor and the like, are steadily pointing, but to which no modern writer of any stripe, unless it be

some obscure student like myself, has ever done anything approaching to justice. . . . Peirce: CP 5.60 Cross-Ref:†† 60. There never was a sounder logical maxim of scientific procedure than Ockham's razor: Entia non sunt multiplicanda praeter necessitatem. That is to say; before you try a complicated hypothesis, you should make quite sure that no simplification of it will explain the facts equally well. No matter if it takes fifty generations of arduous experimentation to explode the simpler hypothesis, and no matter how incredible it may seem that that simpler hypothesis should suffice, still fifty generations are nothing in the life of science, which has all time before it; and in the long run, say in some thousands of generations, time will be economized by proceeding in an orderly manner, and by making it an invariable rule to try the simpler hypothesis first. Indeed, one can never be sure that the simpler hypothesis is not the true one, after all, until its cause has been fought out to the bitter end. But you will mark the limitation of my approval of Ockham's razor. It is a sound maxim of scientific procedure. If the question be what one ought to believe, the logic of the situation must take other factors into account. Speaking strictly, belief is out of place in pure theoretical science, which has nothing nearer to it than the establishment of doctrines, and only the provisional establishment of them, at that.†1 Compared with living belief it is nothing but a ghost. If the captain of a vessel on a lee shore in a terrific storm finds himself in a critical position in which he must instantly either put his wheel to port acting on one hypothesis, or put his wheel to starboard acting on the contrary hypothesis, and his vessel will infallibly be dashed to pieces if he decides the question wrongly, Ockham's razor is not worth the stout belief of any common seaman. For stout belief may happen to save the ship, while Entia non sunt multiplicanda praeter necessitatem would be only a stupid way of spelling Shipwreck. Now in matters of real practical concern we are all in something like the situation of that sea-captain. Peirce: CP 5.61 Cross-Ref:†† 61. Philosophy, as I understand the word, is a positive theoretical science, and a science in an early stage of development. As such it has no more to do with belief than any other science. Indeed, I am bound to confess that it is at present in so unsettled a condition, that if the ordinary theorems of molecular physics and of archaeology are but the ghosts of beliefs, then to my mind, the doctrines of the philosophers are little better than the ghosts of ghosts. I know this is an extremely heretical opinion. The followers of Haeckel are completely in accord with the followers of Hegel in holding that what they call philosophy is a practical science and the best of guides in the formation of what they take to be Religious Beliefs. I simply note the divergence, and pass on to an unquestionable fact; namely, the fact that all modern philosophy is built upon Ockhamism; by which I mean that it is all nominalistic and that it adopts nominalism because of Ockham's razor. And there is no form of modern philosophy of which this is more essentially true than the philosophy of Hegel. But it is not modern philosophers only who are nominalists. The nominalistic Weltanschauung has become incorporated into what I will venture to call the very flesh and blood of the average modern mind. Peirce: CP 5.62 Cross-Ref:†† 62. The third category of which I come now to speak is precisely that whose reality is denied by nominalism. For although nominalism is not credited with any extraordinarily lofty appreciation of the powers of the human soul, yet it attributes to it a power of originating a kind of ideas the like of which Omnipotence has failed to

create as real objects, and those general conceptions which men will never cease to consider the glory of the human intellect must, according to any consistent nominalism, be entirely wanting in the mind of Deity. Leibniz, the modern nominalist par excellence, will not admit that God has the faculty of Reason; and it seems impossible to avoid that conclusion upon nominalistic principles. Peirce: CP 5.63 Cross-Ref:†† 63. But it is not in Nominalism alone that modern thought has attributed to the human mind the miraculous power of originating a category of thought that has no counterpart at all in Heaven or Earth. Already in that strangely influential hodge-podge, the salad of Cartesianism, the doctrine stands out very emphatically that the only force is the force of impact, which clearly belongs to the category of Reaction; and ever since Newton's Principia began to affect the general thought of Europe through the sympathetic spirit of Voltaire, there has been a disposition to deny any kind of action except purely mechanical action. The Corpuscular Philosophy of Boyle -- although the pious Boyle did not himself recognize its character -- was bound to come to that in the last resort; and the idea constantly gained strength throughout the eighteenth century and the nineteenth until the doctrine of the Conservation of Energy, generalized rather loosely by philosophers, led to the theory of psycho-physical parallelism, against which there has, only of recent years, been any very sensible and widespread revolt. Psycho-physical parallelism is merely the doctrine that mechanical action explains all the real facts, except that these facts have an internal aspect which is a little obscure and a little shadowy. Peirce: CP 5.64 Cross-Ref:†† 64. To my way of regarding philosophy, all this movement was perfectly good scientific procedure. For the simpler hypothesis which excluded the influence of ideas upon matter had to be tried and persevered in until it was thoroughly exploded. But I believe that now at last, at any time for the last thirty years, it has been apparent, to every man who sufficiently considered the subject, that there is a mode of influence upon external facts which cannot be resolved into mere mechanical action, so that henceforward it will be a grave error of scientific philosophy to overlook the universal presence in the phenomenon of this third category. Indeed, from the moment that the Idea of Evolution took possession of the minds of men the pure Corpuscular Philosophy together with nominalism had had their doom pronounced. I grew up in Cambridge, [Massachusetts] and was about 21 when the Origin of Species appeared. There was then living here a thinker who left no remains from which one could now gather what an educative influence his was upon the minds of all of us who enjoyed his intimacy, Mr. Chauncey Wright.†1 He had at first been a Hamiltonian but had early passed over into the warmest advocacy of the nominalism of John Stuart Mill; and being a mathematician at a time when dynamics was regarded as the loftiest branch of mathematics, he was also inclined to regard nature from a strictly mechanical point of view. But his interests were wide and he was also a student of Gray.†1 I was away surveying in the wilds of Louisiana when Darwin's great work appeared, and though I learned by letters of the immense sensation it had created, I did not return until early in the following summer when I found Wright all enthusiasm for Darwin, whose doctrines appeared to him as a sort of supplement to those of Mill. I remember well that I then made a remark to him which although he did not assent to it, evidently impressed him enough to perplex him. The remark was that these ideas of development had more vitality by far than any of his other favorite conceptions and that though they might at that moment be in his mind like a little vine clinging to the tree of Associationalism, yet after a time that vine would inevitably kill the tree. He

asked me why I said that and I replied that the reason was that Mill's doctrine was nothing but a metaphysical point of view to which Darwin's, which was nourished by positive observation, must be deadly. Ten or fifteen years later, when Agnosticism was all the go, I prognosticated a short life for it, as philosophies run, for a similar reason. What the true definition of Pragmatism may be, I find it very hard to say; but in my nature it is a sort of instinctive attraction for living facts. Peirce: CP 5.65 Cross-Ref:†† 65. All nature abounds in proofs of other influences than merely mechanical action, even in the physical world. They crowd in upon us at the rate of several every minute. And my observation of men has led me to this little generalization. Speaking only of men who really think for themselves and not of mere reporters, I have not found that it is the men whose lives are mostly passed within the four walls of a physical laboratory who are most inclined to be satisfied with a purely mechanical metaphysics. On the contrary, the more clearly they understand how physical forces work the more incredible it seems to them that such action should explain what happens out of doors. A larger proportion of materialists and agnostics is to be found among the thinking physiologists and other naturalists, and the largest proportion of all among those who derive their ideas of physical science from reading popular books. These last, the Spencers, the Youmanses, and the like, seem to be possessed with the idea that science has got the universe pretty well ciphered down to a fine point; while the Faradays and Newtons seem to themselves like children who have picked up a few pretty pebbles upon the ocean beach. But most of us seem to find it difficult to recognize the greatness and wonder of things familiar to us. As the prophet is not without honor save [in his own country] so it is also with phenomena. Point out to the ordinary man evidence, however conclusive, of other influence than physical action in things he sees every day, and he will say: "Well, I don't see as that frog has got any points about him that's any different from any other frog." For that reason we welcome instances perhaps of less real cogency but which have the merit of being rare and strange. Such, for example, are the right-handed and left-handed screw-structures of the molecules of those bodies which are said to be "optically active." Of every such substance there are two varieties, or as the chemists call them, two modifications, one of which twists a ray of light that passes through it to the right, and the other, by an exactly equal amount, to the left. All the ordinary physical properties of the right-handed and left-handed modifications are identical. Only certain faces of their crystals, often very minute, are differently placed. No chemical process can ever transmute the one modification into the other. And their ordinary chemical behaviour is absolutely the same, so that no strictly chemical process can separate them if they are once mixed. Only the chemical action of one optically active substance upon another is different if they both twist the ray the same way from what it is if they twist the ray different ways. There are certain living organisms which feed on one modification and destroy it while leaving the other one untouched. This is presumably due to such organisms containing in their substance, possibly in very minute proportion, some optically active body. Now I maintain that the original segregation of levo-molecules, or molecules with a left-handed twist, from dextro-molecules, or molecules with a right-handed twist, is absolutely incapable of mechanical explanation. Of course you may suppose that in the original nebula at the very formation of the world right-handed quartz was collected into one place, while left-handed quartz was collected into another place. But to suppose that, is ipso facto to suppose that that segregation was a phenomenon without any mechanical explanation. The three laws of motion draw no dynamical distinction between right-handed and left-handed screws, and a mechanical explanation is an explanation

founded on the three laws of motion. There, then, is a physical phenomenon absolutely inexplicable by mechanical action. This single instance suffices to overthrow the Corpuscular Philosophy.

Peirce: CP 5.66 Cross-Ref:†† LECTURE III †1 THE CATEGORIES CONTINUED

§1. DEGENERATE THIRDNESS †2

66. Category the First is the Idea of that which is such as it is regardless of anything else. That is to say, it is a Quality of Feeling. Peirce: CP 5.66 Cross-Ref:†† Category the Second is the Idea of that which is such as it is as being Second to some First, regardless of anything else, and in particular regardless of any Law, although it may conform to a law. That is to say, it is Reaction as an element of the Phenomenon. Peirce: CP 5.66 Cross-Ref:†† Category the Third is the Idea of that which is such as it is as being a Third, or Medium, between a Second and its First. That is to say, it is Representation as an element of the Phenomenon. Peirce: CP 5.67 Cross-Ref:†† 67. A mere complication of Category the Third, involving no idea essentially different, will give the idea of something which is such as it is by virtue of its relations to any multitude, enumerable, denumeral, or abnumerable or even to any supermultitude of correlates; so that this Category suffices of itself to give the conception of True Continuity, than which no conception yet discovered is higher.†3 Peirce: CP 5.68 Cross-Ref:†† 68. Category the First owing to its Extremely Rudimentary character is not susceptible of any degenerate or weakened modification. Peirce: CP 5.69 Cross-Ref:†† 69. Category the Second has a Degenerate Form, in which there is Secondness indeed, but a weak or Secondary Secondness that is not in the pair in its own quality, but belongs to it only in a certain respect. Moreover, this degeneracy need not be absolute but may be only approximative. Thus a genus characterized by Reaction will by the determination of its essential character split into two species, one a species where the secondness is strong, the other a species where the secondness is weak, and the strong species will subdivide into two that will be similarly related, without any corresponding subdivision of the weak species. For example, Psychological Reaction splits into Willing, where the Secondness is strong, and Sensation, where it is weak;

and Willing again subdivides into Active Willing and Inhibitive Willing, to which last dichotomy nothing in Sensation corresponds. But it must be confessed that subdivision, as such, involves something more than the second category. Peirce: CP 5.70 Cross-Ref:†† 70. Category the Third exhibits two different ways of Degeneracy, where the irreducible idea of Plurality, as distinguished from Duality, is present indeed but in maimed conditions. The First degree of Degeneracy is found in an Irrational Plurality which, as it exists, in contradistinction [to] the form of its representation, is a mere complication of duality. We have just had an example of this in the idea of Subdivision. In pure Secondness, the reacting correlates are Singulars, and as such are Individuals, not capable of further division. Consequently, the conception of Subdivision, say by repeated dichotomy, certainly involves a sort of Thirdness, but it is a thirdness that is conceived to consist in a second secondness. Peirce: CP 5.71 Cross-Ref:†† 71. The most degenerate Thirdness is where we conceive a mere Quality of Feeling, or Firstness, to represent itself to itself as Representation. Such, for example, would be Pure Self-Consciousness, which might be roughly described as a mere feeling that has a dark instinct of being a germ of thought. This sounds nonsensical, I grant. Yet something can be done toward rendering it comprehensible. Peirce: CP 5.71 Cross-Ref:†† I remember a lady's averring that her father had heard a minister, of what complexion she did not say, open a prayer as follows: "O Thou, All-Sufficient, Self-Sufficient, Insufficient God." Now pure Self-consciousness is Self-sufficient, and if it is also regarded as All-sufficient, it would seem to follow that it must be Insufficient. I ought to apologize for introducing such Buffoonery into serious lectures. I do so because I seriously believe that a bit of fun helps thought and tends to keep it pragmatical. Peirce: CP 5.71 Cross-Ref:†† Imagine that upon the soil of a country, that has a single boundary line thus

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view], or [Click here to view], there lies a map of that same country. This map may distort the different provinces of the country to any extent. But I shall suppose that it represents every part of the country that has a single boundary, by a part of the map that has a single boundary, that every part is represented as bounded by such parts as it really is bounded by, that every point of the country is represented by a single point of the map, and that every point of the map represents a single point in the country. Let us further suppose that this map is infinitely minute in its representation so that there is no speck on any grain of sand in the country that could not be seen represented upon the map if we were to examine it under a sufficiently high magnifying power. Since, then, everything on the soil of the country is shown on

the map, and since the map lies on the soil of the country, the map itself will be portrayed in the map, and in this map of the map everything on the soil of the country can be discerned, including the map itself with the map of the map within its boundary. Thus there will be within the map, a map of the map, and within that, a map of the map of the map, and so on ad infinitum. These maps being each within the preceding ones of the series, there will be a point contained in all of them, and this will be the map of itself. Each map which directly or indirectly represents the country is itself mapped in the next; i.e., in the next [it] is represented to be a map of the country. In other words each map is interpreted as such in the next. We may therefore say that each is a representation of the country to the next map; and that point that is in all the maps is in itself the representation of nothing but itself and to nothing but itself. It is therefore the precise analogue of pure self-consciousness. As such it is self-sufficient. It is saved from being insufficient, that is as no representation at all, by the circumstance that it is not all-sufficient, that is, is not a complete representation but is only a point upon a continuous map.†P1 I dare say you may have heard something like this before from Professor Royce, but if so, you will remark an important divergency. The idea itself belongs neither to him nor to me, and was used by me in this connection thirty years ago.†1 Peirce: CP 5.72 Cross-Ref:†† 72. The relatively degenerate forms of the Third category do not fall into a catena, like those of the Second. What we find is this. Taking any class in whose essential idea the predominant element is Thirdness, or Representation, the self-development of that essential idea -- which development, let me say, is not to be compassed by any amount of mere "hard thinking," but only by an elaborate process founded upon experience and reason combined -- results in a trichotomy giving rise to three sub-classes, or genera, involving respectively a relatively genuine thirdness, a relatively reactional thirdness or thirdness of the lesser degree of degeneracy, and a relatively qualitative thirdness or thirdness of the last degeneracy. This last may subdivide, and its species may even be governed by the three categories, but it will not subdivide, in the manner which we are considering, by the essential determinations of its conception. The genus corresponding to the lesser degree of degeneracy, the reactionally degenerate genus, will subdivide after the manner of the Second category, forming a catena; while the genus of relatively genuine Thirdness will subdivide by Trichotomy just like that from which it resulted. Only as the division proceeds, the subdivisions become harder and harder to discern. Peirce: CP 5.73 Cross-Ref:†† 73. The representamen, for example, divides by trichotomy into the general sign or symbol, the index, and the icon.†2 An icon is a representamen which fulfills the function of a representamen by virtue of a character which it possesses in itself, and would possess just the same though its object did not exist. Thus, the statue of a centaur is not, it is true, a representamen if there be no such thing as a centaur. Still, if it represents a centaur, it is by virtue of its shape; and this shape it will have, just as much, whether there be a centaur or not. An index is a representamen which fulfills the function of a representamen by virtue of a character which it could not have if its object did not exist, but which it will continue to have just the same whether it be interpreted as a representamen or not. For instance, an old-fashioned hygrometer is an index. For it is so contrived as to have a physical reaction with dryness and moisture in the air, so that the little man will come out if it is wet, and this would happen just the same if the use of the instrument should be entirely forgotten, so that it ceased actually to convey any information. A symbol is a representamen which fulfills its

function regardless of any similarity or analogy with its object and equally regardless of any factual connection therewith, but solely and simply because it will be interpreted to be a representamen. Such for example is any general word, sentence, or book. Peirce: CP 5.73 Cross-Ref:†† Of these three genera of representamens, the Icon is the Qualitatively degenerate, the Index the Reactionally degenerate, while the Symbol is the relatively genuine genus. Peirce: CP 5.74 Cross-Ref:†† 74. Now the Icon may undoubtedly be divided according to the categories; but the mere completeness of the notion of the icon does not imperatively call for any such division. For a pure icon does not draw any distinction between itself and its object. It represents whatever it may represent, and whatever it is like, it in so far is. It is an affair of suchness only. Peirce: CP 5.75 Cross-Ref:†† 75. It is quite otherwise with the Index. Here is a reactional sign, which is such by virtue of a real connection with its object. Then the question arises is this dual character in the Index, so that it has two elements, by virtue of the one serving as a substitute for the particular object it does, while the other is an involved icon that represents the representamen itself regarded as a quality of the object -- or is there really no such dual character in the index, so that it merely denotes whatever object it happens to be really connected with just as the icon represents whatever object it happens really to resemble? Of the former, the relatively genuine form of Index, the hygrometer, is an example. Its connection with the weather is dualistic, so that by an involved icon, it actually conveys information. On the other hand any mere land-mark by which a particular thing may be recognized because it is as a matter of fact associated with that thing, a proper name without signification, a pointing finger, is a degenerate index. Horatio Greenough, who designed Bunker Hill Monument, tells us in his book †1 that he meant it to say simply "Here!" It just stands on that ground and plainly is not movable. So if we are looking for the battle-field, it will tell us whither to direct our steps. Peirce: CP 5.76 Cross-Ref:†† 76. The Symbol, or relatively genuine form of Representamen, divides by Trichotomy into the Term, the Proposition, and the Argument. The Term corresponds to the Icon and to the degenerate Index. It does excite an icon in the imagination. The proposition conveys definite information like the genuine index, by having two parts of which the function of the one is to indicate the object meant, while that of the other is to represent the representamen by exciting an icon of its quality. The argument is a representamen which does not leave the interpretant to be determined as it may by the person to whom the symbol is addressed, but separately represents what is the interpreting representation that it is intended to determine. This interpreting representation is, of course, the conclusion. It would be interesting to push these illustrations further; but I can linger nowhere. As soon as a subject begins to be interesting I am obliged to pass on to another.

Peirce: CP 5.77 Cross-Ref:†† §2. THE SEVEN SYSTEMS OF METAPHYSICS

77. The three categories furnish an artificial classification of all possible systems of metaphysics which is certainly not without its utility. The scheme is shown in this figure (p. 53). It depends upon what ones of the three categories each system admits as important metaphysico-cosmical elements.†P1 Peirce: CP 5.78 Cross-Ref:†† 78. One very naturally and properly endeavors to give an account of the universe with the fewest and simplest possible categories. Praedicamenta non sunt multiplicanda praeter necessitatem.

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Peirce: CP 5.79 Cross-Ref:†† 79. We ought therefore to admire and extol the efforts of Condillac and the Associationalists to explain everything by means of qualities of feeling [i]. If, however, this turns out to be a failure, the next most admirable hypothesis is that of the corpuscularians, Helmholtz and the like, who would like to explain everything by means of mechanical force, which they do not distinguish from individual reaction [ii]. That again failing, the doctrine of Hegel is to be commended who regards

Category the Third as the only true one [iii]. For in the Hegelian system the other two are only introduced in order to be aufgehoben. All the categories of Hegel's list, from Pure Being up, appear to me very manifestly to involve Thirdness, although he does not appear to recognize it, so immersed is he in this category. Peirce: CP 5.80 Cross-Ref:†† 80. All three of these simplest systems having worked themselves out into absurdity, it is natural next in accordance with the maxim of Parsimony to try explanations of the Universe based on the recognition of two only of the Categories. Peirce: CP 5.81 Cross-Ref:†† 81. The more moderate nominalists who nevertheless apply the epithet mere to thought and to representamens may be said to admit Categories First and Second and to deny the third [i ii]. The Berkeleyans, for whom there are but two kinds of entities, souls, or centres of determinable thought, and ideas in the souls, these ideas being regarded as pure statical entities, little or nothing else than Qualities of Feeling, seem to admit Categories First and Third and to deny Secondness, which they wish to replace by Divine Creative Influence, which certainly has all the flavor of Thirdness [i iii]. So far as one can make out any intelligible aim in that singular hodge-podge, the Cartesian metaphysics, it seems to have been to admit Categories Second and Third as fundamental and to deny the First [ii iii]. Otherwise, I do not know to whom we can attribute this opinion which certainly does not seem to be less acceptable and attractive than several others. But there are other philosophies which seem to do full justice to Categories Second and Third and to minimize the first, and among these perhaps Spinoza and Kant are to be included.

Peirce: CP 5.82 Cross-Ref:†† §3. THE IRREDUCIBILITY OF THE CATEGORIES †1

82. We must begin by asking whether the three categories can be admitted as simple and irreducible conceptions; and afterward go on to ask whether they cannot all be supposed to be real constituents of the universe. For when I say that certain metaphysical schools do not admit them, I do not mean to say that they do not admit them as mere conceptions -- a point to which they do not generally pay much attention, so that their opinions about this are not very marked -- but that they do not admit them as real constituents of the universe. Peirce: CP 5.82 Cross-Ref:†† I do not know that I could add anything material to what I said in my last lecture to show that Category the First must be admitted as an irreducible constituent of the phenomenon. Peirce: CP 5.83 Cross-Ref:†† 83. There would be no question that Category the Second is an irreducible conception were it not for the deplorable condition of the science of logic. This is illustrated by the fact that so flippant and wildly theorizing work as Prantl's Geschichte der Logik should be accepted, as it generally is, even among learned men, as a marvel of patient research. It is true that one or two chapters of it are relatively well done. The account of Aristotle's logic, though not good upon any high standard of completeness or of thorough comprehension, is nevertheless the best account of its

subject that we have. But Prantl, to begin with, does not himself understand logic, meaning by logic the science of which those works treated, of which he gives or he professes to give an account; and yet with the shallowest ideas, he is so puffed up with his own views that he disdains to take the trouble to penetrate their meaning. The crude expressions of contempt in which he continually indulges toward great thinkers ought to put readers on their guard against him. In the next place he belongs to that too well-known class of German critics who get bitten with theories deduced from general conceptions, and who fall in love with these theories because they are their own offspring and treat them as absolute certainties although the complete refutation of them is near at hand. You will understand, of course, that I do not say these things without having read all the chief contributions to the questions on both sides and without having subjected them to careful study and criticism. Prantl's opinions about the Megarian philosophers, about what he calls the Byzantine logic, about the Latin medieval logic, about the Parva Logicalia, are wild theories, utterly untenable, and in several cases easily refuted by an easy examination of the MSS. Moreover, it is not a history of logic but mostly of the most trivial parts of logic. But I shall be asked whether I do not think his reading marvellously extensive. No, I do not. He had the Munich library at his hand. He had only to look into the books, and for the most part he has done little more than merely to look into them. He really often has no idea of what the real substance of the books is; and nothing is more common than to find in his notes passages copied out of one book which are nothing but textual copies of celebrated passages in much older works. I do not deny that the book is useful, because the rest of us haven't access to such a library; but I do not consider it a work of respectable erudition. There is no need of mincing words because he himself not only refers most disrespectfully to such solid students of medieval writings as Charles Thurot, Haureau and others, but frequently descends to what in English we should call the language of Billingsgate in characterising ancient opinions which he may or may not be aware are identical with those held today by analysts of logical forms whose studies are so much more exact than his that they are not to be named in the same day. Peirce: CP 5.84 Cross-Ref:†† 84. Nevertheless, bad as Prantl's history is, it is the best we have, and any person who reads it critically, as every book ought to be read, will easily be able to see that the ancient students of logic, Democritus, Plato, Aristotle, Epicurus, Philoponus, even Chrysippus, were thinkers of the highest order, and that St. Augustine, Abelard, Aquinas, Duns Scotus, Ockham, Paulus Venetus, even Laurentius Valla, were logicians of the most painstaking and subtle types. But when the revival of learning came, the finest minds had their attention turned in quite another direction, and modern mathematics and modern physics drew away still more. The result of all this has been that during the centuries that have elapsed since the appearance of the De Revolutionibus [1543] -- and remember, if you please, that the work of Copernicus was the fruit of the scientific nourishment that he had imbibed in Italy in his youth -- throughout these ages, the chairs of Logic in the Universities have been turned over to a class of men, of whom we should be speaking far too euphemistically if we were to say that they have in no wise represented the Intellectual Level of their age. No, no; let us speak the plain truth -- modern logicians as a class have been distinctly puerile minds, the kind of minds that never mature, and yet never have the élan and originality of youth. First cast your eyes over the pages of a dozen average treatises, dismissing all preconceived estimates of their authors, and see if that is not the impression you derive from them. Why, in the majority of them, the greatest contribution to reasoning that has been generally applied during these

centuries -- the Calculus of Probabilities -- is almost entirely ignored. If it were only the common run of logics that were affected by this state of things, it would not much matter; for if only one per cent of works on the subject were what they should be, we should still be in possession of a splendid and extensive literature. But unfortunately the general standard has been so terribly lowered that even the treatises written by men of real ability have been but half thought out things. Arnauld, for example, was a thinker of considerable force, and yet L'Art de penser, or the Port Royal Logic, is a shameful exhibit of what the two and a half centuries of man's greatest achievements could consider as a good account of how to think. You may retort that the past three centuries seem to have got on nicely without the aid of logic. Yes, I reply, they have, because there is one thing even more vital to science than intelligent methods; and that is, the sincere desire to find out the truth, whatever it may be; and that those centuries have been blessed with. But according to such estimate -- not exactly mere guess-work, although rough enough, no doubt -- as I have been able to form, if logic during those centuries had been studied with half the zeal and genius that has been bestowed upon mathematics, the twentieth century might have opened with the special sciences generally -- particularly such vitally important sciences as molecular physics, chemistry, physiology, psychology, linguistics, and ancient historical criticism -- in a decidedly more advanced condition than there is much promise that they will have reached at the end of 1950. I shouldn't say that human lives were the most precious things in the world; but after all they have their value; and only think how many lives might thus have been saved. We can mention individuals who might probably have done more work; say Abel, Steiner, Gaulois, Sadi Carnot. Think of the labor of a generation of Germany being allowed to flow off into Hegelianism! Think of the extravagant admiration that half a generation of English -- decidedly the best average reasoners of any modern people, bestowed on that silly thing, Hamilton's New Analytic. Look through Vaihinger's commentary to see what an army of students have been entrapped by Kant's view of the relation between his Analytic and Synthetic Judgments -- a view that a study of the logic of relatives would at once have exploded.†1 Peirce: CP 5.85 Cross-Ref:†† 85. Had logic not been sunk since the time of Copernicus into a condition of semi-idiocy, the Logic of Relatives would by this time have been pursued for three centuries by hundreds of students, among whom there would have been no small number who in this direction or in that would have surpassed in ability any of the poor handful of students who have been at work upon it for the last generation or so. And let me tell you that this study would have completely revolutionized men's most general notions about logic -- the very ideas that are today current in the market-place and on the boulevards. One of the early results of such wide study of the logic of relatives must have been to cause the idea of reaction to be solidly fixed in the minds of all men as an irreducible category of Thought†2 -- whatever place might have been accorded to it in metaphysics as a cosmical category. This I venture to say, notwithstanding that the lamented Schröder did not seem to see it so. Schröder followed Sigwart in his most fundamental ideas of logic. Now I entertain a high respect for Sigwart -- the kind of respect that I feel for Rollin as a historian, for Buffon as a zoölogist, for Priestley as a chemist, for Biot as a physicist -- a class of men whom {ohi polloi} always place too high, and scientific specialists too low. He is one of the most critical and least inexact of the inexact logicians. Sigwart, like almost all the stronger logicians of today, present company excepted, makes the fundamental mistake of confounding the logical question with the psychological question.†3 The psychological question is what processes the mind goes through. But the logical

question is whether the conclusion that will be reached, by applying this or that maxim, will or will not accord with the fact. It may be that the mind is so constituted that that which our intellectual instinct approves will be true to the extent to which that instinct approves of it. If so, that is an interesting fact about the human mind; but it has no relevancy for logic whatsoever. Sigwart says that the question of what is good logic and what bad must in the last resort come down to a question of how we feel; it is a matter of Gefühl, that is, a Quality of Feeling. And this he undertakes to demonstrate. For he says if any other criterion be employed, the correctness of this criterion has to be established by reasoning, and in this reasoning antecedent to the establishment of any rational criterion we must rely upon Gefühl; so that Gefühl is that to which any other criterion must ultimately be referred. Good! This is good intelligent work, such as advances philosophy -- a good, square, explicit fallacy that can be squarely met and definitively refuted. It is the more valuable because it is a form of argument of very wide applicability. It is precisely analogous to the reasoning by which the hedonist in ethics, the subjectivist in esthetics, the idealist in metaphysics, attacks the category of reaction. You perceive the analogy between their arguments. The hedonist says that the question of what is good morals and what bad must ultimately come down to a question of pleasure. For, he says, suppose we desire anything but our own pleasure. Then whatever it may be that we desire, we take satisfaction in; and if we did not take satisfaction in it we should not desire it. But this satisfaction is that very Quality of Feeling that we call pleasure; and thus the only thing we ever can desire is pleasure, and all deliberate action must be performed for the sake of our own pleasure. Peirce: CP 5.85 Cross-Ref:†† Every idealist, too, begins with an analogous argument, though he very likely may not remain consistently on the ground it leads to, so far as it leads anywhere. He says: When I perceive anything I am conscious; and when I am conscious of anything, I am immediately conscious and aught else I may be conscious of, I am conscious of through that immediate consciousness. Consequently whatever I learn from perception is merely that I have a feeling together with whatever I infer from that immediate consciousness. Peirce: CP 5.86 Cross-Ref:†† 86. The answer to all such arguments is that no desire can possibly desire its own satisfaction, no judgment can judge itself to be true, and no reasoning can conclude that it is itself sound. For all these propositions stand on the same footing and must stand or fall together. If any judgment judges itself to be true, all judgments -- or at least all assertory judgments -- do so likewise; for there is no ground of discrimination between assertory judgments in this respect. Either therefore the judgment, J, and the judgment "I say that J is true" are the same for all judgments or for none. But if they are identical, their denials are identical. But their denials are respectively "J is not true" and "I do not say that J is true," which are very different. Consequently no judgment judges itself to be true. All that J does is to furnish a premiss which is complete evidence warranting my assertion in another judgment that J is true. It is important to draw this distinction. The judgment J may, for example, be that "Sirius is white." That is a judgment about Sirius. To myself who perceive myself making this judgment, or to another who hears me assert it and admits my veracity, the evidence is complete that I believe Sirius to be white. But the two propositions "Sirius is white" "I judge that Sirius is white" are two distinct propositions. Peirce: CP 5.86 Cross-Ref:††

There are precisely analogous distinctions in the other cases. I may desire that my sick child should recover, and afterwards reflecting upon the intensity of that desire, I probably shall be unable to refrain from desiring that that desire should be gratified. But I cannot desire that a desire of mine should be gratified unless I already have such a desire; and I have no such desire as long as I am yet in the act of forming the desire, so that the desire is not yet complete. I dare say that some people's psychical disposition is such that they have no sooner formed a strong desire than their thoughts take a subjective turn and they forthwith begin to think what satisfaction it would give them if that desire were gratified, and such people find it difficult to conceive that there are other people whose thoughts follow a train of objective suggestions and who think very little about themselves and their gratifications. That is just one of those respects in which different people may be expected to differ widely. But in no case is the desire absolutely the same as the desire of the satisfaction of that desire. Peirce: CP 5.87 Cross-Ref:†† 87. To return, then, to Sigwart's argument, I not only deny what he asserts that when I make an inference I can only do so because of a certain feeling of logical satisfaction that is connected with doing so, but I maintain that I never can draw an inference because of such a feeling. On the contrary, I never know the inference will afford me any such satisfaction except by a subsequent reflexion after I have already drawn it. It may be that on recognizing the satisfaction the inference gives me I shall consider that as an additional reason for believing in it. But this is another inference which in its turn will afford a new gratification if I stop to reflect about it. Peirce: CP 5.87 Cross-Ref:†† In point of fact it is a serious error of reasoning to regard the sense of logicality as anything more than a tolerably strong argument in favor of the soundness of an inference. For although no doubt the sense of logicality carries men right in the main, yet it very frequently deceives them. Peirce: CP 5.87 Cross-Ref:†† But Sigwart's argument is plainly either wholly fallacious or else what it proves is that which he himself distinctly maintains that it proves, namely, that the soundness of an argument consists in nothing but the Gefühl of logicality. Yet this is a downright absurd position for a logician to take; since, if it were true, there could be no such thing as sincere reasoning that was illogical, and logic, as the criticism of arguments and discrimination of the good from the bad, would have no existence at all; and my sincere argument that Sigwart is wholly in the wrong would be a decision from which there would be no appeal. Peirce: CP 5.87 Cross-Ref:†† If the Holy Father by virtue of his infallibility were to command the faithful to believe that everything that any protestant had ever said was ipso facto necessarily true, it could hardly strain one's assent more. Peirce: CP 5.88 Cross-Ref:†† 88. It is certainly hard to believe, until one is forced to the belief, that a conception, so obtrusively complex as Thirdness is, should be an irreducible unanalyzable conception. What, one naturally exclaims, does this man think to convince us that a conception is complex and simple, at the same time! I might answer this by drawing a distinction. It is complex in the sense that different features may be discriminated in it, but the peculiar idea of complexity that it contains,

although it has complexity as its object, is an unanalyzable idea. Of what is the conception of complexity built up? Produce it by construction without using any idea which involves it if you can. Peirce: CP 5.89 Cross-Ref:†† 89. The best way of satisfying oneself whether Thirdness is elementary or not -- at least, it would be the best way for me, who had in the first place a natural aptitude for logical analysis which has been in constant training all my life long (and I rather think it would be the best way for anybody provided he ruminates over his analysis, returns to it again and again, and criticizes it severely and sincerely, until he reaches a complete insight into the analysis) -- the best way, I say, is to take the idea of representation, say the idea of the fact that the object, A, is represented in the representation, B, so as to determine the interpretation, C: to take this idea and endeavor to state what it consists in without introducing the idea of Thirdness at all if possible, or, if you find that impossible, to see what is the minimum or most degenerate form of Thirdness which will answer the purpose. Peirce: CP 5.89 Cross-Ref:†† Then, having exercised yourself on that problem, take another idea in which, according to my views, Thirdness takes a more degenerate form. Try your hand at a logical analysis of the Fact that A gives B to C. Peirce: CP 5.89 Cross-Ref:†† Then pass to a case in which Thirdness takes a still more degenerate form, as for example the idea of "A and B." What is at once A and B involves the idea of three variables. Putting it mathematically, it is Z = XY, which is the equation of the simpler of the two hyperboloids, the two-sheeted one, as it is called. Peirce: CP 5.89 Cross-Ref:†† Whoever wishes to train his logical powers will find those problems furnish capital exercise; and whoever wishes to get a just conception of the universe will find that the solutions of those problems have a more intimate connection with that conception than he could suspect in advance. Peirce: CP 5.90 Cross-Ref:†† 90. I have thus far been intent on repelling attacks upon the categories which should consist in maintaining that the idea of Reaction can be reduced to that of Quality of Feeling, and the idea of Representation to those of Reaction and Quality of Feeling taken together. But meantime may not the enemy have stolen upon my rear, and shall I not suddenly find myself exposed to an attack which shall run as follows: Peirce: CP 5.90 Cross-Ref:†† We fully admit that you have proved, until we begin to doubt it, that Secondness is not involved in Firstness nor Thirdness in Secondness and Firstness. But you have entirely failed to prove that Firstness, Secondness, and Thirdness are independent ideas for the obvious reason that it is as plain as the nose on your face that the idea of a triplet involves the idea of pairs, and the idea of a pair the idea of units. Consequently, Thirdness is the one and sole category. This is substantially the idea of Hegel; and unquestionably it contains a truth. Peirce: CP 5.90 Cross-Ref:†† Not only does Thirdness suppose and involve the ideas of Secondness and Firstness, but never will it be possible to find any Secondness or Firstness in the phenomenon that is not accompanied by Thirdness.

Peirce: CP 5.91 Cross-Ref:†† 91. If the Hegelians confined themselves to that position they would find a hearty friend in my doctrine. Peirce: CP 5.91 Cross-Ref:†† But they do not. Hegel is possessed with the idea that the Absolute is One. Three absolutes he would regard as a ludicrous contradiction in adjecto. Consequently, he wishes to make out that the three categories have not their several independent and irrefutable standings in thought. Firstness and Secondness must somehow be aufgehoben. But it is not true. They are in no way refuted nor refutable. Thirdness it is true involves Secondness and Firstness, in a sense. That is to say, if you have the idea of Thirdness you must have had the ideas of Secondness and Firstness to build upon. But what is required for the idea of a genuine Thirdness is an independent solid Secondness and not a Secondness that is a mere corollary of an unfounded and inconceivable Thirdness; and a similar remark may be made in reference to Firstness. Peirce: CP 5.92 Cross-Ref:†† 92. Let the Universe be an evolution of Pure Reason if you will. Yet if, while you are walking in the street reflecting upon how everything is the pure distillate of Reason, a man carrying a heavy pole suddenly pokes you in the small of the back, you may think there is something in the Universe that Pure Reason fails to account for; and when you look at the color red and ask yourself how Pure Reason could make red to have that utterly inexpressible and irrational positive quality it has, you will be perhaps disposed to think that Quality and Reaction have their independent standing in the Universe.

Peirce: CP 5.93 Cross-Ref:†† LECTURE IV †1 THE REALITY OF THIRDNESS

§1. SCHOLASTIC REALISM

93. I proceed to argue that Thirdness is operative in Nature. Suppose we attack the question experimentally. Here is a stone. Now I place that stone where there will be no obstacle between it and the floor, and I will predict with confidence that as soon as I let go my hold upon the stone it will fall to the floor. I will prove that I can make a correct prediction by actual trial if you like. But I see by your faces that you all think it will be a very silly experiment. Why so? Because you all know very well that I can predict what will happen, and that the fact will verify my prediction. Peirce: CP 5.94 Cross-Ref:†† 94. But how can I know what is going to happen? You certainly do not think

that it is by clairvoyance, as if the future event by its existential reactiveness could affect me directly, as in an experience of it, as an event scarcely past might affect me. You know very well that there is nothing of the sort in this case. Still, it remains true that I do know that that stone will drop, as a fact, as soon as I let go my hold. If I truly know anything, that which I know must be real. It would be quite absurd to say that I could be enabled to know how events are going to be determined over which I can exercise no more control than I shall be able to exercise over this stone after it shall have left my hand, that I can so peer in the future merely on the strength of any acquaintance with any pure fiction. Peirce: CP 5.95 Cross-Ref:†† 95. I know that this stone will fall if it is let go, because experience has convinced me that objects of this kind always do fall; and if anyone present has any doubt on the subject, I should be happy to try the experiment, and I will bet him a hundred to one on the result. Peirce: CP 5.96 Cross-Ref:†† 96. But the general proposition that all solid bodies fall in the absence of any upward forces or pressure, this formula I say, is of the nature of a representation. Our nominalistic friends would be the last to dispute that. They will go so far as to say that it is a mere representation -- the word mere meaning that to be represented and really to be are two very different things; and that this formula has no being except a being represented. It certainly is of the nature of a representation. That is undeniable, I grant. And it is equally undeniable that that which is of the nature of a representation is not ipso facto real. In that respect there is a great contrast between an object of reaction and an object of representation. Whatever reacts is ipso facto real. But an object of representation is not ipso facto real. If I were to predict that on my letting go of the stone it would fly up in the air, that would be mere fiction; and the proof that it was so would be obtained by simply trying the experiment. That is clear. On the other hand, and by the same token, the fact that I know that this stone will fall to the floor when I let it go, as you all must confess, if you are not blinded by theory, that I do know -- and you none of you care to take up my bet, I notice -- is the proof that the formula, or uniformity, as furnishing a safe basis for prediction, is, or if you like it better, corresponds to, a reality. Peirce: CP 5.97 Cross-Ref:†† 97. Possibly at this point somebody may raise an objection and say: You admit, that is one thing really to be and another to be represented; and you further admit that it is of the nature of the law of nature to be represented. Then it follows that it has not the mode of being of a reality. My answer to this would be that it rests upon an ambiguity. When I say that the general proposition as to what will happen, whenever a certain condition may be fulfilled, is of the nature of a representation, I mean that it refers to experiences in futuro, which I do not know are all of them experienced and never can know have been all experienced. But when I say that really to be is different from being represented, I mean that what really is, ultimately consists in what shall be forced upon us in experience, that there is an element of brute compulsion in fact and that fact is not a mere question of reasonableness. Thus, if I say, "I shall wind up my watch every day as long as I live," I never can have a positive experience which certainly covers all that is here promised, because I never shall know for certain that my last day has come. But what the real fact will be does not depend upon what I represent, but upon what the experiential reactions shall be. My assertion that I shall wind up my watch every day of my life may turn out to

accord with facts, even though I be the most irregular of persons, by my dying before nightfall. Peirce: CP 5.97 Cross-Ref:†† If we call that being true by chance, here is a case of a general proposition being entirely true in all its generality by chance. Peirce: CP 5.98 Cross-Ref:†† 98. Every general proposition is limited to a finite number of occasions in which it might conceivably be falsified, supposing that it is an assertion confined to what human beings may experience; and consequently it is conceivable that, although it should be true without exception, it should still only be by chance that it turns out true. Peirce: CP 5.99 Cross-Ref:†† 99. But if I see a man who is very regular in his habits and am led to offer to wager that that man will not miss winding his watch for the next month, you have your choice between two alternative hypotheses only: Peirce: CP 5.99 Cross-Ref:†† 1. You may suppose that some principle or cause is really operative to make him wind his watch daily, which active principle may have more or less strength; or Peirce: CP 5.99 Cross-Ref:†† 2. You may suppose that it is mere chance that his actions have hitherto been regular; and in that case, that regularity in the past affords you not the slightest reason for expecting its continuance in the future, any more than, if he had thrown sixes three times running, that event would render it either more or less likely that his next throw would show sixes. Peirce: CP 5.100 Cross-Ref:†† 100. It is the same with the operations of nature. With overwhelming uniformity, in our past experience, direct and indirect, stones left free to fall have fallen. Thereupon two hypotheses only are open to us. Either Peirce: CP 5.100 Cross-Ref:†† 1. the uniformity with which those stones have fallen has been due to mere chance and affords no ground whatever, not the slightest for any expectation that the next stone that shall be let go will fall; or Peirce: CP 5.100 Cross-Ref:†† 2. the uniformity with which stones have fallen has been due to some active general principle, in which case it would be a strange coincidence that it should cease to act at the moment my prediction was based upon it. Peirce: CP 5.100 Cross-Ref:†† That position, gentlemen, will sustain criticism. It is irrefragable. Peirce: CP 5.101 Cross-Ref:†† 101. Of course, every sane man will adopt the latter hypothesis. If he could doubt it in the case of the stone -- which he can't -- and I may as well drop the stone once for all -- I told you so! -- if anybody doubts this still, a thousand other such inductive predictions are getting verified every day, and he will have to suppose every one of them to be merely fortuitous in order reasonably to escape the conclusion that general principles are really operative in nature. That is the doctrine of scholastic

realism.

Peirce: CP 5.102 Cross-Ref:†† §2. THIRDNESS AND GENERALITY †1

102. You may, perhaps, ask me how I connect generality with Thirdness. Various different replies, each fully satisfactory, may be made to that inquiry. The old definition of a general is Generale est quod natum aptum est dici de multis.†2 This recognizes that the general is essentially predicative and therefore of the nature of a representamen. And by following out that path of suggestion we should obtain a good reply to the inquiry. Peirce: CP 5.103 Cross-Ref:†† 103. In another respect, however, the definition represents a very degenerate sort of generality. None of the scholastic logics fails to explain that sol is a general term; because although there happens to be but one sun yet the term sol aptum natum est dici de multis. But that is most inadequately expressed. If sol is apt to be predicated of many, it is apt to be predicated of any multitude however great, and since there is no maximum multitude,†3 those objects, of which it is fit to be predicated, form an aggregate that exceeds all multitude. Take any two possible objects that might be called suns and, however much alike they may be, any multitude whatsoever of intermediate suns are alternatively possible, and therefore as before these intermediate possible suns transcend all multitude. In short, the idea of a general involves the idea of possible variations which no multitude of existent things could exhaust but would leave between any two not merely many possibilities, but possibilities absolutely beyond all multitude. Peirce: CP 5.104 Cross-Ref:†† 104. Now Thirdness is nothing but the character of an object which embodies Betweenness or Mediation in its simplest and most rudimentary form; and I use it as the name of that element of the phenomenon which is predominant wherever Mediation is predominant, and which reaches its fullness in Representation. Peirce: CP 5.105 Cross-Ref:†† 105. Thirdness, as I use the term, is only a synonym for Representation, to which I prefer the less colored term because its suggestions are not so narrow and special as those of the word Representation. Now it is proper to say that a general principle that is operative in the real world is of the essential nature of a Representation and of a Symbol because its modus operandi is the same as that by which words produce physical effects. Nobody can deny that words do produce such effects. Take, for example, that sentence of Patrick Henry which, at the time of our Revolution, was repeated by every man to his neighbor: Peirce: CP 5.105 Cross-Ref:†† "Three millions of people, armed in the holy cause of Liberty, and in such a country as we possess, are invincible against any force that the enemy can bring against us." Peirce: CP 5.105 Cross-Ref:†† Those words present this character of the general law of nature. They might

have produced effects indefinitely transcending any that circumstances allowed them to produce. It might, for example, have happened that some American schoolboy, sailing as a passenger in the Pacific Ocean, should have idly written down those words on a slip of paper. The paper might have been tossed overboard and might have been picked up by some Jagala on a beach of the island of Luzon; and if he had had them translated to him, they might easily have passed from mouth to mouth there as they did in this country, and with similar effect. Peirce: CP 5.106 Cross-Ref:†† 106. Words then do produce physical effects. It is madness to deny it. The very denial of it involves a belief in it; and nobody can consistently fail to acknowledge it until he sinks to a complete mental paresis. Peirce: CP 5.106 Cross-Ref:†† But how do they produce their effect? They certainly do not, in their character as symbols, directly react upon matter. Such action as they have is merely logical. It is not even psychological. It is merely that one symbol would justify another. However, suppose that first difficulty to have been surmounted, and that they do act upon actual thoughts. That thoughts act on the physical world and conversely, is one of the most familiar of facts. Those who deny it are persons with whom theories are stronger than facts. But how thoughts act on things it is impossible for us, in the present state of our knowledge, so much as to make any very promising guess; although, as I will show you presently,†1 a guess can be made which suffices to show that the problem is not beyond all hope of ultimate solution. Peirce: CP 5.107 Cross-Ref:†† 107. All this is equally true of the manner in which the laws of nature influence matter. A law is in itself nothing but a general formula or symbol. An existing thing is simply a blind reacting thing, to which not merely all generality, but even all representation, is utterly foreign. The general formula may logically determine another, less broadly general. But it will be of its essential nature general, and its being narrower does not in the least constitute any participation in the reacting character of the thing. Here we have that great problem of the principle of individuation which the scholastic doctors after a century of the closest possible analysis were obliged to confess was quite incomprehensible to them. Analogy suggests that the laws of nature are ideas or resolutions in the mind of some vast consciousness, who, whether supreme or subordinate, is a Deity relatively to us. I do not approve of mixing up Religion and Philosophy; but as a purely philosophical hypothesis, that has the advantage of being supported by analogy. Yet I cannot clearly see that beyond that support to the imagination it is of any particular scientific service. . . .

Peirce: CP 5.108 Cross-Ref:†† §3. NORMATIVE JUDGMENTS

108. Reasoning cannot possibly be divorced from logic; because, whenever a man reasons, he thinks that he is drawing a conclusion such as would be justified in every analogous case. He therefore cannot really infer without having a notion of a class of possible inferences, all of which are logically good. That distinction of good

and bad he always has in mind when he infers. Logic proper is the critic of arguments, the pronouncing them to be good or bad. There are, as I am prepared to maintain, operations of the mind which are logically exactly analogous to inferences excepting only that they are unconscious and therefore uncontrollable and therefore not subject to criticism. But that makes all the difference in the world; for inference is essentially deliberate, and self-controlled. Any operation which cannot be controlled, any conclusion which is not abandoned, not merely as soon as criticism has pronounced against it, but in the very act of pronouncing that decree, is not of the nature of rational inference -- is not reasoning. Reasoning as deliberate is essentially critical, and it is idle to criticize as good or bad that which cannot be controlled. Reasoning essentially involves self-control; so that the logica utens†1 is a particular species of morality. Logical goodness and badness, which we shall find is simply the distinction of Truth and Falsity in general, amounts, in the last analysis, to nothing but a particular application of the more general distinction of Moral Goodness and Badness, or Righteousness and Wickedness.†2 Peirce: CP 5.109 Cross-Ref:†† 109. To criticize as logically sound or unsound an operation of thought that cannot be controlled is not less ridiculous than it would be to pronounce the growth of your hair to be morally good or bad. The ridiculousness in both cases consists in the fact that such a critical judgment may be pretended but cannot really be performed in clear thought, for on analysis it will be found absurd. Peirce: CP 5.110 Cross-Ref:†† 110. I am quite aware that this position is open to two serious objections, which I have not time to discuss, but which I have carefully considered and refuted. The first is that this is making logic a question of psychology.†3 But this I deny. Logic does rest on certain facts of experience among which are facts about men, but not upon any theory about the human mind or any theory to explain facts. The other objection is that if the distinction [between] Good and Bad Logic is a special case [of the distinction between] Good and Bad Morals, by the same token the distinction of Good and Bad Morals is a special case of the distinction [between] esthetic Goodness and Badness. Now to admit this is not only to admit hedonism, which no man in his senses, and not blinded by theory or something worse, can admit, but also, having to do with the essentially Dualistic distinction of Good and Bad -- which is manifestly an affair of Category the Second -- it seeks the origin of this distinction in Esthetic Feeling, which belongs to Category the First. Peirce: CP 5.111 Cross-Ref:†† 111. This last objection deceived me for many years. The reply to it involves a very important point which I shall have to postpone to the next lecture. When it first presented itself to me, all I knew of ethics was derived from the study of Jouffroy †1 under Dr. Walker,†2 of Kant, and of a wooden treatise by Whewell;†3 and I was led by this objection to a line of thought which brought me to regard ethics as a mere art, or applied science, and not a pure normative science at all. But when, beginning in 1883, I came to read the works of the great moralists, whose great fertility of thought I found in wonderful contrast to the sterility of the logicians -- I was forced to recognize the dependence of Logic upon Ethics; and then took refuge in the idea that there was no science of esthetics, that, because de gustibus non est disputandum, therefore there is no esthetic truth and falsity or generally valid goodness and badness. But I did not remain of this opinion long. I soon came to see that this whole objection rests upon a fundamental misconception. To say that morality, in the last

resort, comes to an esthetic judgment is not hedonism -- but is directly opposed to hedonism. In the next place, every pronouncement between Good and Bad certainly comes under Category the Second; and for that reason such pronouncement comes out in the voice of conscience with an absoluteness of duality which we do not find even in logic; and although I am still a perfect ignoramus in esthetics, I venture to think that the esthetic state of mind is purest when perfectly naive without any critical pronouncement, and that the esthetic critic founds his judgments upon the result of throwing himself back into such a pure naive state -- and the best critic is the man who has trained himself to do this the most perfectly. Peirce: CP 5.112 Cross-Ref:†† 112. It is a great mistake to suppose that the phenomena of pleasure and pain are mainly phenomena of feeling.†1 Examine pain, which would seem to be a good deal more positive than pleasure. I am unable to recognize with confidence any quality of feeling common to all pains; and if I cannot I am sure it cannot be an easy thing for anybody. For I have gone through a systematic course of training in recognizing my feelings. I have worked with intensity for so many hours a day every day for long years to train myself to this; and it is a training which I would recommend to all of you. The artist has such a training; but most of his effort goes to reproducing in one form or another what he sees or hears, which is in every art a very complicated trade; while I have striven simply to see what it is that I see. That this limitation of the task is a great advantage is proved to me by finding that the great majority of artists are extremely narrow. Their esthetic appreciations are narrow; and this comes from their only having the power of recognizing the qualities of their percepts in certain directions. Peirce: CP 5.112 Cross-Ref:†† But the majority of those who opine that pain is a quality of feeling are not even artists; and even among those who are artists there are extremely few who are artists in pain. But the truth is that there are certain states of mind, especially among states of mind in which Feeling has a large share, which we have an impulse to get rid of. That is the obvious phenomenon; and the ordinary theory is that this impulse is excited by a quality of feeling common to all these states -- a theory which is supported by the fact that this impulse is particularly energetic in regard to states in which Feeling is the predominant element. Now whether this be true or false, it is a theory. It is not the fact that any such common quality in all pains is readily to be recognized. Peirce: CP 5.113 Cross-Ref:†† 113. At any rate, while the whole phenomenon of pain and the whole phenomenon of pleasure are phenomena that arise within the universe of states of mind and attain no great prominence except when they concern states of mind in which Feeling is predominant, yet these phenomena themselves do not mainly consist in any common Feeling-quality of Pleasure and any common Feeling-quality of Pain, even if there are such Qualities of Feeling; but they mainly consist [in a] Pain [which lies] in a Struggle to give a state of mind its quietus, and [in a] Pleasure in a peculiar mode of consciousness allied to the consciousness of making a generalization, in which not Feeling, but rather Cognition is the principal constituent. This may be hard to make out as regards the lower pleasures, but they do not concern the argument we are considering. It is esthetic enjoyment which concerns us; and ignorant as I am of Art, I have a fair share of capacity for esthetic enjoyment; and it seems to me that while in esthetic enjoyment we attend to the totality of Feeling -- and especially to the

total resultant Quality of Feeling presented in the work of art we are contemplating -yet it is a sort of intellectual sympathy, a sense that here is a Feeling that one can comprehend, a reasonable Feeling. I do not succeed in saying exactly what it is, but it is a consciousness belonging to the category of Representation, though representing something in the Category of Quality of Feeling. Peirce: CP 5.113 Cross-Ref:†† In that view of the matter, the objection to the doctrine that the distinction Moral approval and disapproval is ultimately only a species of the distinction Esthetic approval and disapproval seems to be answered. Peirce: CP 5.114 Cross-Ref:†† 114. It appears, then, that Logica utens consisting in self-control, the distinction of logical goodness and badness must begin where control of the processes of cognition begins; and any object that antecedes the distinction, if it has to be named either good or bad, must be named good. For since no fault can be found with it, it must be taken at its own valuation.

Peirce: CP 5.115 Cross-Ref:†† §4. PERCEPTUAL JUDGMENTS †1

115. Where then in the process of cognition does the possibility of controlling it begin? Certainly not before the percept is formed. Peirce: CP 5.115 Cross-Ref:†† Even after the percept is formed there is an operation which seems to me to be quite uncontrollable. It is that of judging what it is that the person perceives. A judgment is an act of formation of a mental proposition combined with an adoption of it or act of assent to it. A percept on the other hand is an image or moving picture or other exhibition. The perceptual judgment, that is, the first judgment of a person as to what is before his senses, bears no more resemblance to the percept than the figure I am going to draw is like a man.

||-|| MAN

Peirce: CP 5.115 Cross-Ref:†† I do not see that it is possible to exercize any control over that operation or to subject it to criticism. If we can criticize it at all, as far as I can see, that criticism would be limited to performing it again and seeing whether, with closer attention, we get the same result. But when we so perform it again, paying now closer attention, the percept is presumably not such as it was before. I do not see what other means we have of knowing whether it is the same as it was before or not, except by comparing the former perceptual judgment and the later one. I should utterly distrust any other method of ascertaining what the character of the percept was. Consequently, until I am better advised, I shall consider the perceptual judgment to be utterly beyond

control. Should I be wrong in this, the Percept, at all events, would seem to be so. Peirce: CP 5.116 Cross-Ref:†† 116. It follows, then, that our perceptual judgments are the first premisses of all our reasonings and that they cannot be called in question. All our other judgments are so many theories whose only justification is that they have been and will be borne out by perceptual judgments. But the perceptual judgments declare one thing to be blue, another yellow -- one sound to be that of A, another that of U, another that of I. These are the Qualities of Feeling which the physicists say are mere illusions because there is no room for them in their theories. If the facts won't agree with the Theory, so much the worse for them. They are bad facts. This sounds to me childish, I confess. It is like an infant that beats an inanimate object that hurts it. Indeed this is true of all fault-finding with others than oneself, and those for whose conduct one is responsible. Reprobation is a silly [business]. Peirce: CP 5.117 Cross-Ref:†† 117. But peradventure I shall be asked whether I do not admit that there is any such thing as an illusion or hallucination. Oh, yes; among artists I have known more than one case of downright hallucinatory imaginations at the beck and call of these {poietai}. Of course, the man knows that such obedient spectres are not real experiences, because experience is that which forces itself upon him, will-he nill-he. Peirce: CP 5.117 Cross-Ref:†† Hallucinations proper -- obsessional hallucinations -- will not down at one's bidding, and people who are subject to them are accustomed to sound the people who are with them in order to ascertain whether the object before them has a being independent of their disease or not. There are also social hallucinations. Peirce: CP 5.117 Cross-Ref:†† In such a case, a photographic camera or other instrument might be of service. Peirce: CP 5.118 Cross-Ref:†† 118. Of course, everybody admits and must admit that these apparitions are entities -- entia; the question is whether these entia belong to the class of realities or not, that is, whether they are such as they are independently of any collection of singular representations that they are so, or whether their mode of being depends upon abnormal conditions. But as for the entire universe of Qualities which the physicist would pronounce Illusory, there is not the smallest shade of just suspicion resting upon their normality. On the contrary, there is considerable evidence that colors, for example, and sounds have the same character for all mankind. Peirce: CP 5.118 Cross-Ref:†† Well, I will skip this. Suffice it to say that there is no reason for suspecting the veracity of the senses, and the presumption is that the physics of the future will find out that they are more real than the present state of scientific theory admits of their being represented as being.†1 Peirce: CP 5.119 Cross-Ref:†† 119. Therefore, if you ask me what part Qualities can play in the economy of the universe, I shall reply that the universe is a vast representamen, a great symbol of God's purpose, working out its conclusions in living realities. Now every symbol must have, organically attached to it, its Indices of Reactions and its Icons of Qualities; and such part as these reactions and these qualities play in an argument that, they of course, play in the universe -- that Universe being precisely an argument. In the little

bit that you or I can make out of this huge demonstration, our perceptual judgments are the premisses for us and these perceptual judgments have icons as their predicates, in which icons Qualities are immediately presented. But what is first for us is not first in nature. The premisses of Nature's own process are all the independent uncaused elements of facts that go to make up the variety of nature which the necessitarian supposes to have been all in existence from the foundation of the world, but which the Tychist supposes are continually receiving new accretions.†1 These premisses of nature, however, though they are not the perceptual facts that are premisses to us, nevertheless must resemble them in being premisses. We can only imagine what they are by comparing them with the premisses for us. As premisses they must involve Qualities. Peirce: CP 5.119 Cross-Ref:†† Now as to their function in the economy of the Universe. The Universe as an argument is necessarily a great work of art, a great poem -- for every fine argument is a poem and a symphony -- just as every true poem is a sound argument. But let us compare it rather with a painting -- with an impressionist seashore piece -- then every Quality in a Premiss is one of the elementary colored particles of the Painting; they are all meant to go together to make up the intended Quality that belongs to the whole as whole. That total effect is beyond our ken; but we can appreciate in some measure the resultant Quality of parts of the whole -- which Qualities result from the combinations of elementary Qualities that belong to the premisses. But I shall endeavor to make this clearer in the next lecture.

Peirce: CP 5.120 Cross-Ref:†† LECTURE V †1 THE THREE KINDS OF GOODNESS

§1. THE DIVISIONS OF PHILOSOPHY †2

120. . . . I have already explained †3 that by Philosophy I mean that department of Positive Science, or Science of Fact, which does not busy itself with gathering facts, but merely with learning what can be learned from that experience which presses in upon every one of us daily and hourly. It does not gather new facts, because it does not need them, and also because new general facts cannot be firmly established without the assumption of a metaphysical doctrine; and this, in turn, requires the coöperation of every department of philosophy; so that such new facts, however striking they may be, afford weaker support to philosophy by far than that common experience which nobody doubts or can doubt, and which nobody ever even pretended to doubt except as a consequence of belief in that experience so entire and perfect that it failed to be conscious of itself; just as an American who has never been abroad fails to perceive the characteristics of Americans; just as a writer is unaware of the peculiarities of his own style; just as none of us can see himself as others see him.

Peirce: CP 5.119 Cross-Ref:†† Now I am going to make a series of assertions which will sound wild; for I cannot stop to argue them, although I cannot omit them if I am to set the supports of pragmatism in their true light. Peirce: CP 5.121 Cross-Ref:†† 121. Philosophy has three grand divisions. The first is Phenomenology, which simply contemplates the Universal Phenomenon and discerns its ubiquitous elements, Firstness, Secondness, and Thirdness, together perhaps with other series of categories. The second grand division is Normative Science, which investigates the universal and necessary laws of the relation of Phenomena to Ends, that is, perhaps, to Truth, Right, and Beauty. The third grand division is Metaphysics, which endeavors to comprehend the Reality of Phenomena. Now Reality is an affair of Thirdness as Thirdness, that is, in its mediation between Secondness and Firstness. Most, if not all of you, are, I doubt not, Nominalists; and I beg you will not take offence at a truth which is just as plain and undeniable to me as is the truth that children do not understand human life. To be a nominalist consists in the undeveloped state in one's mind of the apprehension of Thirdness as Thirdness. The remedy for it consists in allowing ideas of human life to play a greater part in one's philosophy. Metaphysics is the science of Reality. Reality consists in regularity. Real regularity is active law. Active law is efficient reasonableness, or in other words is truly reasonable reasonableness. Reasonable reasonableness is Thirdness as Thirdness. Peirce: CP 5.121 Cross-Ref:†† So then the division of Philosophy into these three grand departments, whose distinctness can be established without stopping to consider the contents of Phenomenology (that is, without asking what the true categories may be), turns out to be a division according to Firstness, Secondness, and Thirdness, and is thus one of the very numerous phenomena I have met with which confirm this list of categories. Peirce: CP 5.122 Cross-Ref:†† 122. Phenomenology treats of the universal Qualities of Phenomena in their immediate phenomenal character, in themselves as phenomena. It, thus, treats of Phenomena in their Firstness. Peirce: CP 5.123 Cross-Ref:†† 123. Normative Science treats of the laws of the relation of phenomena to ends; that is, it treats of Phenomena in their Secondness. Peirce: CP 5.124 Cross-Ref:†† 124. Metaphysics, as I have just remarked, treats of Phenomena in their Thirdness. Peirce: CP 5.125 Cross-Ref:†† 125. If, then, Normative Science does not seem to be sufficiently described by saying that it treats of phenomena in their secondness, this is an indication that our conception of Normative Science is too narrow; and I had come to the conclusion that this is true of even the best modes of conceiving Normative Science which have achieved any renown, many years before I recognized the proper division of philosophy. Peirce: CP 5.125 Cross-Ref:†† I wish I could talk for an hour to you concerning the true conception of

normative science. But I shall only be able to make a few negative assertions which, even if they were proved, would not go far toward developing that conception. Normative Science is not a skill, nor is it an investigation conducted with a view to the production of skill. Coriolis wrote a book on the Analytic Mechanics of the Game of Billiards.†1 If that book does not help people in the least degree to play billiards, that is nothing against it. The book is only intended to be pure theory. In like manner, if Normative Science does not in the least tend to the development of skill, its value as Normative Science remains the same. It is purely theoretical. Of course there are practical sciences of reasoning and investigation, of the conduct of life, and of the production of works of art. They correspond to the Normative Sciences, and may be probably expected to receive aid from them. But they are not integrant parts of these sciences; and the reason that they are not so, thank you, is no mere formalism, but is this, that it will be in general quite different men -- two knots of men not apt to consort the one with the other -- who will conduct the two kinds of inquiry. Nor again is Normative Science a special science, that is, one of those sciences that discover new phenomena. It is not even aided in any appreciable degree by any such science, and let me say that it is no more by psychology than by any other special science. If we were to place six lots each of seven coffee beans in one pan of an equal-armed balance, and forty-two coffee beans in the other pan, and were to find on trial that the two loads nearly balanced one another, this observation might be regarded as adding in some excessively slight measure to the certainty of the proposition that six times seven make forty-two; because it is conceivable that this proposition should be a mistake due to some peculiar insanity affecting the whole human race, and the experiment may possibly evade the effects of that insanity, supposing that we are affected with it. In like manner, and in just about the same degree, the fact that men for the most part show a natural disposition to approve nearly the same arguments that logic approves, nearly the same acts that ethics approves, and nearly the same works of art that esthetics approves, may be regarded as tending to support the conclusions of logic, ethics, and esthetics. But such support is perfectly insignificant; and when it comes to a particular case, to urge that anything is sound and good logically, morally, or esthetically, for no better reason than that men have a natural tendency to think so, I care not how strong and imperious that tendency may be, is as pernicious a fallacy as ever was. Of course it is quite a different thing for a man to acknowledge that he cannot perceive that he doubts what he does not appreciably doubt. Peirce: CP 5.126 Cross-Ref:†† 126. In one of the ways I have indicated, especially the last, Normative Science is by the majority of writers of the present day ranked too low in the scale of the sciences. On the other hand, some students of exact logic rank that normative science, at least, too high, by virtually treating it as on a par with pure mathematics.†1 There are three excellent reasons any one of which ought to rescue them from the error of this opinion. In the first place, the hypotheses from which the deductions of normative science proceed are intended to conform to positive truth of fact and those deductions derive their interest from that circumstance almost exclusively; while the hypotheses of pure mathematics are purely ideal in intention, and their interest is purely intellectual. But in the second place, the procedure of the normative sciences is not purely deductive, as that of mathematics is, nor even principally so. Their peculiar analyses of familiar phenomena, analyses which ought to be guided by the facts of phenomenology in a manner in which mathematics is not at all guided, separate Normative Science from mathematics quite radically. In the third place, there is a most intimate and essential element of Normative Science which

is still more proper to it, and that is its peculiar appreciations, to which nothing at all in the phenomena, in themselves, corresponds. These appreciations relate to the conformity of phenomena to ends which are not immanent within those phenomena. Peirce: CP 5.127 Cross-Ref:†† 127. There are sundry other widely spread misconceptions of the nature of Normative Science. One of these is that the chief, if not the only, problem of Normative Science is to say what is good and what bad, logically, ethically, and esthetically; or what degree of goodness a given description of phenomenon attains. Were this the case, normative science would be, in a certain sense, mathematical, since it would deal entirely with a question of quantity. But I am strongly inclined to think that this view will not sustain critical examination. Logic classifies arguments, and in doing so recognizes different kinds of truth. In ethics, too, qualities of good are admitted by the great majority of moralists. As for esthetics, in that field qualitative differences appear to be so prominent that, abstracted from them, it is impossible to say that there is any appearance which is not esthetically good. Vulgarity and pretension, themselves, may appear quite delicious in their perfection, if we can once conquer our squeamishness about them, a squeamishness which results from a contemplation of them as possible qualities of our own handiwork -- but that is a moral and not an esthetic way of considering them. I hardly need remind you that goodness, whether esthetic, moral, or logical, may either be negative -- consisting in freedom from fault -- or quantitative -- consisting in the degree to which it attains. But in an inquiry, such as we are now engaged upon, negative goodness is the important thing. Peirce: CP 5.128 Cross-Ref:†† 128. A subtle and almost ineradicable narrowness in the conception of Normative Science runs through almost all modern philosophy in making it relate exclusively to the human mind. The beautiful is conceived to be relative to human taste, right and wrong concern human conduct alone, logic deals with human reasoning. Now in the truest sense these sciences certainly are indeed sciences of mind. Only, modern philosophy has never been able quite to shake off the Cartesian idea of the mind, as something that "resides" -- such is the term †1 -- in the pineal gland. Everybody laughs at this nowadays, and yet everybody continues to think of mind in this same general way, as something within this person or that, belonging to him and correlative to the real world. A whole course of lectures would be required to expose this error. I can only hint that if you reflect upon it, without being dominated by preconceived ideas, you will soon begin to perceive that it is a very narrow view of mind. I should think it must appear so to anybody who was sufficiently soaked in the Critic of the Pure Reason.

Peirce: CP 5.129 Cross-Ref:†† §2. ETHICAL AND ESTHETICAL GOODNESS †1

129. I cannot linger more upon the general conception of Normative Science. I must come down to the particular Normative Sciences. These are now commonly said to be logic, ethics, and esthetics. Formerly only logic and ethics were reckoned as such. A few logicians refuse to recognize any other normative science than their own. My own opinions of ethics and esthetics are far less matured than my logical

opinions. It is only since 1883 that I have numbered ethics among my special studies; and until about four years ago, I was not prepared to affirm that ethics was a normative science. As for esthetics, although the first year of my study of philosophy was devoted to this branch exclusively, yet I have since then so completely neglected it that I do not feel entitled to have any confident opinions about it. I am inclined to think that there is such a Normative Science; but I feel by no means sure even of that. Peirce: CP 5.129 Cross-Ref:†† Supposing, however, that normative science divides into esthetics, ethics, and logic, then it is easily perceived, from my standpoint, that this division is governed by the three categories. For Normative Science in general being the science of the laws of conformity of things to ends, esthetics considers those things whose ends are to embody qualities of feeling, ethics those things whose ends lie in action, and logic those things whose end is to represent something. Peirce: CP 5.130 Cross-Ref:†† 130. Just at this point we begin to get upon the trail of the secret of pragmatism, after a long and apparently aimless beating about the bush. Let us glance at the relations of these three sciences to one another. Whatever opinion be entertained in regard to the scope of logic, it will be generally agreed that the heart of it lies in the classification and critic of arguments. Now it is peculiar to the nature of argument that no argument can exist without being referred to some special class of arguments. The act of inference consists in the thought that the inferred conclusion is true because in any analogous case an analogous conclusion would be true. Thus, logic is coeval with reasoning. Whoever reasons ipso facto virtually holds a logical doctrine, his logica utens.†2 This classification is not a mere qualification of the argument. It essentially involves an approval of it -- a qualitative approval. Now such self-approval supposes self-control. Not that we regard our approval as itself a voluntary act, but that we hold the act of inference, which we approve, to be voluntary. That is, if we did not approve, we should not infer. There are mental operations which are as completely beyond our control as the growth of our hair. To approve or disapprove of them would be idle. But when we institute an experiment to test a theory, or when we imagine an extra line to be inserted in a geometrical diagram in order to determine a question in geometry, these are voluntary acts which our logic, whether it be of the natural or the scientific sort, approves. Now, the approval of a voluntary act is a moral approval. Ethics is the study of what ends of action we are deliberately prepared to adopt. That is right action which is in conformity to ends which we are prepared deliberately to adopt. That is all there can be in the notion of righteousness, as it seems to me. The righteous man is the man who controls his passions, and makes them conform to such ends as he is prepared deliberately to adopt as ultimate. If it were in the nature of a man to be perfectly satisfied to make his personal comfort his ultimate aim, no more blame would attach to him for doing so than attaches to a hog for behaving in the same way. A logical reasoner is a reasoner who exercises great self-control in his intellectual operations; and therefore the logically good is simply a particular species of the morally good. Ethics -- the genuine normative science of ethics, as contradistinguished from the branch of anthropology which in our day often passes under the name of ethics -- this genuine ethics is the normative science par excellence, because an end -- the essential object of normative science -- is germane to a voluntary act in a primary way in which it is germane to nothing else. For that reason I have some lingering doubt as to there being any true normative science of the beautiful. On the other hand, an ultimate end of action deliberately adopted -- that is to say, reasonably adopted -- must be a state

of things that reasonably recommends itself in itself aside from any ulterior consideration. It must be an admirable ideal, having the only kind of goodness that such an ideal can have; namely, esthetic goodness. From this point of view the morally good appears as a particular species of the esthetically good. Peirce: CP 5.131 Cross-Ref:†† 131. If this line of thought be sound, the morally good will be the esthetically good specially determined by a peculiar superadded element; and the logically good will be the morally good specially determined by a special superadded element. Now it will be admitted to be, at least, very likely that in order to correct or to vindicate the maxim of pragmatism, we must find out precisely what the logically good consists in; and it would appear from what has been said that, in order to analyze the nature of the logically good, we must first gain clear apprehensions of the nature of the esthetically good and especially that of the morally good. Peirce: CP 5.132 Cross-Ref:†† 132. So, then, incompetent as I am to it, I find the task imposed upon me of defining the esthetically good -- a work which so many philosophical artists have made as many attempts at performing. In the light of the doctrine of categories I should say that an object, to be esthetically good, must have a multitude of parts so related to one another as to impart a positive simple immediate quality to their totality; and whatever does this is, in so far, esthetically good, no matter what the particular quality of the total may be. If that quality be such as to nauseate us, to scare us, or otherwise to disturb us to the point of throwing us out of the mood of esthetic enjoyment, out of the mood of simply contemplating the embodiment of the quality -just, for example, as the Alps affected the people of old times, when the state of civilization was such that an impression of great power was inseparably associated with lively apprehension and terror -- then the object remains none the less esthetically good, although people in our condition are incapacitated from a calm esthetic contemplation of it. Peirce: CP 5.132 Cross-Ref:†† This suggestion must go for what it may be worth, which I dare say may be very little. If it be correct, it will follow that there is no such thing as positive esthetic badness; and since by goodness we chiefly in this discussion mean merely the absence of badness, or faultlessness, there will be no such thing as esthetic goodness. All there will be will be various esthetic qualities; that is, simple qualities of totalities not capable of full embodiment in the parts, which qualities may be more decided and strong in one case than in another. But the very reduction of the intensity may be an esthetic quality; nay, it will be so; and I am seriously inclined to doubt there being any distinction of pure esthetic betterness and worseness. My notion would be that there are innumerable varieties of esthetic quality, but no purely esthetic grade of excellence. Peirce: CP 5.133 Cross-Ref:†† 133. But the instant that an esthetic ideal is proposed as an ultimate end of action, at that instant a categorical imperative pronounces for or against it. Kant, as you know, proposes to allow that categorical imperative to stand unchallenged -- an eternal pronouncement. His position is in extreme disfavor now, and not without reason. Yet I cannot think very highly of the logic of the ordinary attempts at refuting it. The whole question is whether or not this categorical imperative be beyond control. If this voice of conscience is unsupported by ulterior reasons, is it not simply an insistent irrational howl, the hooting of an owl which we may disregard if we can?

Why should we pay any more attention to it than we would to the barking of a cur? If we cannot disregard conscience, all homilies and moral maxims are perfectly idle. But if it can be disregarded, it is, in one sense, not beyond control. It leaves us free to control ourselves. So then, it appears to me that any aim whatever which can be consistently pursued becomes, as soon as it is unfalteringly adopted, beyond all possible criticism, except the quite impertinent criticism of outsiders. An aim which cannot be adopted and consistently pursued is a bad aim. It cannot properly be called an ultimate aim at all. The only moral evil is not to have an ultimate aim. Peirce: CP 5.134 Cross-Ref:†† 134. Accordingly the problem of ethics is to ascertain what end is possible. It might be thoughtlessly supposed that special science could aid in this ascertainment. But that would rest on a misconception of the nature of an absolute aim, which is what would be pursued under all possible circumstances -- that is, even though the contingent facts ascertained by special sciences were entirely different from what they are. Nor, on the other hand, must the definition of such aim be reduced to a mere formalism. Peirce: CP 5.135 Cross-Ref:†† 135. The importance of the matter for pragmatism is obvious. For if the meaning of a symbol consists in how it might cause us to act, it is plain that this "how" cannot refer to the description of mechanical motions that it might cause, but must intend to refer to a description of the action as having this or that aim. In order to understand pragmatism, therefore, well enough to subject it to intelligent criticism, it is incumbent upon us to inquire what an ultimate aim, capable of being pursued in an indefinitely prolonged course of action, can be. Peirce: CP 5.136 Cross-Ref:†† 136. The deduction of this is somewhat intricate, on account of the number of points which have to be taken into account; and of course I cannot go into details. In order that the aim should be immutable under all circumstances, without which it will not be an ultimate aim, it is requisite that it should accord with a free development of the agent's own esthetic quality. At the same time it is requisite that it should not ultimately tend to be disturbed by the reactions upon the agent of that outward world which is supposed in the very idea of action. It is plain that these two conditions can be fulfilled at once only if it happens that the esthetic quality toward which the agent's free development tends and that of the ultimate action of experience upon him are parts of one esthetic total. Whether or not this is really so, is a metaphysical question which it does not fall within the scope of Normative Science to answer. If it is not so, the aim is essentially unattainable. But just as in playing a hand of whist, when only three tricks remain to be played, the rule is to assume that the cards are so distributed that the odd trick can be made, so the rule of ethics will be to adhere to the only possible absolute aim, and to hope that it will prove attainable. Meantime, it is comforting to know that all experience is favorable to that assumption.

Peirce: CP 5.137 Cross-Ref:†† §3. LOGICAL GOODNESS

137. The ground is now cleared for the analysis of logical goodness, or the

goodness of representation. There is a special variety of esthetic goodness that may belong to a representamen, namely, expressiveness. There is also a special moral goodness of representations, namely, veracity. But besides this there is a peculiar mode of goodness which is logical. What this consists in we have to inquire. Peirce: CP 5.138 Cross-Ref:†† 138. The mode of being of a representamen is such that it is capable of repetition. Take, for example, any proverb. "Evil communications corrupt good manners." Every time this is written or spoken in English, Greek, or any other language, and every time it is thought of it is one and the same representamen. It is the same with a diagram or picture. It is the same with a physical sign or symptom. If two weathercocks are different signs, it is only in so far as they refer to different parts of the air. A representamen which should have a unique embodiment, incapable of repetition, would not be a representamen, but a part of the very fact represented. This repetitory character of the representamen involves as a consequence that it is essential to a representamen that it should contribute to the determination of another representamen distinct from itself. For in what sense would it be true that a representamen was repeated if it were not capable of determining some different representamen? "Evil communications corrupt good manners" and {phtheirousin ethe chresth' homiliai kakai} are one and the same representamen. They are so, however, only so far as they are represented as being so; and it is one thing to say that "Evil communications corrupt good manners" and quite a different thing to say that "Evil communications corrupt good manners" and {phtheirousin ethe chresth' homiliai kakai} are two expressions of the same proverb. Thus every representamen must be capable of contributing to the determination of a representamen different from itself. Every conclusion from premisses is an instance in point; and what would be a representamen that was not capable of contributing to any ulterior conclusion? I call a representamen which is determined by another representamen, an interpretant of the latter. Every representamen is related or is capable of being related to a reacting thing, its object, and every representamen embodies, in some sense, some quality, which may be called its signification, what in the case of a common name J.S. Mill calls its connotation, a particularly objectionable expression.†1 Peirce: CP 5.139 Cross-Ref:†† 139. A representamen [as symbol] is either a rhema, a proposition, or an argument. An argument is a representamen which separately shows what interpretant it is intended to determine. A proposition is a representamen which is not an argument, but which separately indicates what object it is intended to represent. A rhema is a simple representation without such separate parts. Peirce: CP 5.140 Cross-Ref:†† 140. Esthetic goodness, or expressiveness, may be possessed, and in some degree must be possessed, by any kind of representamen -- rhema, proposition, or argument. Peirce: CP 5.141 Cross-Ref:†† 141. Moral goodness, or veracity, may be possessed by a proposition or by an argument, but cannot be possessed by a rhema. A mental judgment or inference must possess some degree of veracity. Peirce: CP 5.142 Cross-Ref:†† 142. As to logical goodness, or truth, the statements in the books are faulty; and it is highly important for our inquiry that they should be corrected. The books

distinguish between logical truth, which some of them rightly confine to arguments that do not promise more than they perform, and material truth which belongs to propositions, being that which veracity aims to be; and this is conceived to be a higher grade of truth than mere logical truth. I would correct this conception as follows. In the first place, all our knowledge rests upon perceptual judgments. These are necessarily veracious in greater or less degree according to the effort made, but there is no meaning in saying that they have any other truth than veracity, since a perceptual judgment can never be repeated. At most we can say of a perceptual judgment that its relation to other perceptual judgments is such as to permit a simple theory of the facts. Thus I may judge that I see a clean white surface. But a moment later I may question whether the surface really was clean, and may look again more sharply. If this second more veracious judgment still asserts that I see a clean surface, the theory of the facts will be simpler than if, at my second look, I discern that the surface is soiled. Still, even in this last case, I have no right to say that my first percept was that of a soiled surface. I absolutely have no testimony concerning it, except my perceptual judgment, and although that was careless and had no high degree of veracity, still I have to accept the only evidence in my possession. Now consider any other judgment I may make. That is a conclusion of inferences ultimately based on perceptual judgments, and since these are indisputable, all the truth which my judgment can have must consist in the logical correctness of those inferences. Or I may argue the matter in another way. To say that a proposition is false is not veracious unless the speaker has found out that it is false. Confining ourselves, therefore, to veracious propositions, to say that a proposition is false and that it has been found to be false are equivalent, in the sense of being necessarily either both true or both false. Consequently, to say that a proposition is perhaps false is the same as to say that it will perhaps be found out to be false. Hence to deny one of these is to deny the other. To say that a proposition is certainly true means simply that it never can be found out to be false, or in other words, that it is derived by logically correct arguments from veracious perceptual judgments. Consequently, the only difference between material truth and the logical correctness of argumentation is that the latter refers to a single line of argument and the former to all the arguments which could have a given proposition or its denial as their conclusion. Peirce: CP 5.142 Cross-Ref:†† Let me say to you that this reasoning needs to be scrutinized with the severest and minutest logical criticism, because pragmatism largely depends upon it. Peirce: CP 5.143 Cross-Ref:†† 143. It appears, then, that logical goodness is simply the excellence of argument -- its negative, and more fundamental, goodness being its soundness and weight, its really having the force that it pretends to have and that force being great, while its quantitative goodness consists in the degree in which it advances our knowledge. In what then does the soundness of argument consist? Peirce: CP 5.144 Cross-Ref:†† 144. In order to answer that question it is necessary to recognize three radically different kinds of arguments which I signalized in 1867†1 and which had been recognized by the logicians of the eighteenth century, although [those] logicians quite pardonably failed to recognize the inferential character of one of them. Indeed, I suppose that the three were given by Aristotle in the Prior Analytics, although the unfortunate illegibility of a single word in his MS. and its replacement by a wrong word by his first editor, the stupid [Apellicon], has completely altered the sense of the

chapter on Abduction.†1 At any rate, even if my conjecture is wrong, and the text must stand as it is, still Aristotle, in that chapter on Abduction, was even in that case evidently groping for that mode of inference which I call by the otherwise quite useless name of Abduction -- a word which is only employed in logic to translate the [{apagoge}] of that chapter. Peirce: CP 5.145 Cross-Ref:†† 145. These three kinds of reasoning are Abduction, Induction, and Deduction. Deduction is the only necessary reasoning. It is the reasoning of mathematics. It starts from a hypothesis, the truth or falsity of which has nothing to do with the reasoning; and of course its conclusions are equally ideal. The ordinary use of the doctrine of chances is necessary reasoning, although it is reasoning concerning probabilities. Induction is the experimental testing of a theory. The justification of it is that, although the conclusion at any stage of the investigation may be more or less erroneous, yet the further application of the same method must correct the error. The only thing that induction accomplishes is to determine the value of a quantity. It sets out with a theory and it measures the degree of concordance of that theory with fact. It never can originate any idea whatever. No more can deduction. All the ideas of science come to it by the way of Abduction. Abduction consists in studying facts and devising a theory to explain them. Its only justification is that if we are ever to understand things at all, it must be in that way. Peirce: CP 5.146 Cross-Ref:†† 146. Concerning the relations of these three modes of inference to the categories and concerning certain other details, my opinions, I confess, have wavered. These points are of such a nature that only the closest students of what I have written would remark the discrepancies. Such a student might infer that I have been given to expressing myself without due consideration; but in fact I have never, in any philosophical writing -- barring anonymous contributions to newspapers -- made any statement which was not based on at least half a dozen attempts, in writing, to subject the whole question to a very far more minute and critical examination than could be attempted in print, these attempts being made quite independently of one another, at intervals of many months, but subsequently compared together with the most careful criticism, and being themselves based upon at least two briefs of the state of the question, covering its whole literature, as far as known to me, and carrying the criticism in the strictest logical form to its extreme beginnings, without leaving any loopholes that I was able to discern with my utmost pains, these two briefs being made at an interval of a year or more and as independently as possible, although they were subsequently minutely compared, amended, and reduced to one. My waverings, therefore, have never been due to haste. They may argue stupidity. But I can at least claim that they prove one quality in my favor. That is that so far from my being wedded to opinions as being my own, I have shown rather a decided distrust of any opinion of which I have been an advocate. This perhaps ought to give a slight additional weight to those opinions in which I have never wavered -- although I need not say that the notion of any weight of authority being attached to opinions in philosophy or in science is utterly illogical and unscientific. Among these opinions which I have constantly maintained is this, that while Abductive and Inductive reasoning are utterly irreducible, either to the other or to Deduction, or Deduction to either of them, yet the only rationale of these methods is essentially Deductive or Necessary. If then we can state wherein the validity of Deductive reasoning lies, we shall have defined the foundation of logical goodness of whatever kind.

Peirce: CP 5.147 Cross-Ref:†† 147. Now all necessary reasoning, whether it be good or bad, is of the nature of mathematical reasoning. The philosophers are fond of boasting of the pure conceptual character of their reasoning. The more conceptual it is, the nearer it approaches to verbiage. I am not speaking from surmise. My analyses of reasoning surpass in thoroughness all that has ever been done in print, whether in words or in symbols -- all that DeMorgan, Dedekind, Schröder, Peano, Russell, and others have ever done -- to such a degree as to remind one of the difference between a pencil sketch of a scene and a photograph of it. To say that I analyze the passage from the premisses to the conclusion of a syllogism in Barbara into seven or eight distinct inferential steps gives but a very inadequate idea of the thoroughness of my analysis.†1 Let any responsible person pledge himself to go through the matter and dig it out, point by point, and he shall receive the manuscript. Peirce: CP 5.148 Cross-Ref:†† 148. It is on the basis of such analysis that I declare that all necessary reasoning, be it the merest verbiage of the theologians, so far as there is any semblance of necessity in it, is mathematical reasoning. Now mathematical reasoning is diagrammatic. This is as true of algebra as of geometry. But in order to discern the features of diagrammatic reasoning, it is requisite to begin with examples that are not too simple. In simple cases, the essential features are so nearly obliterated that they can only be discerned when one knows what to look for. But beginning with suitable examples and thence proceeding to others, one finds that the diagram itself, in its individuality, is not what the reasoning is concerned with. I will take an example which recommends itself only by its consideration requiring but a moment. A line abuts upon an ordinary point of another line forming two angles. The sum of these angles is proved by Legendre to be equal to the sum of two right angles by erecting a perpendicular to the second line in the plane of the two and through the point of abuttal. This perpendicular must lie in the one angle or the other. The pupil is supposed to see that. He sees it only in a special case, but he is supposed to perceive that it will be so in any case. The more careful logician may demonstrate that it must fall in one angle or the other; but this demonstration will only consist in substituting a different diagram in place of Legendre's figure. But in any case, either in the new diagram or else, and more usually, in passing from one diagram to the other, the interpreter of the argumentation will be supposed to see something, which will present this little difficulty for the theory of vision, that it is of a general nature. Peirce: CP 5.149 Cross-Ref:†† 149. Mr. Mill's disciples will say that this proves that geometrical reasoning is inductive. I do not wish to speak disparagingly of Mill's treatment †2 of the Pons Asinorum because it penetrates further into the logic of the subject than anybody had penetrated before. Only it does not quite touch bottom. As for such general perceptions being inductive, I might treat the question from a technical standpoint and show that the essential characters of induction are wanting. But besides the interminable length, such a way of dealing with the matter would hardly meet the point. It is better to remark that the "uniformity of nature" is not in question, and that there is no way of applying that principle to supporting the mathematical reasoning that will not enable me to give a precisely analogous instance in every essential particular, except that it will be a fallacy that no good mathematician could overlook. If you admit the principle that logic stops where self-control stops, you will find yourself obliged to admit that a perceptual fact, a logical origin, may involve

generality. This can be shown for ordinary generality. But if you have already convinced yourself that continuity is generality, it will be somewhat easier to show that a perceptual fact may involve continuity than that it can involve non-relative generality. Peirce: CP 5.150 Cross-Ref:†† 150. If you object that there can be no immediate consciousness of generality, I grant that. If you add that one can have no direct experience of the general, I grant that as well. Generality, Thirdness, pours in upon us in our very perceptual judgments, and all reasoning, so far as it depends on necessary reasoning, that is to say, mathematical reasoning, turns upon the perception of generality and continuity at every step.

Peirce: CP 5.151 Cross-Ref:†† LECTURE VI THREE TYPES OF REASONING

§1. PERCEPTUAL JUDGMENTS AND GENERALITY

151. I was remarking at the end of my last lecture that perceptual judgments involve generality. What is the general? The Aristotelian definition is good enough. It is quod aptum natum est praedicari de pluribus;†1 {legö de katholou men ho epi pleionön pephyke katégoreisthai}. De Interp. 7. When logic was studied in a scientific spirit of exactitude it was recognized on all hands that all ordinary judgments contain a predicate and that this predicate is general. There seemed to be some exceptions, of which the only noticeable ones were expository judgments, such as "Tully is Cicero." But the Logic of Relations has now reduced logic to order, and it is seen that a proposition may have any number of subjects but can have but one predicate which is invariably general. Such a proposition as "Tully is Cicero" predicates the general relation of identity of Tully and Cicero.†2 Consequently, it is now clear that if there be any perceptual judgment, or proposition directly expressive of and resulting from the quality of a present percept, or sense-image, that judgment must involve generality in its predicate. Peirce: CP 5.152 Cross-Ref:†† 152. That which is not general is singular; and the singular is that which reacts. The being of a singular may consist in the being of other singulars which are its parts. Thus heaven and earth is a singular; and its being consists in the being of heaven and the being of earth, each of which reacts and is therefore a singular, forming a part of heaven and earth. If I had denied that every perceptual judgment refers, as to its subject, to a singular, and that singular actually reacting upon the mind in forming the judgment, actually reacting too upon the mind in interpreting the judgment, I should have uttered an absurdity. For every proposition whatsoever refers as to its subject to a singular actually reacting upon the utterer of it and actually reacting upon the interpreter of it. All propositions relate to the same ever-reacting

singular; namely, to the totality of all real objects. It is true that when the Arabian romancer tells us that there was a lady named Scherherazade, he does not mean to be understood as speaking of the world of outward realities, and there is a great deal of fiction in what he is talking about. For the fictive is that whose characters depend upon what characters somebody attributes to it; and the story is, of course, the mere creation of the poet's thought. Nevertheless, once he has imagined Scherherazade and made her young, beautiful, and endowed with a gift of spinning stories, it becomes a real fact that so he has imagined her, which fact he cannot destroy by pretending or thinking that he imagined her to be otherwise. What he wishes us to understand is what he might have expressed in plain prose by saying, "I have imagined a lady, Scherherazade by name, young, beautiful and a tireless teller of tales, and I am going on to imagine what tales she told." This would have been a plain expression of professed fact relating to the sum total of realities. Peirce: CP 5.153 Cross-Ref:†† 153. As I said before, propositions usually have more subjects than one; and almost every proposition, if not quite every one, has one or more other singular subjects, to which some propositions do not relate. These are the special parts of the Universe of all Truth †1 to which the given proposition especially refers. It is a characteristic of perceptual judgments that each of them relates to some singular to which no other proposition relates directly, but, if it relates to it at all, does so by relating to that perceptual judgment. When we express a proposition in words, we leave most of its singular subjects unexpressed; for the circumstances of the enunciation sufficiently show what subject is intended and words, owing to their usual generality, are not well adapted to designating singulars. The pronoun, which may be defined as a part of speech intended to fulfill the function of an index, is never intelligible taken by itself apart from the circumstances of its utterance; and the noun, which may be defined as a part of speech put in place of a pronoun, is always liable to be equivocal.†1 Peirce: CP 5.154 Cross-Ref:†† 154. A subject need not be singular. If it is not so, then when the proposition is expressed in the canonical form used by logicians, this subject will present one or other of two imperfections:†2 Peirce: CP 5.154 Cross-Ref:†† On the one hand, it may be indesignative, so that the proposition means that a singular of the universe might replace this subject while the truth was preserved, while failing to designate what singular that is: as when we say, "Some calf has five legs." Peirce: CP 5.154 Cross-Ref:†† Or on the other hand, the subject may be hypothetical, that is may allow any singular to be substituted for it that fulfills certain conditions, without guaranteeing that there is any singular which fulfills these conditions; as when we say, "Any salamander could live in fire," or "Any man who should be stronger than Samson could do all that Samson did." Peirce: CP 5.154 Cross-Ref:†† A subject which has neither of these two imperfections is a singular subject referring to an existing singular collection in its entirety. Peirce: CP 5.155 Cross-Ref:††

155. If a proposition has two or more subjects of which one is indesignative and the other hypothetical, then it makes a difference in what order the replacement by singulars is asserted to be possible. It is, for example, one thing to assert that "Any Catholic there may be adores some woman or other" and quite another thing to assert that "There is some woman whom any Catholic adores." If the first general subject is indesignate, the proposition is called particular. If the first general subject is hypothetical, the proposition is called universal.†3 Peirce: CP 5.155 Cross-Ref:†† A particular proposition asserts the existence of something of a given description. A universal proposition merely asserts the non-existence of anything of a given description. Peirce: CP 5.156 Cross-Ref:†† 156. Had I, therefore, asserted that a perceptual judgment could be a universal proposition, I should have fallen into rank absurdity. For reaction is existence and the perceptual judgment is the cognitive product of a reaction. Peirce: CP 5.156 Cross-Ref:†† But as from the particular proposition that "there is some women whom any Catholic you can find will adore" we can with certainty infer the universal proposition that" any Catholic you can find will adore some woman or other,"†1 so if a perceptual judgment involves any general elements, as it certainly does, the presumption is that a universal proposition can be necessarily deduced from it. Peirce: CP 5.157 Cross-Ref:†† 157. In saying that perceptual judgments involve general elements I certainly never intended to be understood as enunciating any proposition in psychology. For my principles absolutely debar me from making the least use of psychology in logic. I am confined entirely to the unquestionable facts of everyday experience, together with what can be deduced from them. All that I can mean by a perceptual judgment is a judgment absolutely forced upon my acceptance, and that by a process which I am utterly unable to control and consequently am unable to criticize. Nor can I pretend to absolute certainty about any matter of fact. If with the closest scrutiny I am able to give, a judgment appears to have the characters I have described, I must reckon it among perceptual judgments until I am better advised. Now consider the judgment that one event C appears to be subsequent to another event A. Certainly, I may have inferred this; because I may have remarked that C was subsequent to a third event B which was itself subsequent to A. But then these premisses are judgments of the same description. It does not seem possible that I can have performed an infinite series of acts of criticism each of which must require a distinct effort. The case is quite different from that of Achilles and the tortoise because Achilles does not require to make an infinite series of distinct efforts. It therefore appears that I must have made some judgment that one event appeared to be subsequent to another without that judgment having been inferred from any premiss [i.e.] without any controlled and criticized action of reasoning. If this be so, it is a perceptual judgment in the only sense that the logician can recognize. But from that proposition that one event, Z, is subsequent to another event, J, I can at once deduce by necessary reasoning a universal proposition. Namely, the definition of the relation of apparent subsequence is well known, or sufficiently so for our purpose. Z will appear to be subsequent to Y if and only if Z appears to stand in a peculiar relation, R, to Y such that nothing can stand in the relation R to itself, and if, furthermore, whatever event, X, there may be to which Y stands in the relation R, to that same X, Z also stands in the relation R.†1

This being implied in the meaning of subsequence, concerning which there is no room for doubt, it easily follows that whatever is subsequent to C is subsequent to anything, A, to which C is subsequent -- which is a universal proposition. Peirce: CP 5.157 Cross-Ref:†† Thus my assertion at the end of the last lecture appears to be most amply justified. Thirdness pours in upon us through every avenue of sense.

Peirce: CP 5.158 Cross-Ref:†† §2. THE PLAN AND STEPS OF REASONING

158. We may now profitably ask ourselves what logical goodness is. We have seen that any kind of goodness consists in the adaptation of its subject to its end. One might set this down as a truism. Verily, it is scarcely more, although circumstances may have prevented it being clearly apprehended. Peirce: CP 5.158 Cross-Ref:†† If you call this utilitarianism, I shall not be ashamed of the title. For I do not know what other system of philosophy has wrought so much good in the world as that same utilitarianism. Bentham may be a shallow logician; but such truths as he saw, he saw most nobly. As for the vulgar utilitarian, his fault does not lie in his pressing too much the question of what would be the good of this or that. On the contrary his fault is that he never presses the question half far enough, or rather he never really raises the question at all. He simply rests in his present desires as if desire were beyond all dialectic. He wants, perhaps, to go to heaven. But he forgets to ask what would be the good of his going to heaven. He would be happy, there, he thinks. But that is a mere word. It is no real answer to the question. Peirce: CP 5.159 Cross-Ref:†† 159. Our question is, What is the use of thinking? We have already remarked that it is the argument alone which is the primary and direct subject of logical goodness and badness. We have therefore to ask what the end of argumentation is, what it ultimately leads to. Peirce: CP 5.160 Cross-Ref:†† 160. The Germans, whose tendency is to look at everything subjectively and to exaggerate the element of Firstness, maintain that the object is simply to satisfy one's logical feeling and that the goodness of reasoning consists in that esthetic satisfaction alone.†1 This might do if we were gods and not subject to the force of experience. Peirce: CP 5.160 Cross-Ref:†† Or if the force of experience were mere blind compulsion, and we were utter foreigners in the world, then again we might as well think to please ourselves; because we then never could make our thoughts conform to that mere Secondness. Peirce: CP 5.160 Cross-Ref:†† But the saving truth is that there is a Thirdness in experience, an element of Reasonableness to which we can train our own reason to conform more and more. If this were not the case, there could be no such thing as logical goodness or badness; and therefore we need not wait until it is proved that there is a reason operative in

experience to which our own can approximate.†2 We should at once hope that it is so, since in that hope lies the only possibility of any knowledge. Peirce: CP 5.161 Cross-Ref:†† 161. Reasoning is of three types, Deduction, Induction, and Abduction.†3 In deduction, or necessary reasoning, we set out from a hypothetical state of things which we define in certain abstracted respects. Among the characters to which we pay no attention in this mode of argument is whether or not the hypothesis of our premisses conforms more or less to the state of things in the outward world. We consider this hypothetical state of things and are led to conclude that, however it may be with the universe in other respects, wherever and whenever the hypothesis may be realized, something else not explicitly supposed in that hypothesis will be true invariably. Our inference is valid if and only if there really is such a relation between the state of things supposed in the premisses and the state of things stated in the conclusion. Whether this really be so or not is a question of reality, and has nothing at all to do with how we may be inclined to think. If a given person is unable to see the connection, the argument is none the less valid, provided that relation of real facts really subsists. If the entire human race were unable to see the connection, the argument would be none the less sound, although it would not be humanly clear. Let us see precisely how we assure ourselves of the reality of the connection. Here, as everywhere throughout logic, the study of relatives has been of the greatest service. The simple syllogisms, which are alone considered by the old inexact logicians, are such very rudimentary forms that it is practically impossible to discern in them the essential features of deductive inference until our attention has been called to these features in higher forms of deduction. Peirce: CP 5.162 Cross-Ref:†† 162. All necessary reasoning without exception is diagrammatic.†1 That is, we construct an icon of our hypothetical state of things and proceed to observe it. This observation leads us to suspect that something is true, which we may or may not be able to formulate with precision, and we proceed to inquire whether it is true or not. For this purpose it is necessary to form a plan of investigation and this is the most difficult part of the whole operation. We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features. Otherwise, although our conclusions may be correct, they will not be the particular conclusions at which we are aiming. But the greatest point of art consists in the introduction of suitable abstractions. By this I mean such a transformation of our diagrams that characters of one diagram may appear in another as things. A familiar example is where in analysis we treat operations as themselves the subject of operations. Let me say that it would make a grand life-study to give an account of this operation of planning a mathematical demonstration.†2 Sundry sporadic maxims are afloat among mathematicians, and several meritorious books have been written upon the subject, but nothing broad and masterly. With the modern reformed mathematics and with my own and other logical results as a basis, such a theory of the plan of demonstration is no longer a superhuman task. Peirce: CP 5.163 Cross-Ref:†† 163. Having thus determined the plan of the reasoning, we proceed to the reasoning itself, and this I have ascertained can be reduced to three kinds of steps.†3 The first consists in copulating separate propositions into one compound proposition. The second consists in omitting something from a proposition without possibility of

introducing error. The third consists in inserting something into a proposition without introducing error. Peirce: CP 5.164 Cross-Ref:†† 164. You can see precisely what these elementary steps of inference are in Baldwin's Dictionary under Symbolic Logic.†1 As a specimen of what they are like you may take this:

A is a bay horse, Therefore, A is a horse.

Peirce: CP 5.164 Cross-Ref:†† If one asks oneself how one knows that this is certain, one is likely to reply that one imagines a bay horse and on contemplating the image one sees that it is a horse. But that only applies to the single image. How large a horse did this image represent? Would it be the same with a horse of very different size? How old was the horse represented to be; was his tail docked? Would it be so if he had the blind-staggers, and if so are you sure it would be so whatever of the numerous diseases of the horse afflicted him? We are perfectly certain that none of these circumstances could affect the question in the least. It is easy enough to formulate reasons by the dozen; but the difficulty is that they are one and all far less evident than the original inference. I do not see that the logician can do better than to say that he perceives that when a copulative proposition is given, such as "A is a horse and A has a bay color" any member of the copulation may be omitted without changing the proposition from true to false. In a psychological sense I am willing to take the word of the psychologist if he says that such a general truth cannot be perceived. But what better can we do in logic? Peirce: CP 5.165 Cross-Ref:†† 165. Somebody may answer that the copulative proposition contains the conjunction "and" or something equivalent, and that the very meaning of this "and" is that the entire copulation is true if and only if each of the members is singly true; so that it is involved in the very meaning of the copulative proposition that any member may be dropped. Peirce: CP 5.165 Cross-Ref:†† To this I assent with all my heart. But after all, what does it amount to? It is another way of saying that what we call the meaning of a proposition embraces every obvious necessary deduction from it. Considered as the beginning of an analysis of what the meaning of the word "meaning" is, it is a valuable remark. But I ask how it helps us to understand our passing from an accepted judgment A to another judgment C of which we not only feel equally confident but in point of fact are equally sure, barring a possible blunder which could be corrected as soon as attention was called to it, barring another equivalent blunder? Peirce: CP 5.165 Cross-Ref:†† To this the advocate of the explanation by the conception of "meaning" may reply: that is meant which is intended or purposed; that a judgment is a voluntary act, and our intention is not to employ the form of the judgment A, except to the

interpretation of images to which judgments, corresponding in form to C, can be applied. Peirce: CP 5.166 Cross-Ref:†† 166. Perhaps it may reconcile the psychologist to the admission of perceptual judgments involving generality to be told that they are perceptual judgments concerning our own purposes. I certainly think that the certainty of pure mathematics and of all necessary reasoning is due to the circumstance that it relates to objects which are the creations of our own minds, and that mathematical knowledge is to be classed along with knowledge of our own purposes. When we meet with a surprising result in pure mathematics, as we so often do, because a loose reasoning had led us to suppose it impossible, this is essentially the same sort of phenomenon as when in pursuing a purpose we are led to do something that we are quite surprised to find ourselves doing, as being contrary, or apparently contrary, to some weaker purpose. Peirce: CP 5.166 Cross-Ref:†† But if it is supposed that any such considerations afford any logical justification of primary logical principles I must say that, on the contrary, at the very best they beg the question by assuming premisses far less certain than the conclusion to be established.

Peirce: CP 5.167 Cross-Ref:†† §3. INDUCTIVE REASONING †1

167. A generation and a half of evolutionary fashions in philosophy has not sufficed entirely to extinguish the fire of admiration for John Stuart Mill -- that very strong but Philistine philosopher whose inconsistencies fitted him so well to be the leader of a popular school -- and consequently there will still be those who propose to explain the general principles of formal logic, which are now fully shown to be mathematical principles, by means of induction. Anybody who holds to that view today may be assumed to have a very loose notion of induction; so that all he really means is that the general principles in question are derived from images of the imagination by a process which is, roughly speaking, analogous to induction. Understanding him in that way, I heartily agree with him. But he must not expect me in 1903 to have anything more than a historical admiration for conceptions of induction which shed a brilliant light upon the subject in 1843. Induction is so manifestly inadequate to account for the certainty of these principles that it would be a waste of time to discuss such a theory. Peirce: CP 5.168 Cross-Ref:†† 168. However, it is now time for me to pass to the consideration of Inductive Reasoning. When I say that by inductive reasoning I mean a course of experimental investigation, I do not understand experiment in the narrow sense of an operation by which one varies the conditions of a phenomenon almost as one pleases. We often hear students of sciences, which are not in this narrow sense experimental, lamenting that in their departments they are debarred from this aid. No doubt there is much justice in this lament; and yet those persons are by no means debarred from pursuing the same logical method precisely, although not with the same freedom and facility. An experiment, says Stöckhardt, in his excellent School of Chemistry, is a question put to nature.†1 Like any interrogatory, it is based on a supposition. If that

supposition be correct, a certain sensible result is to be expected under certain circumstances which can be created, or at any rate are to be met with. The question is, Will this be the result? If Nature replies "No!" the experimenter has gained an important piece of knowledge. If Nature says "Yes," the experimenter's ideas remain just as they were, only somewhat more deeply engrained. If Nature says "Yes" to the first twenty questions, although they were so devised as to render that answer as surprising as possible, the experimenter will be confident that he is on the right track, since 2 to the 20th power exceeds a million. Peirce: CP 5.169 Cross-Ref:†† 169. Laplace was of the opinion that the affirmative experiments impart a definite probability to the theory; and that doctrine is taught in most books on probability to this day, although it leads to the most ridiculous results, and is inherently self-contradictory. It rests on a very confused notion of what probability is. Probability applies to the question whether a specified kind of event will occur when certain predetermined conditions are fulfilled; and it is the ratio of the number of times in the long run in which that specified result would follow upon the fulfillment of those conditions to the total number of times in which those conditions were fulfilled in the course of experience. It essentially refers to a course of experience, or at least of real events; because mere possibilities are not capable of being counted. You can, for example, ask what the probability is that a given kind of object will be red, provided you define red sufficiently. It is simply the ratio of the number of objects of that kind that are red to the total number of objects of that kind. But to ask in the abstract what the probability is that a shade of color will be red is nonsense, because shades of color are not individuals capable of being counted. You can ask what the probability is that the next chemical element to be discovered will have an atomic weight exceeding a hundred. But you cannot ask what the probability is that the law of universal attraction should be that of the inverse square until you can attach some meaning to statistics of the characters of possible universes. When Leibniz said that this world is the best that was possible, he may have had some glimmer of meaning, but when Quételet †1 says that if a phenomenon has been observed on m occasions, the probability that it will occur on the (m + 1)th occasion is (m+1)/(m+2), he is talking downright nonsense. Mr. F.Y. Edgeworth asserts that of all theories that are started one half are correct. That is not nonsense, but it is ridiculously false. For of theories that have enough to recommend them to be seriously discussed, there are more than two on the average to each general phenomenon to be explained. Poincaré, on the other hand, seems to think that all theories are wrong, and that it is only a question of how wrong they are. Peirce: CP 5.170 Cross-Ref:†† 170. Induction consists in starting from a theory, deducing from it predictions of phenomena, and observing those phenomena in order to see how nearly they agree with the theory. The justification for believing that an experiential theory which has been subjected to a number of experimental tests will be in the near future sustained about as well by further such tests as it has hitherto been, is that by steadily pursuing that method we must in the long run find out how the matter really stands. The reason that we must do so is that our theory, if it be admissible even as a theory, simply consists in supposing that such experiments will in the long run have results of a certain character. But I must not be understood as meaning that experience can be exhausted, or that any approach to exhaustion can be made. What I mean is that if there be a series of objects, say crosses and circles, this series having a beginning but no end, then whatever may be the arrangement or want of arrangement of these

crosses and circles in the entire endless series must be discoverable to an indefinite degree of approximation by examining a sufficient finite number of successive ones beginning at the beginning of the series. This is a theorem capable of strict demonstration. The principle of the demonstration is that whatever has no end can have no mode of being other than that of a law, and therefore whatever general character it may have must be describable, but the only way of describing an endless series is by stating explicitly or implicitly the law of the succession of one term upon another. But every such term has a finite ordinal place from the beginning and therefore, if it presents any regularity for all finite successions from the beginning, it presents the same regularity throughout. Thus the validity of induction depends upon the necessary relation between the general and the singular. It is precisely this which is the support of Pragmatism.

Peirce: CP 5.171 Cross-Ref:†† §4. INSTINCT AND ABDUCTION †1

171. Concerning the validity of Abductive inference, there is little to be said, although that little is pertinent to the problem we have in hand. Peirce: CP 5.171 Cross-Ref:†† Abduction is the process of forming an explanatory hypothesis. It is the only logical operation which introduces any new idea; for induction does nothing but determine a value, and deduction merely evolves the necessary consequences of a pure hypothesis. Peirce: CP 5.171 Cross-Ref:†† Deduction proves that something must be; Induction shows that something actually is operative; Abduction merely suggests that something may be. Peirce: CP 5.171 Cross-Ref:†† Its only justification is that from its suggestion deduction can draw a prediction which can be tested by induction, and that, if we are ever to learn anything or to understand phenomena at all, it must be by abduction that this is to be brought about. Peirce: CP 5.171 Cross-Ref:†† No reason whatsoever can be given for it, as far as I can discover; and it needs no reason, since it merely offers suggestions. Peirce: CP 5.172 Cross-Ref:†† 172. A man must be downright crazy to deny that science has made many true discoveries. But every single item of scientific theory which stands established today has been due to Abduction. Peirce: CP 5.172 Cross-Ref:†† But how is it that all this truth has ever been lit up by a process in which there is no compulsiveness nor tendency toward compulsiveness? Is it by chance? Consider the multitude of theories that might have been suggested. A physicist comes across some new phenomenon in his laboratory. How does he know but the conjunctions of the planets have something to do with it or that it is not perhaps because the dowager empress of China has at that same time a year ago chanced to pronounce some word

of mystical power or some invisible jinnee may be present. Think of what trillions of trillions of hypotheses might be made of which one only is true; and yet after two or three or at the very most a dozen guesses, the physicist hits pretty nearly on the correct hypothesis. By chance he would not have been likely to do so in the whole time that has elapsed since the earth was solidified. You may tell me that astrological and magical hypotheses were resorted to at first and that it is only by degrees that we have learned certain general laws of nature in consequence of which the physicist seeks for the explanation of his phenomenon within the four walls of his laboratory. But when you look at the matter more narrowly, the matter is not to be accounted for in any considerable measure in that way. Take a broad view of the matter. Man has not been engaged upon scientific problems for over twenty thousand years or so. But put it at ten times that if you like. But that is not a hundred thousandth part of the time that he might have been expected to have been searching for his first scientific theory. Peirce: CP 5.172 Cross-Ref:†† You may produce this or that excellent psychological account of the matter. But let me tell you that all the psychology in the world will leave the logical problem just where it was. I might occupy hours in developing that point. I must pass it by. Peirce: CP 5.172 Cross-Ref:†† You may say that evolution accounts for the thing.†1 I don't doubt it is evolution. But as for explaining evolution by chance, there has not been time enough. Peirce: CP 5.173 Cross-Ref:†† 173. However man may have acquired his faculty of divining the ways of Nature, it has certainly not been by a self-controlled and critical logic. Even now he cannot give any exact reason for his best guesses. It appears to me that the clearest statement we can make of the logical situation -- the freest from all questionable admixture -- is to say that man has a certain Insight, not strong enough to be oftener right than wrong, but strong enough not to be overwhelmingly more often wrong than right, into the Thirdnesses, the general elements, of Nature. An Insight, I call it, because it is to be referred to the same general class of operations to which Perceptive Judgments belong. This Faculty is at the same time of the general nature of Instinct, resembling the instincts of the animals in its so far surpassing the general powers of our reason and for its directing us as if we were in possession of facts that are entirely beyond the reach of our senses. It resembles instinct too in its small liability to error; for though it goes wrong oftener than right, yet the relative frequency with which it is right is on the whole the most wonderful thing in our constitution. Peirce: CP 5.174 Cross-Ref:†† 174. One little remark and I will drop this topic. If you ask an investigator why he does not try this or that wild theory, he will say, "It does not seem reasonable." It is curious that we seldom use this word where the strict logic of our procedure is clearly seen. We do [not] say that a mathematical error is not reasonable. We call that opinion reasonable whose only support is instinct. . . .

Peirce: CP 5.175 Cross-Ref:†† §5. THE MEANING OF AN ARGUMENT

175. We have already seen †1 some reason to hold that the idea of meaning is

such as to involve some reference to a purpose. But Meaning is attributed to representamens alone, and the only kind of representamen which has a definite professed purpose is an "argument." The professed purpose of an argument is to determine an acceptance of its conclusion, and it quite accords with general usage to call the conclusion of an argument its meaning. But I may remark that the word meaning has not hitherto been recognized as a technical term of logic, and in proposing it as such (which I have a right to do since I have a new conception to express, that of the conclusion of an argument as its intended interpretant) I should have a recognized right slightly to warp the acceptation of the word "meaning," so as to fit it for the expression of a scientific conception. It seems natural to use the word meaning to denote the intended interpretant of a symbol. Peirce: CP 5.176 Cross-Ref:†† 176. I may presume that you are all familiar with Kant's reiterated insistence that necessary reasoning does nothing but explicate the meaning of its premisses.†2 Now Kant's conception of the nature of necessary reasoning is clearly shown by the logic of relations to be utterly mistaken, and his distinction between analytic and synthetic judgments, which he otherwise and better terms explicatory (erläuternde) and ampliative (erweiternde) judgments, which is based on that conception, is so utterly confused that it is difficult or impossible to do anything with it. But, nevertheless, I think we shall do very well to accept Kant's dictum that necessary reasoning is merely explicatory of the meaning of the terms of the premisses, only reversing the use to be made of it. Namely instead of adopting the conception of meaning from the Wolffian logicians, as he does, and making use of this dictum to express what necessary reasoning can do, about which he was utterly mistaken, we shall do well to understand necessary reasoning as mathematics and the logic of relations compels us to understand it, and to use the dictum, that necessary reasoning only explicates the meanings of the terms of the premisses, to fix our ideas as to what we shall understand by the meaning of a term. Peirce: CP 5.177 Cross-Ref:†† 177. Kant and the logicians with whose writings he was alone acquainted -- he was far from being a thorough student of logic, notwithstanding his great natural power as a logician -- consistently neglected the logic of relations; and the consequence was that the only account they were in condition to give of the meaning of a term, its "signification" as they called it, was that it was composed of all the terms which could be essentially predicated of that term. Consequently, either the analysis of the signification must be capable of [being] pushed on further and further, without limit -- an opinion which Kant †1 expresses in a well-known passage but which he did not develop, or, what was more usual, one ultimately reached certain absolutely simple conceptions such as Being, Quality, Relation, Agency, Freedom, etc., which were regarded as absolutely incapable of definition and of being in the highest degree luminous and clear. It is marvellous what a following this opinion, that those excessively abstracted conceptions were in themselves in the highest degree simple and facile, obtained, notwithstanding its repugnancy to good sense. One of the many important services which the logic of relations has rendered has been that of showing that these so-called simple conceptions, notwithstanding their being unaffected by the particular kind of combination recognized in non-relative logic, are nevertheless capable of analysis in consequence of their implying various modes of relationship. For example, no conceptions are simpler than those of Firstness, Secondness, and Thirdness; but this has not prevented my defining them, and that in a most effective manner, since all the assertions I have made concerning them have

been deduced from those definitions. Peirce: CP 5.178 Cross-Ref:†† 178. Another effect of the neglect of the logic of relations was that Kant imagined that all necessary reasoning was of the type of a syllogism in Barbara. Nothing could be more ridiculously in conflict with well-known facts.†2 For had that been the case, any person with a good logical head would be able instantly to see whether a given conclusion followed from given premisses or not; and moreover the number of conclusions from a small number of premisses would be very moderate. Now it is true that when Kant wrote, Legendre and Gauss had not shown what a countless multitude of theorems are deducible from the very few premisses of arithmetic. I suppose we must excuse him, therefore, for not knowing this. But it is difficult to understand what the state of mind on this point could have been of logicians who were at the same time mathematicians, such as Euler, Lambert, and Ploucquet. Euler invented the logical diagrams which go under his name; for the claims that have been made in favor of predecessors may be set down as baseless;†1 and Lambert used an equivalent system.†2 Now I need not say that both of these men were mathematicians of great power. One is simply astounded that they should seem to say that all the reasonings of mathematics could be represented in any such ways. One may suppose that Euler never paid much attention to logic. But Lambert wrote a large book in two volumes on the subject, and a pretty superficial affair it is. One has a difficulty in realizing that the author of it was the same man who came so near to the discovery of the non-Euclidean geometry. The logic of relatives is now able to exhibit in strict logical form the reasoning of mathematics. You will find an example of it -- although too simple a one to put all the features into prominence -- in that chapter †3 of Schröder's logic in which he remodels the reasoning of Dedekind in his brochure Was sind und was sollen die Zahlen; and if it be objected that this analysis was chiefly the work of Dedekind who did not employ the machinery of the logic of relations, I reply that Dedekind's whole book is nothing but an elaboration of a paper published by me several years previously in the American Journal of Mathematics†4 which paper was the direct result of my logical studies. These analyses show that although most of the steps of the reasoning have considerable resemblance to Barbara, yet the difference of effect is very great indeed. Peirce: CP 5.179 Cross-Ref:†† 179. On the whole, then, if by the meaning of a term, proposition, or argument, we understand the entire general intended interpretant, then the meaning of an argument is explicit. It is its conclusion; while the meaning of a proposition or term is all that that proposition or term could contribute to the conclusion of a demonstrative argument. But while this analysis will be found useful, it is by no means sufficient to cut off all nonsense or to enable us to judge of the maxim of pragmatism. What we need is an account of the ultimate meaning of a term. To this problem we have to address ourselves.

Peirce: CP 5.180 Cross-Ref:†† LECTURE VII †1 PRAGMATISM AND ABDUCTION

§1. THE THREE COTARY PROPOSITIONS

180. At the end of my last lecture I had just enunciated three propositions which seem to me to give to pragmatism its peculiar character. In order to be able to refer to them briefly this evening, I will call them, for the nonce, my cotary propositions. Cos, cotis, is a whetstone. They appear to me to put the edge on the maxim of pragmatism. Peirce: CP 5.181 Cross-Ref:†† 181. These cotary propositions are as follows: Peirce: CP 5.181 Cross-Ref:†† (1) Nihil est in intellectu quod non prius fuerit in sensu. I take this in a sense somewhat different from that which Aristotle intended.†2 By intellectus, I understand the meaning of any representation in any kind of cognition, virtual, symbolic, or whatever it may be. Berkeley †3 and nominalists of his stripe deny that we have any idea at all of a triangle in general, which is neither equilateral, isosceles, nor scalene. But he cannot deny that there are propositions about triangles in general, which propositions are either true or false; and as long as that is the case, whether we have an idea of a triangle in some psychological sense or not, I do not, as a logician, care. We have an intellectus, a meaning, of which the triangle in general is an element. As for the other term, in sensu, that I take in the sense of in a perceptual judgment, the starting point or first premiss of all critical and controlled thinking. I will state presently what I conceive to be the evidence of the truth of this first cotary proposition. But I prefer to begin by recalling to you what all three of them are. Peirce: CP 5.181 Cross-Ref:†† (2) The second is that perceptual judgments contain general elements, so that universal propositions are deducible from them in the manner in which the logic of relations shows that particular propositions usually, not to say invariably, allow universal propositions to be necessarily inferred from them. This I sufficiently argued in my last lecture. This evening I shall take the truth of it for granted. Peirce: CP 5.181 Cross-Ref:†† (3) The third cotary proposition is that abductive inference shades into perceptual judgment without any sharp line of demarcation between them; or, in other words, our first premisses, the perceptual judgments, are to be regarded as an extreme case of abductive inferences, from which they differ in being absolutely beyond criticism. The abductive suggestion comes to us like a flash. It is an act of insight, although of extremely fallible insight. It is true that the different elements of the hypothesis were in our minds before; but it is the idea of putting together what we had never before dreamed of putting together which flashes the new suggestion before our contemplation. Peirce: CP 5.181 Cross-Ref:†† On its side, the perceptive judgment is the result of a process, although of a process not sufficiently conscious to be controlled, or, to state it more truly, not

controllable and therefore not fully conscious. If we were to subject this subconscious process to logical analysis, we should find that it terminated in what that analysis would represent as an abductive inference, resting on the result of a similar process which a similar logical analysis would represent to be terminated by a similar abductive inference, and so on ad infinitum. This analysis would be precisely analogous to that which the sophism of Achilles and the Tortoise applies to the chase of the Tortoise by Achilles, and it would fail to represent the real process for the same reason. Namely, just as Achilles does not have to make the series of distinct endeavors which he is represented as making, so this process of forming the perceptual judgment, because it is sub-conscious and so not amenable to logical criticism, does not have to make separate acts of inference, but performs its act in one continuous process.

Peirce: CP 5.182 Cross-Ref:†† §2. ABDUCTION AND PERCEPTUAL JUDGMENTS

182. I have already put in my brief in favor of my second cotary proposition, and in what I am about to say I shall treat that as already sufficiently proved. In arguing it I avoided all resort to anything like special phenomena, upon which I do not think that philosophy ought to rest, at all. Still, there is no harm in using special observations merely in an abductive way to throw a light upon doctrines otherwise established, and to aid the mind in grasping them; and there are some phenomena which, I think, do aid us to see what is meant by asserting that perceptual judgments contain general elements, and which will also naturally lead up to a consideration of the third cossal proposition. Peirce: CP 5.183 Cross-Ref:†† 183. I will show you a figure which I remember my father [Benjamin Peirce] drawing in one of his lectures. I do not remember what it was supposed to show; but I cannot imagine what else it could have been but my cotary proposition No. 2. If so, in maintaining that proposition I am substantially treading in his footprints, though he would doubtless have put the proposition into a shape very different from mine. Here is the figure (though I cannot draw it as skillfully as he did). It consists of a serpentine line. But when it is completely drawn, it appears to be a stone wall.

[Click here to view] The point is that there are two ways of conceiving the matter. Both, I beg you to remark, are general ways of classing the line, general classes under which the line is subsumed. But the very decided preference of our perception for one mode of classing the percept shows that this classification is contained in the perceptual judgment. So it is with that well-known unshaded outline figure of a pair of steps seen in perspective. We seem at first to be looking at the steps from above; but some unconscious part of the mind seems to tire of putting that construction upon it and suddenly we seem to see the steps from below, and so the perceptive judgment, and the percept itself, seems to keep shifting from one general aspect to the other and back again. Peirce: CP 5.183 Cross-Ref:†† In all such visual illusions of which two or three dozen are well known, the most striking thing is that a certain theory of interpretation of the figure has all the appearance of being given in perception. The first time it is shown to us, it seems as completely beyond the control of rational criticism as any percept is; but after many repetitions of the now familiar experiment, the illusion wears off, becoming first less decided, and ultimately ceasing completely. This shows that these phenomena are true connecting links between abductions and perceptions. Peirce: CP 5.184 Cross-Ref:†† 184. If the percept or perceptual judgment were of a nature entirely unrelated to abduction, one would expect that the percept would be entirely free from any characters that are proper to interpretations, while it can hardly fail to have such characters if it be merely a continuous series of what, discretely and consciously performed, would be abductions. We have here then almost a crucial test of my third cotary proposition. Now, then, how is the fact? The fact is that it is not necessary to go beyond ordinary observations of common life to find a variety of widely different

ways in which perception is interpretative. Peirce: CP 5.185 Cross-Ref:†† 185. The whole series of hypnotic phenomena, of which so many fall within the realm of ordinary everyday observation -- such as our waking up at the hour we wish to wake much nearer than our waking selves could guess it -- involve the fact that we perceive what we are adjusted for interpreting, though it be far less perceptible than any express effort could enable us to perceive; while that, to the interpretation of which our adjustments are not fitted, we fail to perceive although it exceed in intensity what we should perceive with the utmost ease, if we cared at all for its interpretation. It is a marvel to me that the clock in my study strikes every half hour in the most audible manner, and yet I never hear it. I should not know at all whether the striking part were going, unless it is out of order and strikes the wrong hour. If it does that, I am pretty sure to hear it. Another familiar fact is that we perceive, or seem to perceive, objects differently from how they really are, accommodating them to their manifest intention. Proofreaders get high salaries because ordinary people miss seeing misprints, their eyes correcting them. We can repeat the sense of a conversation, but we are often quite mistaken as to what words were uttered. Some politicians think it a clever thing to convey an idea which they carefully abstain from stating in words. The result is that a reporter is ready to swear quite sincerely that a politician said something to him which the politician was most careful not to say. Peirce: CP 5.185 Cross-Ref:†† I should tire you if I dwelt further on anything so familiar, especially to every psychological student, as the interpretativeness of the perceptive judgment. It is plainly nothing but the extremest case of Abductive Judgments. Peirce: CP 5.186 Cross-Ref:†† 186. If this third cotary proposition be admitted, the second, that the perceptual judgment contains general elements, must be admitted; and as for the first, that all general elements are given in perception, that loses most of its significance. For if a general element were given otherwise than in the perceptual judgment, it could only first appear in an abductive suggestion, and that is now seen to amount substantially to the same thing. I not only opine, however, that every general element of every hypothesis, however wild or sophisticated it may be, [is] given somewhere in perception, but I will venture so far as to assert that every general form of putting concepts together is, in its elements, given in perception. In order to decide whether this be so or not, it is necessary to form a clear notion of the precise difference between abductive judgment and the perceptual judgment which is its limiting case. The only symptom by which the two can be distinguished is that we cannot form the least conception of what it would be to deny the perceptual judgment. If I judge a perceptual image to be red, I can conceive of another man's not having that same percept. I can also conceive of his having this percept but never having thought whether it was red or not. I can conceive that while colors are among his sensations, he shall never have had his attention directed to them. Or I can conceive that, instead of redness, a somewhat different conception should arise in his mind; that he should, for example, judge that this percept has a warmth of color. I can imagine that the redness of my percept is excessively faint and dim so that one can hardly make sure whether it is red or not. But that any man should have a percept similar to mine and should ask himself the question whether this percept be red, which would imply that he had already judged some percept to be red, and that he should, upon careful

attention to this percept, pronounce it to be decidedly and clearly not red, when I judge it to be prominently red, that I cannot comprehend at all. An abductive suggestion, however, is something whose truth can be questioned or even denied. Peirce: CP 5.187 Cross-Ref:†† 187. We thus come to the test of inconceivability as the only means of distinguishing between an abduction and a perceptual judgment. Now I fully assent to all that Stuart Mill so forcibly said in his Examination of Hamilton as to the utter untrustworthiness of the test of inconceivability.†1 That which is inconceivable to us today, may prove tomorrow to be conceivable and even probable; so that we never can be absolutely sure that a judgment is perceptual and not abductive; and this may seem to constitute a difficulty in the way of satisfying ourselves that the first cotary proposition is true. Peirce: CP 5.187 Cross-Ref:†† I should easily show you that this difficulty, however formidable theoretically, amounts practically to little or nothing for a person skilled in shaping such inquiries. But this is unnecessary, since the objection founded upon it has no logical force whatever. Peirce: CP 5.188 Cross-Ref:†† 188. No doubt, in regard to the first cotary proposition, [that proposition] follows as a necessary consequence of the possibility that what are really abductions have been mistaken for perceptions. For the question is whether that which really is an abductive result can contain elements foreign to its premisses. It must be remembered that abduction, although it is very little hampered by logical rules, nevertheless is logical inference, asserting its conclusion only problematically or conjecturally, it is true, but nevertheless having a perfectly definite logical form. Peirce: CP 5.189 Cross-Ref:†† 189. Long before I first classed abduction as an inference it was recognized by logicians that the operation of adopting an explanatory hypothesis -- which is just what abduction is -- was subject to certain conditions. Namely, the hypothesis cannot be admitted, even as a hypothesis, unless it be supposed that it would account for the facts or some of them. The form of inference, therefore, is this:

The surprising fact, C, is observed; But if A were true, C would be a matter of course, Hence, there is reason to suspect that A is true.

Peirce: CP 5.189 Cross-Ref:†† Thus, A cannot be abductively inferred, or if you prefer the expression, cannot be abductively conjectured until its entire content is already present in the premiss, "If A were true, C would be a matter of course." Peirce: CP 5.190 Cross-Ref:†† 190. Whether this be a correct account of the matter or not, the mere suggestion of it as a possibility shows that the bare fact that abductions may be mistaken for perceptions does not necessarily affect the force of an argument to show

[that] quite new conceptions cannot be obtained from abduction. Peirce: CP 5.191 Cross-Ref:†† 191. But when the account just given of abduction is proposed as a proof that all conceptions must be given substantially in perception, three objections will be started. Namely, in the first place, it may be said that even if this be the normative form of abduction, the form to which abduction ought to conform, yet it may be that new conceptions arise in a manner which puts the rules of logic at defiance. In the second place, waiving this objection, it may be said that the argument would prove too much; for if it were valid, it would follow that no hypothesis could be so fantastic as not to have presented itself entire in experience. In the third place, it may be said that granting that the abductive conclusion, "A is true" rests upon the premiss, "If A is true, C is true," still it would be contrary to common knowledge to assert that the antecedents of all conditional judgments are given in perception, and thus it remains almost certain that some conceptions have a different origin. Peirce: CP 5.192 Cross-Ref:†† 192. In answer to the first of these objections, it is to be remarked that it is only in deduction that there is no difference between a valid argument and a strong one. An argument is valid if it possesses the sort of strength that it professes and tends toward the establishment of the conclusion in the way in which it pretends to do this. But the question of its strength does not concern the comparison of the due effect of the argument with its pretensions, but simply upon how great its due effect is. An argument is none the less logical for being weak, provided it does not pretend to a strength that it does not possess. It is, I suppose, in view of this that the best modern logicians outside the English school never say a word about fallacies. They assume that there is no such thing as an argument illogical in itself. An argument is fallacious only so far as it is mistakenly, though not illogically, inferred to have professed what it did not perform. Perhaps it may be said that if all our reasonings conform to the laws of logic, this is, at any rate, nothing but a proposition in psychology which my principles ought to forbid my recognizing. But I do not offer it as a principle of psychology only. For a principle of psychology is a contingent truth, while this, as I contend, is a necessary truth. Namely, if a fallacy involves nothing in its conclusion which was not in its premisses, that is nothing that was not in any previous knowledge that aided in suggesting it, then the forms of logic will invariably and necessarily enable us logically to account for it as due to a mistake arising from the use of a logical but weak argumentation.†1 In most cases it is due to an abduction. The conclusion of an abduction is problematic or conjectural, but is not necessarily at the weakest grade of surmise, and what we call assertoric judgments are, accurately, problematic judgments of a high grade of hopefulness. There is therefore no difficulty in maintaining that fallacies are merely due to mistakes which are logically valid, though weak argumentations. If, however, a fallacy contains something in the conclusion which was not in the premisses at all, that is, was in no previous knowledge or none that influenced the result, then again a mistake, due as before to weak inference, has been committed; only in this case the mistake consists in taking that to be an inference which, in respect to this new element, is not an inference, at all. That part of the conclusion which inserts the wholly new element can be separated from the rest with which it has no logical connection nor appearance of logical connection. The first emergence of this new element into consciousness must be regarded as a perceptive judgment. We are irresistibly led to judge that we are conscious of it. But the connection of this perception with other elements must be an ordinary logical inference, subject to error like all inference.

Peirce: CP 5.193 Cross-Ref:†† 193. As for the second objection that, according to my account of abduction, every hypothesis, however fantastic, must have presented itself entire in perception, I have only to say that this could only arise in a mind entirely unpractised in the logic of relations, and apparently quite oblivious of any other mode of inference than abduction. Deduction accomplishes first the simple colligation of different perceptive judgments into a copulative whole, and then, with or without the aid of other modes of inference, is quite capable of transforming this copulative proposition so as to bring certain of its parts into more intimate connection. Peirce: CP 5.194 Cross-Ref:†† 194. But the third objection is the really serious one. In it lies the whole nodus of the question; and its full refutation would be quite a treatise. If the antecedent is not given in a perceptive judgment, then it must first emerge in the conclusion of an inference. At this point we are obliged to draw the distinction between the matter and the logical form. With the aid of the logic of relations it would be easy to show that the entire logical matter of a conclusion must in any mode of inference be contained, piecemeal, in the premisses. Ultimately therefore it must come from the uncontrolled part of the mind, because a series of controlled acts must have a first. But as to the logical form, it would be, at any rate, extremely difficult to dispose of it in the same way. An induction, for example, concludes a ratio of frequency; but there is nothing about any such ratio in the single instances on which it is based. Where do the conceptions of deductive necessity, of inductive probability, of abductive expectability come from? Where does the conception of inference itself come from? That is the only difficulty. But self-control is the character which distinguishes reasonings from the processes by which perceptual judgments are formed, and self-control of any kind is purely inhibitory. It originates nothing. Therefore it cannot be in the act of adoption of an inference, in the pronouncing of it to be reasonable, that the formal conceptions in question can first emerge. It must be in the first perceiving that so one might conceivably reason. And what is the nature of that? I see that I have instinctively described the phenomenon as a "perceiving." I do not wish to argue from words; but a word may furnish a valuable suggestion. What can our first acquaintance with an inference, when it is not yet adopted, be but a perception of the world of ideas? In the first suggestion of it, the inference must be thought of as an inference, because when it is adopted there is always the thought that so one might reason in a whole class of cases. But the mere act of inhibition cannot introduce this conception. The inference must, then, be thought of as an inference in the first suggestion of it. Now when an inference is thought of as an inference, the conception of inference becomes a part of the matter of thought. Therefore, the same argument which we used in regard to matter in general applies to the conception of inference. But I am prepared to show in detail, and indeed virtually have shown, that all the forms of logic can be reduced to combinations of the conception of inference, the conception of otherness, and the conception of a character.†1 These are obviously simply forms of Thirdness, Secondness, and Firstness of which the last two are unquestionably given in perception. Consequently the whole logical form of thought is so given in its elements.

Peirce: CP 5.195 Cross-Ref:†† §3. PRAGMATISM -- THE LOGIC OF ABDUCTION

195. It appears to me, then, that my three cotary propositions are satisfactorily grounded. Nevertheless, since others may not regard them as so certain as I myself do, I propose in the first instance to disregard them, and to show that, even if they are put aside as doubtful, a maxim practically little differing in most of its applications from that of pragmatism ought to be acknowledged and followed; and after this has been done, I will show how the recognition of the cotary propositions will affect the matter. . . . Peirce: CP 5.196 Cross-Ref:†† 196. If you carefully consider the question of pragmatism you will see that it is nothing else than the question of the logic of abduction. That is, pragmatism proposes a certain maxim which, if sound, must render needless any further rule as to the admissibility of hypotheses to rank as hypotheses, that is to say, as explanations of phenomena held as hopeful suggestions; and, furthermore, this is all that the maxim of pragmatism really pretends to do, at least so far as it is confined to logic, and is not understood as a proposition in psychology. For the maxim of pragmatism is that a conception can have no logical effect or import differing from that of a second conception except so far as, taken in connection with other conceptions and intentions, it might conceivably modify our practical conduct differently from that second conception. Now it is indisputable that no rule of abduction would be admitted by any philosopher which should prohibit on any formalistic grounds any inquiry as to how we ought in consistency to shape our practical conduct. Therefore, a maxim which looks only to possibly practical considerations will not need any supplement in order to exclude any hypotheses as inadmissible. What hypotheses it admits all philosophers would agree ought to be admitted. On the other hand, if it be true that nothing but such considerations has any logical effect or import whatever, it is plain that the maxim of pragmatism cannot cut off any kind of hypothesis which ought to be admitted. Thus, the maxim of pragmatism, if true, fully covers the entire logic of abduction. It remains to inquire whether this maxim may not have some further logical effect. If so, it must in some way affect inductive or deductive inference. But that pragmatism cannot interfere with induction is evident; because induction simply teaches us what we have to expect as a result of experimentation, and it is plain that any such expectation may conceivably concern practical conduct. In a certain sense it must affect deduction. Anything which gives a rule to abduction and so puts a limit upon admissible hypotheses will cut down the premisses of deduction, and thereby will render a reductio ad absurdum and other equivalent forms of deduction possible which would not otherwise have been possible. But here three remarks may be made. First, to affect the premisses of deduction is not to affect the logic of deduction. For in the process of deduction itself, no conception is introduced to which pragmatism could be supposed to object, except the acts of abstraction. Concerning that I have only time to say that pragmatism ought not to object to it. Secondly, no effect of pragmatism which is consequent upon its effect on abduction can go to show that pragmatism is anything more than a doctrine concerning the logic of abduction. Thirdly, if pragmatism is the doctrine that every conception is a conception of conceivable practical effects, it makes conception reach far beyond the practical. It allows any flight of imagination, provided this imagination ultimately alights upon a possible practical effect; and thus many hypotheses may seem at first glance to be excluded by the pragmatical maxim that are not really so excluded.

Peirce: CP 5.197 Cross-Ref:†† 197. Admitting, then, that the question of Pragmatism is the question of Abduction, let us consider it under that form. What is good abduction? What should an explanatory hypothesis be to be worthy to rank as a hypothesis? Of course, it must explain the facts. But what other conditions ought it to fulfill to be good? The question of the goodness of anything is whether that thing fulfills its end. What, then, is the end of an explanatory hypothesis? Its end is, through subjection to the test of experiment, to lead to the avoidance of all surprise and to the establishment of a habit of positive expectation that shall not be disappointed. Any hypothesis, therefore, may be admissible, in the absence of any special reasons to the contrary, provided it be capable of experimental verification, and only insofar as it is capable of such verification. This is approximately the doctrine of pragmatism. But just here a broad question opens out before us. What are we to understand by experimental verification? The answer to that involves the whole logic of induction. Peirce: CP 5.198 Cross-Ref:†† 198. Let me point out to you the different opinions which we actually find men holding today -- perhaps not consistently, but thinking that they hold them -upon this subject. In the first place, we find men who maintain that no hypothesis ought to be admitted, even as a hypothesis, any further than its truth or its falsity is capable of being directly perceived. This, as well as I can make out, is what was in the mind of Auguste Comte,†1 who is generally assumed to have first formulated this maxim. Of course, this maxim of abduction supposes that, as people say, we "are to believe only what we actually see"; and there are well-known writers, and writers of no little intellectual force, who maintain that it is unscientific to make predictions -unscientific, therefore, to expect anything. One ought to restrict one's opinions to what one actually perceives. I need hardly say that that position cannot be consistently maintained. It refutes itself, for it is itself an opinion relating to more than is actually in the field of momentary perception. Peirce: CP 5.199 Cross-Ref:†† 199. In the second place, there are those who hold that a theory which has sustained a number of experimental tests may be expected to sustain a number of other similar tests, and to have a general approximate truth, the justification of this being that this kind of inference must prove correct in the long run, as I explained in a previous lecture.†2 But these logicians refuse to admit that we can ever have a right to conclude definitely that a hypothesis is exactly true, that is that it should be able to sustain experimental tests in endless series; for, they urge, no hypothesis can be subjected to an endless series of tests. They are willing we should say that a theory is true, because, all our ideas being more or less vague and approximate, what we mean by saying that a theory is true can only be that it is very near true. But they will not allow us to say that anything put forth as an anticipation of experience should assert exactitude, because exactitude in experience would imply experiences in endless series, which is impossible. Peirce: CP 5.200 Cross-Ref:†† 200. In the third place, the great body of scientific men hold that it is too much to say that induction must be restricted to that for which there can be positive experimental evidence. They urge that the rationale of induction as it is understood by logicians of the second group, themselves, entitles us to hold a theory, provided it be such that if it involve any falsity, experiment must some day detect that falsity. We, therefore, have a right, they will say, to infer that something never will happen,

provided it be of such a nature that it could not occur without being detected. Peirce: CP 5.201 Cross-Ref:†† 201. I wish to avoid in the present lecture arguing any such points, because the substance of all sound argumentation about pragmatism has, as I conceive it, been already given in previous lectures, and there is no end to the forms in which it might be stated. I must, however, except from this statement the logical principles which I intend to state in tomorrow evening's lecture on multitude and continuity;†1 and for the sake of making the relation clear between this third position and the fourth and fifth, I must anticipate a little what I shall further explain tomorrow. Peirce: CP 5.202 Cross-Ref:†† 202. What ought persons, who hold this third position, to say to the Achilles sophism? Or rather . . . what would they be obliged to say to Achilles overtaking the tortoise (Achilles and the tortoise being geometrical points) supposing that our only knowledge was derived inductively from observations of the relative positions of Achilles and the tortoise at those stages of the progress that the sophism supposes, and supposing that Achilles really moves twice as fast as the tortoise? They ought to say that if it could not happen that Achilles in one of those stages of his progress should at length reach a certain finite distance behind the tortoise which he would be unable to halve, without our learning that fact, then we should have a right to conclude that he could halve every distance and consequently that he could make his distance behind the tortoise less than all fractions having a power of two for the denominator. Therefore unless these logicians were to suppose a distance less than any measurable distance, which would be contrary to their principles, they would be obliged to say that Achilles could reduce his distance behind the tortoise to zero. Peirce: CP 5.203 Cross-Ref:†† 203. The reason why it would be contrary to their principles to admit any distance less than a measurable distance, is that their way of supporting induction implies that they differ from the logicians of the second class, in that these third class logicians admit that we can infer a proposition implying an infinite multitude and therefore implying the reality of the infinite multitude itself, while their mode of justifying induction would exclude every infinite multitude except the lowest grade, that of the multitude of all integer numbers. Because with reference to a greater multitude than that, it would not be true that what did not occur in a finite ordinal place in a series could not occur anywhere within the infinite series -- which is the only reason they admit for the inductive conclusion. Peirce: CP 5.203 Cross-Ref:†† But now let us look at something else that those logicians would be obliged to admit. Namely, suppose any regular polygon to have all its vertices joined by straight radii to its centre. Then if there were any particular finite number of sides for a regular polygon with radii so drawn, which had the singular property that it should be impossible to bisect all the angles by new radii equal to the others and by connecting the extremities of each new radius to those of the two adjacent old radii to make a new polygon of double the number of angles -- if, I say, there were any finite number of sides for which this could not be done -- it may be admitted that we should be able to find it out. The question I am asking supposes arbitrarily that they admit that. Therefore these logicians of the third class would have to admit that all such polygons could so have their sides doubled and that consequently there would be a polygon of an infinite multitude of sides which could be, on their principles, nothing else than the circle. But it is easily proved that the perimeter of that polygon, that is, the

circumference of the circle, would be incommensurable, so that an incommensurable measure is real, and thence it easily follows that all such lengths are real or possible. But these exceed in multitude the only multitude those logicians admit. Without any geometry, the same result could be reached, supposing only that we have an indefinitely bisectible quantity. Peirce: CP 5.204 Cross-Ref:†† 204. We are thus led to a fourth opinion very common among mathematicians, who generally hold that any one irrational real quantity (say of length, for example) whether algebraical or transcendental in its general expression, is just as possible and admissible as any rational quantity, but who generally reason that if the distance between two points is less than any assignable quantity, that is, less than any finite quantity, then it is nothing at all. If that be the case, it is possible for us to conceive, with mathematical precision, a state of things in favor of whose actual reality there would seem to be no possible sound argument, however weak. For example, we can conceive that the diagonal of a square is incommensurable with its side. That is to say, if you first name any length commensurable with the side, the diagonal will differ from that by a finite quantity (and a commensurable quantity), yet however accurately we may measure the diagonal of an apparent square, there will always be a limit to our accuracy and the measure will always be commensurable. So we never could have any reason to think it otherwise. Moreover, if there be, as they seem to hold, no other points on a line than such as are at distances assignable to an indefinite approximation, it will follow that if a line has an extremity, that extreme point may be conceived to be taken away so as to leave the line without any extremity, while leaving all the other points just as they were. In that case, all the points stand discrete and separate; and the line might be torn apart at any number of places without disturbing the relations of the points to one another. Each point has, on that view, its own independent existence, and there can be no merging of one into another. There is no continuity of points in the sense in which continuity implies generality. Peirce: CP 5.205 Cross-Ref:†† 205. In the fifth place it may be held that we can be justified in inferring true generality, true continuity. But I do not see in what way we ever can be justified in doing so unless we admit the cotary propositions, and in particular that such continuity is given in perception; that is, that whatever the underlying psychical process may be, we seem to perceive a genuine flow of time, such that instants melt into one another without separate individuality. Peirce: CP 5.205 Cross-Ref:†† It would not be necessary for me to deny a psychical theory which should make this to be illusory, in such [a] sense as [one might say] that anything beyond all logical criticism is illusory, but I confess I should strongly suspect that such a psychological theory involved a logical inconsistency; and at best it could do nothing at all toward solving the logical question.

Peirce: CP 5.206 Cross-Ref:†† §4. THE TWO FUNCTIONS OF PRAGMATISM

206. There are two functions which we may properly require that Pragmatism should perform; or if not pragmatism, whatever the true doctrine of the Logic of

Abduction may be, ought to do these two services. Peirce: CP 5.206 Cross-Ref:†† Namely, it ought, in the first place, to give us an expeditious riddance of all ideas essentially unclear. In the second place, it ought to lend support, and help to render distinct, ideas essentially clear, but more or less difficult of apprehension; and in particular, it ought to take a satisfactory attitude toward the element of thirdness. Peirce: CP 5.207 Cross-Ref:†† 207. Of these two offices of Pragmatism, there is at the present day not so crying a need of the first as there was a quarter of a century ago when I enunciated the maxim. The state of logical thought is very much improved. Thirty years ago †1 when, in consequence of my study of the logic of relations, I told philosophers that all conceptions ought to be defined, with the sole exception of the familiar concrete conceptions of everyday life, my opinion was considered in every school to be utterly incomprehensible. The doctrine then was, as it remains in nineteen out of every score of logical treatises that are appearing in these days, that there is no way of defining a term except by enumerating all its universal predicates, each of which is more abstracted and general than the term defined. So unless this process can go on endlessly, which was a doctrine little followed, the explication of a concept must stop at such ideas as Pure Being, Agency, Substance and the like, which were held to be ideas so perfectly simple that no explanation whatever could be given of them. This grotesque doctrine was shattered by the logic of relations, which showed that the simplest conceptions, such as Quality, Relation, Self-consciousness could be defined and that such definitions would be of the greatest service in dealing with them.†1 By this time, although few really study the logic of relations, one seldom meets with a philosopher who continues to think the most general relations are particularly simple in any except a technical sense; and of course, the only alternative is to regard as the simplest the practically applied notions of familiar life. We should hardly find today a man of Kirchhoff's rank in science saying that we know exactly what energy does but what energy is we do not know in the least.†2 For the answer would be that energy being a term in a dynamical equation, if we know how to apply that equation, we thereby know what energy is, although we may suspect that there is some more fundamental law underlying the laws of motion. Peirce: CP 5.208 Cross-Ref:†† 208. In the present situation of philosophy, it is far more important that thirdness should be adequately dealt with by our logical maxim of abduction. The urgent pertinence of the question of thirdness, at this moment of the breakup of agnostic calm, when we see that the chief difference between philosophers is in regard to the extent to which they allow elements of thirdness a place in their theories, is too plain to be insisted upon. Peirce: CP 5.209 Cross-Ref:†† 209. I shall take it for granted that as far as thought goes, I have sufficiently shown that thirdness is an element not reducible to secondness and firstness. But even if so much be granted, three attitudes may be taken: Peirce: CP 5.209 Cross-Ref:†† (1) That thirdness, though an element of the mental phenomenon, ought not to be admitted into a theory of the real, because it is not experimentally verifiable; Peirce: CP 5.209 Cross-Ref:††

(2) That thirdness is experimentally verifiable, that is, is inferable by induction, [abduction?] although it cannot be directly perceived; Peirce: CP 5.209 Cross-Ref:†† (3) That it is directly perceived, from which the other cotary propositions can hardly be separated. Peirce: CP 5.210 Cross-Ref:†† 210. The man who takes the first position ought to admit no general law as really operative. Above all, therefore, he ought not to admit the law of laws, the law of the uniformity of nature. He ought to abstain from all prediction, however qualified by a confession of fallibility. But that position can practically not be maintained. Peirce: CP 5.211 Cross-Ref:†† 211. The man who takes the second position will hold thirdness to be an addition which the operation of abduction introduces over and above what its premisses in any way contain, and further that this element, though not perceived in experiment, is justified by experiment. Then his conception of reality must be such as completely to sunder the real from perception; and the puzzle for him will be why perception should be allowed such authority in regard to what is real. Peirce: CP 5.211 Cross-Ref:†† I do not think that man can consistently hold that there is room in time for an event between any two events separate in time. But even if he could, he would (if he could grasp the reasons) be forced to acknowledge that the contents of time consists of separate, independent, unchanging states, and nothing else. There would not be even a determinate order of sequence among these states. He might insist that one order of sequence was more readily grasped by us; but nothing more. Every man is fully satisfied that there is such a thing as truth, or he would not ask any question. That truth consists in a conformity to something independent of his thinking it to be so, or of any man's opinion on that subject. But for the man who holds this second opinion, the only reality, there could be, would be conformity to the ultimate result of inquiry. But there would not be any course of inquiry possible except in the sense that it would be easier for him to interpret the phenomenon; and ultimately he would be forced to say that there was no reality at all except that he now at this instant finds a certain way of thinking easier than any other. But that violates the very idea of reality and of truth. Peirce: CP 5.212 Cross-Ref:†† 212. The man who takes the third position and accepts the cotary propositions will hold, with firmest of grasps, to the recognition that logical criticism is limited to what we can control. In the future we may be able to control more but we must consider what we can now control. Some elements we can control in some limited measure. But the content of the perceptual judgment cannot be sensibly controlled now, nor is there any rational hope that it ever can be. Concerning that quite uncontrolled part of the mind, logical maxims have as little to do as with the growth of hair and nails. We may be dimly able to see that, in part, it depends on the accidents of the moment, in part on what is personal or racial, in part is common to all nicely adjusted organisms whose equilibrium has narrow ranges of stability, in part on whatever is composed of vast collections of independently variable elements, in part on whatever reacts, and in part on whatever has any mode of being. But the sum of it all is that our logically controlled thoughts compose a small part of the mind, the mere blossom of a vast complexus, which we may call the instinctive mind, in which this

man will not say that he has faith, because that implies the conceivability of distrust, but upon which he builds as the very fact to which it is the whole business of his logic to be true. Peirce: CP 5.212 Cross-Ref:†† That he will have no difficulty with Thirdness is clear enough, because he will hold that the conformity of action to general intentions is as much given in perception as is the element of action itself, which cannot really be mentally torn away from such general purposiveness. There can be no doubt that he will allow hypotheses fully all the range they ought to be allowed. The only question will be whether he succeeds in excluding from hypotheses everything unclear and nonsensical. It will be asked whether he will not have a shocking leaning toward anthropomorphic conceptions. I fear I must confess that he will be inclined to see an anthropomorphic, or even a zoömorphic, if not a physiomorphic element in all our conceptions. But against unclear and nonsensical hypotheses, [of] whatever ægis [he will be protected]. Pragmatism will be more essentially significant for him than for any other logician, for the reason that it is in action that logical energy returns to the uncontrolled and uncriticizable parts of the mind. His maxim will be this: Peirce: CP 5.212 Cross-Ref:†† The elements of every concept enter into logical thought at the gate of perception and make their exit at the gate of purposive action; and whatever cannot show its passports at both those two gates is to be arrested as unauthorized by reason. Peirce: CP 5.212 Cross-Ref:†† The digestion of such thoughts is slow, ladies and gentlemen; but when you come in the future to reflect upon all that I have said, I am confident you will find the seven hours, you have spent in listening to these ideas, have not been altogether wasted.

Peirce: CP 5 Book 2 Question 1 BOOK II

PUBLISHED PAPERS I

QUESTIONS CONCERNING CERTAIN FACULTIES CLAIMED FOR MANP†1

QUESTION 1. Whether by the simple contemplation of a cognition, independently of any previous knowledge and without reasoning from signs, we are enabled rightly to judge whether that cognition has been determined by a previous cognition or whether it refers immediately to its object.

Peirce: CP 5.213 Cross-Ref:†† 213. Throughout this paper, the term intuition will be taken as signifying a cognition not determined by a previous cognition of the same object, and therefore so determined by something out of the consciousness.†P1 Let me request the reader to note this. Intuition here will be nearly the same as "premiss not itself a conclusion"; the only difference being that premisses and conclusions are judgments, whereas an intuition may, as far as its definition states, be any kind of cognition whatever. But just as a conclusion (good or bad) is determined in the mind of the reasoner by its premiss, so cognitions not judgments may be determined by previous cognitions; and a cognition not so determined, and therefore determined directly by the transcendental object, is to be termed an intuition. Peirce: CP 5.214 Cross-Ref:†† 214. Now, it is plainly one thing to have an intuition and another to know intuitively that it is an intuition, and the question is whether these two things, distinguishable in thought, are, in fact, invariably connected, so that we can always intuitively distinguish between an intuition and a cognition determined by another. Every cognition, as something present, is, of course, an intuition of itself. But the determination of a cognition by another cognition or by a transcendental object is not, at least so far as appears obviously at first, a part of the immediate content of that cognition, although it would appear to be an element of the action or passion of the transcendental ego, which is not, perhaps, in consciousness immediately; and yet this transcendental action or passion may invariably determine a cognition of itself, so that, in fact, the determination or non-determination of the cognition by another may be a part of the cognition. In this case, I should say that we had an intuitive power of distinguishing an intuition from another cognition. Peirce: CP 5.214 Cross-Ref:†† There is no evidence that we have this faculty, except that we seem to feel that we have it. But the weight of that testimony depends entirely on our being supposed to have the power of distinguishing in this feeling whether the feeling be the result of education, old associations, etc., or whether it is an intuitive cognition; or, in other words, it depends on presupposing the very matter testified to. Is this feeling infallible? And is this judgment concerning it infallible, and so on, ad infinitum? Supposing that a man really could shut himself up in such a faith, he would be, of course, impervious to the truth, "evidence-proof." Peirce: CP 5.215 Cross-Ref:†† 215. But let us compare the theory with the historic facts. The power of intuitively distinguishing intuitions from other cognitions has not prevented men from disputing very warmly as to which cognitions are intuitive. In the middle ages, reason and external authority were regarded as two coördinate sources of knowledge, just as reason and the authority of intuition are now; only the happy device of considering the enunciations of authority to be essentially indemonstrable had not yet been hit upon. All authorities were not considered as infallible, any more than all reasons; but when Berengarius said that the authoritativeness of any particular authority must rest upon reason, the proposition was scouted as opinionated, impious, and absurd.†1 Thus, the credibility of authority was regarded by men of that time simply as an ultimate premiss, as a cognition not determined by a previous cognition of the same object, or, in our terms, as an intuition. It is strange that they should have thought so,

if, as the theory now under discussion supposes, by merely contemplating the credibility of the authority, as a Fakir does his God, they could have seen that it was not an ultimate premiss! Now, what if our internal authority should meet the same fate, in the history of opinions, as that external authority has met? Can that be said to be absolutely certain which many sane, well-informed, and thoughtful men already doubt?†P1 Peirce: CP 5.216 Cross-Ref:†† 216. Every lawyer knows how difficult it is for witnesses to distinguish between what they have seen and what they have inferred. This is particularly noticeable in the case of a person who is describing the performances of a spiritual medium or of a professed juggler. The difficulty is so great that the juggler himself is often astonished at the discrepancy between the actual facts and the statement of an intelligent witness who has not understood the trick. A part of the very complicated trick of the Chinese rings consists in taking two solid rings linked together, talking about them as though they were separate -- taking it for granted, as it were -- then pretending to put them together, and handing them immediately to the spectator that he may see that they are solid. The art of this consists in raising, at first, the strong suspicion that one is broken. I have seen McAlister do this with such success, that a person sitting close to him, with all his faculties straining to detect the illusion, would have been ready to swear that he saw the rings put together, and, perhaps, if the juggler had not professedly practised deception, would have considered a doubt of it as a doubt of his own veracity. This certainly seems to show that it is not always very easy to distinguish between a premiss and a conclusion, that we have no infallible power of doing so, and that in fact our only security in difficult cases is in some signs from which we can infer that a given fact must have been seen or must have been inferred. In trying to give an account of a dream, every accurate person must often have felt that it was a hopeless undertaking to attempt to disentangle waking interpretations and fillings out from the fragmentary images of the dream itself. Peirce: CP 5.217 Cross-Ref:†† 217. The mention of dreams suggests another argument. A dream, as far as its own content goes, is exactly like an actual experience. It is mistaken for one. And yet all the world believes that dreams are determined, according to the laws of the association of ideas, etc., by previous cognitions. If it be said that the faculty of intuitively recognizing intuitions is asleep, I reply that this is a mere supposition, without other support. Besides, even when we wake up, we do not find that the dream differed from reality, except by certain marks, darkness and fragmentariness. Not unfrequently a dream is so vivid that the memory of it is mistaken for the memory of an actual occurrence. Peirce: CP 5.218 Cross-Ref:†† 218. A child has, as far as we know, all the perceptive powers of a man. Yet question him a little as to how he knows what he does. In many cases, he will tell you that he never learned his mother-tongue; he always knew it, or he knew it as soon as he came to have sense. It appears, then, that he does not possess the faculty of distinguishing, by simple contemplation, between an intuition and a cognition determined by others. Peirce: CP 5.219 Cross-Ref:†† 219. There can be no doubt that before the publication of Berkeley's book on Vision,†1 it had generally been believed that the third dimension of space was immediately intuited, although, at present, nearly all admit that it is known by

inference. We had been contemplating the object since the very creation of man, but this discovery was not made until we began to reason about it. Peirce: CP 5.220 Cross-Ref:†† 220. Does the reader know of the blind spot on the retina? Take a number of this journal, turn over the cover so as to expose the white paper, lay it sideways upon the table before which you must sit, and put two cents upon it, one near the left-hand edge, and the other to the right. Put your left hand over your left eye, and with the right eye look steadily at the left-hand cent. Then, with your right hand, move the right-hand cent (which is now plainly seen) towards the left hand. When it comes to a place near the middle of the page it will disappear -- you cannot see it without turning your eye. Bring it nearer to the other cent, or carry it further away, and it will reappear; but at that particular spot it cannot be seen. Thus it appears that there is a blind spot nearly in the middle of the retina; and this is confirmed by anatomy. It follows that the space we immediately see (when one eye is closed) is not, as we had imagined, a continuous oval, but is a ring, the filling up of which must be the work of the intellect. What more striking example could be desired of the impossibility of distinguishing intellectual results from intuitional data, by mere contemplation? Peirce: CP 5.221 Cross-Ref:†† 221. A man can distinguish different textures of cloth by feeling; but not immediately, for he requires to move his fingers over the cloth, which shows that he is obliged to compare the sensations of one instant with those of another. Peirce: CP 5.222 Cross-Ref:†† 222. The pitch of a tone depends upon the rapidity of the succession of the vibrations which reach the ear. Each of those vibrations produces an impulse upon the ear. Let a single such impulse be made upon the ear, and we know, experimentally, that it is perceived. There is, therefore, good reason to believe that each of the impulses forming a tone is perceived. Nor is there any reason to the contrary. So that this is the only admissible supposition. Therefore, the pitch of a tone depends upon the rapidity with which certain impressions are successively conveyed to the mind. These impressions must exist previously to any tone; hence, the sensation of pitch is determined by previous cognitions. Nevertheless, this would never have been discovered by the mere contemplation of that feeling. Peirce: CP 5.223 Cross-Ref:†† 223. A similar argument may be urged in reference to the perception of two dimensions of space. This appears to be an immediate intuition. But if we were to see immediately an extended surface, our retinas must be spread out in an extended surface. Instead of that, the retina consists of innumerable needles pointing towards the light, and whose distances from one another are decidedly greater than the minimum visibile. Suppose each of those nerve-points conveys the sensation of a little colored surface. Still, what we immediately see must even then be, not a continuous surface, but a collection of spots. Who could discover this by mere intuition? But all the analogies of the nervous system are against the supposition that the excitation of a single nerve can produce an idea as complicated as that of a space, however small. If the excitation of no one of these nerve points can immediately convey the impression of space, the excitation of all cannot do so. For, the excitation of each produces some impression (according to the analogies of the nervous system), hence, the sum of these impressions is a necessary condition of any perception produced by the excitation of all; or, in other terms, a perception produced by the excitation of all is determined by the mental impressions produced by the excitation

of every one. This argument is confirmed by the fact that the existence of the perception of space can be fully accounted for by the action of faculties known to exist, without supposing it to be an immediate impression. For this purpose, we must bear in mind the following facts of physio-psychology: 1. The excitation of a nerve does not of itself inform us where the extremity of it is situated. If, by a surgical operation, certain nerves are displaced, our sensations from those nerves do not inform us of the displacement. 2. A single sensation does not inform us how many nerves or nerve-points are excited. 3. We can distinguish between the impressions produced by the excitations of different nerve-points. 4. The differences of impressions produced by different excitations of similar nerve-points are similar. Let a momentary image be made upon the retina. By No. 2, the impression thereby produced will be indistinguishable from what might be produced by the excitation of some conceivable single nerve. It is not conceivable that the momentary excitation of a single nerve should give the sensation of space. Therefore, the momentary excitation of all the nerve-points of the retina cannot, immediately or mediately, produce the sensation of space. The same argument would apply to any unchanging image on the retina. Suppose, however, that the image moves over the retina. Then the peculiar excitation which at one instant affects one nerve-point, at a later instant will affect another. These will convey impressions which are very similar by 4, and yet which are distinguishable by 3. Hence, the conditions for the recognition of a relation between these impressions are present. There being, however, a very great number of nerve-points affected by a very great number of successive excitations, the relations of the resulting impressions will be almost inconceivably complicated. Now, it is a known law of mind, that when phenomena of an extreme complexity are presented, which yet would be reduced to order or mediate simplicity by the application of a certain conception, that conception sooner or later arises in application to those phenomena. In the case under consideration, the conception of extension would reduce the phenomena to unity, and, therefore, its genesis is fully accounted for. It remains only to explain why the previous cognitions which determine it are not more clearly apprehended. For this explanation, I shall refer to a paper upon a new list of categories, Section 5,†P1 merely adding that just as we are able to recognize our friends by certain appearances, although we cannot possibly say what those appearances are and are quite unconscious of any process of reasoning, so in any case when the reasoning is easy and natural to us, however complex may be the premisses, they sink into insignificance and oblivion proportionately to the satisfactoriness of the theory based upon them. This theory of space is confirmed by the circumstance that an exactly similar theory is imperatively demanded by the facts in reference to time. That the course of time should be immediately felt is obviously impossible. For, in that case, there must be an element of this feeling at each instant. But in an instant there is no duration and hence no immediate feeling of duration. Hence, no one of these elementary feelings is an immediate feeling of duration; and, hence the sum of all is not. On the other hand, the impressions of any moment are very complicated -- containing all the images (or the elements of the images) of sense and memory, which complexity is reducible to mediate simplicity by means of the conception of time.†P2 Peirce: CP 5.224 Cross-Ref:†† 224. We have, therefore, a variety of facts, all of which are most readily explained on the supposition that we have no intuitive faculty of distinguishing intuitive from mediate cognitions. Some arbitrary hypothesis may otherwise explain any one of these facts; this is the only theory which brings them to support one another. Moreover, no facts require the supposition of the faculty in question.

Whoever has studied the nature of proof will see, then, that there are here very strong reasons for disbelieving the existence of this faculty. These will become still stronger when the consequences of rejecting it have, in this paper and in a following one, been more fully traced out.

Peirce: CP 5 Book 2 Question 2 QUESTION 2. Whether we have an intuitive self-consciousness. Peirce: CP 5.225 Cross-Ref:†† 225. Self-consciousness, as the term is here used, is to be distinguished both from consciousness generally, from the internal sense, and from pure apperception. Any cognition is a consciousness of the object as represented; by self-consciousness is meant a knowledge of ourselves. Not a mere feeling of subjective conditions of consciousness, but of our personal selves. Pure apperception is the self-assertion of THE ego; the self-consciousness here meant is the recognition of my private self. I know that I (not merely the I) exist. The question is, how do I know it; by a special intuitive faculty, or is it determined by previous cognitions? Peirce: CP 5.226 Cross-Ref:†† 226. Now, it is not self-evident that we have such an intuitive faculty, for it has just been shown that we have no intuitive power of distinguishing an intuition from a cognition determined by others. Therefore, the existence or non-existence of this power is to be determined upon evidence, and the question is whether self-consciousness can be explained by the action of known faculties under conditions known to exist, or whether it is necessary to suppose an unknown cause for this cognition, and, in the latter case, whether an intuitive faculty of self-consciousness is the most probable cause which can be supposed. Peirce: CP 5.227 Cross-Ref:†† 227. It is first to be observed that there is no known self-consciousness to be accounted for in extremely young children. It has already been pointed out by Kant †P1 that the late use of the very common word "I" with children indicates an imperfect self-consciousness in them, and that, therefore, so far as it is admissible for us to draw any conclusion in regard to the mental state of those who are still younger, it must be against the existence of any self-consciousness in them. Peirce: CP 5.228 Cross-Ref:†† 228. On the other hand, children manifest powers of thought much earlier. Indeed, it is almost impossible to assign a period at which children do not already exhibit decided intellectual activity in directions in which thought is indispensable to their well-being. The complicated trigonometry of vision, and the delicate adjustments of coördinated movement, are plainly mastered very early. There is no reason to question a similar degree of thought in reference to themselves. Peirce: CP 5.229 Cross-Ref:†† 229. A very young child may always be observed to watch its own body with great attention. There is every reason why this should be so, for from the child's point of view this body is the most important thing in the universe. Only what it touches has any actual and present feeling; only what it faces has any actual color; only what is on its tongue has any actual taste. Peirce: CP 5.230 Cross-Ref:††

230. No one questions that, when a sound is heard by a child, he thinks, not of himself as hearing, but of the bell or other object as sounding. How when he wills to move a table? Does he then think of himself as desiring, or only of the table as fit to be moved? That he has the latter thought, is beyond question; that he has the former, must, until the existence of an intuitive self-consciousness is proved, remain an arbitrary and baseless supposition. There is no good reason for thinking that he is less ignorant of his own peculiar condition than the angry adult who denies that he is in a passion. Peirce: CP 5.231 Cross-Ref:†† 231. The child, however, must soon discover by observation that things which are thus fit to be changed are apt actually to undergo this change, after a contact with that peculiarly important body called Willy or Johnny. This consideration makes this body still more important and central, since it establishes a connection between the fitness of a thing to be changed and a tendency in this body to touch it before it is changed. Peirce: CP 5.232 Cross-Ref:†† 232. The child learns to understand the language; that is to say, a connection between certain sounds and certain facts becomes established in his mind. He has previously noticed the connection between these sounds and the motions of the lips of bodies somewhat similar to the central one, and has tried the experiment of putting his hand on those lips and has found the sound in that case to be smothered. He thus connects that language with bodies somewhat similar to the central one. By efforts, so unenergetic that they should be called rather instinctive, perhaps, than tentative, he learns to produce those sounds. So he begins to converse. Peirce: CP 5.233 Cross-Ref:†† 233. It must be about this time that he begins to find that what these people about him say is the very best evidence of fact. So much so, that testimony is even a stronger mark of fact than the facts themselves, or rather than what must now be thought of as the appearances themselves. (I may remark, by the way, that this remains so through life; testimony will convince a man that he himself is mad.) A child hears it said that the stove is hot. But it is not, he says; and, indeed, that central body is not touching it, and only what that touches is hot or cold. But he touches it, and finds the testimony confirmed in a striking way. Thus, he becomes aware of ignorance, and it is necessary to suppose a self in which this ignorance can inhere. So testimony gives the first dawning of self-consciousness. Peirce: CP 5.234 Cross-Ref:†† 234. But, further, although usually appearances are either only confirmed or merely supplemented by testimony, yet there is a certain remarkable class of appearances which are continually contradicted by testimony. These are those predicates which we know to be emotional, but which he distinguishes by their connection with the movements of that central person, himself (that the table wants moving, etc.) These judgments are generally denied by others. Moreover, he has reason to think that others, also, have such judgments which are quite denied by all the rest. Thus, he adds to the conception of appearance as the actualization of fact, the conception of it as something private and valid only for one body. In short, error appears, and it can be explained only by supposing a self which is fallible. Peirce: CP 5.235 Cross-Ref:†† 235. Ignorance and error are all that distinguish our private selves from the

absolute ego of pure apperception. Peirce: CP 5.236 Cross-Ref:†† 236. Now, the theory which, for the sake of perspicuity, has thus been stated in a specific form, may be summed up as follows: At the age at which we know children to be self-conscious, we know that they have been made aware of ignorance and error; and we know them to possess at that age powers of understanding sufficient to enable them to infer from ignorance and error their own existence. Thus we find that known faculties, acting under conditions known to exist, would rise to self-consciousness. The only essential defect in this account of the matter is, that while we know that children exercise as much understanding as is here supposed, we do not know that they exercise it in precisely this way. Still the supposition that they do so is infinitely more supported by facts, than the supposition of a wholly peculiar faculty of the mind. Peirce: CP 5.237 Cross-Ref:†† 237. The only argument worth noticing for the existence of an intuitive self-consciousness is this. We are more certain of our own existence than of any other fact; a premiss cannot determine a conclusion to be more certain than it is itself; hence, our own existence cannot have been inferred from any other fact. The first premiss must be admitted, but the second premiss is founded on an exploded theory of logic. A conclusion cannot be more certain than that some one of the facts which support it is true, but it may easily be more certain than any one of those facts. Let us suppose, for example, that a dozen witnesses testify to an occurrence. Then my belief in that occurrence rests on the belief that each of those men is generally to be believed upon oath. Yet the fact testified to is made more certain than that any one of those men is generally to be believed. In the same way, to the developed mind of man, his own existence is supported by every other fact, and is, therefore, incomparably more certain than any one of these facts. But it cannot be said to be more certain than that there is another fact, since there is no doubt perceptible in either case. Peirce: CP 5.237 Cross-Ref:†† It is to be concluded, then, that there is no necessity of supposing an intuitive self-consciousness, since self-consciousness may easily be the result of inference.

Peirce: CP 5 Book 2 Question 3 QUESTION 3. Whether we have an intuitive power of distinguishing between the subjective elements of different kinds of cognitions. Peirce: CP 5.238 Cross-Ref:†† 238. Every cognition involves something represented, or that of which we are conscious, and some action or passion of the self whereby it becomes represented. The former shall be termed the objective, the latter the subjective, element of the cognition. The cognition itself is an intuition of its objective element, which may therefore be called, also, the immediate object. The subjective element is not necessarily immediately known, but it is possible that such an intuition of the subjective element of a cognition of its character, whether that of dreaming, imagining, conceiving, believing, etc., should accompany every cognition. The question is whether this is so. Peirce: CP 5.239 Cross-Ref:†† 239. It would appear, at first sight, that there is an overwhelming array of

evidence in favor of the existence of such a power. The difference between seeing a color and imagining it is immense. There is a vast difference between the most vivid dream and reality. And if we had no intuitive power of distinguishing between what we believe and what we merely conceive, we never, it would seem, could in any way distinguish them; since if we did so by reasoning, the question would arise whether the argument itself was believed or conceived, and this must be answered before the conclusion could have any force. And thus there would be a regressus ad infinitum. Besides, if we do not know that we believe, then, from the nature of the case, we do not believe. Peirce: CP 5.240 Cross-Ref:†† 240. But be it noted that we do not intuitively know the existence of this faculty. For it is an intuitive one, and we cannot intuitively know that a cognition is intuitive. The question is, therefore, whether it is necessary to suppose the existence of this faculty, or whether then the facts can be explained without this supposition. Peirce: CP 5.241 Cross-Ref:†† 241. In the first place, then, the difference between what is imagined or dreamed and what is actually experienced, is no argument in favor of the existence of such a faculty. For it is not questioned that there are distinctions in what is present to the mind, but the question is, whether independently of any such distinctions in the immediate objects of consciousness, we have any immediate power of distinguishing different modes of consciousness. Now, the very fact of the immense difference in the immediate objects of sense and imagination, sufficiently accounts for our distinguishing those faculties; and instead of being an argument in favor of the existence of an intuitive power of distinguishing the subjective elements of consciousness, it is a powerful reply to any such argument, so far as the distinction of sense and imagination is concerned. Peirce: CP 5.242 Cross-Ref:†† 242. Passing to the distinction of belief and conception, we meet the statement that the knowledge of belief is essential to its existence. Now, we can unquestionably distinguish a belief from a conception, in most cases, by means of a peculiar feeling of conviction; and it is a mere question of words whether we define belief as that judgment which is accompanied by this feeling, or as that judgment from which a man will act. We may conveniently call the former sensational, the latter active, belief. That neither of these necessarily involves the other, will surely be admitted without any recital of facts. Taking belief in the sensational sense, the intuitive power of reorganizing it will amount simply to the capacity for the sensation which accompanies the judgment. This sensation, like any other, is an object of consciousness; and therefore the capacity for it implies no intuitive recognition of subjective elements of consciousness. If belief is taken in the active sense, it may be discovered by the observation of external facts and by inference from the sensation of conviction which usually accompanies it. Peirce: CP 5.243 Cross-Ref:†† 243. Thus, the arguments in favor of this peculiar power of consciousness disappear, and the presumption is again against such a hypothesis. Moreover, as the immediate objects of any two faculties must be admitted to be different, the facts do not render such a supposition in any degree necessary.

Peirce: CP 5 Book 2 Question 4 QUESTION 4. Whether we have any power of introspection, or whether our whole knowledge of the internal world is derived from the observation of external facts. Peirce: CP 5.244 Cross-Ref:†† 244. It is not intended here to assume the reality of the external world. Only, there is a certain set of facts which are ordinarily regarded as external, while others are regarded as internal. The question is whether the latter are known otherwise than by inference from the former. By introspection, I mean a direct perception of the internal world, but not necessarily a perception of it as internal. Nor do I mean to limit the signification of the word to intuition, but would extend it to any knowledge of the internal world not derived from external observation. Peirce: CP 5.245 Cross-Ref:†† 245. There is one sense in which any perception has an internal object, namely, that every sensation is partly determined by internal conditions. Thus, the sensation of redness is as it is, owing to the constitution of the mind; and in this sense it is a sensation of something internal. Hence, we may derive a knowledge of the mind from a consideration of this sensation, but that knowledge would, in fact, be an inference from redness as a predicate of something external. On the other hand, there are certain other feelings -- the emotions, for example -- which appear to arise in the first place, not as predicates at all, and to be referable to the mind alone. It would seem, then, that by means of these, a knowledge of the mind may be obtained, which is not inferred from any character of outward things. The question is whether this is really so. Peirce: CP 5.246 Cross-Ref:†† 246. Although introspection is not necessarily intuitive, it is not self-evident that we possess this capacity; for we have no intuitive faculty of distinguishing different subjective modes of consciousness. The power, if it exists, must be known by the circumstance that the facts cannot be explained without it. Peirce: CP 5.247 Cross-Ref:†† 247. In reference to the above argument from the emotions, it must be admitted that if a man is angry, his anger implies, in general, no determinate and constant character in its object. But, on the other hand, it can hardly be questioned that there is some relative character in the outward thing which makes him angry, and a little reflection will serve to show that his anger consists in his saying to himself, "this thing is vile, abominable, etc." and that it is rather a mark of returning reason to say, "I am angry." In the same way any emotion is a predication concerning some object, and the chief difference between this and an objective intellectual judgment is that while the latter is relative to human nature or to mind in general, the former is relative to the particular circumstances and disposition of a particular man at a particular time. What is here said of emotions in general, is true in particular of the sense of beauty and of the moral sense. Good and bad are feelings which first arise as predicates, and therefore are either predicates of the not-I, or are determined by previous cognitions (there being no intuitive power of distinguishing subjective elements of consciousness). Peirce: CP 5.248 Cross-Ref:†† 248. It remains, then, only to inquire whether it is necessary to suppose a

particular power of introspection for the sake of accounting for the sense of willing. Now, volition, as distinguished from desire, is nothing but the power of concentrating the attention, of abstracting. Hence, the knowledge of the power of abstracting may be inferred from abstract objects, just as the knowledge of the power of seeing is inferred from colored objects. Peirce: CP 5.249 Cross-Ref:†† 249. It appears, therefore, that there is no reason for supposing a power of introspection; and, consequently, the only way of investigating a psychological question is by inference from external facts.

Peirce: CP 5 Book 2 Question 5 QUESTION 5. Whether we can think without signs. Peirce: CP 5.250 Cross-Ref:†† 250. This is a familiar question, but there is, to this day, no better argument in the affirmative than that thought must precede every sign. This assumes the impossibility of an infinite series. But Achilles, as a fact, will overtake the tortoise. How this happens, is a question not necessary to be answered at present, as long as it certainly does happen. Peirce: CP 5.251 Cross-Ref:†† 251. If we seek the light of external facts, the only cases of thought which we can find are of thought in signs. Plainly, no other thought can be evidenced by external facts. But we have seen that only by external facts can thought be known at all. The only thought, then, which can possibly be cognized is thought in signs. But thought which cannot be cognized does not exist. All thought, therefore, must necessarily be in signs. Peirce: CP 5.252 Cross-Ref:†† 252. A man says to himself, "Aristotle is a man; therefore, he is fallible." Has he not, then, thought what he has not said to himself, that all men are fallible? The answer is, that he has done so, so far as this is said in his therefore. According to this, our question does not relate to fact, but is a mere asking for distinctness of thought. Peirce: CP 5.253 Cross-Ref:†† 253. From the proposition that every thought is a sign, it follows that every thought must address itself to some other, must determine some other, since that is the essence of a sign. This, after all, is but another form of the familiar axiom, that in intuition, i.e., in the immediate present, there is no thought, or, that all which is reflected upon has past. Hinc loquor inde est. That, since any thought, there must have been a thought, has its analogue in the fact that, since any past time, there must have been an infinite series of times. To say, therefore, that thought cannot happen in an instant, but requires a time, is but another way of saying that every thought must be interpreted in another, or that all thought is in signs.

Peirce: CP 5 Book 2 Question 6 QUESTION 6. Whether a sign can have any meaning, if by its definition it is the sign of something absolutely incognizable.

Peirce: CP 5.254 Cross-Ref:†† 254. It would seem that it can, and that universal and hypothetical propositions are instances of it. Thus, the universal proposition, "all ruminants are cloven-hoofed," speaks of a possible infinity of animals, and no matter how many ruminants may have been examined, the possibility must remain that there are others which have not been examined. In the case of a hypothetical proposition, the same thing is still more manifest; for such a proposition speaks not merely of the actual state of things, but of every possible state of things, all of which are not knowable, inasmuch as only one can so much as exist. Peirce: CP 5.255 Cross-Ref:†† 255. On the other hand, all our conceptions are obtained by abstractions and combinations of cognitions first occurring in judgments of experience. Accordingly, there can be no conception of the absolutely incognizable, since nothing of that sort occurs in experience. But the meaning of a term is the conception which it conveys. Hence, a term can have no such meaning. Peirce: CP 5.256 Cross-Ref:†† 256. If it be said that the incognizable is a concept compounded of the concept not and cognizable, it may be replied that not is a mere syncategorematic term and not a concept by itself. Peirce: CP 5.257 Cross-Ref:†† 257. If I think "white," I will not go so far as Berkeley †1 and say that I think of a person seeing, but I will say that what I think is of the nature of a cognition, and so of anything else which can be experienced. Consequently, the highest concept which can be reached by abstractions from judgments of experience -- and therefore, the highest concept which can be reached at all -- is the concept of something of the nature of a cognition. Not, then, or what is other than, if a concept, is a concept of the cognizable. Hence, not-cognizable, if a concept, is a concept of the form "A, not-A," and is, at least, self-contradictory. Thus, ignorance and error can only be conceived as correlative to a real knowledge and truth, which latter are of the nature of cognitions. Over against any cognition, there is an unknown but knowable reality; but over against all possible cognition, there is only the self-contradictory. In short, cognizability (in its widest sense) and being are not merely metaphysically the same, but are synonymous terms. Peirce: CP 5.258 Cross-Ref:†† 258. To the argument from universal and hypothetical propositions, the reply is, that though their truth cannot be cognized with absolute certainty, it may be probably known by induction.

Peirce: CP 5 Book 2 Question 7 QUESTION 7. Whether there is any cognition not determined by a previous cognition. Peirce: CP 5.259 Cross-Ref:†† 259. It would seem that there is or has been; for since we are in possession of cognitions, which are all determined by previous ones, and these by cognitions earlier still, there must have been a first in this series or else our state of cognition at any time is completely determined, according to logical laws, by our state at any previous

time. But there are many facts against the last supposition, and therefore in favor of intuitive cognitions. Peirce: CP 5.260 Cross-Ref:†† 260. On the other hand, since it is impossible to know intuitively that a given cognition is not determined by a previous one, the only way in which this can be known is by hypothetic inference from observed facts. But to adduce the cognition by which a given cognition has been determined is to explain the determinations of that cognition. And it is the only way of explaining them. For something entirely out of consciousness which may be supposed to determine it, can, as such, only be known and only adduced in the determinate cognition in question. So, that to suppose that a cognition is determined solely by something absolutely external, is to suppose its determinations incapable of explanation. Now, this is a hypothesis which is warranted under no circumstances, inasmuch as the only possible justification for a hypothesis is that it explains the facts, and to say that they are explained and at the same time to suppose them inexplicable is self-contradictory. Peirce: CP 5.261 Cross-Ref:†† 261. If it be objected that the peculiar character of red is not determined by any previous cognition, I reply that that character is not a character of red as a cognition; for if there be a man to whom red things look as blue ones do to me and vice versa, that man's eyes teach him the same facts that they would if he were like me. Peirce: CP 5.262 Cross-Ref:†† 262. Moreover, we know of no power by which an intuition could be known. For, as the cognition is beginning, and therefore in a state of change, at only the first instant would it be intuition. And, therefore, the apprehension of it must take place in no time and be an event occupying no time.†P1 Besides, all the cognitive faculties we know of are relative, and consequently their products are relations. But the cognition of a relation is determined by previous cognitions. No cognition not determined by a previous cognition, then, can be known. It does not exist, then, first, because it is absolutely incognizable, and second, because a cognition only exists so far as it is known. Peirce: CP 5.263 Cross-Ref:†† 263. The reply to the argument that there must be a first is as follows: In retracing our way from conclusions to premisses, or from determined cognitions to those which determine them, we finally reach, in all cases, a point beyond which the consciousness in the determined cognition is more lively than in the cognition which determines it. We have a less lively consciousness in the cognition which determines our cognition of the third dimension than in the latter cognition itself; a less lively consciousness in the cognition which determines our cognition of a continuous surface (without a blind spot) than in this latter cognition itself; and a less lively consciousness of the impressions which determine the sensation of tone than of that sensation itself. Indeed, when we get near enough to the external this is the universal rule. Now let any horizontal line represent a cognition, and let the length of the line serve to measure (so to speak) the liveliness of consciousness in that cognition. A point, having no length, will, on this principle, represent an object quite out of consciousness. Let one horizontal line below another represent a cognition which determines the cognition represented by that other and which has the same object as the latter. Let the finite distance between two such lines represent that they are two different cognitions. With this aid to thinking, let us see whether "there must be a

first." Suppose an inverted triangle [Click here to view] to be gradually dipped into water. At any date or instant, the surface of the water makes a horizontal line across that triangle. This line represents a cognition. At a subsequent date, there is a sectional line so made, higher upon the triangle. This represents another cognition of the same object determined by the former, and having a livelier consciousness. The apex of the triangle represents the object external to the mind which determines both these cognitions. The state of the triangle before it reaches the water, represents a state of cognition which contains nothing which determines these subsequent cognitions. To say, then, that if there be a state of cognition by which all subsequent cognitions of a certain object are not determined, there must subsequently be some cognition of that object not determined by previous cognitions of the same object, is to say that when that triangle is dipped into the water there must be a sectional line made by the surface of the water lower than which no surface line had been made in that way. But draw the horizontal line where you will, as many horizontal lines as you please can be assigned at finite distances below it and below one another. For any such section is at some distance above the apex, otherwise it is not a line. Let this distance be a. Then there have been similar sections at the distances 1/2a, 1/4a, 1/8a, 1/16a, above the apex, and so on as far as you please. So that it is not true that there must be a first. Explicate the logical difficulties of this paradox (they are identical with those of the Achilles) in whatever way you may. I am content with the result, as long as your principles are fully applied to the particular case of cognitions determining one another. Deny motion, if it seems proper to do so; only then deny the process of determination of one cognition by another. Say that instants and lines are fictions; only say, also, that states of cognition and judgments are fictions. The point here insisted on is not this or that logical solution of the difficulty, but merely that cognition arises by a process of beginning, as any other change comes to pass. Peirce: CP 5.263 Cross-Ref:†† In a subsequent paper, I shall trace the consequences of these principles, in reference to the questions of reality, of individuality, and of the validity of the laws of logic.

Peirce: CP 5.264 Cross-Ref:†† II SOME CONSEQUENCES OF FOUR INCAPACITIES†1

§1. THE SPIRIT OF CARTESIANISME

264. Descartes is the father of modern philosophy, and the spirit of Cartesianism -- that which principally distinguishes it from the scholasticism which it displaced -- may be compendiously stated as follows:

Peirce: CP 5.264 Cross-Ref:†† 1. It teaches that philosophy must begin with universal doubt; whereas scholasticism had never questioned fundamentals. Peirce: CP 5.264 Cross-Ref:†† 2. It teaches that the ultimate test of certainty is to be found in the individual consciousness; whereas scholasticism had rested on the testimony of sages and of the Catholic Church. Peirce: CP 5.264 Cross-Ref:†† 3. The multiform argumentation of the middle ages is replaced by a single thread of inference depending often upon inconspicuous premisses. Peirce: CP 5.264 Cross-Ref:†† 4. Scholasticism had its mysteries of faith, but undertook to explain all created things. But there are many facts which Cartesianism not only does not explain but renders absolutely inexplicable, unless to say that "God makes them so" is to be regarded as an explanation. Peirce: CP 5.265 Cross-Ref:†† 265. In some, or all of these respects, most modern philosophers have been, in effect, Cartesians. Now without wishing to return to scholasticism, it seems to me that modern science and modern logic require us to stand upon a very different platform from this. Peirce: CP 5.265 Cross-Ref:†† 1. We cannot begin with complete doubt. We must begin with all the prejudices which we actually have when we enter upon the study of philosophy. These prejudices are not to be dispelled by a maxim, for they are things which it does not occur to us can be questioned. Hence this initial skepticism will be a mere self-deception, and not real doubt; and no one who follows the Cartesian method will ever be satisfied until he has formally recovered all those beliefs which in form he has given up. It is, therefore, as useless a preliminary as going to the North Pole would be in order to get to Constantinople by coming down regularly upon a meridian. A person may, it is true, in the course of his studies, find reason to doubt what he began by believing; but in that case he doubts because he has a positive reason for it, and not on account of the Cartesian maxim. Let us not pretend to doubt in philosophy what we do not doubt in our hearts. Peirce: CP 5.265 Cross-Ref:†† 2. The same formalism appears in the Cartesian criterion, which amounts to this: "Whatever I am clearly convinced of, is true." If I were really convinced, I should have done with reasoning and should require no test of certainty. But thus to make single individuals absolute judges of truth is most pernicious. The result is that metaphysicians will all agree that metaphysics has reached a pitch of certainty far beyond that of the physical sciences; -- only they can agree upon nothing else. In sciences in which men come to agreement, when a theory has been broached it is considered to be on probation until this agreement is reached. After it is reached, the question of certainty becomes an idle one, because there is no one left who doubts it. We individually cannot reasonably hope to attain the ultimate philosophy which we pursue; we can only seek it, therefore, for the community of philosophers. Hence, if disciplined and candid minds carefully examine a theory and refuse to accept it, this ought to create doubts in the mind of the author of the theory himself.

Peirce: CP 5.265 Cross-Ref:†† 3. Philosophy ought to imitate the successful sciences in its methods, so far as to proceed only from tangible premisses which can be subjected to careful scrutiny, and to trust rather to the multitude and variety of its arguments than to the conclusiveness of any one. Its reasoning should not form a chain which is no stronger than its weakest link, but a cable whose fibers may be ever so slender, provided they are sufficiently numerous and intimately connected. Peirce: CP 5.265 Cross-Ref:†† 4. Every unidealistic philosophy supposes some absolutely inexplicable, unanalyzable ultimate; in short, something resulting from mediation itself not susceptible of mediation. Now that anything is thus inexplicable can only be known by reasoning from signs. But the only justification of an inference from signs is that the conclusion explains the fact. To suppose the fact absolutely inexplicable, is not to explain it, and hence this supposition is never allowable. Peirce: CP 5.265 Cross-Ref:†† In the last number of this journal will be found a piece entitled "Questions concerning certain Faculties claimed for Man," [Paper No. I] which has been written in this spirit of opposition to Cartesianism. That criticism of certain faculties resulted in four denials, which for convenience may here be repeated: Peirce: CP 5.265 Cross-Ref:†† 1. We have no power of Introspection, but all knowledge of the internal world is derived by hypothetical reasoning from our knowledge of external facts. Peirce: CP 5.265 Cross-Ref:†† 2. We have no power of Intuition, but every cognition is determined logically by previous cognitions. Peirce: CP 5.265 Cross-Ref:†† 3. We have no power of thinking without signs. Peirce: CP 5.265 Cross-Ref:†† 4. We have no conception of the absolutely incognizable. These propositions cannot be regarded as certain; and, in order to bring them to a further test, it is now proposed to trace them out to their consequences. We may first consider the first alone; then trace the consequences of the first and second; then see what else will result from assuming the third also; and, finally, add the fourth to our hypothetical premisses.

Peirce: CP 5.266 Cross-Ref:†† §2. MENTAL ACTIONE

266. In accepting the first proposition, we must put aside all prejudices derived from a philosophy which bases our knowledge of the external world on our self-consciousness. We can admit no statement concerning what passes within us except as a hypothesis necessary to explain what takes place in what we commonly call the external world. Moreover when we have upon such grounds assumed one faculty or mode of action of the mind, we cannot, of course, adopt any other

hypothesis for the purpose of explaining any fact which can be explained by our first supposition, but must carry the latter as far as it will go. In other words, we must, as far as we can do so without additional hypotheses, reduce all kinds of mental action to one general type. Peirce: CP 5.267 Cross-Ref:†† 267. The class of modifications of consciousness with which we must commence our inquiry must be one whose existence is indubitable, and whose laws are best known, and, therefore (since this knowledge comes from the outside), which most closely follows external facts; that is, it must be some kind of cognition. Here we may hypothetically admit the second proposition of the former paper, according to which there is no absolutely first cognition of any object, but cognition arises by a continuous process. We must begin, then, with a process of cognition, and with that process whose laws are best understood and most closely follow external facts. This is no other than the process of valid inference, which proceeds from its premiss, A, to its conclusion, B, only if, as a matter of fact, such a proposition as B is always or usually true when such a proposition as A is true. It is a consequence, then, of the first two principles whose results we are to trace out, that we must, as far as we can, without any other supposition than that the mind reasons, reduce all mental action to the formula of valid reasoning. Peirce: CP 5.268 Cross-Ref:†† 268. But does the mind in fact go through the syllogistic process? It is certainly very doubtful whether a conclusion -- as something existing in the mind independently, like an image -- suddenly displaces two premisses existing in the mind in a similar way. But it is a matter of constant experience, that if a man is made to believe in the premisses, in the sense that he will act from them and will say that they are true, under favorable conditions he will also be ready to act from the conclusion and to say that that is true. Something, therefore, takes place within the organism which is equivalent to the syllogistic process. Peirce: CP 5.269 Cross-Ref:†† 269. A valid inference is either complete or incomplete.†1 An incomplete inference is one whose validity depends upon some matter of fact not contained in the premisses. This implied fact might have been stated as a premiss, and its relation to the conclusion is the same whether it is explicitly posited or not, since it is at least virtually taken for granted; so that every valid incomplete argument is virtually complete. Complete arguments are divided into simple and complex.†2 A complex argument is one which from three or more premisses concludes what might have been concluded by successive steps in reasonings each of which is simple. Thus, a complex inference comes to the same thing in the end as a succession of simple inferences. Peirce: CP 5.270 Cross-Ref:†† 270. A complete, simple, and valid argument, or syllogism, is either apodictic or probable.†1 An apodictic or deductive syllogism is one whose validity depends unconditionally upon the relation of the fact inferred to the facts posited in the premisses. A syllogism whose validity should depend not merely upon its premisses, but upon the existence of some other knowledge, would be impossible; for either this other knowledge would be posited, in which case it would be a part of the premisses, or it would be implicitly assumed, in which case the inference would be incomplete. But a syllogism whose validity depends partly upon the non-existence of some other knowledge, is a probable syllogism.

Peirce: CP 5.271 Cross-Ref:†† 271. A few examples will render this plain. The two following arguments are apodictic or deductive: Peirce: CP 5.271 Cross-Ref:†† 1. No series of days of which the first and last are different days of the week exceeds by one a multiple of seven days; now the first and last days of any leap-year are different days of the week, and therefore no leap-year consists of a number of days one greater than a multiple of seven. Peirce: CP 5.271 Cross-Ref:†† 2. Among the vowels there are no double letters; but one of the double letters (w) is compounded of two vowels: hence, a letter compounded of two vowels is not necessarily itself a vowel. Peirce: CP 5.271 Cross-Ref:†† In both these cases, it is plain that as long as the premisses are true, however other facts may be, the conclusions will be true. On the other hand, suppose that we reason as follows: "A certain man had the Asiatic cholera. He was in a state of collapse, livid, quite cold, and without perceptible pulse. He was bled copiously. During the process he came out of collapse, and the next morning was well enough to be about. Therefore, bleeding tends to cure the cholera." This is a fair probable inference, provided that the premisses represent our whole knowledge of the matter. But if we knew, for example, that recoveries from cholera were apt to be sudden, and that the physician who had reported this case had known of a hundred other trials of the remedy without communicating the result, then the inference would lose all its validity. Peirce: CP 5.272 Cross-Ref:†† 272. The absence of knowledge which is essential to the validity of any probable argument relates to some question which is determined by the argument itself. This question, like every other, is whether certain objects have certain characters. Hence, the absence of knowledge is either whether besides the objects which, according to the premisses, possess certain characters, any other objects possess them; or, whether besides the characters which, according to the premisses, belong to certain objects, any other characters not necessarily involved in these belong to the same objects. In the former case, the reasoning proceeds as though all the objects which have certain characters were known, and this is induction; in the latter case, the inference proceeds as though all the characters requisite to the determination of a certain object or class were known, and this is hypothesis. This distinction, also, may be made more plain by examples. Peirce: CP 5.273 Cross-Ref:†† 273. Suppose we count the number of occurrences of the different letters in a certain English book, which we may call A. Of course, every new letter which we add to our count will alter the relative number of occurrences of the different letters; but as we proceed with our counting, this change will be less and less. Suppose that we find that as we increase the number of letters counted, the relative number of e's approaches nearly 11 1/4 per cent. of the whole, that of the t's 8 1/2 per cent., that of the a's 8 per cent., that of the s's 7 1/2 per cent., etc. Suppose we repeat the same observations with half a dozen other English writings (which we may designate as B, C, D, E, F, G) with the like result. Then we may infer that in every English writing of

some length, the different letters occur with nearly those relative frequencies. Peirce: CP 5.273 Cross-Ref:†† Now this argument depends for its validity upon our not knowing the proportion of letters in any English writing besides A, B, C, D, E, F and G. For if we know it in respect to H, and it is not nearly the same as in the others, our conclusion is destroyed at once; if it is the same, then the legitimate inference is from A, B, C, D, E, F, G and H, and not from the first seven alone. This, therefore, is an induction. Peirce: CP 5.273 Cross-Ref:†† Suppose, next, that a piece of writing in cipher is presented to us, without the key. Suppose we find that it contains something less than 26 characters, one of which occurs about 11 per cent. of all the times, another 8 1/2 per cent., another 8 per cent., and another 7 1/2 per cent. Suppose that when we substitute for these e, t, a and s, respectively, we are able to see how single letters may be substituted for each of the other characters so as to make sense in English, provided, however, that we allow the spelling to be wrong in some cases. If the writing is of any considerable length, we may infer with great probability that this is the meaning of the cipher. Peirce: CP 5.273 Cross-Ref:†† The validity of this argument depends upon there being no other known characters of the writing in cipher which would have any weight in the matter; for if there are -- if we know, for example, whether or not there is any other solution of it -this must be allowed its effect in supporting or weakening the conclusion. This, then, is hypothesis. Peirce: CP 5.274 Cross-Ref:†† 274. All valid reasoning is either deductive, inductive, or hypothetic; or else it combines two or more of these characters. Deduction is pretty well treated in most logical textbooks; but it will be necessary to say a few words about induction and hypothesis in order to render what follows more intelligible. Peirce: CP 5.275 Cross-Ref:†† 275. Induction may be defined as an argument which proceeds upon the assumption that all the members of a class or aggregate have all the characters which are common to all those members of this class concerning which it is known, whether they have these characters or not; or, in other words, which assumes that that is true of a whole collection which is true of a number of instances taken from it at random. This might be called statistical argument. In the long run, it must generally afford pretty correct conclusions from true premisses. If we have a bag of beans partly black and partly white, by counting the relative proportions of the two colors in several different handfuls, we can approximate more or less to the relative proportions in the whole bag, since a sufficient number of handfuls would constitute all the beans in the bag. The central characteristic and key to induction is, that by taking the conclusion so reached as major premiss of a syllogism, and the proposition stating that such and such objects are taken from the class in question as the minor premiss, the other premiss of the induction will follow from them deductively.†1 Thus, in the above example we concluded that all books in English have about 11 1/4 per cent. of their letters e's. From that as major premiss, together with the proposition that A, B, C, D, E, F and G are books in English, it follows deductively that A, B, C, D, E, F and G have about 11 1/4 per cent. of their letters e's. Accordingly, induction has been defined by Aristotle †2 as the inference of the major premiss of a syllogism from its minor premiss and conclusion. The function of an induction is to substitute for a

series of many subjects, a single one which embraces them and an indefinite number of others. Thus it is a species of "reduction of the manifold to unity." Peirce: CP 5.276 Cross-Ref:†† 276. Hypothesis may be defined as an argument which proceeds upon the assumption that a character which is known necessarily to involve a certain number of others, may be probably predicated of any object which has all the characters which this character is known to involve. Just as induction may be regarded as the inference of the major premiss of a syllogism, so hypothesis may be regarded as the inference of the minor premiss, from the other two propositions. Thus, the example taken above consists of two such inferences of the minor premisses of the following syllogisms: Peirce: CP 5.276 Cross-Ref:†† 1. Every English writing of some length in which such and such characters denote e, t, a, and s, has about 11 1/4 per cent. of the first sort of marks, 8 1/2 of the second, 8 of the third, and 7 1/2 of the fourth. Peirce: CP 5.276 Cross-Ref:†† This secret writing is an English writing of some length, in which such and such characters denote e, t, a, and s, respectively:

.·. This secret writing has about 11 1/4 per cent. of its characters of the first kind, 8 1/2 of the second, 8 of the third, and 7 1/2 of the fourth. Peirce: CP 5.276 Cross-Ref:†† 2. A passage written with such an alphabet makes sense when such and such letters are severally substituted for such and such characters. This secret writing is written with such an alphabet. .·. This secret writing makes sense when such and such substitutions are made. Peirce: CP 5.276 Cross-Ref:†† The function of hypothesis is to substitute for a great series of predicates forming no unity in themselves, a single one (or small number) which involves them all, together (perhaps) with an indefinite number of others. It is, therefore, also a reduction of a manifold to unity.†P1 Every deductive syllogism may be put into the form

If A, then B; But A: .·. B.

And as the minor premiss in this form appears as antecedent or reason of a hypothetical proposition, hypothetic inference may be called reasoning from consequent to antecedent.

Peirce: CP 5.277 Cross-Ref:†† 277. The argument from analogy, which a popular writer †1 upon logic calls reasoning from particulars to particulars, derives its validity from its combining the characters of induction and hypothesis, being analyzable either into a deduction or an induction, or a deduction and a hypothesis.†1 Peirce: CP 5.278 Cross-Ref:†† 278. But though inference is thus of three essentially different species, it also belongs to one genus. We have seen that no conclusion can be legitimately derived which could not have been reached by successions of arguments having two premisses each, and implying no fact not asserted. Peirce: CP 5.279 Cross-Ref:†† 279. Either of these premisses is a proposition asserting that certain objects have certain characters. Every term of such a proposition stands either for certain objects or for certain characters. The conclusion may be regarded as a proposition substituted in place of either premiss, the substitution being justified by the fact stated in the other premiss. The conclusion is accordingly derived from either premiss by substituting either a new subject for the subject of the premiss, or a new predicate for the predicate of the premiss, or by both substitutions. Now the substitution of one term for another can be justified only so far as the term substituted represents only what is represented in the term replaced. If, therefore, the conclusion be denoted by the formula,

S is P;

and this conclusion be derived, by a change of subject, from a premiss which may on this account be expressed by the formula,

M is P,

then the other premiss must assert that whatever thing is represented by S is represented by M, or that

Every S is an M;

while, if the conclusion, S is P, is derived from either premiss by a change of predicate, that premiss may be written

S is M;

and the other premiss must assert that whatever characters are implied in P are implied in M, or that

Whatever is M is P.

In either case, therefore, the syllogism must be capable of expression in the form,

S is M; M is P: .·. S is P.

Peirce: CP 5.279 Cross-Ref:†† Finally, if the conclusion differs from either of its premisses, both in subject and predicate, the form of statement of conclusion and premiss may be so altered that they shall have a common term. This can always be done, for if P is the premiss and C the conclusion, they may be stated thus:

The state of things represented in P is real, and The state of things represented in C is real.

Peirce: CP 5.279 Cross-Ref:†† In this case the other premiss must in some form virtually assert that every state of things such as is represented by C is the state of things represented in P. Peirce: CP 5.279 Cross-Ref:†† All valid reasoning, therefore, is of one general form; and in seeking to reduce all mental action to the formulæ of valid inference, we seek to reduce it to one single type. Peirce: CP 5.280 Cross-Ref:†† 280. An apparent obstacle to the reduction of all mental action to the type of valid inferences is the existence of fallacious reasoning. Every argument implies the truth of a general principle of inferential procedure (whether involving some matter of fact concerning the subject of argument, or merely a maxim relating to a system of signs), according to which it is a valid argument. If this principle is false, the argument is a fallacy; but neither a valid argument from false premisses, nor an exceedingly weak, but not altogether illegitimate, induction or hypothesis, however its force may be over-estimated, however false its conclusion, is a fallacy. Peirce: CP 5.281 Cross-Ref:†† 281. Now words, taken just as they stand, if in the form of an argument, thereby do imply whatever fact may be necessary to make the argument conclusive; so that to the formal logician, who has to do only with the meaning of the words

according to the proper principles of interpretation, and not with the intention of the speaker as guessed at from other indications, the only fallacies should be such as are simply absurd and contradictory, either because their conclusions are absolutely inconsistent with their premisses, or because they connect propositions by a species of illative conjunction, by which they cannot under any circumstances be validly connected. Peirce: CP 5.282 Cross-Ref:†† 282. But to the psychologist an argument is valid only if the premisses from which the mental conclusion is derived would be sufficient, if true, to justify it, either by themselves, or by the aid of other propositions which had previously been held for true. But it is easy to show that all inferences made by man, which are not valid in this sense, belong to four classes, viz.: 1. Those whose premisses are false; 2. Those which have some little force, though only a little; 3. Those which result from confusion of one proposition with another; 4. Those which result from the indistinct apprehension, wrong application, or falsity, of a rule of inference. For, if a man were to commit a fallacy not of either of these classes, he would, from true premisses conceived with perfect distinctness, without being led astray by any prejudice or other judgment serving as a rule of inference, draw a conclusion which had really not the least relevancy. If this could happen, calm consideration and care could be of little use in thinking, for caution only serves to insure our taking all the facts into account, and to make those which we do take account of, distinct; nor can coolness do anything more than to enable us to be cautious, and also to prevent our being affected by a passion in inferring that to be true which we wish were true, or which we fear may be true, or in following some other wrong rule of inference. But experience shows that the calm and careful consideration of the same distinctly conceived premisses (including prejudices) will insure the pronouncement of the same judgment by all men. Now if a fallacy belongs to the first of these four classes and its premisses are false, it is to be presumed that the procedure of the mind from these premisses to the conclusion is either correct, or errs in one of the other three ways; for it cannot be supposed that the mere falsity of the premisses should affect the procedure of reason when that falsity is not known to reason. If the fallacy belongs to the second class and has some force, however little, it is a legitimate probable argument, and belongs to the type of valid inference. If it is of the third class and results from the confusion of one proposition with another, this confusion must be owing to a resemblance between the two propositions; that is to say, the person reasoning, seeing that one proposition has some of the characters which belong to the other, concludes that it has all the essential characters of the other, and is equivalent to it. Now this is a hypothetic inference, which though it may be weak, and though its conclusion happens to be false, belongs to the type of valid inferences; and, therefore, as the nodus of the fallacy lies in this confusion, the procedure of the mind in these fallacies of the third class conforms to the formula of valid inference. If the fallacy belongs to the fourth class, it either results from wrongly applying or misapprehending a rule of inference, and so is a fallacy of confusion, or it results from adopting a wrong rule of inference. In this latter case, this rule is in fact taken as a premiss, and therefore the false conclusion is owing merely to the falsity of a premiss. In every fallacy, therefore, possible to the mind of man, the procedure of the mind conforms to the formula of valid inference.

Peirce: CP 5.283 Cross-Ref:†† §3. THOUGHT-SIGNSE

283. The third principle whose consequences we have to deduce is, that, whenever we think, we have present to the consciousness some feeling, image, conception, or other representation, which serves as a sign. But it follows from our own existence (which is proved by the occurrence of ignorance and error †1) that everything which is present to us is a phenomenal manifestation of ourselves. This does not prevent its being a phenomenon of something without us, just as a rainbow is at once a manifestation both of the sun and of the rain. When we think, then, we ourselves, as we are at that moment, appear as a sign. Now a sign has, as such, three references: first, it is a sign to some thought which interprets it; second, it is a sign for some object to which in that thought it is equivalent; third, it is a sign, in some respect or quality, which brings it into connection with its object. Let us ask what the three correlates are to which a thought-sign refers. Peirce: CP 5.284 Cross-Ref:†† 284. (1) When we think, to what thought does that thought-sign which is ourself address itself? It may, through the medium of outward expression, which it reaches perhaps only after considerable internal development, come to address itself to thought of another person. But whether this happens or not, it is always interpreted by a subsequent thought of our own. If, after any thought, the current of ideas flows on freely, it follows the law of mental association. In that case, each former thought suggests something to the thought which follows it, i.e., is the sign of something to this latter. Our train of thought may, it is true, be interrupted. But we must remember that, in addition to the principal element of thought at any moment, there are a hundred things in our mind to which but a small fraction of attention or consciousness is conceded. It does not, therefore, follow, because a new constituent of thought gets the uppermost that the train of thought which it displaces is broken off altogether. On the contrary, from our second principle, that there is no intuition or cognition not determined by previous cognitions, it follows that the striking in of a new experience is never an instantaneous affair, but is an event occupying time, and coming to pass by a continuous process. Its prominence in consciousness, therefore, must probably be the consummation of a growing process; and if so, there is no sufficient cause for the thought which had been the leading one just before, to cease abruptly and instantaneously. But if a train of thought ceases by gradually dying out, it freely follows its own law of association as long as it lasts, and there is no moment at which there is a thought belonging to this series, subsequently to which there is not a thought which interprets or repeats it. There is no exception, therefore, to the law that every thought-sign is translated or interpreted in a subsequent one, unless it be that all thought comes to an abrupt and final end in death. Peirce: CP 5.285 Cross-Ref:†† 285. (2) The next question is: For what does the thought-sign stand -- what does it name -- what is its suppositum? The outward thing, undoubtedly, when a real outward thing is thought of. But still, as the thought is determined by a previous thought of the same object, it only refers to the thing through denoting this previous thought. Let us suppose, for example, that Toussaint is thought of, and first thought of as a negro, but not distinctly as a man. If this distinctness is afterwards added, it is through the thought that a negro is a man; that is to say, the subsequent thought, man, refers to the outward thing by being predicated of that previous thought, negro, which has been had of that thing. If we afterwards think of Toussaint as a general, then we think that this negro, this man, was a general. And so in every case the

subsequent thought denotes what was thought in the previous thought. Peirce: CP 5.286 Cross-Ref:†† 286. (3) The thought-sign stands for its object in the respect which is thought; that is to say, this respect is the immediate object of consciousness in the thought, or, in other words, it is the thought itself, or at least what the thought is thought to be in the subsequent thought to which it is a sign. Peirce: CP 5.287 Cross-Ref:†† 287. We must now consider two other properties of signs which are of great importance in the theory of cognition. Since a sign is not identical with the thing signified, but differs from the latter in some respects, it must plainly have some characters which belong to it in itself, and have nothing to do with its representative function. These I call the material qualities of the sign. As examples of such qualities, take in the word "man," its consisting of three letters -- in a picture, its being flat and without relief. In the second place, a sign must be capable of being connected (not in the reason but really) with another sign of the same object, or with the object itself. Thus, words would be of no value at all unless they could be connected into sentences by means of a real copula which joins signs of the same thing. The usefulness of some signs -- as a weathercock, a tally, etc. -- consists wholly in their being really connected with the very things they signify. In the case of a picture such a connection is not evident, but it exists in the power of association which connects the picture with the brain-sign which labels it. This real, physical connection of a sign with its object, either immediately or by its connection with another sign, I call the pure demonstrative application of the sign. Now the representative function of a sign lies neither in its material quality nor in its pure demonstrative application; because it is something which the sign is, not in itself or in a real relation to its object, but which it is to a thought, while both of the characters just defined belong to the sign independently of its addressing any thought. And yet if I take all the things which have certain qualities and physically connect them with another series of things, each to each, they become fit to be signs. If they are not regarded as such they are not actually signs, but they are so in the same sense, for example, in which an unseen flower can be said to be red, this being also a term relative to a mental affection. Peirce: CP 5.288 Cross-Ref:†† 288. Consider a state of mind which is a conception. It is a conception by virtue of having a meaning, a logical comprehension; and if it is applicable to any object, it is because that object has the characters contained in the comprehension of this conception. Now the logical comprehension of a thought is usually said to consist of the thoughts contained in it; but thoughts are events, acts of the mind. Two thoughts are two events separated in time, and one cannot literally be contained in the other. It may be said that all thoughts exactly similar are regarded as one; and that to say that one thought contains another, means that it contains one exactly similar to that other. But how can two thoughts be similar? Two objects can only be regarded as similar if they are compared and brought together in the mind. Thoughts have no existence except in the mind; only as they are regarded do they exist. Hence, two thoughts cannot be similar unless they are brought together in the mind. But, as to their existence, two thoughts are separated by an interval of time. We are too apt to imagine that we can frame a thought similar to a past thought, by matching it with the latter, as though this past thought were still present to us. But it is plain that the knowledge that one thought is similar to or in any way truly representative of another, cannot be derived from immediate perception, but must be an hypothesis

(unquestionably fully justifiable by facts), and that therefore the formation of such a representing thought must be dependent upon a real effective force behind consciousness, and not merely upon a mental comparison. What we must mean, therefore, by saying that one concept is contained in another, is that we normally represent one to be in the other; that is, that we form a particular kind of judgment,†P1 of which the subject signifies one concept and the predicate the other. Peirce: CP 5.289 Cross-Ref:†† 289. No thought in itself, then, no feeling in itself, contains any others, but is absolutely simple and unanalyzable; and to say that it is composed of other thoughts and feelings, is like saying that a movement upon a straight line is composed of the two movements of which it is the resultant; that is to say, it is a metaphor, or fiction, parallel to the truth. Every thought, however artificial and complex, is, so far as it is immediately present, a mere sensation without parts, and therefore, in itself, without similarity to any other, but incomparable with any other and absolutely sui generis.†P2 Whatever is wholly incomparable with anything else is wholly inexplicable, because explanation consists in bringing things under general laws or under natural classes. Hence every thought, in so far as it is a feeling of a peculiar sort, is simply an ultimate, inexplicable fact. Yet this does not conflict with my postulate that that fact should be allowed to stand as inexplicable; for, on the one hand, we never can think, "This is present to me," since, before we have time to make the reflection, the sensation is past, and, on the other hand, when once past, we can never bring back the quality of the feeling as it was in and for itself, or know what it was like in itself, or even discover the existence of this quality except by a corollary from our general theory of ourselves, and then not in its idiosyncrasy, but only as something present. But, as something present, feelings are all alike and require no explanation, since they contain only what is universal. So that nothing which we can truly predicate of feelings is left inexplicable, but only something which we cannot reflectively know. So that we do not fall into the contradiction of making the Mediate immediable. Finally, no present actual thought (which is a mere feeling) has any meaning, any intellectual value; for this lies not in what is actually thought, but in what this thought may be connected with in representation by subsequent thoughts; so that the meaning of a thought is altogether something virtual.†1 It may be objected, that if no thought has any meaning, all thought is without meaning. But this is a fallacy similar to saying, that, if in no one of the successive spaces which a body fills there is room for motion, there is no room for motion throughout the whole. At no one instant in my state of mind is there cognition or representation, but in the relation of my states of mind at different instants there is.†P1 In short, the Immediate (and therefore in itself unsusceptible of mediation -- the Unanalyzable, the Inexplicable, the Unintellectual) runs in a continuous stream through our lives; it is the sum total of consciousness, whose mediation, which is the continuity of it, is brought about by a real effective force behind consciousness. Peirce: CP 5.290 Cross-Ref:†† 290. Thus, we have in thought three elements: first, the representative function which makes it a representation; second, the pure denotative application, or real connection, which brings one thought into relation with another; and third, the material quality, or how it feels, which gives thought its quality.†P1 Peirce: CP 5.291 Cross-Ref:†† 291. That a sensation is not necessarily an intuition, or first impression of sense, is very evident in the case of the sense of beauty; and has been shown [in 222],

in the case of sound. When the sensation beautiful is determined by previous cognitions, it always arises as a predicate; that is, we think that something is beautiful. Whenever a sensation thus arises in consequence of others, induction shows that those others are more or less complicated. Thus, the sensation of a particular kind of sound arises in consequence of impressions upon the various nerves of the ear being combined in a particular way, and following one another with a certain rapidity. A sensation of color depends upon impressions upon the eye following one another in a regular manner, and with a certain rapidity. The sensation of beauty arises upon a manifold of other impressions. And this will be found to hold good in all cases. Secondly, all these sensations are in themselves simple, or more so than the sensations which give rise to them. Accordingly, a sensation is a simple predicate taken in place of a complex predicate; in other words, it fulfills the function of an hypothesis. But the general principle that every thing to which such and such a sensation belongs, has such and such a complicated series of predicates, is not one determined by reason (as we have seen), but is of an arbitrary nature. Hence, the class of hypothetic inferences which the arising of a sensation resembles, is that of reasoning from definition to definitum, in which the major premiss is of an arbitrary nature. Only in this mode of reasoning, this premiss is determined by the conventions of language, and expresses the occasion upon which a word is to be used; and in the formation of a sensation, it is determined by the constitution of our nature, and expresses the occasions upon which sensation, or a natural mental sign, arises. Thus, the sensation, so far as it represents something, is determined, according to a logical law, by previous cognitions; that is to say, these cognitions determine that there shall be a sensation. But so far as the sensation is a mere feeling of a particular sort, it is determined only by an inexplicable, occult power; and so far, it is not a representation, but only the material quality of a representation. For just as in reasoning from definition to definitum, it is indifferent to the logician how the defined word shall sound, or how many letters it shall contain, so in the case of this constitutional word, it is not determined by an inward law how it shall feel in itself. A feeling, therefore, as a feeling, is merely the material quality of a mental sign. Peirce: CP 5.292 Cross-Ref:†† 292. But there is no feeling which is not also a representation, a predicate of something determined logically by the feelings which precede it. For if there are any such feelings not predicates, they are the emotions. Now every emotion has a subject. If a man is angry, he is saying to himself that this or that is vile and outrageous. If he is in joy, he is saying "this is delicious." If he is wondering, he is saying "this is strange." In short, whenever a man feels, he is thinking of something. Even those passions which have no definite object -- as melancholy -- only come to consciousness through tinging the objects of thought. That which makes us look upon the emotions more as affections of self than other cognitions, is that we have found them more dependent upon our accidental situation at the moment than other cognitions; but that is only to say that they are cognitions too narrow to be useful. The emotions, as a little observation will show, arise when our attention is strongly drawn to complex and inconceivable circumstances. Fear arises when we cannot predict our fate; joy, in the case of certain indescribable and peculiarly complex sensations. If there are some indications that something greatly for my interest, and which I have anticipated would happen, may not happen; and if, after weighing probabilities, and inventing safeguards, and straining for further information, I find myself unable to come to any fixed conclusion in reference to the future, in the place of that intellectual hypothetic inference which I seek, the feeling of anxiety arises. When something happens for which I cannot account, I wonder. When I endeavor to realize to myself

what I never can do, a pleasure in the future, I hope. "I do not understand you," is the phrase of an angry man. The indescribable, the ineffable, the incomprehensible, commonly excite emotion; but nothing is so chilling as a scientific explanation. Thus an emotion is always a simple predicate substituted by an operation of the mind for a highly complicated predicate.†1 Now if we consider that a very complex predicate demands explanation by means of an hypothesis, that that hypothesis must be a simpler predicate substituted for that complex one; and that when we have an emotion, an hypothesis, strictly speaking, is hardly possible -- the analogy of the parts played by emotion and hypothesis is very striking. There is, it is true, this difference between an emotion and an intellectual hypothesis, that we have reason to say in the case of the latter, that to whatever the simple hypothetic predicate can be applied, of that the complex predicate is true; whereas, in the case of an emotion this is a proposition for which no reason can be given, but which is determined merely by our emotional constitution. But this corresponds precisely to the difference between hypothesis and reasoning from definition to definitum, and thus it would appear that emotion is nothing but sensation. There appears to be a difference, however, between emotion and sensation, and I would state it as follows: Peirce: CP 5.293 Cross-Ref:†† 293. There is some reason to think that, corresponding to every feeling within us, some motion takes place in our bodies. This property of the thought-sign, since it has no rational dependence upon the meaning of the sign, may be compared with what I have called the material quality of the sign; but it differs from the latter inasmuch as it is not essentially necessary that it should be felt in order that there should be any thought-sign. In the case of a sensation, the manifold of impressions which precede and determine it are not of a kind, the bodily motion corresponding to which comes from any large ganglion or from the brain, and probably for this reason the sensation produces no great commotion in the bodily organism; and the sensation itself is not a thought which has a very strong influence upon the current of thought except by virtue of the information it may serve to afford. An emotion, on the other hand, comes much later in the development of thought -- I mean, further from the first beginning of the cognition of its object -- and the thoughts which determine it already have motions corresponding to them in the brain, or the chief ganglion; consequently, it produces large movements in the body, and independently of its representative value, strongly affects the current of thought. The animal motions to which I allude, are, in the first place and obviously, blushing, blenching, staring, smiling, scowling, pouting, laughing, weeping, sobbing, wriggling, flinching, trembling, being petrified, sighing, sniffing, shrugging, groaning, heartsinking, trepidation, swelling of the heart, etc., etc. To these may, perhaps, be added, in the second place, other more complicated actions, which nevertheless spring from a direct impulse and not from deliberation. Peirce: CP 5.294 Cross-Ref:†† 294. That which distinguishes both sensations proper and emotions from the feeling of a thought, is that in the case of the two former the material quality is made prominent, because the thought has no relation of reason to the thoughts which determine it, which exists in the last case and detracts from the attention given to the mere feeling. By there being no relation of reason to the determining thoughts, I mean that there is nothing in the content of the thought which explains why it should arise only on occasion of these determining thoughts. If there is such a relation of reason, if the thought is essentially limited in its application to these objects, then the thought comprehends a thought other than itself; in other words, it is then a complex thought. An incomplex thought can, therefore, be nothing but a sensation or emotion, having

no rational character. This is very different from the ordinary doctrine, according to which the very highest and most metaphysical conceptions are absolutely simple. I shall be asked how such a conception of a being is to be analyzed, or whether I can ever define one, two, and three, without a diallelon. Now I shall admit at once that neither of these conceptions can be separated into two others higher than itself; and in that sense, therefore, I fully admit that certain very metaphysical and eminently intellectual notions are absolutely simple. But though these concepts cannot be defined by genus and difference, there is another way in which they can be defined. All determination is by negation; we can first recognize any character only by putting an object which possesses it into comparison with an object which possesses it not. A conception, therefore, which was quite universal in every respect would be unrecognizable and impossible. We do not obtain the conception of Being, in the sense implied in the copula, by observing that all the things which we can think of have something in common, for there is no such thing to be observed. We get it by reflecting upon signs -- words or thoughts; we observe that different predicates may be attached to the same subject, and that each makes some conception applicable to the subject; then we imagine that a subject has something true of it merely because a predicate (no matter what) is attached to it -- and that we call Being. The conception of being is, therefore, a conception about a sign -- a thought, or word; and since it is not applicable to every sign, it is not primarily universal, although it is so in its mediate application to things. Being, therefore, may be defined; it may be defined, for example, as that which is common to the objects included in any class, and to the objects not included in the same class.†1 But it is nothing new to say that metaphysical conceptions are primarily and at bottom thoughts about words, or thoughts about thoughts; it is the doctrine both of Aristotle (whose categories are parts of speech) and of Kant (whose categories are the characters of different kinds of propositions). Peirce: CP 5.295 Cross-Ref:†† 295. Sensation and the power of abstraction or attention may be regarded as, in one sense, the sole constituents of all thought. Having considered the former, let us now attempt some analysis of the latter. By the force of attention, an emphasis is put upon one of the objective elements of consciousness. This emphasis is, therefore, not itself an object of immediate consciousness; and in this respect it differs entirely from a feeling. Therefore, since the emphasis, nevertheless, consists in some effect upon consciousness, and so can exist only so far as it affects our knowledge; and since an act cannot be supposed to determine that which precedes it in time, this act can consist only in the capacity which the cognition emphasized has for producing an effect upon memory, or otherwise influencing subsequent thought. This is confirmed by the fact that attention is a matter of continuous quantity; for continuous quantity, so far as we know it, reduces itself in the last analysis to time. Accordingly, we find that attention does, in fact, produce a very great effect upon subsequent thought. In the first place, it strongly affects memory, a thought being remembered for a longer time the greater the attention originally paid to it. In the second place, the greater the attention, the closer the connection and the more accurate the logical sequence of thought. In the third place, by attention a thought may be recovered which has been forgotten. From these facts, we gather that attention is the power by which thought at one time is connected with and made to relate to thought at another time; or, to apply the conception of thought as a sign, that it is the pure demonstrative application of a thought-sign. Peirce: CP 5.296 Cross-Ref:††

296. Attention is roused when the same phenomenon presents itself repeatedly on different occasions, or the same predicate in different subjects. We see that A has a certain character, that B has the same, C has the same; and this excites our attention, so that we say, "These have this character." Thus attention is an act of induction; but it is an induction which does not increase our knowledge, because our "these" covers nothing but the instances experienced. It is, in short, an argument from enumeration. Peirce: CP 5.297 Cross-Ref:†† 297. Attention produces effects upon the nervous system. These effects are habits, or nervous associations.†1 A habit arises, when, having had the sensation of performing a certain act, m, on several occasions a, b, c, we come to do it upon every occurrence of the general event, l, of which a, b and c are special cases. That is to say, by the cognition that Peirce: CP 5.297 Cross-Ref:†† Every case of a, b, or c, is a case of m, is determined the cognition that Every case of l is a case of m.

Thus the formation of a habit is an induction, and is therefore necessarily connected with attention or abstraction. Voluntary actions result from the sensations produced by habits, as instinctive actions result from our original nature. Peirce: CP 5.298 Cross-Ref:†† 298. We have thus seen that every sort of modification of consciousness -Attention, Sensation, and Understanding -- is an inference. But the objection may be made that inference deals only with general terms, and that an image, or absolutely singular representation, cannot therefore be inferred. Peirce: CP 5.299 Cross-Ref:†† 299. "Singular" and "individual" are equivocal terms.†1 A singular may mean that which can be but in one place at one time. In this sense it is not opposed to general. The sun is a singular in this sense, but, as is explained in every good treatise on logic, it is a general term. I may have a very general conception of Hermolaus Barbarus, but still I conceive him only as able to be in one place at one time. When an image is said to be singular, it is meant that it is absolutely determinate in all respects. Every possible character, or the negative thereof, must be true of such an image. In the words of the most eminent expounder of the doctrine, the image of a man "must be either of a white, or a black, or a tawny; a straight or a crooked; a tall, or a low, or a middle-sized man."†2 It must be of a man with his mouth open or his mouth shut, whose hair is precisely of such and such a shade, and whose figure has precisely such and such proportions. No statement of Locke has been so scouted by all friends of images as his denial that the "idea" of a triangle must be either of an obtuse-angled, right-angled, or acute-angled triangle. In fact, the image of a triangle must be of one, each of whose angles is of a certain number of degrees, minutes, and seconds. Peirce: CP 5.300 Cross-Ref:†† 300. This being so, it is apparent that no man has a true image of the road to his office, or of any other real thing. Indeed he has no image of it at all unless he can not only recognize it, but imagines it (truly or falsely) in all its infinite details. This being the case, it becomes very doubtful whether we ever have any such thing as an image in our imagination. Please, reader, to look at a bright red book, or other

brightly colored object, and then to shut your eyes and say whether you see that color, whether brightly or faintly -- whether, indeed, there is anything like sight there. Hume and the other followers of Berkeley maintain that there is no difference between the sight and the memory of the red book except in "their different degrees of force and vivacity." "The colors which the memory employs," says Hume, "are faint and dull compared with those in which our original perceptions are clothed."†1 If this were a correct statement of the difference, we should remember the book as being less red than it is; whereas, in fact, we remember the color with very great precision for a few moments (please to test this point, reader), although we do not see anything like it. We carry away absolutely nothing of the color except the consciousness that we could recognize it. As a further proof of this, I will request the reader to try a little experiment. Let him call up, if he can, the image of a horse -- not of one which he has ever seen, but of an imaginary one -- and before reading further let him by contemplation †P1 fix the image in his memory . . . [sic]. Has the reader done as requested? for I protest that it is not fair play to read further without doing so. -- Now, the reader can say in general of what color that horse was, whether grey, bay, or black. But he probably cannot say precisely of what shade it was. He cannot state this as exactly as he could just after having seen such a horse. But why, if he had an image in his mind which no more had the general color than it had the particular shade, has the latter vanished so instantaneously from his memory while the former still remains? It may be replied, that we always forget the details before we do the more general characters; but that this answer is insufficient is, I think, shown by the extreme disproportion between the length of time that the exact shade of something looked at is remembered as compared with that instantaneous oblivion to the exact shade of the thing imagined, and the but slightly superior vividness of the memory of the thing seen as compared with the memory of the thing imagined. Peirce: CP 5.301 Cross-Ref:†† 301. The nominalists, I suspect, confound together thinking a triangle without thinking that it is either equilateral, isosceles, or scalene, and thinking a triangle without thinking whether it is equilateral, isosceles, or scalene. Peirce: CP 5.302 Cross-Ref:†† 302. It is important to remember that we have no intuitive power of distinguishing between one subjective mode of cognition and another;†1 and hence often think that something is presented to us as a picture, while it is really constructed from slight data by the understanding. This is the case with dreams, as is shown by the frequent impossibility of giving an intelligible account of one without adding something which we feel was not in the dream itself. Many dreams, of which the waking memory makes elaborate and consistent stories, must probably have been in fact mere jumbles of these feelings of the ability to recognize this and that which I have just alluded to. Peirce: CP 5.303 Cross-Ref:†† 303. I will now go so far as to say that we have no images even in actual perception. It will be sufficient to prove this in the case of vision; for if no picture is seen when we look at an object, it will not be claimed that hearing, touch, and the other senses, are superior to sight in this respect. That the picture is not painted on the nerves of the retina is absolutely certain, if, as physiologists inform us, these nerves are needlepoints pointing to the light and at distances considerably greater than the minimum visibile. The same thing is shown by our not being able to perceive that there is a large blind spot near the middle of the retina. If, then, we have a picture

before us when we see, it is one constructed by the mind at the suggestion of previous sensations. Supposing these sensations to be signs, the understanding by reasoning from them could attain all the knowledge of outward things which we derive from sight, while the sensations are quite inadequate to forming an image or representation absolutely determinate. If we have such an image or picture, we must have in our minds a representation of a surface which is only a part of every surface we see, and we must see that each part, however small, has such and such a color. If we look from some distance at a speckled surface, it seems as if we did not see whether it were speckled or not; but if we have an image before us, it must appear to us either as speckled, or as not speckled. Again, the eye by education comes to distinguish minute differences of color; but if we see only absolutely determinate images, we must, no less before our eyes are trained than afterwards, see each color as particularly such and such a shade. Thus to suppose that we have an image before us when we see, is not only a hypothesis which explains nothing whatever, but is one which actually creates difficulties which require new hypotheses in order to explain them away. Peirce: CP 5.304 Cross-Ref:†† 304. One of these difficulties arises from the fact that the details are less easily distinguished than, and forgotten before, the general circumstances. Upon this theory, the general features exist in the details: the details are, in fact, the whole picture. It seems, then, very strange that that which exists only secondarily in the picture should make more impression than the picture itself. It is true that in an old painting the details are not easily made out; but this is because we know that the blackness is the result of time, and is no part of the picture itself. There is no difficulty in making out the details of the picture as it looks at present; the only difficulty is in guessing what it used to be. But if we have a picture on the retina, the minutest details are there as much as, nay, more than, the general outline and significancy of it. Yet that which must actually be seen, it is extremely difficult to recognize; while that which is only abstracted from what is seen is very obvious. Peirce: CP 5.305 Cross-Ref:†† 305. But the conclusive argument against our having any images, or absolutely determinate representations in perception, is that in that case we have the materials in each such representation for an infinite amount of conscious cognition, which we yet never become aware of. Now there is no meaning in saying that we have something in our minds which never has the least effect on what we are conscious of knowing. The most that can be said is, that when we see we are put in a condition in which we are able to get a very large and perhaps indefinitely great amount of knowledge of the visible qualities of objects. Peirce: CP 5.306 Cross-Ref:†† 306. Moreover, that perceptions are not absolutely determinate and singular is obvious from the fact that each sense is an abstracting mechanism. Sight by itself informs us only of colors and forms. No one can pretend that the images of sight are determinate in reference to taste. They are, therefore, so far general that they are neither sweet nor non-sweet, bitter nor non-bitter, having savor nor insipid. Peirce: CP 5.307 Cross-Ref:†† 307. The next question is whether we have any general conceptions except in judgments. In perception, where we know a thing as existing, it is plain that there is a judgment that the thing exists, since a mere general concept of a thing is in no case a cognition of it as existing. It has usually been said, however, that we can call up any concept without making any judgment; but it seems that in this case we only

arbitrarily suppose ourselves to have an experience. In order to conceive the number 7, I suppose, that is, I arbitrarily make the hypothesis or judgment, that there are certain points before my eyes, and I judge that these are seven. This seems to be the most simple and rational view of the matter, and I may add that it is the one which has been adopted by the best logicians. If this be the case, what goes by the name of the association of images is in reality an association of judgments. The association of ideas is said to proceed according to three principles -- those of resemblance, of contiguity, and of causality. But it would be equally true to say that signs denote what they do on the three principles of resemblance, contiguity, and causality. There can be no question that anything is a sign of whatever is associated with it by resemblance, by contiguity, or by causality: nor can there be any doubt that any sign recalls the thing signified. So, then, the association of ideas consists in this, that a judgment occasions another judgment, of which it is the sign. Now this is nothing less nor more than inference. Peirce: CP 5.308 Cross-Ref:†† 308. Everything in which we take the least interest creates in us its own particular emotion, however slight this may be. This emotion is a sign and a predicate of the thing. Now, when a thing resembling this thing is presented to us, a similar emotion arises; hence, we immediately infer that the latter is like the former. A formal logician of the old school may say, that in logic no term can enter into the conclusion which had not been contained in the premisses, and that therefore the suggestion of something new must be essentially different from inference. But I reply that that rule of logic applies only to those arguments which are technically called completed. We can and do reason --

Elias was a man; .·.

He was mortal.

And this argument is just as valid as the full syllogism, although it is so only because the major premiss of the latter happens to be true. If to pass from the judgment "Elias was a man" to the judgment "Elias was mortal," without actually saying to one's self that "All men are mortal," is not inference, then the term "inference" is used in so restricted a sense that inferences hardly occur outside of a logic-book. Peirce: CP 5.309 Cross-Ref:†† 309. What is here said of association by resemblance is true of all association. All association is by signs. Everything has its subjective or emotional qualities, which are attributed either absolutely or relatively, or by conventional imputation to anything which is a sign of it. And so we reason,

The sign is such and such; .·.

The sign is that thing.

This conclusion receiving, however, a modification, owing to other considerations, so

as to become -Peirce: CP 5.309 Cross-Ref:†† The sign is almost (is representative of) that thing.

Peirce: CP 5.310 Cross-Ref:†† §4. MAN, A SIGNE

310. We come now to the consideration of the last of the four principles whose consequences we were to trace; namely, that the absolutely incognizable is absolutely inconceivable. That upon Cartesian principles the very realities of things can never be known in the least, most competent persons must long ago have been convinced. Hence the breaking forth of idealism, which is essentially anti-Cartesian, in every direction, whether among empiricists (Berkeley, Hume), or among noologists (Hegel, Fichte). The principle now brought under discussion is directly idealistic; for, since the meaning of a word is the conception it conveys, the absolutely incognizable has no meaning because no conception attaches to it. It is, therefore, a meaningless word; and, consequently, whatever is meant by any term as "the real" is cognizable in some degree, and so is of the nature of a cognition, in the objective sense of that term. Peirce: CP 5.311 Cross-Ref:†† 311. At any moment we are in possession of certain information, that is, of cognitions which have been logically derived by induction and hypothesis from previous cognitions which are less general, less distinct, and of which we have a less lively consciousness. These in their turn have been derived from others still less general, less distinct, and less vivid; and so on back to the ideal †P1 first, which is quite singular, and quite out of consciousness. This ideal first is the particular thing-in-itself. It does not exist as such. That is, there is no thing which is in-itself in the sense of not being relative to the mind, though things which are relative to the mind doubtless are, apart from that relation. The cognitions which thus reach us by this infinite series of inductions and hypotheses (which though infinite a parte ante logice, is yet as one continuous process not without a beginning in time) are of two kinds, the true and the untrue, or cognitions whose objects are real and those whose objects are unreal. And what do we mean by the real? It is a conception which we must first have had when we discovered that there was an unreal, an illusion; that is, when we first corrected ourselves. Now the distinction for which alone this fact logically called, was between an ens relative to private inward determinations, to the negations belonging to idiosyncrasy, and an ens such as would stand in the long run. The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of a definite increase of knowledge.†1 And so those two series of cognition -- the real and the unreal -- consist of those which, at a time sufficiently future, the community will always continue to re-affirm; and of those which, under the same conditions, will ever after be denied. Now, a proposition whose falsity can never be discovered, and the error of which therefore is absolutely incognizable, contains, upon our principle, absolutely no error. Consequently, that which is thought in these cognitions is the real, as it really is. There is nothing, then, to prevent our knowing outward things as

they really are, and it is most likely that we do thus know them in numberless cases, although we can never be absolutely certain of doing so in any special case. Peirce: CP 5.312 Cross-Ref:†† 312. But it follows that since no cognition of ours is absolutely determinate, generals must have a real existence. Now this scholastic realism is usually set down as a belief in metaphysical fictions. But, in fact, a realist is simply one who knows no more recondite reality than that which is represented in a true representation. Since, therefore, the word "man" is true of something, that which "man" means is real. The nominalist must admit that man is truly applicable to something; but he believes that there is beneath this a thing in itself, an incognizable reality. His is the metaphysical figment. Modern nominalists are mostly superficial men, who do not know, as the more thorough Roscellinus and Occam did, that a reality which has no representation is one which has no relation and no quality. The great argument for nominalism is that there is no man unless there is some particular man. That, however, does not affect the realism of Scotus; for although there is no man of whom all further determination can be denied, yet there is a man, abstraction being made of all further determination. There is a real difference between man irrespective of what the other determinations may be, and man with this or that particular series of determinations, although undoubtedly this difference is only relative to the mind and not in re. Such is the position of Scotus.†P1 Occam's great objection is, there can be no real distinction which is not in re, in the thing-in-itself; but this begs the question for it is itself based only on the notion that reality is something independent of representative relation †P1 Peirce: CP 5.313 Cross-Ref:†† 313.†1 Such being the nature of reality in general, in what does the reality of the mind consist? We have seen that the content of consciousness, the entire phenomenal manifestation of mind, is a sign resulting from inference. Upon our principle, therefore, that the absolutely incognizable does not exist, so that the phenomenal manifestation of a substance is the substance, we must conclude that the mind is a sign developing according to the laws of inference. What distinguishes a man from a word? There is a distinction doubtless. The material qualities, the forces which constitute the pure denotative application, and the meaning of the human sign, are all exceedingly complicated in comparison with those of the word. But these differences are only relative. What other is there? It may be said that man is conscious, while a word is not. But consciousness is a very vague term. It may mean that emotion which accompanies the reflection that we have animal life. This is a consciousness which is dimmed when animal life is at its ebb in old age, or sleep, but which is not dimmed when the spiritual life is at its ebb; which is the more lively the better animal a man is, but which is not so, the better man he is. We do not attribute this sensation to words, because we have reason to believe that it is dependent upon the possession of an animal body. But this consciousness, being a mere sensation, is only a part of the material quality of the man-sign. Again, consciousness is sometimes used to signify the I think, or unity in thought; but the unity is nothing but consistency, or the recognition of it. Consistency belongs to every sign, so far as it is a sign; and therefore every sign, since it signifies primarily that it is a sign, signifies its own consistency. The man-sign acquires information, and comes to mean more than he did before. But so do words. Does not electricity mean more now than it did in the days of Franklin? Man makes the word, and the word means nothing which the man has not made it mean, and that only to some man. But since man can think only by means of words or other external symbols, these might turn round and say: "You mean nothing which we have not taught you, and then only so far as you address

some word as the interpretant of your thought." In fact, therefore, men and words reciprocally educate each other; each increase of a man's information involves and is involved by, a corresponding increase of a word's information. Peirce: CP 5.314 Cross-Ref:†† 314. Without fatiguing the reader by stretching this parallelism too far, it is sufficient to say that there is no element whatever of man's consciousness which has not something corresponding to it in the word; and the reason is obvious. It is that the word or sign which man uses is the man himself. For, as the fact that every thought is a sign, taken in conjunction with the fact that life is a train of thought, proves that man is a sign; so, that every thought is an external sign, proves that man is an external sign. That is to say, the man and the external sign are identical, in the same sense in which the words homo and man are identical. Thus my language is the sum total of myself; for the man is the thought. Peirce: CP 5.315 Cross-Ref:†† 315. It is hard for man to understand this, because he persists in identifying himself with his will, his power over the animal organism, with brute force. Now the organism is only an instrument of thought. But the identity of a man consists in the consistency of what he does and thinks, and consistency is the intellectual character of a thing; that is, is its expressing something. Peirce: CP 5.316 Cross-Ref:†† 316. Finally, as what anything really is, is what it may finally come to be known to be in the ideal state of complete information, so that reality depends on the ultimate decision of the community; so thought is what it is, only by virtue of its addressing a future thought which is in its value as thought identical with it, though more developed. In this way, the existence of thought now depends on what is to be hereafter; so that it has only a potential existence, dependent on the future thought of the community. Peirce: CP 5.317 Cross-Ref:†† 317. The individual man, since his separate existence is manifested only by ignorance and error, so far as he is anything apart from his fellows, and from what he and they are to be, is only a negation. This is man,

". . . proud man, Most ignorant of what he's most assured, His glassy essence."

Peirce: CP 5.318 Cross-Ref:†† III

GROUNDS OF VALIDITY OF THE LAWS OF LOGIC: FURTHER CONSEQUENCES OF FOUR INCAPACITIES †1

§1. OBJECTIONS TO THE SYLLOGISME

318. If, as I maintained in an article in the last number of this Journal,†2 every judgment results from inference, to doubt every inference is to doubt everything. It has often been argued that absolute scepticism is self-contradictory; but this is a mistake: and even if it were not so, it would be no argument against the absolute sceptic, inasmuch as he does not admit that no contradictory propositions are true. Indeed, it would be impossible to move such a man, for his scepticism consists in considering every argument and never deciding upon its validity; he would, therefore, act in this way in reference to the arguments brought against him. Peirce: CP 5.318 Cross-Ref:†† But then there are no such beings as absolute sceptics. Every exercise of the mind consists in inference, and so, though there are inanimate objects without beliefs, there may be †3 no intelligent beings in that condition. Peirce: CP 5.318 Cross-Ref:†† Yet it is quite possible that a person should doubt every principle of inference. He may not have studied logic, and though a logical formula may sound very obviously true to him, he may feel a little uncertain whether some subtile deception may not lurk in it. Indeed, I certainly shall have, among the most cultivated and respected of my readers, those who deny that those laws of logic, which men generally admit, have universal validity. But I address myself, also, to those who have no such doubts, for even to them it may be interesting to consider how it is that these principles come to be true. Finally, having put forth in former numbers of this Journal some rather heretical principles of philosophical research, one of which is nothing can be admitted to be absolutely inexplicable,†1 it behooves me to take up a challenge which has been given me to show how upon my principles the validity of the laws of logic can be other than inexplicable. Peirce: CP 5.319 Cross-Ref:†† 319. I shall be arrested, at the outset, by a sweeping objection to my whole undertaking. It will be said that my deduction of logical principles, being itself an argument, depends for its whole virtue upon the truth of the very principles in question; so that whatever my proof may be, it must take for granted the very things to be proved. But to this I reply, that I am neither addressing absolute sceptics, nor men in any state of fictitious doubt whatever. I require the reader to be candid; and if he becomes convinced of a conclusion, to admit it. There is nothing to prevent a man's perceiving the force of certain special arguments, although he does not yet know that a certain general law of arguments holds good; for the general rule may hold good in some cases and not in others. A man may reason well without understanding the principles of reasoning, just as he may play billiards well without understanding analytical mechanics. If you, the reader, actually find that my arguments have a convincing force with you, it is a mere pretence to call them illogical.†2 Peirce: CP 5.320 Cross-Ref:†† 320. That if one sign denotes generally everything denoted by a second, and this second denotes generally everything denoted by a third, then the first denotes

generally everything denoted by the third, is not doubted by anybody who distinctly apprehends the meaning of these words. The deduction of the general form of syllogism, therefore, will consist only of an explanation of the suppositio communis.†P1 Now, what the formal logician means by an expression of the form, "Every M is P," is that anything of which M is predicable is P; thus, if S is M, that S is P. The premiss that "Every M is P" may, therefore, be denied; but to admit it, unambiguously, in the sense intended, is to admit that the inference is good that S is P if S is M. He, therefore, who does not deny that S is P -- M, S, P, being any terms such that S is M and every M is P -- denies nothing that the formal logician maintains in reference to this matter; and he who does deny this, simply is deceived by an ambiguity of language. How we come to make any judgments in the sense of the above "Every M is P," may be understood from the theory of reality put forth in the article in the last number. It was there shown that real things are of a cognitive and therefore significative nature, so that the real is that which signifies something real.†1 Consequently, to predicate anything of anything real is to predicate it of that of which that subject (the real) is itself predicated; for to predicate one thing of another is to state that the former is a sign of the latter. Peirce: CP 5.321 Cross-Ref:†† 321. These considerations show the reason of the validity of the formula,

S is M; M is P: .·. S is P.

They hold good whatever S and P may be, provided that they be such that any middle term between them can be found. That P should be a negative term, therefore, or that S should be a particular term, would not interfere at all with the validity of this formula. Hence, the following formulæ are also valid:

S is M; M is not P: .·. S is not P.

Some S is M; M is P: .·. Some S is P.

Some S is M; M is not P: .·. Some S is not P.

Peirce: CP 5.322 Cross-Ref:†† 322. Moreover, as all that class of inferences which depend upon the introduction of relative terms can be reduced to the general form, they also are shown

to be valid. Thus, it is proved to be correct to reason thus: Peirce: CP 5.322 Cross-Ref:†† Every relation of a subject to its predicate is a relation of the relative "not X'd, except by the X of some," to its correlate, where X is any relative I please. Peirce: CP 5.322 Cross-Ref:†† Every relation of "man" to "animal" is a relation of a subject to its predicate. .·. Every relation of "man" to "animal" is a relation of the relative "not X'd, except by the X of some," to its correlate, where X is any relative I please. Peirce: CP 5.322 Cross-Ref:†† Every relation of the relative "not X'd, except by the X of some," to its correlate, where X is any relative I please, is a relation of the relative "not headed, except by the head of some," to its correlate. .·. Every relation of "man" to "animal" is a relation of the relative "not headed, except by the head of some," to its correlate.†P1 Peirce: CP 5.323 Cross-Ref:†† 323. At the same time, as will be seen from this example, the proof of the validity of these inferences depends upon the assumption of the truth of certain general statements concerning relatives. These formulæ can all be deduced from the principle, that in a system of signs in which no sign is taken in two different senses, two signs which differ only in their manner of representing their object, but which are equivalent in meaning, can always be substituted for one another. Any case of the falsification of this principle would be a case of the dependence of the mode of existence of the thing represented upon the mode of this or that representation of it, which, as has been shown in the article in the last number, is contrary to the nature of reality.†1 Peirce: CP 5.324 Cross-Ref:†† 324. The next formula of syllogism to be considered is the following:

S is other than P; M is P: .·. S is other than M.

The meaning of "not" or "other than" seems to have greatly perplexed the German logicians, and it may be, therefore, that it is used in different senses. If so, I propose to defend the validity of the above formula only when other than is used in a particular sense. By saying that one thing or class is other than a second, I mean that any third whatever is identical with the class which is composed of that third and of whatever is, at once, the first and second. For example, if I say that rats are not mice, I mean that any third class as dogs is identical with dogs plus†2 rats-which-are-mice; that is to say, the addition of rats-which-are-mice, to anything, leaves the latter just what it was before. This being all that I mean by S is other than P, I mean absolutely the same thing when I say that S is other than P, that I do when I say that P is other than S; and the same when I say that S is other than M, that I do when I say that M is other than S. Hence the above formula is only another way of writing the following:

M is P; P is not S: .·. M is not S.

But we have already seen that this is valid. Peirce: CP 5.325 Cross-Ref:†† 325. A very similar formula to the above is the following:

S is M; some S is P: .·. Some M is P.

By saying that some of a class is of any character, I mean simply that no statement which implies that none of that class is of that character is true. But to say that none of that class is of that character, is, as I take the word "not," to say that nothing of that character is of that class. Consequently, to say that some of A is B, is, as I understand words and in the only sense in which I defend this formula, to say that some B is A. In this way the formula is reduced to the following, which has already been shown to be valid:

Some P is S; S is M: .·. Some P is M.

Peirce: CP 5.326 Cross-Ref:†† 326. The only demonstrative syllogisms which are not included among the above forms are the Theophrastean moods, which are all easily reduced by means of simple conversions.†1 Peirce: CP 5.327 Cross-Ref:†† 327. Let us now consider what can be said against all this, and let us take up the objections which have actually been made to the syllogistic formulae, beginning with those which are of a general nature and then examining those sophisms which have been pronounced irresolvable by the rules of ordinary logic. Peirce: CP 5.327 Cross-Ref:†† It is a very ancient notion that no proof can be of any value, because it rests on premisses which themselves equally require proof, which again must rest on other premisses, and so back to infinity. This really does show that nothing can be proved beyond the possibility†2 of a doubt; that no argument could be legitimately used against an absolute sceptic; and that inference is only a transition from one cognition to another, and not the creation of a cognition. But the objection is intended to go much further than this, and to show (as it certainly seems to do) that inference not

only cannot produce infallible cognition, but that it cannot produce cognition at all. It is true, that since some judgment precedes every judgment inferred, either the first premisses were not inferred, or there have been no first premisses. But it does not follow that because there has been no first in a series, therefore that series has had no beginning in time; for the series may be continuous,†P1 and may have begun gradually, as was shown in an article in this volume,†3 where this difficulty has already been resolved. Peirce: CP 5.328 Cross-Ref:†† 328. A somewhat similar objection has been made by Locke †1 and others,†2 to the effect that the ordinary demonstrative syllogism is a petitio principii, inasmuch as the conclusion is already implicitly stated in the major premiss. Take, for example, the syllogism,

All men are mortal; Socrates is a man: .·.

Socrates is mortal.

This attempt to prove that Socrates is mortal begs the question, it is said, since if the conclusion is denied by anyone, he thereby denies that all men are mortal. But what such considerations really prove is that the syllogism is demonstrative. To call it a petitio principii is a mere confusion of language. It is strange that philosophers, who are so suspicious of the words virtual and potential, should have allowed this "implicit" to pass unchallenged. A petitio principii consists in reasoning from the unknown to the unknown.†3 Hence, a logician who is simply engaged in stating what general forms of argument are valid, can, at most, have nothing more to do with the consideration of this fallacy than to note those cases in which from logical principles a premiss of a certain form cannot be better known than a conclusion of the corresponding form. But it is plainly beyond the province of the logician, who has only proposed to state what forms of facts involve what others, to inquire whether man can have a knowledge of universal propositions without a knowledge of every particular contained under them, by means of natural insight, divine revelation, induction, or testimony. The only petitio principii, therefore, which he can notice is the assumption of the conclusion itself in the premiss; and this, no doubt, those who call the syllogism a petitio principii believe is done in that formula. But the proposition "All men are mortal" does not in itself involve the statement that Socrates is mortal, but only that "whatever has man truly predicated of it is mortal." In other words, the conclusion is not involved in the meaning of the premiss, but only the validity of the syllogism. So that this objection merely amounts to arguing that the syllogism is not valid, because it is demonstrative.†P1 Peirce: CP 5.329 Cross-Ref:†† 329. A much more interesting objection is that a syllogism is a purely mechanical process. It proceeds according to a bare rule or formula; and a machine might be constructed which would so transpose the terms of premisses. This being so (and it is so), it is argued that this cannot be thought; that there is no life in it. Swift has ridiculed the syllogism in the "Voyage to Laputa," by describing a machine for making science:

Peirce: CP 5.329 Cross-Ref:†† "By this contrivance, the most ignorant person, at a reasonable charge, and with little bodily labor, might write books in philosophy, poetry, politics, laws, mathematics, and theology, without the least assistance from genius or study." Peirce: CP 5.329 Cross-Ref:†† The idea involved in this objection seems to be that it requires mind to apply any formula or use any machine. If, then, this mind is itself only another formula, it requires another mind behind it to set it into operation, and so on ad infinitum. This objection fails in much the same way that the first one which we considered failed. It is as though a man should address a land surveyor as follows: "You do not make a true representation of the land; you only measure lengths from point to point -- that is to say, lines. If you observe angles, it is only to solve triangles and obtain the lengths of their sides. And when you come to make your map, you use a pencil which can only make lines, again. So, you have to do solely with lines. But the land is a surface; and no number of lines, however great, will make any surface, however small. You, therefore, fail entirely to represent the land." The surveyor, I think, would reply, "Sir, you have proved that my lines cannot make up the land, and that, therefore, my map is not the land. I never pretended that it was. But that does not prevent it from truly representing the land, as far as it goes. It cannot, indeed, represent every blade of grass; but it does not represent that there is not a blade of grass where there is. To abstract from a circumstance is not to deny it." Suppose the objector were, at this point, to say, "To abstract from a circumstance is to deny it. Wherever your map does not represent a blade of grass, it represents there is no blade of grass. Let us take things on their own valuation." Would not the surveyor reply: "This map is my description of the country. Its own valuation can be nothing but what I say, and all the world understands, that I mean by it. Is it very unreasonable that I should demand to be taken as I mean, especially when I succeed in making myself understood?" What the objector's reply to this question would be, I leave it to anyone to say who thinks his position well taken. Now this line of objection is parallel to that which is made against the syllogism. It is shown that no number of syllogisms can constitute the sum total of any mental action, however restricted. This may be freely granted, and yet it will not follow that the syllogism does not truly represent the mental action, as far as it purports to represent it at all. There is reason to believe that the act on of the mind is, as it were, a continuous movement. Now the doctrine embodied in syllogistic formulae (so far as it applies to the mind at all) is, that if two successive positions, occupied by the mind in this movement, be taken, they will be found to have certain relations. It is true that no number of successions of positions can make up a continuous movement; and this, I suppose, is what is meant by saying that a syllogism is a dead formula, while thinking is a living process. But the reply is that the syllogism is not intended to represent the mind, as to its life or deadness, but only as to the relation of its different judgments concerning the same thing. And it should be added that the relation between syllogism and thought does not spring from considerations of formal logic, but from those of psychology. All that the formal logician has to say is, that if facts capable of expression in such and such forms of words are true, another fact whose expression is related in a certain way to the expression of these others is also true. Peirce: CP 5.330 Cross-Ref:†† 330. Hegel taught that ordinary reasoning is "one-sided." A part of what he meant was that by such inference a part only of all that is true of an object can be

learned, owing to the generality or abstractedness of the predicates inferred. This objection is, therefore, somewhat similar to the last; for the point of it is that no number of syllogisms would give a complete knowledge of the object. This, however, presents a difficulty which the other did not; namely, that if nothing incognizable exists, and all knowledge is by mental action, by mental action everything is cognizable. So that if by syllogism everything is not cognizable, syllogism does not exhaust the modes of mental action. But grant the validity of this argument and it proves too much; for it makes, not the syllogism particularly, but all finite knowledge to be worthless. However much we know, more may come to be found out. Hence, all can never be known. This seems to contradict the fact that nothing is absolutely incognizable; and it would really do so if our knowledge were something absolutely limited. For, to say that all can never be known, means that information may increase beyond any assignable point; that is, that an absolute termination of all increase of knowledge is absolutely incognizable, and therefore does not exist. In other words, the proposition merely means that the sum of all that will be known up to any time, however advanced, into the future, has a ratio less than any assignable ratio to all that may be known at a time still more advanced. This, however,†1 does not, in the least,†2 contradict the fact that everything is cognizable; it only contradicts a proposition, which no one can maintain, that it is possible to cognize everything,†P1 that is, that at some time all things will be known.†3 It may, however, very justly be said that the difficulty still remains, how at every future time, however late, there can be something yet to happen. It is no longer a contradiction, but it is a difficulty; that is to say, lengths of time are shown not to afford an adequate conception of futurity in general; and the question arises, in what other way we are to conceive of it. I might indeed, perhaps, fairly drop the question here, and say that the difficulty had become so entirely removed from the syllogism in particular, that the formal logician need not feel himself specially called on to consider it. The solution, however, is very simple. It is that we conceive of the future, as a whole, by considering that this word, like any other general term, as "inhabitant of St. Louis," may be taken distributively or collectively. We conceive of the infinite, therefore, not directly or on the side of its infinity, but by means of a consideration concerning words or a second intention. Peirce: CP 5.331 Cross-Ref:†† 331. Another objection to the syllogism is that its "therefore" is merely subjective; that, because a certain conclusion syllogistically follows from a premiss, it does not follow that the fact denoted by the conclusion really depends upon the fact denoted by the premiss, so that the syllogism does not represent things as they really are. But it has been fully shown that if the facts are as the premisses represent, they are also as the conclusion represents. Now this is a purely objective statement: therefore, there is a real connection between the facts stated as premisses and those stated as conclusion. It is true that there is often an appearance of reasoning deductively from effects to causes. Thus we may reason as follows: "There is smoke; there is never smoke without fire: hence, there has been fire." Yet smoke is not the cause of fire, but the effect of it. Indeed, it is evident, that in many cases an event is a demonstrative sign of a certain previous event having occurred. Hence, we can reason deductively from relatively future to relatively past, whereas causation †P1 really determines events in the direct order of time. Nevertheless, if we can thus reason against the stream of time, it is because there really are such facts as that "If there is smoke, there has been fire," in which the following event is the antecedent. Indeed, if we consider the manner in which such a proposition became known to us, we shall find that what it really means is that "If we find smoke, we shall find evidence on the whole that there has been fire"; and this, if reality consists in the agreement that the

whole community would eventually come to, is the very same thing as to say that there really has been fire. In short, the whole present difficulty is resolved instantly by this theory of reality, because it makes all reality something which is constituted by an event indefinitely future. Peirce: CP 5.332 Cross-Ref:†† 332. Another objection, for which I am quite willing to allow a great German philosopher the whole credit, is that sometimes the conclusion is false, although both the premisses and the syllogistic form are correct.†P1 Of this he gives the following examples. From the middle term that a wall has been painted blue, it may correctly be concluded that it is blue; but notwithstanding this syllogism it may be green if it has also received a coat of yellow, from which last circumstance by itself it would follow that it is yellow. If from the middle term of the sensuous faculty it be concluded that man is neither good nor bad, since neither can be predicated of the sensuous, the syllogism is correct; but the conclusion is false, since of man in the concrete, spirituality is equally true, and may serve as middle term in an opposite syllogism. From the middle term of the gravitation of the planets, satellites, and comets, towards the sun, it follows correctly that these bodies fall into the sun; but they do not fall into it, because (!) they equally gravitate to their own centres, or, in other words (!!), they are supported by centrifugal force. Now, does Hegel mean to say that these syllogisms satisfy the rules for syllogism given by those who defend syllogism? or does he mean to grant that they do not satisfy those rules, but to set up some rules of his own for syllogism which shall insure its yielding false conclusions from true premisses? If the latter, he ignores the real issue, which is whether the syllogism as defined by the rules of formal logic is correct, and not whether the syllogism as represented by Hegel is correct. But if he means that the above examples satisfy the usual definition of a true syllogism, he is mistaken. The first, stated in form, is as follows:

Whatever has been painted blue is blue; This wall has been painted blue: .·.

This wall is blue.

Now "painted blue" may mean painted with blue paint, or painted so as to be blue. If, in the example, the former were meant, the major premiss would be false. As he has stated that it is true, the latter meaning of "painted blue" must be the one intended. Again, "blue" may mean blue at some time, or blue at this time. If the latter be meant, the major premiss is plainly false; therefore, the former is meant. But the conclusion is said to contradict the statement that the wall is yellow. If blue were here taken in the more general sense, there would be no such contradiction. Hence, he means in the conclusion that this wall is now blue; that is to say, he reasons thus:

Whatever has been made blue has been blue; This has been made blue:

.·.

This is blue now.

Now substituting letters for the subjects and predicates, we get the form,

M is P; S is M: .·.

S is Q.

This is not a syllogism in the ordinary sense of that term, or in any sense in which anybody maintains that the syllogism is valid. Peirce: CP 5.332 Cross-Ref:†† The second example given by Hegel, when written out in full, is as follows:

Sensuality is neither good nor bad; Man has (not is) sensuality: .·.

Man is neither good nor bad.

Or, the same argument may be stated as follows:

The sensuous, as such, is neither good nor bad; Man is sensuous: .·.

Man is neither good nor bad.

Peirce: CP 5.332 Cross-Ref:†† When letters are substituted for subject and predicate in either of these arguments, it takes the form,

M is P; S is N: .·.

S is P.

This, again, bears but a very slight resemblance to a syllogism.

The third example, when stated at full length, is as follows

Whatever tends towards the sun, on the whole, falls into the sun; The planets tend toward the sun: .·.

The planets fall into the sun. This is a fallacy similar to the last.

Peirce: CP 5.332 Cross-Ref:†† I wonder that this eminent logician did not add to his list of examples of correct syllogism the following:

It either rains, or it does not rain; It does not rain: .·.

It rains.

This is fully as deserving of serious consideration as any of those which he has brought forward. The rainy day and the pleasant day are both, in the first place, day. Secondly, each is the negation of a day. It is indifferent which be regarded as the positive. The pleasant is Other to the rainy, and the rainy is in like manner Other to the pleasant. Thus, both are equally Others. Both are Others of each other, or each is Other for itself. So this day being other than rainy, that to which it is Other is itself. But it is Other than itself. Hence, it is itself Rainy.

Peirce: CP 5.333 Cross-Ref:†† §2. THE THREE KINDS OF SOPHISMSE

333. Some sophisms have, however, been adduced, mostly by the Eleatics and Sophists, which really are extremely difficult to resolve by syllogistic rules; and according to some modern authors this is actually impossible. These sophisms fall into three classes: first, those which relate to continuity; second, those which relate to consequences of supposing things to be other than they are; third, those which relate to propositions which imply their own falsity. Of the first class, the most celebrated are Zeno's arguments concerning motion. One of these is, that if Achilles overtakes a tortoise in any finite time, and the tortoise has the start of him by a distance which may be called a, then Achilles has to pass over the sum of distances represented by the polynomial

1/2a + 1/4a + 1/8a + 1/16a + 1/32a etc.

up to infinity. Every term of this polynomial is finite, and it has an infinite number of terms; consequently, Achilles must in a finite time pass over a distance equal to the sum of an infinite number of finite distances. Now this distance must be infinite, because no finite distance, however small, can be multiplied by an infinite number without giving an infinite distance. So that even if none of these finite distances were larger than the smallest (which is finite since all are finite), the sum of the whole would be infinite. But Achilles cannot pass over an infinite distance in a finite time; therefore, he cannot overtake the tortoise in any time, however great. Peirce: CP 5.333 Cross-Ref:†† The solution of this fallacy is as follows: The conclusion is supposed to follow from the undoubted †1 fact that Achilles cannot overtake the tortoise without passing over an infinite number of terms of that series of finite distances. That is, no case of his overtaking the tortoise would be a case of his not passing over a non-finite number of terms; that is (by simple conversion), no case of his not passing over a non-finite number of terms would be a case of his overtaking the tortoise. But if he does not pass over a non-finite number of terms, he either passes over a finite number, or he passes over none; and conversely. Consequently, nothing more has been said than that every case of his passing over only a finite number of terms, or of his not passing over any, is a case of his not overtaking the tortoise. Consequently, nothing more can be concluded than that he passes over a distance greater than the sum of any finite number of the above series of terms. But because a quantity is greater than any quantity of a certain series, it does not follow that it is greater than any quantity.†P1 Peirce: CP 5.333 Cross-Ref:†† In fact, the reasoning in this sophism may be exhibited as follows: We start with the series of numbers,

1/2a 1/2a+1/4a 1/2a+1/4a+1/8a 1/2a+1/4a+1/8a+1/16a etc. etc. etc.

Then, the implied argument is Any number of this series is less than a; But any number you please is less than the number of terms of this series: Hence, any number you please is less than a. This involves an obvious confusion between the number of terms and the value of the greatest term.

Peirce: CP 5.334 Cross-Ref:†† 334. Another argument by Zeno against motion, is that a body fills a space no larger than itself. In that place there is no room for motion. Hence, while in the place where it is, it does not move. But it never is other than in the place where it is. Hence, it never moves. Putting this into form, it will read:

No body in a place no larger than itself is moving; But every body is a body in a place no larger than itself: .·.

No body is moving.

The error of this consists in the fact that the minor premiss is only true in the sense that during a time sufficiently short the space occupied by a body is as little larger than itself as you please. All that can be inferred from this is, that during no time a body will move no distance. Peirce: CP 5.335 Cross-Ref:†† 335. All the arguments of Zeno depend on supposing that a continuum has ultimate parts. But a continuum is precisely that, every part of which has parts, in the same sense.†1 Hence, he makes out his contradictions only by making a self-contradictory supposition. In ordinary and mathematical language, we allow ourselves to speak of such parts -- points -- and whenever we are led into contradiction thereby, we have simply to express ourselves more accurately to resolve the difficulty. Peirce: CP 5.336 Cross-Ref:†† 336. Suppose a piece of glass to be laid on a sheet of paper so as to cover half of it. Then, every part of the paper is covered, or not covered; for "not" means merely outside of, or other than. But is the line under the edge of the glass covered or not? It is no more on one side of the edge than it is on the other. Therefore, it is either on both sides, or neither side. It is not on neither side; for if it were it would be not on either side, therefore not on the covered side, therefore not covered, therefore on the uncovered side. It is not partly on one side and partly on the other, because it has no width. Hence, it is wholly on both sides, or both covered and not covered. Peirce: CP 5.336 Cross-Ref:†† The solution of this is, that we have supposed a part too narrow to be partly uncovered and partly covered; that is to say, a part which has no parts in a continuous surface, which by definition has no such parts. The reasoning, therefore, simply serves to reduce this supposition to an absurdity. Peirce: CP 5.336 Cross-Ref:†† It may be said that there really is such a thing as a line. If a shadow falls on a surface, there really is a division between the light and the darkness. That is true. But it does not follow that because we attach a definite meaning to the part of a surface being covered, therefore we know what we mean when we say that a line is covered. We may define a covered line as one which separates two surfaces both of which are covered, or as one which separates two surfaces either of which is covered. In the former case, the line under the edge is uncovered; in the latter case, it is covered.

Peirce: CP 5.337 Cross-Ref:†† 337. In the sophisms thus far considered, the appearance of contradiction depends mostly upon an ambiguity; in those which we are now to consider, two true propositions really do in form conflict with one another. We are apt to think that formal logic forbids this, whereas a familiar argument, the reductio ad absurdum, depends on showing that contrary predicates are true of a subject, and that therefore that subject does not exist. Many logicians, it is true, make affirmative propositions assert the existence of their subjects.†P1 The objection to this is that it cannot be extended to hypotheticals. The proposition

If A then B

may conveniently be regarded as equivalent to

Every case of the truth of A is a case of the truth of B.

But this cannot be done if the latter proposition asserts the existence of its subject; that is, asserts that A really happens. If, however, a categorical affirmative be regarded as asserting the existence of its subject, the principle of the reductio ad absurdum is that two propositions of the forms,

If A were true, B would not be true, and If A were true, B would be true,

may both be true at once; and that if they are so, A is not true. It will be well, perhaps, to illustrate this point. No man of common sense would deliberately upset his inkstand if there were ink in it; that is, if any ink would run out. Hence, by simple conversion,

If he were deliberately to upset his inkstand, no ink would be spilt.

But suppose there is ink in it. Then, it is also true, that

If he were deliberately to upset his inkstand, the ink would be spilt.

These propositions are both true, and the law of contradiction is not violated which asserts only that nothing has contradictory predicates: only, it follows from these propositions that the man will not deliberately overturn his inkstand. Peirce: CP 5.338 Cross-Ref:†† 338. There are two ways in which deceptive sophisms may result from this circumstance. In the first place, contradictory propositions are never both true. Now, as a universal proposition may be true when the subject does not exist, it follows that the contradictory of a universal -- that is, a particular -- cannot be taken in such a sense as to be true when the subject does not exist. But a particular simply asserts a part of what is asserted in the universal over it; therefore, the universal over it asserts the subject to exist. Consequently, there are two kinds of universals, those which do not assert the subject to exist, and these have no particular propositions under them, and those which do assert that the subject exists, and these strictly speaking have no contradictories. For example, there is no use of such a form of proposition as "Some griffins would be dreadful animals," as particular under the useful form "The griffin would be a dreadful animal"; and the apparent contradictories "All of John Smith's family are ill," and "Some of John Smith's family are not ill," are both false at once if John Smith has no family. Here, though an inference from a universal to the particular under it is always valid, yet a procedure which greatly resembles this would be sophistical if the universal were one of those propositions which does not assert the existence of its subject. The following sophism depends upon this; I call it the True Gorgias: Gorgias. What say you, Socrates, of black? Is any black, white? Socrates. No, by Zeus! Gor. Do you say, then, that no black is white? Soc. None at all. Gor. But is everything either black or non-black? Soc. Of course. Gor. And everything either white or non-white? Soc. Yes. Gor. And everything either rough or smooth? Soc. Yes. Gor. And everything either real or unreal? Soc. Oh, bother! yes. Gor. Do you say, then, that all black is either rough black or smooth black? Soc. Yes. Gor. And that all white is either real white or unreal white? Soc. Yes. Gor. And yet is no black, white? Soc. None at all. Gor. Nor no white, black? Soc. By no means. Gor. What? Is no smooth black, white? Soc. No; you cannot prove that, Gorgias. Gor. Nor no rough black, white? Soc. Neither. Gor. Nor no real white, black? Soc. No.

Gor. Nor no unreal white, black? Soc. No, I say. No white at all is black. Gor. What if black is smooth, is it not white? Soc. Not in the least. Gor. And if the last is false, is the first false? Soc. It follows. Gor. If, then, black is white, does it follow, that black is not smooth? Soc. It does. Gor. Black-white is not smooth? Soc. What do you mean? Gor. Can any dead man speak? Soc. No, indeed. Gor. And is any speaking man dead? Soc. I say, no. Gor. And is any good king tyrannical? Soc. No. Gor. And is any tyrannical king good? Soc. I just said no. Gor. And you said, too, that no rough black is white, did you not? Soc. Yes. Gor. Then, is any black-white, rough? Soc. No. Gor. And is any unreal black, white? Soc. No. Gor. Then, is any black-white unreal? Soc. No. Gor. No black-white is rough? Soc. None. Gor. All black-white, then, is non-rough? Soc. Yes. Gor. And all black-white, non-unreal? Soc. Yes. Gor. All black-white is then smooth? Soc. Yes. Gor. And all real? Soc. Yes. Gor. Some smooth, then, is black-white? Soc. Of course. Gor. And some real is black-white? Soc. So it seems. Gor. Some black-white smooth is black-white? Soc. Yes. Gor. Some black smooth is black-white? Soc. Yes. Gor. Some black smooth is white? Soc. Yes. Gor. Some black real is black-white? Soc. Yes. Gor. Some black real is white? Soc. Yes. Gor. Some real black is white? Soc. Yes. Gor. And some smooth black is white? Soc. Yes. Gor. Then, some black is white? Soc. I think so myself.

Peirce: CP 5.339 Cross-Ref:††

339. The principle of the reductio ad absurdum also occasions deceptions in another way, owing to the fact that we have many words, such as can, may, must, etc., which imply more or less vaguely an otherwise unexpressed condition, so that these propositions are in fact hypotheticals. Accordingly, if the unexpressed condition is some state of things which does not actually come to pass, the two propositions may appear to be contrary to one another. Thus, the moralist says, "You ought to do this, and you can do it." This "You can do it" is principally hortatory in its force: so far as it is a statement of fact, it means merely, "If you try, you will do it." Now, if the act is an outward one and the act is not performed, the scientific man, in view of the fact that every event in the physical world depends exclusively on physical antecedents, says that in this case the laws of nature prevented the thing from being done, and that therefore, "Even if you had tried, you would not have done it." Yet the reproachful conscience still says you might have done it; that is, that "If you had tried, you would have done it." This is called the paradox of freedom and fate; and it is usually supposed that one of these propositions must be true and the other false.†1 But since, in fact, you have not tried, there is no reason why the supposition that you have tried should not be reduced to an absurdity. In the same way, if you had tried and had performed the action, the conscience might say, "If you had not tried, you would not have done it"; while the understanding would say, "Even if you had not tried, you would have done it." These propositions are perfectly consistent, and only serve to reduce the supposition that you did not try to an absurdity.†P1 Peirce: CP 5.340 Cross-Ref:†† 340.†1 The third class of sophisms consists of the so-called Insolubilia. Here is an example of one of them with its resolution:

THIS PROPOSITION IS NOT TRUE IS IT TRUE OR NOT? Suppose it true.

|

Then,

Suppose it not true.

|

Then,

The proposition is true;

| It is not true.

But, that it is not true

|

is the proposition:

|.·. It is true that it is

|

not true.

.·. That it is not true is .·. It is not true.

| But, the proposition is that it is true; | .·. The proposition is true.

Besides,

|

It is true.

Besides,

| The proposition is not true. |

.·. It is true that it is true,| But that it is not true is the | proposition.

| not true.

| .·. It is not true that it is | .·. That it is not true, is not not true;

|

true.

| But, the proposition is that it| .·. That it is true, is true. is not true,

| |

.·. The proposition is not true.

| .·. It is true.

|

.·. Whether it is true or not, it is both true and not.

.·. It is both true and not, which is absurd.

Peirce: CP 5.340 Cross-Ref:†† Since the conclusion is false, the reasoning is bad, or the premisses are not all true. But the reasoning is a dilemma; either, then, the disjunctive principle that it is either true or not is false, or the reasoning under one or the other branch is bad, or the reasoning is altogether valid. If the principle that it is either true or not is false, it is other than true and other than not true; that is, not true and not not true; that is, not true and true. But this is absurd. Hence, the disjunctive principle is valid. There are two arguments under each horn of the dilemma; both the arguments under one or the other branch must be false. But, in each case, the second argument involves all the premisses and forms of inference involved in the first; hence, if the first is false, the second necessarily is so. We may, therefore, confine our attention to the first arguments in the two branches. The forms of argument contained in these are two: first, the simple syllogism in Barbara, and, second, the consequence from the truth of a proposition to the proposition itself. These are both correct. Hence, the whole form of reasoning is correct, and nothing remains to be false but a premiss. But since the repetition of an alternative supposition is not a premiss, there is, properly speaking, but one premiss in the whole. This is that the proposition is the same as that that proposition is not true. This, then, must be false. Hence the proposition signifies either less or more than this. If it does not signify as much as this, it signifies nothing, and hence it is not true, and hence another proposition which says of it what it says of itself is true. But if the proposition in question signifies something more than that it is itself not true, then the premiss that

Whatever is said in the proposition is that it is not true,

is not true. And as a proposition is true only if whatever is said in it is true, but is false if anything said in it is false, the first argument on the second side of the dilemma contains a false premiss, and the second an undistributed middle. But the first argument on the first side remains good. Hence, if the proposition means more than that it is not true, it is not true, and another proposition which repeats this of it is true. Hence, whether the proposition does or does not mean that it is not true, it is not true, and a proposition which repeats this of it is true. Peirce: CP 5.340 Cross-Ref:†† Since this repeating proposition is true, it has a meaning. Now, a proposition has a meaning if any part of it has a meaning. Hence, the original proposition (a part of which repeated has a meaning) has itself a meaning. Hence, it must imply something besides that which it explicitly states. But it has no particular determination to any further implication. Hence, what more it signifies it must signify by virtue of being a proposition at all. That is to say, every proposition must imply something analogous to what this implies. Now, the repetition of this proposition does not contain this implication, for otherwise it could not be true; hence, what every proposition implies must be something concerning itself. What every proposition implies concerning itself must be something which is false of the proposition now under discussion, for the whole falsity of this proposition lies therein, since all that it explicitly lays down is true. It must be something which would not be false if the proposition were true, for in that case some true proposition would be false. Hence, it must be that it is itself true. That it is, every proposition asserts its own truth. Peirce: CP 5.340 Cross-Ref:†† The proposition in question, therefore, is true in all other respects but its implication of its own truth.†P1

Peirce: CP 5.341 Cross-Ref:†† §3. THE SOCIAL THEORY OF LOGICE

341. The difficulty of showing how the law of deductive reasoning is true depends upon our inability to conceive of its not being true. In the case of probable reasoning the difficulty is of quite another kind; here, where we see precisely what the procedure is, we wonder how such a process can have any validity at all. How magical it is that by examining a part of a class we can know what is true of the whole of the class, and by study of the past can know the future; in short, that we can know what we have not experienced! Peirce: CP 5.341 Cross-Ref:†† Is not this an intellectual intuition! Is it not that besides ordinary experience which is dependent on there being a certain physical connection between our organs and the thing experienced, there is a second avenue of truth dependent only on there being a certain intellectual connection between our previous knowledge and what we learn in that way? Yes, this is true. Man has this faculty, just as opium has a somnific virtue; but some further questions may be asked, nevertheless. How is the existence of this faculty accounted for? In one sense, no doubt, by natural selection. Since it is absolutely essential to the preservation of so delicate an organism as man's, no race which had it not has been able to sustain itself. This accounts for the prevalence of

this faculty, provided it was only a possible one. But how can it be possible? What could enable the mind to know physical things which do not physically influence it and which it does not influence? The question cannot be answered by any statement concerning the human mind, for it is equivalent to asking what makes the facts usually to be, as inductive and hypothetic conclusions from true premisses represent them to be? Facts of a certain kind are usually true when facts having certain relations to them are true; what is the cause of this? That is the question. Peirce: CP 5.342 Cross-Ref:†† 342. The usual reply is that nature is everywhere regular; as things have been, so they will be; as one part of nature is, so is every other. But this explanation will not do. Nature is not regular. No disorder would be less orderly than the existing arrangement. It is true that the special laws and regularities are innumerable; but nobody thinks of the irregularities, which are infinitely more frequent. Every fact true of any one thing in the universe is related to every fact true of every other. But the immense majority of these relations are fortuitous and irregular. A man in China bought a cow three days and five minutes after a Greenlander had sneezed. Is that abstract circumstance connected with any regularity whatever? And are not such relations infinitely more frequent than those which are regular? But if a very large number of qualities were to be distributed among a very large number of things in almost any way, there would chance to be some few regularities. If, for example, upon a checker-board of an enormous number of squares, painted all sorts of colors, myriads of dice were to be thrown, it could hardly fail to happen, that upon some color, or shade of color, out of so many, some one of the six numbers should not be uppermost on any die. This would be a regularity; for, the universal proposition would be true that upon that color that number is never turned up. But suppose this regularity abolished, then a far more remarkable regularity would be created, namely, that on every color every number is turned up. Either way, therefore, a regularity must occur. Indeed, a little reflection will show that, although we have here only variations of color and of the numbers of the dice, many regularities must occur. And the greater the number of objects, the more respects in which they vary, and the greater the number of varieties in each respect, the greater will be the number of regularities. Now, in the universe, all these numbers are infinite. Therefore, however disorderly the chaos, the number of regularities must be infinite. The orderliness of the universe, therefore, if it exists, must consist in the large proportion of relations which present a regularity to those which are quite irregular. But this proportion in the actual universe is, as we have seen, as small as it can be; and, therefore, the orderliness of the universe is as little as that of any arrangement whatever. Peirce: CP 5.343 Cross-Ref:†† 343. But even if there were such an orderliness in things, it never could be discovered. For it would belong to things either collectively or distributively. If it belonged to things collectively, that is to say, if things formed a system, the difficulty would be that a system can only be known by seeing some considerable proportion of the whole. Now we never can know how great a part of the whole of nature we have discovered. If the order were distributive, that is, belonged to all things only by belonging to each thing, the difficulty would be that a character can only be known by comparing something which has it †1 with something which has it not. Being, quality, relation, and other universals are not known except as characters of words or other signs, attributed by a figure of speech to things. Thus, in neither case could the order of things be known. But the order of things would not help the validity of our reasoning -- that is, would not help us to reason correctly -- unless we knew what the

order of things required the relation between the known reason from to the unknown reasoned to, to be. Peirce: CP 5.344 Cross-Ref:†† 344. But even if this order both existed and were known, the knowledge would be of no use except as a general principle, from which things could be deduced. It would not explain how knowledge could be increased (in contradistinction to being rendered more distinct), and so it would not explain how it could itself have been acquired. Peirce: CP 5.345 Cross-Ref:†† 345. Finally, if the validity of induction and hypothesis were dependent on a particular constitution of the universe, we could imagine a universe in which these modes of inference should not be valid, just as we can imagine a universe in which there would be no attraction, but things should merely drift about. Accordingly, J.S. Mill, who explains the validity of induction by the uniformity of nature,†P1 maintains that he can imagine a universe without any regularity, so that no probable inference would be valid in it.†P2 In the universe as it is, probable arguments sometimes fail, nor can any definite proportion of cases be stated in which they hold good; all that can be said is that in the long run they prove approximately correct. Can a universe be imagined in which this would not be the case? It must be a universe where probable argument can have some application, in order that it may fail half the time. It must, therefore, be a universe experienced. Of the finite number of propositions true of a finite amount of experience of such a universe, no one would be universal in form, unless the subject of it were an individual. For if there were a plural universal proposition, inferences by analogy from one particular to another would hold good invariably in reference to that subject. So that these arguments might be no better than guesses in reference to other parts of the universe, but they would invariably hold good in a finite proportion of it, and so would on the whole be somewhat better than guesses. There could, also, be no individuals in that universe, for there must be some general class -- that is, there must be some things more or less alike -- or probable argument would find no premisses there; therefore, there must be two mutually exclusive classes, since every class has a residue outside of it; hence, if there were any individual, that individual would be wholly excluded from one or other of these classes. Hence, the universal plural proposition would be true, that no one of a certain class was that individual. Hence, no universal proposition would be true. Accordingly, every combination of characters would occur in such a universe. But this would not be disorder, but the simplest order; it would not be unintelligible, but, on the contrary, everything conceivable would be found in it with equal frequency. The notion, therefore, of a universe in which probable arguments should fail as often as hold true, is absurd.†1 We can suppose it in general terms, but we cannot specify how it should be other than self-contradictory.†P1 Peirce: CP 5.346 Cross-Ref:†† 346. Since we cannot conceive of probable inferences as not generally holding good, and since no special supposition will serve to explain their validity, many logicians have sought to base this validity on that of deduction, and that in a variety of ways. The only attempt of this sort, however, which deserves to be noticed is that which seeks to determine the probability of a future event by the theory of probabilities, from the fact that a certain number of similar events have been observed. Whether this can be done or not depends on the meaning assigned to the word probability. But if this word is to be taken in such a sense that a form of

conclusion which is probable is valid; since the validity of an inference (or its correspondence with facts) consists solely in this, that when such premisses are true, such a conclusion is generally true, then probability can mean nothing but the ratio of the frequency of occurrence of a specific event to a general one over it. In this sense of the term, it is plain that the probability of an inductive conclusion cannot be deduced from the premisses; for from the inductive premisses

S', S'', S''' are M, S', S'', S''' are P,

nothing follows deductively, except that any M, which is S', or S'', or S''' is P; or, less explicitly, that some M is P. Peirce: CP 5.347 Cross-Ref:†† 347. Thus, we seem to be driven to this point. On the one hand, no determination of things, no fact, can result in the validity of probable argument; nor, on the other hand, is such argument reducible to that form which holds good, however the facts may be. This seems very much like a reduction to absurdity of the validity of such reasoning; and a paradox of the greatest difficulty is presented for solution. Peirce: CP 5.348 Cross-Ref:†† 348. There can be no doubt of the importance of this problem. According to Kant, the central question of philosophy is "How are synthetical judgments a priori possible?" But antecedently to this comes the question how synthetical judgments in general, and still more generally, how synthetical reasoning is possible at all. When the answer to the general problem has been obtained, the particular one will be comparatively simple. This is the lock upon the door of philosophy.†1 Peirce: CP 5.349 Cross-Ref:†† 349. All probable inference, whether induction or hypothesis, is inference from the parts to the whole. It is essentially the same, therefore, as statistical inference. Out of a bag of black and white beans I take a few handfuls, and from this sample I can judge approximately the proportions of black and white in the whole. This is identical with induction. Now we know upon what the validity of this inference depends. It depends upon the fact that in the long run, any one bean would be taken out as often as any other. For were this not so, the mean of a large number of results of such testings of the contents of the bag would not be precisely the ratio of the numbers of the two colors of beans in the bag. Now we may divide the question of the validity of induction into two parts: first, why of all inductions premisses for which occur, the generality should hold good, and second, why men are not fated always to light upon the small proportion of worthless inductions. Then, the first of these two questions is readily answered. For since all the members of any class are the same as all that are to be known; and since from any part of those which are to be known an induction is competent to the rest, in the long run any one member of a class will occur as the subject of a premiss of a possible induction as often as any other, and, therefore, the validity of induction depends simply upon the fact that the parts make up and constitute the whole. This in its turn depends simply upon there being such a state of things that any general terms are possible. But it has been shown in 311 that being at all is being in general. And thus this part of the validity of

induction depends merely on there being any reality. Peirce: CP 5.350 Cross-Ref:†† 350. From this it appears that we cannot say that the generality of inductions are true, but only that in the long run they approximate to the truth. This is the truth of the statement, that the universality of an inference from induction is only the analogue of true universality. Hence, also, it cannot be said that we know an inductive conclusion to be true, however loosely we state it; we only know that by accepting inductive conclusions, in the long run our errors balance one another. In fact, insurance companies proceed upon induction; -- they do not know what will happen to this or that policyholder; they only know that they are secure in the long run. Peirce: CP 5.351 Cross-Ref:†† 351. The other question relative to the validity of induction, is why men are not fated always to light upon those inductions which are highly deceptive. The explanation of the former branch of the problem we have seen to be that there is something real. Now, since if there is anything real, then (on account of this reality consisting in the ultimate agreement of all men, and on account of the fact that reasoning from parts to whole, is the only kind of synthetic reasoning which men possess) it follows necessarily that a sufficiently long succession of inferences from parts to whole will lead men to a knowledge of it, so that in that case they cannot be fated on the whole to be thoroughly unlucky in their inductions. This second branch of the problem is in fact equivalent to asking why there is anything real, and thus its solution will carry the solution of the former branch one step further. Peirce: CP 5.352 Cross-Ref:†† 352.†1 The answer to this question may be put into a general and abstract, or a special detailed form. If men were not to be able to learn from induction, it must be because as a general rule, when they had made an induction, the order of things (as they appear in experience), would then undergo a revolution. Just herein would the unreality of such a universe consist; namely, that the order of the universe should depend on how much men should know of it. But this general rule would be capable of being itself discovered by induction; and so it must be a law of such a universe, that when this was discovered it would cease to operate. But this second law would itself be capable of discovery. And so in such a universe there would be nothing which would not sooner or later be known; and it would have an order capable of discovery by a sufficiently long course of reasoning. But this is contrary to the hypothesis, and therefore that hypothesis is absurd. This is the particular answer. But we may also say, in general, that if nothing real exists, then, since every question supposes that something exists -- for it maintains its own urgency -- it supposes only illusions to exist. But the existence even of an illusion is a reality; for an illusion affects all men, or it does not. In the former case, it is a reality according to our theory of reality; in the latter case, it is independent of the state of mind of any individuals except those whom it happens to affect. So that the answer to the question, Why is anything real? is this: That question means, "supposing anything to exist, why is something real?" The answer is, that that very existence is reality by definition. Peirce: CP 5.352 Cross-Ref:†† All that has here been said, particularly of induction, applies to all inference from parts to whole, and therefore to hypothesis, and so to all probable inference. Peirce: CP 5.352 Cross-Ref:†† Thus, I claim to have shown, in the first place, that it is possible to hold a

consistent theory of the validity of the laws of ordinary logic. Peirce: CP 5.353 Cross-Ref:†† 353. But now let us suppose the idealistic theory of reality, which I have in this paper taken for granted to be false. In that case, inductions would not be true unless the world were so constituted that every object should be presented in experience as often as any other; and further, unless we were so constituted that we had no more tendency to make bad inductions than good ones. These facts might be explained by the benevolence of the Creator; but, as has already been argued, they could not explain, but are absolutely refuted by the fact that no state of things can be conceived in which probable arguments should not lead to the truth. This affords a most important argument in favor of that theory of reality, and thus of those denials of certain faculties from which it was deduced, as well as of the general style of philosophizing by which those denials were reached. Peirce: CP 5.354 Cross-Ref:†† 354. Upon our theory of reality and of logic, it can be shown that no inference of any individual can be thoroughly logical without certain determinations of his mind which do not concern any one inference immediately; for we have seen that that mode of inference which alone can teach us anything, or carry us at all beyond what was implied in our premisses -- in fact, does not give us to know any more than we knew before; only, we know that, by faithfully adhering to that mode of inference, we shall, on the whole, approximate to the truth. Each of us is an insurance company, in short. But, now, suppose that an insurance company, among its risks, should take one exceeding in amount the sum of all the others. Plainly, it would then have no security whatever. Now, has not every single man such a risk? What shall it profit a man if he shall gain the whole world and lose his own soul? If a man has a transcendent personal interest infinitely outweighing all others, then, upon the theory of validity of inference just developed, he is devoid of all security, and can make no valid inference whatever. What follows? That logic rigidly requires, before all else, that no determinate fact, nothing which can happen to a man's self, should be of more consequence to him than everything else. He who would not sacrifice his own soul to save the whole world, is illogical in all his inferences, collectively. So the social principle is rooted intrinsically in logic.†1 Peirce: CP 5.355 Cross-Ref:†† 355. That being the case, it becomes interesting to inquire how it is with men as a matter of fact. There is a psychological theory that man cannot act without a view to his own pleasure. This theory is based on a falsely assumed subjectivism. Upon our principles of the objectivity of knowledge, it could not be based; and if they are correct, it is reduced to an absurdity. It seems to me that the usual opinion of the selfishness of man is based in large measure upon this false theory. I do not think that the facts bear out the usual opinion. The immense self-sacrifices which the most wilful men often make, show that wilfulness is a very different thing from selfishness. The care that men have for what is to happen after they are dead, cannot be selfish. And finally and chiefly, the constant use of the word "we" -- as when we speak of our possessions on the Pacific -- our destiny as a republic -- in cases in which no personal interests at all are involved, show conclusively that men do not make their personal interests their only ones, and therefore may, at least, subordinate them to the interests of the community. Peirce: CP 5.356 Cross-Ref:†† 356. But just the revelation of the possibility of this complete self-sacrifice in

man, and the belief in its saving power, will serve to redeem the logicality of all men. For he who recognizes the logical necessity of complete self-identification of one's own interests with those of the community, and its potential existence in man, even if he has it not himself, will perceive that only the inferences of that man who has it are logical, and so views his own inferences as being valid only so far as they would be accepted by that man. But so far as he has this belief, he becomes identified with that man. And that ideal perfection of knowledge by which we have seen that reality is constituted must thus belong to a community in which this identification is complete. Peirce: CP 5.357 Cross-Ref:†† 357. This would serve as a complete establishment of private logicality, were it not that the assumption, that man or the community (which may be wider than man) shall ever arrive at a state of information greater than some definite finite information, is entirely unsupported by reasons. There cannot be a scintilla of evidence to show that at some time all living beings shall not be annihilated at once, and that forever after there shall be throughout the universe any intelligence whatever. Indeed, this very assumption involves itself a transcendent and supreme interest, and therefore from its very nature is unsusceptible of any support from reasons. This infinite hope which we all have (for even the atheist will constantly betray his calm expectation that what is Best will come about) is something so august and momentous, that all reasoning in reference to it is a trifling impertinence. We do not want to know what are the weights of reasons pro and con -- that is, how much odds we should wish to receive on such a venture in the long run -- because there is no long run in the case; the question is single and supreme, and ALL is at stake upon it. We are in the condition of a man in a life and death struggle; if he have not sufficient strength, it is wholly indifferent to him how he acts, so that the only assumption upon which he can act rationally is the hope of success. So this sentiment is rigidly demanded by logic. If its object were any determinate fact, any private interest, it might conflict with the results of knowledge and so with itself; but when its object is of a nature as wide as the community can turn out to be, it is always a hypothesis uncontradicted by facts and justified by its indispensableness for making any action rational.

Peirce: CP 5.358 Cross-Ref:†† IV

THE FIXATION OF BELIEF†1

§1. SCIENCE AND LOGICE

358. Few persons care to study logic, because everybody conceives himself to be proficient enough in the art of reasoning already. But I observe that this satisfaction is limited to one's own ratiocination, and does not extend to that of other men. Peirce: CP 5.359 Cross-Ref:††

359. We come to the full possession of our power of drawing inferences, the last of all our faculties; for it is not so much a natural gift as a long and difficult art. The history of its practice would make a grand subject for a book. The medieval schoolman, following the Romans, made logic the earliest of a boy's studies after grammar, as being very easy. So it was as they understood it. Its fundamental principle, according to them, was, that all knowledge rests either on †1 authority or reason; but that whatever is deduced by reason depends ultimately on a premiss derived from authority. Accordingly, as soon as a boy was perfect in the syllogistic procedure, his intellectual kit of tools was held to be complete. Peirce: CP 5.360 Cross-Ref:†† 360. To Roger Bacon,†2 that remarkable mind who in the middle of the thirteenth century was almost a scientific man, the schoolmen's conception of reasoning appeared only an obstacle to truth. He saw that experience alone teaches anything -- a proposition which to us seems easy to understand, because a distinct conception of experience has been handed down to us from former generations; which to him likewise †3 seemed perfectly clear, because its difficulties had not yet unfolded themselves. Of all kinds of experience, the best, he thought, was interior illumination, which teaches many things about Nature which the external senses could never discover, such as the transubstantiation of bread. Peirce: CP 5.361 Cross-Ref:†† 361. Four centuries later, the more celebrated Bacon, in the first book of his Novum Organum, gave his clear account of experience as something which must be open to verification and reexamination. But, superior as Lord Bacon's conception is to earlier notions, a modern reader who is not in awe of his grandiloquence is chiefly struck by the inadequacy of his view of scientific procedure. That we have only to make some crude experiments, to draw up briefs of the results in certain blank forms, to go through these by rule, checking off everything disproved and setting down the alternatives, and that thus in a few years physical science would be finished up -what an idea! "He wrote on science like a Lord Chancellor,"†P1 indeed, as Harvey, a genuine man of science said.†4 Peirce: CP 5.362 Cross-Ref:†† 362. The early scientists, Copernicus, Tycho Brahe, Kepler, Galileo, Harvey †5, and Gilbert, had methods more like those of their modern brethren. Kepler undertook to draw a curve through the places of Mars,†P2 and to state the times occupied by the planet in describing the different parts of that curve;†1 but perhaps †2 his greatest service to science was in impressing on men's minds that this was the thing to be done if they wished to improve astronomy; that they were not to content themselves with inquiring whether one system of epicycles was better than another but that they were to sit down to the figures and find out what the curve, in truth, was. He accomplished this by his incomparable energy and courage, blundering along in the most inconceivable way (to us), from one irrational hypothesis to another, until, after trying twenty-two of these, he fell, by the mere exhaustion of his invention, upon the orbit which a mind well furnished with the weapons of modern logic would have tried almost at the outset.†P1 Peirce: CP 5.363 Cross-Ref:†† 363. In the same way, every work of science great enough to be well †3 remembered for a few generations affords some exemplification of the defective state of the art of reasoning of the time when it was written; and each chief step in science has been a lesson in logic. It was so when Lavoisier and his contemporaries took up

the study of Chemistry. The old chemist's maxim had been, "Lege, lege, lege, labora, ora, et relege." Lavoisier's method was not to read and pray, but to dream that some long and complicated chemical process would have a certain effect, to put it into practice with dull patience, after its inevitable failure, to dream that with some modification it would have another result, and to end by publishing the last dream as a fact: his way was to carry his mind into his laboratory, and literally †4 to make of his alembics and cucurbits instruments of thought, giving a new conception of reasoning as something which was to be done with one's eyes open, in †5 manipulating real things instead of words and fancies. Peirce: CP 5.364 Cross-Ref:†† 364. The Darwinian controversy is, in large part, a question of logic. Mr. Darwin proposed to apply the statistical method to biology.†P1 The same thing has been done in a widely different branch of science, the theory of gases. Though unable to say what the movements of any particular molecule of gas would be on a certain hypothesis regarding the constitution of this class of bodies, Clausius and Maxwell were yet able, eight years before the publication of Darwin's immortal work,†1 by the application of the doctrine of probabilities, to predict that in the long run such and such a proportion of the molecules would, under given circumstances, acquire such and such velocities; that there would take place, every second, such and such a relative †2 number of collisions, etc.; and from these propositions were able to deduce certain properties of gases, especially in regard to their heat-relations. In like manner, Darwin, while unable to say what the operation of variation and natural selection in any individual case will be, demonstrates that in the long run they will, or †3 would,†4 adapt animals to their circumstances. Whether or not existing animal forms are due to such action, or what position the theory ought to take, forms the subject of a discussion in which questions of fact and questions of logic are curiously interlaced.

Peirce: CP 5.365 Cross-Ref:†† §2. GUIDING PRINCIPLESE

365. The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know. Consequently, reasoning is good if it be such as to †P2 give a true conclusion from true premisses, and not otherwise. Thus, the question of validity is purely one of fact and not of thinking. A being the facts stated in the †5 premisses and B being that concluded,†1 the question is, whether these facts are really so related that if A were B would generally be.†2 If so, the inference is valid; if not, not. It is not in the least the question whether, when the premisses are accepted by the mind, we feel an impulse to accept the conclusion also. It is true that we do generally reason correctly by nature. But that is an accident; the true conclusion would remain true if we had no impulse to accept it; and the false one would remain false, though we could not resist the tendency to believe in it. Peirce: CP 5.366 Cross-Ref:†† 366. We are, doubtless, in the main logical animals, but we are not perfectly so. Most of us, for example, are naturally more sanguine and hopeful than logic would justify. We seem to be so constituted that in the absence of any facts to go upon we are happy and self-satisfied; so that the effect of experience is continually to contract our hopes and aspirations. Yet a lifetime of the application of this corrective does not

usually eradicate our sanguine disposition. Where hope is unchecked by any experience, it is likely that our optimism is extravagant. Logicality in regard to practical matters (if this be understood, not in the old sense, but as consisting in a wise union of security with fruitfulness of reasoning †3) is the most useful quality an animal can possess, and might, therefore, result from the action of natural selection; but outside of these it is probably of more advantage to the animal to have his mind filled with pleasing and encouraging visions, independently of their truth; and thus, upon unpractical subjects, natural selection might occasion a fallacious tendency of thought.†P1 Peirce: CP 5.367 Cross-Ref:†† 367. That which determines us, from given premisses, to draw one inference rather than another, is some habit of mind, whether it be constitutional or acquired. The habit is good or otherwise, according as it produces true conclusions from true premisses or not; and an inference is regarded as valid or not, without reference to the truth or falsity of its conclusion specially, but according as the habit which determines it is such as to produce true conclusions in general or not. The particular habit of mind which governs this or that inference may be formulated in a proposition whose truth depends on the validity of the inferences which the habit determines; and such a formula is called a guiding principle of inference. Suppose, for example, that we observe that a rotating disk of copper quickly comes to rest when placed between the poles of a magnet, and we infer that this will happen with every disk of copper. The guiding principle is, that what is true of one piece of copper is true of another. Such a guiding principle with regard to copper would be much safer than with regard to many other substances -- brass, for example. Peirce: CP 5.368 Cross-Ref:†† 368. A book might be written to signalize all the most important of these guiding principles of reasoning. It would probably be, we must confess, of no service to a person whose thought is directed wholly to practical subjects, and whose activity moves along thoroughly-beaten paths. The problems that †1 present themselves to such a mind are matters of routine which he has learned once for all to handle in learning his business. But let a man venture into an unfamiliar field, or where his results are not continually checked by experience, and all history shows that the most masculine intellect will ofttimes lose his orientation and waste his efforts in directions which bring him no nearer to his goal, or even carry him entirely astray. He is like a ship in the open sea, with no one on board who understands the rules of navigation. And in such a case some general study of the guiding principles of reasoning would be sure to be found useful. Peirce: CP 5.369 Cross-Ref:†† 369. The subject could hardly be treated, however, without being first limited; since almost any fact may serve as a guiding principle. But it so happens that there exists a division among facts, such that in one class are all those which are absolutely essential as guiding principles, while in the others are all which have any other interest as objects of research. This division is between those which are necessarily taken for granted in asking why †2 a certain conclusion is thought to follow †1 from certain premisses, and those which are not implied in such a †2 question. A moment's thought will show that a variety of facts are already assumed when the logical question is first asked. It is implied, for instance, that there are such states of mind as doubt and belief -- that a passage from one to the other is possible, the object of thought remaining the same, and that this transition is subject to some rules by †3

which all minds are alike bound. As these are facts which we must already know before we can have any clear conception of reasoning at all, it cannot be supposed to be any longer of much interest to inquire into their truth or falsity. On the other hand, it is easy to believe that those rules of reasoning which are deduced from the very idea of the process are the ones which are the most essential; and, indeed, that so long as it conforms to these it will, at least, not lead to false conclusions from true premisses. In point of fact, the importance of what may be deduced from the assumptions involved in the logical question turns out to be greater than might be supposed, and this for reasons which it is difficult to exhibit at the outset. The only one which I shall here mention is, that conceptions which are really products of logical reflection, without being readily seen to be so, mingle with our ordinary thoughts, and are frequently the causes of great confusion. This is the case, for example, with the conception of quality. A quality, as such, is never an object of observation. We can see that a thing is blue or green, but the quality of being blue and the quality of being green are not things which we see; they are products of logical reflections. The truth is, that common-sense, or thought as it first emerges above the level of the narrowly practical, is deeply imbued with that bad logical quality to which the epithet metaphysical is commonly applied; and nothing can clear it up but a severe course of logic.

Peirce: CP 5.370 Cross-Ref:†† §3. DOUBT AND BELIEFE

370. We generally know when we wish to ask a question and when we wish to pronounce a judgment, for there is a dissimilarity between the sensation of doubting and that of believing. Peirce: CP 5.371 Cross-Ref:†† 371. But this is not all which distinguishes doubt from belief. There is a practical difference. Our beliefs guide our desires and shape our actions. The Assassins, or followers of the Old Man of the Mountain, used to rush into death at his least command, because they believed that obedience to him would insure everlasting felicity. Had they doubted this, they would not have acted as they did. So it is with every belief, according to its degree. The feeling of believing is a more or less sure indication of there being established in our nature some habit which will determine our actions.†P1 Doubt never has such an effect. Peirce: CP 5.372 Cross-Ref:†† 372. Nor must we overlook a third point of difference. Doubt is an uneasy and dissatisfied state from which we struggle to free ourselves and pass into the state of belief;†P2 while the latter is a calm and satisfactory state which we do not wish to avoid, or to change to a belief in anything else.†P3 On the contrary, we cling tenaciously, not merely to believing, but to believing just what we do believe. Peirce: CP 5.373 Cross-Ref:†† 373. Thus, both doubt and belief have positive effects upon us, though very different ones. Belief does not make us act at once, but puts us into such a condition that we shall behave in some †1 certain way, when the occasion arises. Doubt has not the least such †2 active †3 effect,†4 but stimulates us to inquiry †5 until it is destroyed. This reminds us of the irritation of a nerve and the reflex action produced

thereby; while for the analogue of belief, in the nervous system, we must look to what are called nervous associations †6 -- for example, to that habit of the nerves in consequence of which the smell of a peach will make the mouth water.†P1

Peirce: CP 5.374 Cross-Ref:†† §4. THE END OF INQUIRYE

374. The irritation of doubt causes a struggle to attain a state of belief. I shall term this struggle Inquiry, though it must be admitted that this is sometimes not a very apt designation. Peirce: CP 5.375 Cross-Ref:†† 375. The irritation of doubt is the only immediate motive for the struggle to attain belief. It is certainly best for us that our beliefs should be such as may truly guide our actions so as to satisfy our desires; and this reflection will make us reject every †1 belief which does not seem to have been so formed as to insure this result. But it will only do so by creating a doubt in the place of that belief.†P1 With the doubt, therefore, the struggle begins, and with the cessation of doubt it ends. Hence, the sole object of inquiry is the settlement of opinion. We may fancy that this is not enough for us, and that we seek, not merely an opinion, but a true opinion. But put this fancy to the test, and it proves groundless; for as soon as a firm belief is reached we are entirely satisfied, whether the belief be true or false. And it is clear that nothing out of the sphere of our knowledge can be our object, for nothing which does not affect the mind can be the motive for mental effort. The most that can be maintained is, that we seek for a belief that we shall think to be true. But we think each one of our beliefs to be true, and, indeed, it is mere tautology to say so.†P2 Peirce: CP 5.375 Cross-Ref:†† That the settlement of opinion is the sole end of inquiry is a very important proposition. It sweeps away, at once, various vague and erroneous conceptions of proof. A few of these may be noticed here. Peirce: CP 5.376 Cross-Ref:†† 376. 1. Some philosophers have imagined that to start an inquiry it was only necessary to utter a question whether orally or by setting †2 it down upon paper, and have even recommended us to begin our studies with questioning everything! But the mere putting of a proposition into the interrogative form does not stimulate the mind to any struggle after belief. There must be a real and living doubt, and without this all discussion is idle.†P3 Peirce: CP 5.376 Cross-Ref:†† 2. It is a very common idea that a demonstration must rest on some ultimate and absolutely indubitable propositions. These, according to one school, are first principles of a general nature; according to another, are first sensations. But, in point of fact, an inquiry, to have that completely satisfactory result called demonstration, has only to start with propositions perfectly free from all actual doubt. If the premisses are not in fact doubted at all, they cannot be more satisfactory than they are.†P1 Peirce: CP 5.376 Cross-Ref:††

3. Some people seem to love to argue a point after all the world is fully convinced of it. But no further advance can be made. When doubt ceases, mental action on the subject comes to an end; and, if it did go on, it would be without a purpose.†P2

Peirce: CP 5.377 Cross-Ref:†† §5. METHODS OF FIXING BELIEFE

377. If the settlement of opinion is the sole object of inquiry, and if belief is of the nature of a habit, why should we not attain the desired end, by taking as †1 answer to a question any †2 we may fancy, and constantly reiterating it to ourselves, dwelling on all which may conduce to that belief, and learning to turn with contempt and hatred from anything that †3 might disturb it? This simple and direct method is really pursued by many men. I remember once being entreated not to read a certain newspaper lest it might change my opinion upon free-trade. "Lest I might be entrapped by its fallacies and misstatements," was the form of expression. "You are not," my friend said, "a special student of political economy. You might, therefore, easily be deceived by fallacious arguments upon the subject. You might, then, if you read this paper, be led to believe in protection. But you admit that free-trade is the true doctrine; and you do not wish to believe what is not true." I have often known this system to be deliberately adopted. Still oftener, the instinctive dislike of an undecided state of mind, exaggerated into a vague dread of doubt, makes men cling spasmodically to the views they already take. The man feels that, if he only holds to his belief without wavering, it will be entirely satisfactory. Nor can it be denied that a steady and immovable faith yields great peace of mind. It may, indeed, give rise to inconveniences, as if a man should resolutely continue to believe that fire would not burn him, or that he would be eternally damned if he received his ingesta otherwise than through a stomach-pump. But then the man who adopts this method will not allow that its inconveniences are greater than its advantages. He will say, "I hold steadfastly to the truth, and the truth is always wholesome." And in many cases it may very well be that the pleasure he derives from his calm faith overbalances any inconveniences resulting from its deceptive character. Thus, if it be true that death is annihilation, then the man who believes that he will certainly go straight to heaven when he dies, provided he have fulfilled certain simple observances in this life, has a cheap pleasure which will not be followed by the least disappointment.†P1 A similar consideration seems to have weight with many persons in religious topics, for we frequently hear it said, "Oh, I could not believe so-and-so, because I should be wretched if I did." When an ostrich buries its head in the sand as danger approaches, it very likely takes the happiest course. It hides the danger, and then calmly says there is no danger; and, if it feels perfectly sure there is none, why should it raise its head to see? A man may go through life, systematically keeping out of view all that might cause a change in his opinions, and if he only succeeds -- basing his method, as he does, on two fundamental psychological laws -- I do not see what can be said against his doing so. It would be an egotistical impertinence to object that his procedure is irrational, for that only amounts to saying that his method of settling belief is not ours. He does not propose to himself to be rational, and, indeed, will often talk with scorn of man's weak and illusive reason. So let him think as he pleases. Peirce: CP 5.378 Cross-Ref:††

378. But this method of fixing belief, which may be called the method of tenacity, will be unable to hold its ground in practice. The social impulse is against it. The man who adopts it will find that other men think differently from him, and it will be apt to occur to him, in some saner moment, that their opinions are quite as good as his own, and this will shake his confidence in his belief. This conception, that another man's thought or sentiment may be equivalent to one's own, is a distinctly new step, and a highly important one. It arises from an impulse too strong in man to be suppressed, without danger of destroying the human species. Unless we make ourselves hermits, we shall necessarily influence each other's opinions; so that the problem becomes how to fix belief, not in the individual merely, but in the community. Peirce: CP 5.379 Cross-Ref:†† 379. Let the will of the state act, then, instead of that of the individual. Let an institution be created which shall have for its object to keep correct doctrines before the attention of the people, to reiterate them perpetually, and to teach them to the young; having at the same time power to prevent contrary doctrines from being taught, advocated, or expressed. Let all possible causes of a change of mind be removed from men's apprehensions. Let them be kept ignorant, lest they should learn of some reason to think otherwise than they do. Let their passions be enlisted, so that they may regard private and unusual opinions with hatred and horror. Then, let all men who reject the established belief be terrified into silence. Let the people turn out and tar-and-feather such men, or let inquisitions be made into the manner of thinking of suspected persons, and when they are found guilty of forbidden beliefs, let them be subjected to some signal punishment. When complete agreement could not otherwise be reached, a general massacre of all who have not thought in a certain way has proved a very effective means of settling opinion in a country. If the power to do this be wanting, let a list of opinions be drawn up, to which no man of the least independence of thought can assent, and let the faithful be required to accept all these propositions, in order to segregate them as radically as possible from the influence of the rest of the world. Peirce: CP 5.379 Cross-Ref:†† This method has, from the earliest times, been one of the chief means of upholding correct theological and political doctrines, and of preserving their universal or catholic character. In Rome, especially, it has been practised from the days of Numa Pompilius to those of Pius Nonus. This is the most perfect example in history; but wherever there is a priesthood -- and no religion has been without one -- this method has been more or less made use of. Wherever there is an aristocracy, or a guild, or any association of a class of men whose interests depend, or are supposed to depend, on certain propositions, there will be inevitably found some traces of this natural product of social feeling. Cruelties always accompany this system; and when it is consistently carried out, they become atrocities of the most horrible kind in the eyes of any rational man. Nor should this occasion surprise, for the officer of a society does not feel justified in surrendering the interests of that society for the sake of mercy, as he might his own private interests. It is natural, therefore, that sympathy and fellowship should thus produce a most ruthless power. Peirce: CP 5.380 Cross-Ref:†† 380. In judging this method of fixing belief, which may be called the method of authority, we must, in the first place, allow its immeasurable mental and moral superiority to the method of tenacity. Its success is proportionately greater; and, in

fact, it has over and over again worked the most majestic results. The mere structures of stone which it has caused to be put together -- in Siam, for example, in Egypt, and in Europe -- have many of them a sublimity hardly more than rivaled by the greatest works of Nature. And, except the geological epochs, there are no periods of time so vast as those which are measured by some of these organized faiths.†P1 If we scrutinize the matter closely, we shall find that there has not been one of their creeds which has remained always the same; yet the change is so slow as to be imperceptible during one person's life, so that individual belief remains sensibly fixed. For the mass of mankind, then, there is perhaps no better method than this. If it is their highest impulse to be intellectual slaves, then slaves they ought to remain. Peirce: CP 5.381 Cross-Ref:†† 381. But no institution can undertake to regulate opinions upon every subject. Only the most important ones can be attended to, and on the rest men's minds must be left to the action of natural causes. This imperfection will be no source of weakness so long as men are in such a state of culture that one opinion does not influence another -- that is, so long as they cannot put two and two together. But in the most priest-ridden states some individuals will be found who are raised above that condition. These men possess a wider sort of social feeling; they see that men in other countries and in other ages have held to very different doctrines from those which they themselves have been brought up to believe; and they cannot help seeing that it is the mere accident of their having been taught as they have, and of their having been surrounded with the manners and associations they have, that has caused them to believe as they do and not far differently. Nor can their candour †1 resist the reflection that there is no reason to rate their own views at a higher value than those of other nations and other centuries; thus giving †2 rise to doubts in their minds. Peirce: CP 5.382 Cross-Ref:†† 382. They will further perceive that such doubts as these must exist in their minds with reference to every belief which seems to be determined by the caprice either of themselves or of those who originated the popular opinions. The willful adherence to a belief, and the arbitrary forcing of it upon others, must, therefore, both be given up. A different †3 new method of settling opinions must be adopted, that †4 shall not only produce an impulse to believe, but shall also decide what proposition it is which is to be believed. Let the action of natural preferences be unimpeded, then, and under their influence let men, conversing together and regarding matters in different lights, gradually develop beliefs in harmony with natural causes. This method resembles that by which conceptions of art have been brought to maturity. The most perfect example of it is to be found in the history of metaphysical philosophy. Systems of this sort have not usually rested upon any observed facts, at least not in any great degree. They have been chiefly adopted because their fundamental propositions seemed "agreeable to reason." This is an apt expression; it does not mean that which agrees with experience, but that which we find ourselves inclined to believe. Plato, for example, finds it agreeable to reason that the distances of the celestial spheres from one another should be proportional to the different lengths of strings which produce harmonious chords. Many philosophers have been led to their main conclusions by considerations like this;†P1 but this is the lowest and least developed form which the method takes, for it is clear that another man might find Kepler's theory, that the celestial spheres are proportional to the inscribed and circumscribed spheres of the different regular solids, more agreeable to his reason. But the shock of opinions will soon lead men to rest on preferences of a far more universal nature. Take, for example, the doctrine that man only acts selfishly -- that is,

from the consideration that acting in one way will afford him more pleasure than acting in another. This rests on no fact in the world, but it has had a wide acceptance as being the only reasonable theory.†P1 Peirce: CP 5.383 Cross-Ref:†† 383. This method is far more intellectual and respectable from the point of view of reason than either of the others which we have noticed. Indeed, as long as no better method can be applied, it ought to be followed, since it is then the expression of instinct which must be the ultimate cause of belief in all cases.†1 But its failure has been the most manifest. It makes of inquiry something similar to the development of taste; but taste, unfortunately, is always more or less a matter of fashion, and accordingly metaphysicians have never come to any fixed agreement, but the pendulum has swung backward and forward between a more material and a more spiritual philosophy, from the earliest times to the latest. And so from this, which has been called the a priori method, we are driven, in Lord Bacon's phrase, to a true induction. We have examined into this a priori method as something which promised to deliver our opinions from their accidental and capricious element. But development, while it is a process which eliminates the effect of some casual circumstances, only magnifies that of others. This method, therefore, does not differ in a very essential way from that of authority. The government may not have lifted its finger to influence my convictions; I may have been left outwardly quite free to choose, we will say, between monogamy and polygamy, and, appealing to my conscience only, I may have concluded that the latter practice is in itself licentious. But when I come to see that the chief obstacle to the spread of Christianity among a people of as high culture as the Hindoos has been a conviction of the immorality of our way of treating women, I cannot help seeing that, though governments do not interfere, sentiments in their development will be very greatly determined by accidental causes. Now, there are some people, among whom I must suppose that my reader is to be found, who, when they see that any belief of theirs is determined by any circumstance extraneous to the facts, will from that moment not merely admit in words that that belief is doubtful, but will experience a real doubt of it, so that it ceases in some degree at least †1 to be a belief. Peirce: CP 5.384 Cross-Ref:†† 384. To satisfy our doubts, therefore, it is necessary that a method should be found by which our beliefs may be determined †2 by nothing human, but by some external permanency -- by something upon which our thinking has no effect.†P1 Some mystics imagine that they have such a method in a private inspiration from on high. But that is only a form of the method of tenacity, in which the conception of truth as something public is not yet developed. Our external permanency would not be external, in our sense, if it was restricted in its influence to one individual. It must be something which affects, or might affect, every man. And, though these affections are necessarily as various as are individual conditions, yet the method must be such that the ultimate conclusion of every man shall be the same.†P1 Such is the method of science. Its fundamental hypothesis, restated in more familiar language, is this: There are Real things, whose characters are entirely independent of our opinions about them; those Reals †1 affect our senses according to regular laws, and, though our sensations are as different as are †2 our relations to the objects, yet, by taking advantage of the laws of perception, we can ascertain by reasoning how things really and †3 truly †4 are; and any man, if he have sufficient experience and he †5 reason enough about it, will be led to the one True conclusion. The new conception here involved is that of Reality. It may be asked how I know that there are any Reals.†6 If

this hypothesis is the sole support of my method of inquiry, my method of inquiry must not be used to support my hypothesis. The reply †7 is this: 1. If investigation cannot be regarded as proving that there are Real things, it at least does not lead to a contrary conclusion; but the method and the conception on which it is based remain ever in harmony. No doubts of the method, therefore, necessarily arise from its practice, as is the case with all the others. 2. The feeling which gives rise to any method of fixing belief is a dissatisfaction at two repugnant propositions. But here already is a vague concession that there is some one thing which a proposition should represent.†8 Nobody, therefore, can really doubt that there are Reals,†9 for,†10 if he did, doubt would not be a source of dissatisfaction. The hypothesis, therefore, is one which every mind admits. So that the social impulse does not cause men †11 to doubt it. 3. Everybody uses the scientific method about a great many things, and only ceases to use it when he does not know how to apply it. 4. Experience of the method has not led us †12 to doubt it, but, on the contrary, scientific investigation has had the most wonderful triumphs in the way of settling opinion. These afford the explanation of my not doubting the method or the hypothesis which it supposes; and not having any doubt, nor believing that anybody else whom I could influence has, it would be the merest babble for me to say more about it. If there be anybody with a living doubt upon the subject, let him consider it.†P1 Peirce: CP 5.385 Cross-Ref:†† 385. To describe the method of scientific investigation is the object of this series of papers. At present I have only room to notice some points of contrast between it and other methods of fixing belief. Peirce: CP 5.385 Cross-Ref:†† This is the only one of the four methods which presents any distinction of a right and a wrong way. If I adopt the method of tenacity, and shut myself out from all influences, whatever I think necessary to doing this, is necessary according to that method. So with the method of authority: the state may try to put down heresy by means which, from a scientific point of view, seem very ill-calculated to accomplish its purposes; but the only test on that method is what the state thinks; so that it cannot pursue the method wrongly. So with the a priori method. The very essence of it is to think as one is inclined to think. All metaphysicians will be sure to do that, however they may be inclined to judge each other to be perversely wrong. The Hegelian system recognizes every natural tendency of thought as logical, although it be certain to be abolished by counter-tendencies. Hegel thinks there is a regular system in the succession of these tendencies, in consequence of which, after drifting one way and the other for a long time, opinion will at last go right. And it is true that metaphysicians do †1 get the right ideas at last; Hegel's system of Nature represents tolerably the science of his †2 day; and one may be sure that whatever scientific investigation shall have †3 put out of doubt will presently receive a priori demonstration on the part of the metaphysicians. But with the scientific method the case is different. I may start with known and observed facts to proceed to the unknown; and yet the rules which I follow in doing so may not be such as investigation would approve. The test of whether I am truly following the method is not an immediate appeal to my feelings and purposes, but, on the contrary, itself involves the application of the method. Hence it is that bad reasoning as well as good reasoning is possible; and this fact is the foundation of the practical side of logic. Peirce: CP 5.386 Cross-Ref:†† 386. It is not to be supposed that the first three methods of settling opinion

present no advantage whatever over the scientific method. On the contrary, each has some peculiar convenience of its own. The a priori method is distinguished for its comfortable conclusions. It is the nature of the process to adopt whatever belief we are inclined to, and there are certain flatteries to the vanity of man which we all believe by nature, until we are awakened from our pleasing dream by †4 rough facts. The method of authority will always govern the mass of mankind; and those who wield the various forms of organized force in the state will never be convinced that dangerous reasoning ought not to be suppressed in some way. If liberty of speech is to be untrammeled from the grosser forms of constraint, then uniformity of opinion will be secured by a moral terrorism to which the respectability of society will give its thorough approval. Following the method of authority is the path of peace. Certain non-conformities are permitted; certain others (considered unsafe) are forbidden. These are different in different countries and in different ages; but, wherever you are, let it be known that you seriously hold a tabooed belief, and you may be perfectly sure of being treated with a cruelty less brutal but more refined than hunting you like a wolf. Thus, the greatest intellectual benefactors of mankind have never dared, and dare not now, to utter the whole of their thought; and thus a shade of prima facie doubt is cast upon every proposition which is considered essential to the security of society. Singularly enough, the persecution does not all come from without; but a man torments himself and is oftentimes most distressed at finding himself believing propositions which he has been brought up to regard with aversion. The peaceful and sympathetic man will, therefore, find it hard to resist the temptation to submit his opinions to authority. But most of all I admire the method of tenacity for its strength, simplicity, and directness. Men who pursue it are distinguished for their decision of character, which becomes very easy with such a mental rule. They do not waste time in trying to make up their minds what they want, but, fastening like lightning upon whatever alternative comes first, they hold to it to the end, whatever happens, without an instant's irresolution. This is one of the splendid qualities which generally accompany brilliant, unlasting success. It is impossible not to envy the man who can dismiss reason, although we know how it must turn out at last. Peirce: CP 5.387 Cross-Ref:†† 387. Such are the advantages which the other methods of settling opinion have over scientific investigation. A man should consider well of them; and then he should consider that, after all, he wishes his opinions to coincide with the fact, and that there is no reason why the results of those three first †1 methods should do so. To bring about this effect is the prerogative of the method of science. Upon such considerations he has to make his choice -- a choice which is far more than the adoption of any intellectual opinion, which is one of the ruling decisions of his life, to which, when once made, he is bound to adhere. The force of habit will sometimes cause a man to hold on to old beliefs, after he is in a condition to see that they have no sound basis. But reflection upon the state of the case will overcome these habits, and he ought to allow reflection its full weight. People sometimes shrink from doing this, having an idea that beliefs are wholesome which they cannot help feeling rest on nothing. But let such persons suppose an analogous though different case from their own. Let them ask themselves what they would say to a reformed Mussulman who should hesitate to give up his old notions in regard to the relations of the sexes; or to a reformed Catholic who should still shrink from reading the Bible. Would they not say that these persons ought to consider the matter fully, and clearly understand the new doctrine, and then ought to embrace it, in its entirety? But, above all, let it be considered that what is more wholesome than any particular belief is integrity of belief, and that to avoid looking into the support of any belief from a fear that it may turn out rotten is

quite as immoral as it is disadvantageous. The person who confesses that there is such a thing as truth, which is distinguished from falsehood simply by this, that if acted on it should, on full consideration, carry †1 us to the point we aim at and not astray, and then, though convinced of this, dares not know the truth and seeks to avoid it, is in a sorry state of mind indeed. Peirce: CP 5.387 Cross-Ref:†† †P1 Yes, the other methods do have their merits: a clear logical conscience does cost something -- just as any virtue, just as all that we cherish, costs us dear. But we should not desire it to be otherwise. The genius of a man's logical method should be loved and reverenced as his bride, whom he has chosen from all the world. He need not contemn the others; on the contrary, he may honor them deeply, and in doing so he only honors her the more. But she is the one that he has chosen, and he knows that he was right in making that choice. And having made it, he will work and fight for her, and will not complain that there are blows to take, hoping that there may be as many and as hard to give, and will strive to be the worthy knight and champion of her from the blaze of whose splendors he draws his inspiration and his courage.

Peirce: CP 5.388 Cross-Ref:†† V HOW TO MAKE OUR IDEAS CLEAR.

§1. CLEARNESS AND DISTINCTNESSE

388. Whoever has looked into a modern treatise on logic of the common sort,†P1 will doubtless remember the two distinctions between clear and obscure conceptions, and between distinct and confused conceptions. They have lain in the books now for nigh two centuries, unimproved and unmodified, and are generally reckoned by logicians as among the gems of their doctrine. Peirce: CP 5.389 Cross-Ref:†† 389. A clear idea is defined as one which is so apprehended that it will be recognized wherever it is met with, and so that no other will be mistaken for it. If it fails of this clearness, it is said to be obscure. Peirce: CP 5.389 Cross-Ref:†† This is rather a neat bit of philosophical terminology; yet, since it is clearness that they were defining, I wish the logicians had made their definition a little more plain. Never to fail to recognize an idea, and under no circumstances to mistake another for it, let it come in how recondite a form it may, would indeed imply such prodigious force and clearness of intellect as is seldom met with in this world. On the other hand, merely to have such an acquaintance with the idea as to have become familiar with it, and to have lost all hesitancy in recognizing it in ordinary cases, hardly seems to deserve the name of clearness of apprehension, since after all it only amounts to a subjective feeling of mastery which may be entirely mistaken. I take it, however, that when the logicians speak of "clearness," they mean nothing more than

such a familiarity with an idea, since they regard the quality as but a small merit, which needs to be supplemented by another, which they call distinctness. Peirce: CP 5.390 Cross-Ref:†† 390. A distinct idea is defined as one which contains nothing which is not clear. This is technical language; by the contents of an idea logicians understand whatever is contained in its definition. So that an idea is distinctly apprehended, according to them, when we can give a precise definition of it, in abstract terms. Here the professional logicians leave the subject; and I would not have troubled the reader with what they have to say, if it were not such a striking example of how they have been slumbering through ages of intellectual activity, listlessly disregarding the enginery of modern thought, and never dreaming of applying its lessons to the improvement of logic. It is easy to show that the doctrine that familiar use and abstract distinctness make the perfection of apprehension has its only true place in philosophies which have long been extinct; and it is now time to formulate the method of attaining to a more perfect clearness of thought, such as we see and admire in the thinkers of our own time. Peirce: CP 5.391 Cross-Ref:†† 391. When Descartes set about the reconstruction of philosophy, his first step was to (theoretically) permit scepticism and to discard the practice of the schoolmen of looking to authority as the ultimate source of truth. That done, he sought a more natural fountain of true principles, and thought he found †1 it in the human mind; thus passing, in the directest way, from the method of authority to that of apriority, as described in my first paper.†1 Self-consciousness was to furnish us with our fundamental truths, and to decide what was agreeable to reason. But since, evidently, not all ideas are true, he was led to note, as the first condition of infallibility, that they must be clear. The distinction between an idea seeming clear and really being so, never occurred to him. Trusting to introspection, as he did, even for a knowledge of external things, why should he question its testimony in respect to the contents of our own minds? But then, I suppose, seeing men, who seemed to be quite clear and positive, holding opposite opinions upon fundamental principles, he was further led to say that clearness of ideas is not sufficient, but that they need also to be distinct, i.e., to have nothing unclear about them. What he probably meant by this (for he did not explain himself with precision) was, that they must sustain the test of dialectical examination; that they must not only seem clear at the outset, but that discussion must never be able to bring to light points of obscurity connected with them. Peirce: CP 5.392 Cross-Ref:†† 392. Such was the distinction of Descartes, and one sees that it was precisely on the level of his philosophy. It was somewhat developed by Leibnitz. This great and singular genius was as remarkable for what he failed to see as for what he saw. That a piece of mechanism could not do work perpetually without being fed with power in some form, was a thing perfectly apparent to him; yet he did not understand that the machinery of the mind can only transform knowledge, but never originate it, unless it be fed with facts of observation. He thus missed the most essential point of the Cartesian philosophy, which is, that to accept propositions which seem perfectly evident to us is a thing which, whether it be logical or illogical, we cannot help doing. Instead of regarding the matter in this way, he sought to reduce the first principles of science to two †2 classes, those which cannot be denied without self-contradiction, and those which result from the principle of sufficient reason (of which more anon),†1 and was apparently unaware of the great difference between his position and

that of Descartes.†P1 So he reverted to the old trivialities †2 of logic; and, above all, abstract definitions played a great part in his philosophy. It was quite natural, therefore, that on observing that the method of Descartes labored under the difficulty that we may seem to ourselves to have clear apprehensions of ideas which in truth are very hazy, no better remedy occurred to him than to require an abstract definition of every important term. Accordingly, in adopting the distinction of clear and distinct notions, he described the latter quality as the clear apprehension of everything contained in the definition; and the books have ever since copied his words.†1 There is no danger that his chimerical scheme will ever again be over-valued. Nothing new can ever be learned by analyzing definitions. Nevertheless, our existing beliefs can be set in order by this process, and order is an essential element of intellectual economy, as of every other. It may be acknowledged, therefore, that the books are right in making familiarity with a notion the first step toward clearness of apprehension, and the defining of it the second. But in omitting all mention of any higher perspicuity of thought, they simply mirror a philosophy which was exploded a hundred years ago. That much-admired "ornament of logic" -- the doctrine of clearness and distinctness -may be pretty enough, but it is high time to relegate to our cabinet of curiosities the antique bijou, and to wear about us something better adapted to modern uses. Peirce: CP 5.393 Cross-Ref:†† 393.†P1 The very first lesson that we have a right to demand that logic shall teach us is, how to make our ideas clear; and a most important one it is, depreciated only by minds who stand in need of it. To know what we think, to be masters of our own meaning, will make a solid foundation for great and weighty thought. It is most easily learned by those whose ideas are meagre and restricted; and far happier they than such as wallow helplessly in a rich mud of conceptions. A nation, it is true, may, in the course of generations, overcome the disadvantage of an excessive wealth of language and its natural concomitant, a vast, unfathomable deep of ideas. We may see it in history, slowly perfecting its literary forms, sloughing at length its metaphysics, and, by virtue of the untirable patience which is often a compensation, attaining great excellence in every branch of mental acquirement. The page of history is not yet unrolled that †2 is to tell us whether such a people will or will not in the long run prevail over one whose ideas (like the words of their language) are few, but which possesses a wonderful mastery over those which it has. For an individual, however, there can be no question that a few clear ideas are worth more than many confused ones. A young man would hardly be persuaded to sacrifice the greater part of his thoughts to save the rest; and the muddled head is the least apt to see the necessity of such a sacrifice. Him we can usually only commiserate, as a person with a congenital defect. Time will help him, but intellectual maturity with regard to clearness is apt to †1 come rather late. This seems †2 an unfortunate arrangement of Nature, inasmuch as clearness is of less use to a man settled in life, whose errors have in great measure had their effect, than it would be to one whose path lay †3 before him. It is terrible to see how a single unclear idea, a single formula without meaning, lurking in a young man's head, will sometimes act like an obstruction of inert matter in an artery, hindering the nutrition of the brain, and condemning its victim to pine away in the fullness of his intellectual vigor and in the midst of intellectual plenty. Many a man has cherished for years as his hobby some vague shadow of an idea, too meaningless to be positively false; he has, nevertheless, passionately loved it, has made it his companion by day and by night, and has given to it his strength and his life, leaving all other occupations for its sake, and in short has lived with it and for it, until it has become, as it were, flesh of his flesh and bone of his bone; and then he has waked up some bright morning to find it gone, clean vanished away like the beautiful Melusina

of the fable, and the essence of his life gone with it. I have myself known such a man; and who can tell how many histories of circle-squarers, metaphysicians, astrologers, and what not, may not be told in the old German [French!] story?

Peirce: CP 5.394 Cross-Ref:†† §2. THE PRAGMATIC MAXIME

394. The principles set forth in the first part of this essay †4 lead, at once, to a method of reaching a clearness of thought of †5 higher grade than the "distinctness" of the logicians. It was there noticed †6 that the action of thought is excited by the irritation of doubt, and ceases when belief is attained; so that the production of belief is the sole function of thought.†1 All these words, however, are too strong for my purpose. It is as if I had described the phenomena as they appear under a mental microscope. Doubt and Belief, as the words are commonly employed, relate to religious or other grave discussions. But here I use them to designate the starting of any question, no matter how small or how great, and the resolution of it. If, for instance, in a horse-car, I pull out my purse and find a five-cent nickel and five coppers, I decide, while my hand is going to the purse, in which way I will pay my fare. To call such a question Doubt, and my decision Belief, is certainly to use words very disproportionate to the occasion. To speak of such a doubt as causing an irritation which needs to be appeased, suggests a temper which is uncomfortable to the verge of insanity. Yet, looking at the matter minutely, it must be admitted that, if there is the least hesitation as to whether I shall pay the five coppers or the nickel (as there will be sure to be, unless I act from some previously contracted habit in the matter), though irritation is too strong a word, yet I am excited to such small mental activity as may be necessary to deciding how I shall act. Most frequently doubts arise from some indecision, however momentary, in our action. Sometimes it is not so. I have, for example, to wait in a railway-station, and to pass the time I read the advertisements on the walls. I compare the advantages of different trains and different routes which I never expect to take, merely fancying myself to be in a state of hesitancy, because I am bored with having nothing to trouble me. Feigned hesitancy, whether feigned for mere amusement or with a lofty purpose, plays a great part in the production of scientific inquiry. However the doubt may originate, it stimulates the mind to an activity which may be slight or energetic, calm or turbulent. Images pass rapidly through consciousness, one incessantly melting into another, until at last, when all is over -- it may be in a fraction of a second, in an hour, or after long years -we find ourselves decided as to how we should act under such circumstances as those which occasioned our hesitation. In other words, we have attained belief. Peirce: CP 5.395 Cross-Ref:†† 395. In this process we observe two sorts of elements of consciousness, the distinction between which may best be made clear by means of an illustration. In a piece of music there are the separate notes, and there is the air. A single tone may be prolonged for an hour or a day, and it exists as perfectly in each second of that time as in the whole taken together; so that, as long as it is sounding, it might be present to a sense from which everything in the past was as completely absent as the future itself. But it is different with the air, the performance of which occupies a certain time, during the portions of which only portions of it are played. It consists in an orderliness in the succession of sounds which strike the ear at different times; and to

perceive it there must be some continuity of consciousness which makes the events of a lapse of time present to us. We certainly only perceive the air by hearing the separate notes; yet we cannot be said to directly hear it, for we hear only what is present at the instant, and an orderliness of succession cannot exist in an instant. These two sorts of objects, what we are immediately conscious of and what we are mediately conscious of, are found in all consciousness. Some elements (the sensations) are completely present at every instant so long as they last, while others (like thought) are actions having beginning, middle, and end, and consist in a congruence in the succession of sensations which flow through the mind. They cannot be immediately present to us, but must cover some portion of the past or future. Thought is a thread of melody running through the succession of our sensations. Peirce: CP 5.396 Cross-Ref:†† 396. We may add that just as a piece of music may be written in parts, each part having its own air, so various systems of relationship of succession subsist together between the same sensations. These different systems are distinguished by having different motives, ideas, or functions. Thought is only one such system, for its sole motive, idea, and function is to produce belief, and whatever does not concern that purpose belongs to some other system of relations. The action of thinking may incidentally have other results; it may serve to amuse us, for example, and among dilettanti it is not rare to find those who have so perverted thought to the purposes of pleasure that it seems to vex them to think that the questions upon which they delight to exercise it may ever get finally settled; and a positive discovery which takes a favorite subject out of the arena of literary debate is met with ill-concealed dislike. This disposition is the very debauchery of thought. But the soul and meaning of thought, abstracted from the other elements which accompany it, though it may be voluntarily thwarted, can never be made to direct itself toward anything but the production of belief. Thought in action has for its only possible motive the attainment of thought at rest; and whatever does not refer to belief is no part of the thought itself. Peirce: CP 5.397 Cross-Ref:†† 397. And what, then, is belief? It is the demi-cadence which closes a musical phrase in the symphony of our intellectual life. We have seen that it has just three properties: First, it is something that we are aware of; second, it appeases the irritation of doubt; and, third, it involves the establishment in our nature of a rule of action, or, say for short, a habit. As it appeases the irritation of doubt, which is the motive for thinking, thought relaxes, and comes to rest for a moment when belief is reached. But, since belief is a rule for action, the application of which involves further doubt and further thought, at the same time that it is a stopping-place, it is also a new starting-place for thought. That is why I have permitted myself to call it thought at rest, although thought is essentially an action. The final upshot of thinking is the exercise of volition, and of this thought no longer forms a part; but belief is only a stadium of mental action, an effect upon our nature due to thought, which will influence future thinking. Peirce: CP 5.398 Cross-Ref:†† 398. The essence of belief is the establishment of a habit; and different beliefs are distinguished by the different modes of action to which they give rise. If beliefs do not differ in this respect, if they appease the same doubt by producing the same rule of action, then no mere differences in the manner of consciousness of them can make them different beliefs, any more than playing a tune in different keys is playing different tunes. Imaginary distinctions are often drawn between beliefs which differ

only in their mode of expression; -- the wrangling which ensues is real enough, however. To believe that any objects are arranged among themselves †1 as in Fig. 1, and to believe that they are arranged [as] in Fig. 2, are one and the same belief; yet it is conceivable that a man should assert one proposition and deny the other. Such false distinctions do as much harm as the confusion of beliefs really different, and are among the pitfalls of which we ought constantly to beware, especially when we are upon metaphysical ground. One singular deception of this sort, which often occurs, is to mistake the sensation

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produced by our own unclearness of thought for a character of the object we are thinking. Instead of perceiving that the obscurity is purely subjective, we fancy that we contemplate a quality of the object which is essentially mysterious; and if our conception be afterward presented to us in a clear form we do not recognize it as the same, owing to the absence of the feeling of unintelligibility. So long as this deception lasts, it obviously puts an impassable barrier in the way of perspicuous thinking; so that it equally interests the opponents of rational thought to perpetuate it, and its adherents to guard against it. Peirce: CP 5.399 Cross-Ref:††

399. Another such deception is to mistake a mere difference in the grammatical construction of two words for a distinction between the ideas they express. In this pedantic age, when the general mob of writers attend so much more to words than to things, this error is common enough. When I just said that thought is an action, and that it consists in a relation, although a person performs an action but not a relation, which can only be the result of an action, yet there was no inconsistency in what I said, but only a grammatical vagueness. Peirce: CP 5.400 Cross-Ref:†† 400. From all these sophisms we shall be perfectly safe so long as we reflect that the whole function of thought is to produce habits of action; and that whatever there is connected with a thought, but irrelevant to its purpose, is an accretion to it, but no part of it. If there be a unity among our sensations which has no reference to how we shall act on a given occasion, as when we listen to a piece of music, why we do not call that thinking. To develop its meaning, we have, therefore, simply to determine what habits it produces, for what a thing means is simply what habits it involves. Now, the identity of a habit depends on how it might lead us to act, not merely under such circumstances as are likely to arise, but under such as might possibly occur, no matter how improbable they may be.†1 What the habit is depends on when and how it causes us to act. As for the when, every stimulus to action is derived from perception; as for the how, every purpose of action is to produce some sensible result. Thus, we come down to what is tangible and †2 conceivably †3 practical, as the root of every real distinction of thought, no matter how subtile it may be; and there is no distinction of meaning so fine as to consist in anything but a possible difference of practice. Peirce: CP 5.401 Cross-Ref:†† 401. To see what this principle leads to, consider in the light of it such a doctrine as that of transubstantiation. The Protestant churches generally hold that the elements of the sacrament are flesh and blood only in a tropical sense; they nourish our souls as meat and the juice of it would our bodies. But the Catholics maintain that they are literally just meat and blood †4; although they possess all the sensible qualities of wafercakes and diluted wine. But we can have no conception of wine except what may enter into a belief, either -Peirce: CP 5.401 Cross-Ref:†† 1. That this, that, or the other, is wine; or, 2. That wine possesses certain properties. Such beliefs are nothing but self-notifications that we should, upon occasion, act in regard to such things as we believe to be wine according to the qualities which we believe wine to possess. The occasion of such action would be some sensible perception, the motive of it to produce some sensible result. Thus our action has exclusive reference to what affects the senses, our habit has the same bearing as our action, our belief the same as our habit, our conception the same as our belief; and we can consequently mean nothing by wine but what has certain effects, direct or indirect, upon our senses; and to talk of something as having all the sensible characters of wine, yet being in reality blood, is senseless jargon. Now, it is not my object to pursue the theological question; and having used it as a logical example I drop it, without caring to anticipate the theologian's reply. I only desire to point out how impossible it is that we should have an idea in our minds which relates to anything but conceived sensible effects of things. Our idea of anything is our idea of its sensible effects; and if we fancy that we have any other we deceive ourselves, and mistake a mere sensation accompanying the thought for a part of the thought itself. It

is absurd to say that thought has any meaning unrelated to its only function. It is foolish for Catholics and Protestants to fancy themselves in disagreement about the elements of the sacrament, if they agree in regard to all their sensible effects, here and †1 hereafter.†2 Peirce: CP 5.402 Cross-Ref:†† 402. It appears, then, that the rule for attaining the third grade of clearness of apprehension is as follows: Consider what effects, that †3 might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.†P1†P2†P3

Peirce: CP 5.403 Cross-Ref:†† §3. SOME APPLICATIONS OF THE PRAGMATIC MAXIME

403. Let us illustrate this rule by some examples; and, to begin with the simplest one possible, let us ask what we mean by calling a thing hard. Evidently that it will not be scratched by many other substances. The whole conception of this quality, as of every other, lies in its conceived effects. There is absolutely no difference between a hard thing and a soft thing so long as they are not brought to the test. Suppose, then, that a diamond could be crystallized in the midst of a cushion of soft cotton, and should remain there until it was finally burned up. Would it be false to say that that diamond was soft? This seems a foolish question, and would be so, in fact, except in the realm of logic. There such questions are often of the greatest utility as serving to bring logical principles into sharper relief than real discussions ever could. In studying logic we must not put them aside with hasty answers, but must consider them with attentive care, in order to make out the principles involved. We may, in the present case, modify our question, and ask what prevents us from saying that all hard bodies remain perfectly soft until they are touched, when their hardness increases with the pressure until they are scratched. Reflection will show that the reply is this: there would be no falsity in such modes of speech. They would involve a modification of our present usage of speech with regard to the words hard and soft, but not of their meanings. For they represent no fact to be different from what it is; only they involve arrangements of facts which would be exceedingly maladroit.†1 This leads us to remark that the question of what would occur under circumstances which do not actually arise is not a question of fact, but only of the most perspicuous arrangement of them. For example, the question of free-will and fate in its simplest form, stripped of verbiage, is something like this: I have done something of which I am ashamed; could I, by an effort of the will, have resisted the temptation, and done otherwise? The philosophical reply is, that this is not a question of fact, but only of the arrangement of facts.†2 Arranging them so as to exhibit what is particularly pertinent to my question -- namely, that I ought to blame myself for having done wrong -- it is perfectly true to say that, if I had willed to do otherwise than I did, I should have done otherwise. On the other hand, arranging the facts so as to exhibit another important consideration, it is equally true that, when a temptation has once been allowed to work, it will, if it has a certain force, produce its effect, let me struggle how I may. There is no objection to a contradiction in what would result from a false supposition. The reductio ad absurdum consists in showing that contradictory results would follow from a hypothesis which is consequently judged to be false. Many questions are involved in the free-will discussion, and I am far from

desiring to say that both sides are equally right. On the contrary, I am of opinion that one side denies important facts, and that the other does not. But what I do say is, that the above single question was the origin of the whole doubt; that, had it not been for this question, the controversy would never have arisen; and that this question is perfectly solved in the manner which I have indicated. Peirce: CP 5.403 Cross-Ref:†† Let us next seek a clear idea of Weight. This is another very easy case. To say that a body is heavy means simply that, in the absence of opposing force, it will fall. This (neglecting certain specifications of how it will fall, etc., which exist in the mind of the physicist who uses the word) is evidently the whole conception of weight. It is a fair question whether some particular facts may not account for gravity; but what we mean by the force itself is completely involved in its effects. Peirce: CP 5.404 Cross-Ref:†† 404. This leads us to undertake an account of the idea of Force in general. This is the great conception which, developed in the early part of the seventeenth century from the rude idea of a cause, and constantly improved upon since, has shown us how to explain all the changes of motion which bodies experience, and how to think about all physical phenomena; which has given birth to modern science, and changed the face of the globe; and which, aside from its more special uses, has played a principal part in directing the course of modern thought, and in furthering modern social development. It is, therefore, worth some pains to comprehend it. According to our rule, we must begin by asking what is the immediate use of thinking about force; and the answer is, that we thus account for changes of motion. If bodies were left to themselves, without the intervention of forces, every motion would continue unchanged both in velocity and in direction. Furthermore, change of motion never takes place abruptly; if its direction is changed, it is always through a curve without angles; if its velocity alters, it is by degrees. The gradual changes which are constantly taking place are conceived by geometers to be compounded together according to the rules of the parallelogram of forces. If the reader does not already know what this is, he will find it, I hope, to his advantage to endeavor to follow the following explanation; but if mathematics are insupportable to him, pray let him skip three paragraphs rather than that we should part company here. Peirce: CP 5.404 Cross-Ref:†† A path is a line whose beginning and end are distinguished. Two paths are considered to be equivalent, which, beginning at the same point, lead to the same point. Thus the two paths, A B C D E and A F G H E (Fig. 3), are equivalent. Paths which do not begin at the same point are considered to be equivalent, provided that, on moving either of them without turning it, but keeping it always parallel to its original position, when its beginning coincides with that of the other path, the ends also coincide. Paths are considered as geometrically added together, when one begins where the other ends; thus

[Click here to view]

[Click here to view] Fig. 3

Fig. 4

the path A E is conceived to be a sum of A B, B C, C D, and D E. In the parallelogram of Fig. 4 the diagonal A C is the sum of A B and B C; or, since A D is geometrically equivalent to B C, A C is the geometrical sum of A B and A D. Peirce: CP 5.404 Cross-Ref:†† All this is purely conventional. It simply amounts to this: that we choose to call paths having the relations I have described equal or added. But, though it is a convention, it is a convention with a good reason. The rule for geometrical addition may be applied not only to paths, but to any other things which can be represented by

paths. Now, as a path is determined by the varying direction and distance of the point which moves over it from the starting-point, it follows that anything which from its beginning to its end is determined by a varying direction and a varying magnitude is capable of being represented by a line. Accordingly, velocities may be represented by lines, for they have only directions and rates. The same thing is true of accelerations, or changes of velocities. This is evident enough in the case of velocities; and it becomes evident for accelerations if we consider that precisely what velocities are to positions -- namely, states of change of them -- that accelerations are to velocities. Peirce: CP 5.404 Cross-Ref:†† The so-called "parallelogram of forces" is simply a rule for compounding accelerations. The rule is, to represent the accelerations by paths, and then to geometrically add the paths. The geometers, however, not only use the "parallelogram of forces" to compound different accelerations, but also to resolve one acceleration into a sum of several. Let A B (Fig. 5

[Click here to view]) be the path which represents a certain acceleration -- say, such a change in the motion of a body that at the end of one second the body will, under the influence of that change, be in a position different from what it would have had if its motion had continued unchanged such that a path equivalent to A B would lead from the latter position to the former. This acceleration may be considered as the sum of the accelerations represented by A C and C B. It may also be considered as the sum of the very different accelerations represented by A D and D B, where A D is almost the opposite of A C. And it is clear that there is an immense variety of ways in which A B might be resolved into the sum of two accelerations. Peirce: CP 5.404 Cross-Ref:†† After this tedious explanation, which I hope, in view of the extraordinary interest of the conception of force, may not have exhausted the reader's patience, we are prepared at last to state the grand fact which this conception embodies. This fact is that if the actual changes of motion which the different particles of bodies experience are each resolved in its appropriate way, each component acceleration is precisely such as is prescribed by a certain law of Nature, according to which bodies, in the relative positions which the bodies in question actually have at the moment,†P1 always receive certain accelerations, which, being compounded by geometrical addition, give the acceleration which the body actually experiences.

Peirce: CP 5.404 Cross-Ref:†† This is the only fact which the idea of force represents, and whoever will take the trouble clearly to apprehend what this fact is, perfectly comprehends what force is. Whether we ought to say that a force is an acceleration, or that it causes an acceleration, is a mere question of propriety of language, which has no more to do with our real meaning than the difference between the French idiom "Il fait froid" and its English equivalent "It is cold." Yet it is surprising to see how this simple affair has muddled men's minds. In how many profound treatises is not force spoken of as a "mysterious entity," which seems to be only a way of confessing that the author despairs of ever getting a clear notion of what the word means! In a recent admired work on Analytic Mechanics†1 it is stated that we understand precisely the effect of force, but what force itself is we do not understand! This is simply a self-contradiction. The idea which the word force excites in our minds has no other function than to affect our actions, and these actions can have no reference to force otherwise than through its effects. Consequently, if we know what the effects of force are, we are acquainted with every fact which is implied in saying that a force exists, and there is nothing more to know. The truth is, there is some vague notion afloat that a question may mean something which the mind cannot conceive; and when some hair-splitting philosophers have been confronted with the absurdity of such a view, they have invented an empty distinction between positive and negative conceptions, in the attempt to give their non-idea a form not obviously nonsensical. The nullity of it is sufficiently plain from the considerations given a few pages back; and, apart from those considerations, the quibbling character of the distinction must have struck every mind accustomed to real thinking.

Peirce: CP 5.405 Cross-Ref:†† §4. REALITYE

405. Let us now approach the subject of logic, and consider a conception which particularly concerns it, that of reality. Taking clearness in the sense of familiarity, no idea could be clearer than this. Every child uses it with perfect confidence, never dreaming that he does not understand it. As for clearness in its second grade, however, it would probably puzzle most men, even among those of a reflective turn of mind, to give an abstract definition of the real. Yet such a definition may perhaps be reached by considering the points of difference between reality and its opposite, fiction. A figment is a product of somebody's imagination; it has such characters as his thought impresses upon it. That those characters are independent of how you or I think is an external reality. There are, however, phenomena within our own minds, dependent upon our thought, which are at the same time real in the sense that we really think them. But though their characters depend on how we think, they do not depend on what we think those characters to be. Thus, a dream has a real existence as a mental phenomenon, if somebody has really dreamt it; that he dreamt so and so, does not depend on what anybody thinks was dreamt, but is completely independent of all opinion on the subject. On the other hand, considering, not the fact of dreaming, but the thing dreamt, it retains its peculiarities by virtue of no other fact than that it was dreamt to possess them. Thus we may define the real as that whose characters are independent of what anybody may think them to be.

Peirce: CP 5.406 Cross-Ref:†† 406. But, however satisfactory such a definition may be found, it would be a great mistake to suppose that it makes the idea of reality perfectly clear. Here, then, let us apply our rules. According to them, reality, like every other quality, consists in the peculiar sensible effects which things partaking of it produce. The only effect which real things have is to cause belief, for all the sensations which they excite emerge into consciousness in the form of beliefs. The question therefore is, how is true belief (or belief in the real) distinguished from false belief (or belief in fiction). Now, as we have seen in the former paper,†1 the ideas of truth and falsehood, in their full development, appertain exclusively to the experiential †2 method of settling opinion. A person who arbitrarily chooses the propositions which he will adopt can use the word truth only to emphasize the expression of his determination to hold on to his choice. Of course, the method of tenacity †3 never prevailed exclusively; reason is too natural to men for that. But in the literature of the dark ages we find some fine examples of it. When Scotus Erigena is commenting upon a poetical passage in which hellebore is spoken of as having caused the death of Socrates, he does not hesitate to inform the inquiring reader that Helleborus and Socrates were two eminent Greek philosophers, and that the latter, having been overcome in argument by the former, took the matter to heart and died of it! What sort of an idea of truth could a man have who could adopt and teach, without the qualification of a perhaps, an opinion taken so entirely at random? The real spirit of Socrates, who I hope would have been delighted to have been "overcome in argument," because he would have learned something by it, is in curious contrast with the naive idea of the glossist, for whom (as for "the born missionary" of today)†1 discussion would seem to have been simply a struggle. When philosophy began to awake from its long slumber, and before theology completely dominated it, the practice seems to have been for each professor to seize upon any philosophical position he found unoccupied and which seemed a strong one, to intrench himself in it, and to sally forth from time to time to give battle to the others. Thus, even the scanty records we possess of those disputes enable us to make out a dozen or more opinions held by different teachers at one time concerning the question of nominalism and realism. Read the opening part of the Historia Calamitatum of Abelard,†2 who was certainly as philosophical as any of his contemporaries, and see the spirit of combat which it breathes. For him, the truth is simply his particular stronghold. When the method of authority †3 prevailed, the truth meant little more than the Catholic faith. All the efforts of the scholastic doctors are directed toward harmonizing their faith in Aristotle and their faith in the Church, and one may search their ponderous folios through without finding an argument which goes any further. It is noticeable that where different faiths flourish side by side, renegades are looked upon with contempt even by the party whose belief they adopt; so completely has the idea of loyalty replaced that of truth-seeking. Since the time of Descartes, the defect in the conception of truth has been less apparent. Still, it will sometimes strike a scientific man that the philosophers have been less intent on finding out what the facts are, than on inquiring what belief is most in harmony with their system. It is hard to convince a follower of the a priori method by adducing facts; but show him that an opinion he is defending is inconsistent with what he has laid down elsewhere, and he will be very apt to retract it. These minds do not seem to believe that disputation is ever to cease; they seem to think that the opinion which is natural for one man is not so for another, and that belief will, consequently, never be settled. In contenting themselves with fixing their own opinions by a method which would lead another man to a different result, they betray their feeble hold of the conception of what truth is.

Peirce: CP 5.407 Cross-Ref:†† 407. On the other hand, all the followers of science are animated by a cheerful hope †1 that the processes of investigation, if only pushed far enough, will give one certain solution to each †2 question to which they apply it †3. One man may investigate the velocity of light by studying the transits of Venus and the aberration of the stars; another by the oppositions of Mars and the eclipses of Jupiter's satellites; a third by the method of Fizeau; a fourth by that of Foucault; a fifth by the motions of the curves of Lissajoux; a sixth, a seventh, an eighth, and a ninth, may follow the different methods of comparing the measures of statical and dynamical electricity. They may at first obtain different results, but, as each perfects his method and his processes, the results are found to move †4 steadily together toward a destined centre. So with all scientific research. Different minds may set out with the most antagonistic views, but the progress of investigation carries them by a force outside of themselves to one and the same conclusion. This activity of thought by which we are carried, not where we wish, but to a fore-ordained goal, is like the operation of destiny. No modification of the point of view taken, no selection of other facts for study, no natural bent of mind even, can enable a man to escape the predestinate opinion. This great hope †5 is embodied in the conception of truth and reality. The opinion which is fated †P1 to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real. That is the way I would explain reality. Peirce: CP 5.408 Cross-Ref:†† 408. But it may be said that this view is directly opposed to the abstract definition which we have given of reality, inasmuch as it makes the characters of the real depend on what is ultimately thought about them. But the answer to this is that, on the one hand, reality is independent, not necessarily of thought in general, but only of what you or I or any finite number of men may think about it; and that, on the other hand, though the object of the final opinion depends on what that opinion is, yet what that opinion is does not depend on what you or I or any man thinks. Our perversity and that of others may indefinitely postpone the settlement of opinion; it might even conceivably cause an arbitrary proposition to be universally accepted as long as the human race should last. Yet even that would not change the nature of the belief, which alone could be the result of investigation carried sufficiently far; and if, after the extinction of our race, another should arise with faculties and disposition for investigation, that true opinion must be the one which they would ultimately come to. "Truth crushed to earth shall rise again," and the opinion which would finally result from investigation does not depend on how anybody may actually think. But the reality of that which is real does depend on the real fact that investigation is destined to lead, at last, if continued long enough, to a belief in it. Peirce: CP 5.409 Cross-Ref:†† 409. But I may be asked what I have to say to all the minute facts of history, forgotten never to be recovered, to the lost books of the ancients, to the buried secrets.

"Full many a gem of purest ray serene The dark, unfathomed caves of ocean bear; Full many a flower is born to blush unseen,

And waste its sweetness on the desert air."

Do these things not really exist because they are hopelessly beyond the reach of our knowledge? And then, after the universe is dead (according to the prediction of some scientists), and all life has ceased forever, will not the shock of atoms continue though there will be no mind to know it? To this I reply that, though in no possible state of knowledge can any number be great enough to express the relation between the amount of what rests unknown to the amount of the known, yet it is unphilosophical to suppose that, with regard to any given question (which has any clear meaning), investigation would not bring forth a solution of it, if it were carried far enough. Who would have said, a few years ago, that we could ever know of what substances stars are made whose light may have been longer in reaching us than the human race has existed? Who can be sure of what we shall not know in a few hundred years? Who can guess what would be the result of continuing the pursuit of science for ten thousand years, with the activity of the last hundred? And if it were to go on for a million, or a billion, or any number of years you please, how is it possible to say that there is any question which might not ultimately be solved? Peirce: CP 5.409 Cross-Ref:†† But it may be objected, "Why make so much of these remote considerations, especially when it is your principle that only practical distinctions have a meaning?" Well, I must confess that it makes very little difference whether we say that a stone on the bottom of the ocean, in complete darkness, is brilliant or not -- that is to say, that it probably makes no difference, remembering always that that stone may be fished up tomorrow. But that there are gems at the bottom of the sea, flowers in the untraveled desert, etc., are propositions which, like that about a diamond being hard when it is not pressed, concern much more the arrangement of our language than they do the meaning of our ideas. Peirce: CP 5.410 Cross-Ref:†† 410. It seems to me, however, that we have, by the application of our rule, reached so clear an apprehension of what we mean by reality, and of the fact which the idea rests on, that we should not, perhaps, be making a pretension so presumptuous as it would be singular, if we were to offer a metaphysical theory of existence for universal acceptance among those who employ the scientific method of fixing belief. However, as metaphysics is a subject much more curious than useful, the knowledge of which, like that of a sunken reef, serves chiefly to enable us to keep clear of it, I will not trouble the reader with any more Ontology at this moment. I have already been led much further into that path than I should have desired; and I have given the reader such a dose of mathematics, psychology, and all that is most abstruse, that I fear he may already have left me, and that what I am now writing is for the compositor and proof-reader exclusively. I trusted to the importance of the subject. There is no royal road to logic, and really valuable ideas can only be had at the price of close attention. But I know that in the matter of ideas the public prefer the cheap and nasty; and in my next paper †1 I am going to return to the easily intelligible, and not wander from it again. The reader who has been at the pains of wading through this paper, shall be rewarded in the next one by seeing how beautifully what has been developed in this tedious way can be applied to the ascertainment of the rules of scientific reasoning.

Peirce: CP 5.410 Cross-Ref:†† We have, hitherto, not crossed the threshold of scientific logic. It is certainly important to know how to make our ideas clear, but they may be ever so clear without being true. How to make them so, we have next to study. How to give birth to those vital and procreative ideas which multiply into a thousand forms and diffuse themselves everywhere, advancing civilization and making the dignity of man, is an art not yet reduced to rules, but of the secret of which the history of science affords some hints.

Peirce: CP 5.411 Cross-Ref:†† VI

WHAT PRAGMATISM IS†1

§1. THE EXPERIMENTALISTS' VIEW OF ASSERTIONE

411. The writer of this article has been led by much experience to believe that every physicist, and every chemist, and, in short, every master in any department of experimental science, has had his mind moulded by his life in the laboratory to a degree that is little suspected. The experimentalist himself can hardly be fully aware of it, for the reason that the men whose intellects he really knows about are much like himself in this respect. With intellects of widely different training from his own, whose education has largely been a thing learned out of books, he will never become inwardly intimate, be he on ever so familiar terms with them; for he and they are as oil and water, and though they be shaken up together, it is remarkable how quickly they will go their several mental ways, without having gained more than a faint flavor from the association. Were those other men only to take skillful soundings of the experimentalist's mind -- which is just what they are unqualified to do, for the most part -- they would soon discover that, excepting perhaps upon topics where his mind is trammelled by personal feeling or by his bringing up, his disposition is to think of everything just as everything is thought of in the laboratory, that is, as a question of experimentation. Of course, no living man possesses in their fullness all the attributes characteristic of his type: it is not the typical doctor whom you will see every day driven in buggy or coupe, nor is it the typical pedagogue that will be met with in the first schoolroom you enter. But when you have found, or ideally constructed upon a basis of observation, the typical experimentalist, you will find that whatever assertion you may make to him, he will either understand as meaning that if a given prescription for an experiment ever can be and ever is carried out in act, an experience of a given description will result, or else he will see no sense at all in what you say. If you talk to him as Mr. Balfour talked not long ago to the British Association †1 saying that "the physicist . . . seeks for something deeper than the laws connecting possible objects of experience," that "his object is physical reality" unrevealed in experiments, and that the existence of such non-experiential reality "is the unalterable

faith of science," to all such ontological meaning you will find the experimentalist mind to be color-blind. What adds to that confidence in this, which the writer owes to his conversations with experimentalists, is that he himself may almost be said to have inhabited a laboratory from the age of six until long past maturity; and having all his life associated mostly with experimentalists, it has always been with a confident sense of understanding them and of being understood by them. Peirce: CP 5.412 Cross-Ref:†† 412. That laboratory life did not prevent the writer (who here and in what follows simply exemplifies the experimentalist type) from becoming interested in methods of thinking; and when he came to read metaphysics, although much of it seemed to him loosely reasoned and determined by accidental prepossessions, yet in the writings of some philosophers, especially Kant, Berkeley, and Spinoza, he sometimes came upon strains of thought that recalled the ways of thinking of the laboratory, so that he felt he might trust to them; all of which has been true of other laboratory-men. Peirce: CP 5.412 Cross-Ref:†† Endeavoring, as a man of that type naturally would, to formulate what he so approved, he framed the theory that a conception, that is, the rational purport of a word or other expression, lies exclusively in its conceivable bearing upon the conduct of life; so that, since obviously nothing that might not result from experiment can have any direct bearing upon conduct, if one can define accurately all the conceivable experimental phenomena which the affirmation or denial of a concept could imply, one will have therein a complete definition of the concept, and there is absolutely nothing more in it. For this doctrine he invented the name pragmatism. Some of his friends wished him to call it practicism or practicalism (perhaps on the ground that {praktikos} is better Greek than {pragmatikos}. But for one who had learned philosophy out of Kant, as the writer, along with nineteen out of every twenty experimentalists who have turned to philosophy, had done, and who still thought in Kantian terms most readily, praktisch and pragmatisch were as far apart as the two poles, the former belonging in a region of thought where no mind of the experimentalist type can ever make sure of solid ground under his feet, the latter expressing relation to some definite human purpose. Now quite the most striking feature of the new theory was its recognition of an inseparable connection between rational cognition and rational purpose; and that consideration it was which determined the preference for the name pragmatism.

Peirce: CP 5.413 Cross-Ref:†† §2. PHILOSOPHICAL NOMENCLATURE †1E

413. Concerning the matter of philosophical nomenclature, there are a few plain considerations, which the writer has for many years longed to submit to the deliberate judgment of those few fellow-students of philosophy, who deplore the present state of that study, and who are intent upon rescuing it therefrom and bringing it to a condition like that of the natural sciences, where investigators, instead of contemning each the work of most of the others as misdirected from beginning to end, coöperate, stand upon one another's shoulders, and multiply incontestible results;

where every observation is repeated, and isolated observations go for little; where every hypothesis that merits attention is subjected to severe but fair examination, and only after the predictions to which it leads have been remarkably borne out by experience is trusted at all, and even then only provisionally; where a radically false step is rarely taken, even the most faulty of those theories which gain wide credence being true in their main experiential predictions. To those students, it is submitted that no study can become scientific in the sense described, until it provides itself with a suitable technical nomenclature, whose every term has a single definite meaning universally accepted among students of the subject, and whose vocables have no such sweetness or charms as might tempt loose writers to abuse them -- which is a virtue of scientific nomenclature too little appreciated. It is submitted that the experience of those sciences which have conquered the greatest difficulties of terminology, which are unquestionably the taxonomic sciences, chemistry, mineralogy, botany, zoölogy, has conclusively shown that the one only way in which the requisite unanimity and requisite ruptures with individual habits and preferences can be brought about is so to shape the canons of terminology that they shall gain the support of moral principle and of every man's sense of decency; and that, in particular (under defined restrictions), the general feeling shall be that he who introduces a new conception into philosophy is under an obligation to invent acceptable terms to express it, and that when he has done so, the duty of his fellow-students is to accept those terms, and to resent any wresting of them from their original meanings, as not only a gross discourtesy to him to whom philosophy was indebted for each conception, but also as an injury to philosophy itself; and furthermore, that once a conception has been supplied with suitable and sufficient words for its expression, no other technical terms denoting the same things, considered in the same relations, should be countenanced. Should this suggestion find favor, it might be deemed needful that the philosophians in congress assembled should adopt, after due deliberation, convenient canons to limit the application of the principle. Thus, just as is done in chemistry, it might be wise to assign fixed meanings to certain prefixes and suffixes. For example, it might be agreed, perhaps, that the prefix prope- should mark a broad and rather indefinite extension of the meaning of the term to which it was prefixed; the name of a doctrine would naturally end in -ism, while -icism might mark a more strictly defined acception of that doctrine, etc. Then again, just as in biology no account is taken of terms antedating Linnæus, so in philosophy it might be found best not to go back of the scholastic terminology. To illustrate another sort of limitation, it has probably never happened that any philosopher has attempted to give a general name to his own doctrine without that name's soon acquiring in common philosophical usage, a signification much broader than was originally intended. Thus, special systems go by the names Kantianism, Benthamism, Comteanism, Spencerianism, etc., while transcendentalism, utilitarianism, positivism, evolutionism, synthetic philosophy, etc., have irrevocably and very conveniently been elevated to broader governments.

Peirce: CP 5.414 Cross-Ref:†† §3. PRAGMATICISME

414. After awaiting in vain, for a good many years, some particularly opportune conjuncture of circumstances that might serve to recommend his notions of the ethics of terminology, the writer has now, at last, dragged them in over head and

shoulders, on an occasion when he has no specific proposal to offer nor any feeling but satisfaction at the course usage has run without any canons or resolutions of a congress. His word "pragmatism" has gained general recognition in a generalized sense that seems to argue power of growth and vitality. The famed psychologist, James, first took it up,†1 seeing that his "radical empiricism" substantially answered to the writer's definition of pragmatism, albeit with a certain difference in the point of view. Next, the admirably clear and brilliant thinker, Mr. Ferdinand C.S. Schiller, casting about for a more attractive name for the "anthropomorphism" of his Riddle of the Sphinx, lit, in that most remarkable paper of his on Axioms as Postulates,†2 upon the same designation "pragmatism," which in its original sense was in generic agreement with his own doctrine, for which he has since found the more appropriate specification "humanism," while he still retains "pragmatism" in a somewhat wider sense. So far all went happily. But at present, the word begins to be met with occasionally in the literary journals, where it gets abused in the merciless way that words have to expect when they fall into literary clutches. Sometimes the manners of the British have effloresced in scolding at the word as ill-chosen -- ill-chosen, that is, to express some meaning that it was rather designed to exclude. So then, the writer, finding his bantling "pragmatism" so promoted, feels that it is time to kiss his child good-by and relinquish it to its higher destiny; while to serve the precise purpose of expressing the original definition, he begs to announce the birth of the word "pragmaticism," which is ugly enough to be safe from kidnappers.†P1 Peirce: CP 5.415 Cross-Ref:†† 415. Much as the writer has gained from the perusal of what other pragmatists have written, he still thinks there is a decisive advantage in his original conception of the doctrine. From this original form every truth that follows from any of the other forms can be deduced, while some errors can be avoided into which other pragmatists have fallen. The original view appears, too, to be a more compact and unitary conception than the others. But its capital merit, in the writer's eyes, is that it more readily connects itself with a critical proof of its truth. Quite in accord with the logical order of investigation, it usually happens that one first forms an hypothesis that seems more and more reasonable the further one examines into it, but that only a good deal later gets crowned with an adequate proof. The present writer having had the pragmatist theory under consideration for many years longer than most of its adherents, would naturally have given more attention to the proof of it. At any rate, in endeavoring to explain pragmatism, he may be excused for confining himself to that form of it that he knows best. In the present article there will be space only to explain just what this doctrine (which, in such hands as it has now fallen into, may probably play a pretty prominent part in the philosophical discussions of the next coming years), really consists in. Should the exposition be found to interest readers of The Monist, they would certainly be much more interested in a second article which would give some samples of the manifold applications of pragmaticism (assuming it to be true) to the solution of problems of different kinds. After that, readers might be prepared to take an interest in a proof that the doctrine is true -- a proof which seems to the writer to leave no reasonable doubt on the subject, and to be the one contribution of value that he has to make to philosophy. For it would essentially involve the establishment of the truth of synechism.†1 Peirce: CP 5.416 Cross-Ref:†† 416. The bare definition of pragmaticism could convey no satisfactory comprehension of it to the most apprehensive of minds, but requires the commentary to be given below. Moreover, this definition takes no notice of one or two other

doctrines without the previous acceptance (or virtual acceptance) of which pragmaticism itself would be a nullity. They are included as a part of the pragmatism of Schiller, but the present writer prefers not to mingle different propositions. The preliminary propositions had better be stated forthwith. Peirce: CP 5.416 Cross-Ref:†† The difficulty in doing this is that no formal list of them has ever been made. They might all be included under the vague maxim, "Dismiss make-believes." Philosophers of very diverse stripes propose that philosophy shall take its start from one or another state of mind in which no man, least of all a beginner in philosophy, actually is. One proposes that you shall begin by doubting everything, and says that there is only one thing that you cannot doubt, as if doubting were "as easy as lying." Another proposes that we should begin by observing "the first impressions of sense," forgetting that our very percepts are the results of cognitive elaboration. But in truth, there is but one state of mind from which you can "set out," namely, the very state of mind in which you actually find yourself at the time you do "set out" -- a state in which you are laden with an immense mass of cognition already formed, of which you cannot divest yourself if you would; and who knows whether, if you could, you would not have made all knowledge impossible to yourself? Do you call it doubting to write down on a piece of paper that you doubt? If so, doubt has nothing to do with any serious business. But do not make believe; if pedantry has not eaten all the reality out of you, recognize, as you must, that there is much that you do not doubt, in the least. Now that which you do not at all doubt, you must and do regard as infallible, absolute truth. Here breaks in Mr. Make Believe: "What! Do you mean to say that one is to believe what is not true, or that what a man does not doubt is ipso facto true?" No, but unless he can make a thing white and black at once, he has to regard what he does not doubt as absolutely true. Now you, per hypothesiu, are that man. "But you tell me there are scores of things I do not doubt. I really cannot persuade myself that there is not some one of them about which I am mistaken." You are adducing one of your make-believe facts, which, even if it were established, would only go to show that doubt has a limen, that is, is only called into being by a certain finite stimulus. You only puzzle yourself by talking of this metaphysical "truth" and metaphysical "falsity," that you know nothing about. All you have any dealings with are your doubts and beliefs,†P1 with the course of life that forces new beliefs upon you and gives you power to doubt old beliefs. If your terms "truth" and "falsity" are taken in such senses as to be definable in terms of doubt and belief and the course of experience (as for example they would be, if you were to define the "truth" as that to a belief in which belief would tend if it were to tend indefinitely toward absolute fixity), well and good: in that case, you are only talking about doubt and belief. But if by truth and falsity you mean something not definable in terms of doubt and belief in any way, then you are talking of entities of whose existence you can know nothing, and which Ockham's razor would clean shave off. Your problems would be greatly simplified, if, instead of saying that you want to know the "Truth," you were simply to say that you want to attain a state of belief unassailable by doubt. Peirce: CP 5.417 Cross-Ref:†† 417. Belief is not a momentary mode of consciousness; it is a habit of mind essentially enduring for some time, and mostly (at least) unconscious; and like other habits, it is (until it meets with some surprise that begins its dissolution) perfectly self-satisfied. Doubt is of an altogether contrary genus. It is not a habit, but the privation of a habit. Now a privation of a habit, in order to be anything at all, must be a condition of erratic activity that in some way must get superseded by a habit.

Peirce: CP 5.418 Cross-Ref:†† 418. Among the things which the reader, as a rational person, does not doubt, is that he not merely has habits, but also can exert a measure of self-control over his future actions; which means, however, not that he can impart to them any arbitrarily assignable character, but, on the contrary, that a process of self-preparation will tend to impart to action (when the occasion for it shall arise), one fixed character, which is indicated and perhaps roughly measured by the absence (or slightness) of the feeling of self-reproach, which subsequent reflection will induce. Now, this subsequent reflection is part of the self-preparation for action on the next occasion. Consequently, there is a tendency, as action is repeated again and again, for the action to approximate indefinitely toward the perfection of that fixed character, which would be marked by entire absence of self-reproach. The more closely this is approached, the less room for self-control there will be; and where no self-control is possible there will be no self-reproach. Peirce: CP 5.419 Cross-Ref:†† 419. These phenomena seem to be the fundamental characteristics which distinguish a rational being. Blame, in every case, appears to be a modification, often accomplished by a transference, or "projection," of the primary feeling of self-reproach. Accordingly, we never blame anybody for what had been beyond his power of previous self-control. Now, thinking is a species of conduct which is largely subject to self-control. In all their features (which there is no room to describe here), logical self-control is a perfect mirror of ethical self-control -- unless it be rather a species under that genus.†1 In accordance with this, what you cannot in the least help believing is not, justly speaking, wrong belief. In other words, for you it is the absolute truth. True, it is conceivable that what you cannot help believing today, you might find you thoroughly disbelieve tomorrow. But then there is a certain distinction between things you "cannot" do, merely in the sense that nothing stimulates you to the great effort and endeavors that would be required, and things you cannot do because in their own nature they are insusceptible of being put into practice. In every stage of your excogitations, there is something of which you can only say, "I cannot think otherwise," and your experientially based hypothesis is that the impossibility is of the second kind. Peirce: CP 5.420 Cross-Ref:†† 420. There is no reason why "thought," in what has just been said, should be taken in that narrow sense in which silence and darkness are favorable to thought. It should rather be understood as covering all rational life, so that an experiment shall be an operation of thought. Of course, that ultimate state of habit to which the action of self-control ultimately tends, where no room is left for further self-control, is, in the case of thought, the state of fixed belief, or perfect knowledge. Peirce: CP 5.421 Cross-Ref:†† 421. Two things here are all-important to assure oneself of and to remember. The first is that a person is not absolutely an individual. His thoughts are what he is "saying to himself," that is, is saying to that other self that is just coming into life in the flow of time. When one reasons, it is that critical self that one is trying to persuade; and all thought whatsoever is a sign, and is mostly of the nature of language. The second thing to remember is that the man's circle of society (however widely or narrowly this phrase may be understood), is a sort of loosely compacted person, in some respects of higher rank than the person of an individual organism. It is these two things alone that render it possible for you -- but only in the abstract, and

in a Pickwickian sense -- to distinguish between absolute truth and what you do not doubt. Peirce: CP 5.422 Cross-Ref:†† 422. Let us now hasten to the exposition of pragmaticism itself. Here it will be convenient to imagine that somebody to whom the doctrine is new, but of rather preternatural perspicacity, asks questions of a pragmaticist. Everything that might give a dramatic illusion must be stripped off, so that the result will be a sort of cross between a dialogue and a catechism, but a good deal liker the latter -- something rather painfully reminiscent of Mangnall's Historical Questions. Peirce: CP 5.422 Cross-Ref:†† Questioner: I am astounded at your definition of your pragmatism, because only last year I was assured by a person above all suspicion of warping the truth -himself a pragmatist -- that your doctrine precisely was "that a conception is to be tested by its practical effects." You must surely, then, have entirely changed your definition very recently. Peirce: CP 5.422 Cross-Ref:†† Pragmatist: If you will turn to Vols. VI and VII of the Revue Philosophique, or to the Popular Science Monthly for November 1877 and January 1878 [Papers No. IV and V], you will be able to judge for yourself whether the interpretation you mention was not then clearly excluded. The exact wording of the English enunciation, (changing only the first person into the second), was: "Consider what effects that might conceivably have practical bearing you conceive the object of your conception to have. Then your conception of those effects is the WHOLE of your conception of the object."†1 Peirce: CP 5.422 Cross-Ref:†† Questioner: Well, what reason have you for asserting that this is so? Peirce: CP 5.422 Cross-Ref:†† Pragmatist: That is what I specially desire to tell you. But the question had better be postponed until you clearly understand what those reasons profess to prove. Peirce: CP 5.423 Cross-Ref:†† 423. Questioner: What, then, is the raison d'être of the doctrine? What advantage is expected from it? Peirce: CP 5.423 Cross-Ref:†† Pragmatist: It will serve to show that almost every proposition of ontological metaphysics is either meaningless gibberish -- one word being defined by other words, and they by still others, without any real conception ever being reached -- or else is downright absurd; so that all such rubbish being swept away, what will remain of philosophy will be a series of problems capable of investigation by the observational methods of the true sciences -- the truth about which can be reached without those interminable misunderstandings and disputes which have made the highest of the positive sciences a mere amusement for idle intellects, a sort of chess -idle pleasure its purpose, and reading out of a book its method. In this regard, pragmaticism is a species of prope-positivism. But what distinguishes it from other species is, first, its retention of a purified philosophy; secondly, its full acceptance of the main body of our instinctive beliefs; and thirdly, its strenuous insistence upon the truth of scholastic realism (or a close approximation to that, well-stated by the late Dr.

Francis Ellingwood Abbot in the Introduction to his Scientific Theism). So, instead of merely jeering at metaphysics, like other prope-positivists, whether by long drawn-out parodies or otherwise, the pragmaticist extracts from it a precious essence, which will serve to give life and light to cosmology and physics. At the same time, the moral applications of the doctrine are positive and potent; and there are many other uses of it not easily classed. On another occasion, instances may be given to show that it really has these effects. Peirce: CP 5.424 Cross-Ref:†† 424. Questioner: I hardly need to be convinced that your doctrine would wipe out metaphysics. Is it not as obvious that it must wipe out every proposition of science and everything that bears on the conduct of life? For you say that the only meaning that, for you, any assertion bears is that a certain experiment has resulted in a certain way: Nothing else but an experiment enters into the meaning. Tell me, then, how can an experiment, in itself, reveal anything more than that something once happened to an individual object and that subsequently some other individual event occurred? Peirce: CP 5.424 Cross-Ref:†† Pragmatist: That question is, indeed, to the purpose -- the purpose being to correct any misapprehensions of pragmaticism. You speak of an experiment in itself, emphasising "in itself." You evidently think of each experiment as isolated from every other. It has not, for example, occurred to you, one might venture to surmise, that every connected series of experiments constitutes a single collective experiment. What are the essential ingredients of an experiment? First, of course, an experimenter of flesh and blood. Secondly, a verifiable hypothesis. This is a proposition †P1 relating to the universe environing the experimenter, or to some well-known part of it and affirming or denying of this only some experimental possibility or impossibility. The third indispensable ingredient is a sincere doubt in the experimenter's mind as to the truth of that hypothesis. Peirce: CP 5.424 Cross-Ref:†† Passing over several ingredients on which we need not dwell, the purpose, the plan, and the resolve, we come to the act of choice by which the experimenter singles out certain identifiable objects to be operated upon. The next is the external (or quasi-external) ACT by which he modifies those objects. Next, comes the subsequent reaction of the world upon the experimenter in a perception; and finally, his recognition of the teaching of the experiment. While the two chief parts of the event itself are the action and the reaction, yet the unity of essence of the experiment lies in its purpose and plan, the ingredients passed over in the enumeration. Peirce: CP 5.425 Cross-Ref:†† 425. Another thing: in representing the pragmaticist as making rational meaning to consist in an experiment (which you speak of as an event in the past), you strikingly fail to catch his attitude of mind. Indeed, it is not in an experiment, but in experimental phenomena, that rational meaning is said to consist. When an experimentalist speaks of a phenomenon, such as "Hall's phenomenon," "Zeemann's phenomenon" and its modification, "Michelson's phenomenon," or "the chessboard phenomenon," he does not mean any particular event that did happen to somebody in the dead past, but what surely will happen to everybody in the living future who shall fulfill certain conditions. The phenomenon consists in the fact that when an experimentalist shall come to act according to a certain scheme that he has in mind, then will something else happen, and shatter the doubts of sceptics, like the celestial

fire upon the altar of Elijah. Peirce: CP 5.426 Cross-Ref:†† 426. And do not overlook the fact that the pragmaticist maxim says nothing of single experiments or of single experimental phenomena (for what is conditionally true in futuro can hardly be singular), but only speaks of general kinds of experimental phenomena. Its adherent does not shrink from speaking of general objects as real, since whatever is true represents a real. Now the laws of nature are true. Peirce: CP 5.427 Cross-Ref:†† 427. The rational meaning of every proposition lies in the future. How so? The meaning of a proposition is itself a proposition. Indeed, it is no other than the very proposition of which it is the meaning: it is a translation of it. But of the myriads of forms into which a proposition may be translated, what is that one which is to be called its very meaning? It is, according to the pragmaticist, that form in which the proposition becomes applicable to human conduct, not in these or those special circumstances, nor when one entertains this or that special design, but that form which is most directly applicable to self-control under every situation, and to every purpose. This is why he locates the meaning in future time; for future conduct is the only conduct that is subject to self-control. But in order that that form of the proposition which is to be taken as its meaning should be applicable to every situation and to every purpose upon which the proposition has any bearing, it must be simply the general description of all the experimental phenomena which the assertion of the proposition virtually predicts. For an experimental phenomenon is the fact asserted by the proposition that action of a certain description will have a certain kind of experimental result; and experimental results are the only results that can affect human conduct. No doubt, some unchanging idea may come to influence a man more than it had done; but only because some experience equivalent to an experiment has brought its truth home to him more intimately than before. Whenever a man acts purposively, he acts under a belief in some experimental phenomenon. Consequently, the sum of the experimental phenomena that a proposition implies makes up its entire bearing upon human conduct. Your question, then, of how a pragmaticist can attribute any meaning to any assertion other than that of a single occurrence is substantially answered. Peirce: CP 5.428 Cross-Ref:†† 428. Questioner: I see that pragmaticism is a thorough-going phenomenalism. Only why should you limit yourself to the phenomena of experimental science rather than embrace all observational science? Experiment, after all, is an uncommunicative informant. It never expiates †1: it only answers "yes" or "no"; or rather it usually snaps out "No!" or, at best only utters an inarticulate grunt for the negation of its "no." The typical experimentalist is not much of an observer. It is the student of natural history to whom nature opens the treasury of her confidence, while she treats the cross-examining experimentalist with the reserve he merits. Why should your phenomenalism sound the meagre jews-harp of experiment rather than the glorious organ of observation? Peirce: CP 5.428 Cross-Ref:†† Pragmaticist: Because pragmaticism is not definable as "thorough-going phenomenalism," although the latter doctrine may be a kind of pragmatism. The richness of phenomena lies in their sensuous quality. Pragmaticism does not intend to define the phenomenal equivalents of words and general ideas, but, on the contrary,

eliminates their sential element, and endeavors to define the rational purport, and this it finds in the purposive bearing of the word or proposition in question. Peirce: CP 5.429 Cross-Ref:†† 429. Questioner: Well, if you choose so to make Doing the Be-all and the End-all of human life, why do you not make meaning to consist simply in doing? Doing has to be done at a certain time upon a certain object. Individual objects and single events cover all reality, as everybody knows, and as a practicalist ought to be the first to insist. Yet, your meaning, as you have described it, is general. Thus, it is of the nature of a mere word and not a reality. You say yourself that your meaning of a proposition is only the same proposition in another dress. But a practical man's meaning is the very thing he means. What do you make to be the meaning of "George Washington"? Peirce: CP 5.429 Cross-Ref:†† Pragmaticist: Forcibly put! A good half dozen of your points must certainly be admitted. It must be admitted, in the first place, that if pragmaticism really made Doing to be the Be-all and the End-all of life, that would be its death. For to say that we live for the mere sake of action, as action, regardless of the thought it carries out, would be to say that there is no such thing as rational purport.†1 Secondly, it must be admitted that every proposition professes to be true of a certain real individual object, often the environing universe. Thirdly, it must be admitted that pragmaticism fails to furnish any translation or meaning of a proper name, or other designation of an individual object. Fourthly, the pragmaticistic meaning is undoubtedly general; and it is equally indisputable that the general is of the nature of a word or sign. Fifthly, it must be admitted that individuals alone exist; and sixthly, it may be admitted that the very meaning of a word or significant object ought to be the very essence of reality of what it signifies. But when those admissions have been unreservedly made, you find the pragmaticist still constrained most earnestly to deny the force of your objection, you ought to infer that there is some consideration that has escaped you. Putting the admissions together, you will perceive that the pragmaticist grants that a proper name (although it is not customary to say that it has a meaning), has a certain denotative function peculiar, in each case, to that name and its equivalents; and that he grants that every assertion contains such a denotative or pointing-out function. In its peculiar individuality, the pragmaticist excludes this from the rational purport of the assertion, although the like of it, being common to all assertions, and so, being general and not individual, may enter into the pragmaticistic purport. Whatever exists, ex-sists, that is, really acts upon other existents, so obtains a self-identity, and is definitely individual. As to the general, it will be a help to thought to notice that there are two ways of being general. A statue of a soldier on some village monument, in his overcoat and with his musket, is for each of a hundred families the image of its uncle, its sacrifice to the Union. That statue, then, though it is itself single, represents any one man of whom a certain predicate may be true. It is objectively general. The word "soldier," whether spoken or written, is general in the same way; while the name, "George Washington," is not so. But each of these two terms remains one and the same noun, whether it be spoken or written, and whenever and wherever it be spoken or written. This noun is not an existent thing: it is a type,†1 or form, to which objects, both those that are externally existent and those which are imagined, may conform, but which none of them can exactly be. This is subjective generality. The pragmaticistic purport is general in both ways. Peirce: CP 5.430 Cross-Ref:††

430. As to reality, one finds it defined in various ways; but if that principle of terminological ethics that was proposed be accepted, the equivocal language will soon disappear. For realis and realitas are not ancient words. They were invented to be terms of philosophy in the thirteenth century,†2 and the meaning they were intended to express is perfectly clear. That is real which has such and such characters, whether anybody thinks it to have those characters or not. At any rate, that is the sense in which the pragmaticist uses the word. Now, just as conduct controlled by ethical reason tends toward fixing certain habits of conduct, the nature of which (as to illustrate the meaning, peaceable habits and not quarrelsome habits) does not depend upon any accidental circumstances, and in that sense may be said to be destined; so, thought, controlled by a rational experimental logic, tends to the fixation of certain opinions, equally destined, the nature of which will be the same in the end, however the perversity of thought of whole generations may cause the postponement of the ultimate fixation. If this be so, as every man of us virtually assumes that it is, in regard to each matter the truth of which he seriously discusses, then, according to the adopted definition of "real," the state of things which will be believed in that ultimate opinion is real. But, for the most part, such opinions will be general. Consequently, some general objects are real. (Of course, nobody ever thought that all generals were real; but the scholastics used to assume that generals were real when they had hardly any, or quite no, experiential evidence to support their assumption; and their fault lay just there, and not in holding that generals could be real.) One is struck with the inexactitude of thought even of analysts of power, when they touch upon modes of being. One will meet, for example, the virtual assumption that what is relative to thought cannot be real. But why not, exactly? Red is relative to sight, but the fact that this or that is in that relation to vision that we call being red is not itself relative to sight; it is a real fact. Peirce: CP 5.431 Cross-Ref:†† 431. Not only may generals be real, but they may also be physically efficient, not in every metaphysical sense, but in the common-sense acception in which human purposes are physically efficient.†1 Aside from metaphysical nonsense, no sane man doubts that if I feel the air in my study to be stuffy, that thought may cause the window to be opened. My thought, be it granted, was an individual event. But what determined it to take the particular determination it did, was in part the general fact that stuffy air is unwholesome, and in part other Forms, concerning which Dr. Carus †2 has caused so many men to reflect to advantage -- or rather, by which, and the general truth concerning which Dr. Carus's mind was determined to the forcible enunciation of so much truth. For truths, on the average, have a greater tendency to get believed than falsities have. Were it otherwise, considering that there are myriads of false hypotheses to account for any given phenomenon, against one sole true one (or if you will have it so, against every true one), the first step toward genuine knowledge must have been next door to a miracle. So, then, when my window was opened, because of the truth that stuffy air is malsain, a physical effort was brought into existence by the efficiency of a general and non-existent truth. This has a droll sound because it is unfamiliar; but exact analysis is with it and not against it; and it has besides, the immense advantage of not blinding us to great facts -- such as that the ideas "justice" and "truth" are, notwithstanding the iniquity of the world, the mightiest of the forces that move it. Generality is, indeed, an indispensable ingredient of reality; for mere individual existence or actuality without any regularity whatever is a nullity. Chaos is pure nothing. Peirce: CP 5.432 Cross-Ref:††

432. That which any true proposition asserts is real, in the sense of being as it is regardless of what you or I may think about it. Let this proposition be a general conditional proposition as to the future, and it is a real general such as is calculated really to influence human conduct; and such the pragmaticist holds to be the rational purport of every concept. Peirce: CP 5.433 Cross-Ref:†† 433. Accordingly, the pragmaticist does not make the summum bonum to consist in action, but makes it to consist in that process of evolution whereby the existent comes more and more to embody those generals which were just now said to be destined, which is what we strive to express in calling them reasonable. In its higher stages, evolution takes place more and more largely through self-control, and this gives the pragmaticist a sort of justification for making the rational purport to be general. Peirce: CP 5.434 Cross-Ref:†† 434. There is much more in elucidation of pragmaticism that might be said to advantage, were it not for the dread of fatiguing the reader. It might, for example, have been well to show clearly that the pragmaticist does not attribute any different essential mode of being to an event in the future from that which he would attribute to a similar event in the past, but only that the practical attitude of the thinker toward the two is different.†1 It would also have been well to show that the pragmaticist does not make Forms to be the only realities in the world,†2 any more than he makes the reasonable purport of a word to be the only kind of meaning there is.†3 These things are, however, implicitly involved in what has been said. There is only one remark concerning the pragmaticist's conception of the relation of his formula to the first principles of logic which need detain the reader. Peirce: CP 5.435 Cross-Ref:†† 435. Aristotle's definition of universal predication,†1 which is usually designated (like a papal bull or writ of court, from its opening words), as the Dictum de omni, may be translated as follows: "We call a predication (be it affirmative or negative), universal, when, and only when, there is nothing among the existent individuals to which the subject affirmatively belongs, but to which the predicate will not likewise be referred (affirmatively or negatively, according as the universal predication is affirmative or negative)." The Greek is: {legomen de to kata pantos katégoreisthai hotan méden éi labein tön tou hypokeimenon kath' ohy thateron ou lechthésetai kai to kata médenos hösantös}. The important words "existent individuals" have been introduced into the translation (which English idiom would not here permit to be literal); but it is plain that existent individuals were what Aristotle meant. The other departures from literalness only serve to give modern English forms of expression. Now, it is well known that propositions in formal logic go in pairs, the two of one pair being convertible into another by the interchange of the ideas of antecedent and consequent, subject and predicate, etc.†2 The parallelism extends so far that it is often assumed to be perfect; but it is not quite so. The proper mate of this sort to the Dictum de omni is the following definition of affirmative predication: We call a predication affirmative (be it universal or particular) when, and only when, there is nothing among the sensational effects that belong universally to the predicate which will not be (universally or particularly, according as the affirmative predication is universal or particular), said to belong to the subject. Now, this is substantially the essential proposition of pragmaticism. Of course, its parallelism to the Dictum de omni will only be admitted by a person who admits the

truth of pragmaticism.

Peirce: CP 5.436 Cross-Ref:†† §4. PRAGMATICISM AND HEGELIAN ABSOLUTE IDEALISME

436. Suffer me to add one word more on this point. For if one cares at all to know what the pragmaticist theory consists in, one must understand that there is no other part of it to which the pragmaticist attaches quite as much importance as he does to the recognition in his doctrine of the utter inadequacy of action or volition or even of resolve or actual purpose, as materials out of which to construct a conditional purpose or the concept of conditional purpose. Had a purposed article concerning the principle of continuity and synthetising the ideas of the other articles of a series †1 in the early volumes of The Monist ever been written,†2 it would have appeared how, with thorough consistency, that theory involved the recognition that continuity is an indispensable element of reality, and that continuity is simply what generality becomes in the logic of relatives, and thus, like generality, and more than generality, is an affair of thought, and is the essence of thought. Yet even in its truncated condition, an extra-intelligent reader might discern that the theory of those cosmological articles made reality to consist in something more than feeling and action could supply, inasmuch as the primeval chaos, where those two elements were present, was explicitly shown to be pure nothing. Now, the motive for alluding to that theory just here is, that in this way one can put in a strong light a position which the pragmaticist holds and must hold, whether that cosmological theory be ultimately sustained or exploded, namely, that the third category -- the category of thought, representation, triadic relation, mediation, genuine thirdness, thirdness as such -- is an essential ingredient of reality, yet does not by itself constitute reality, since this category (which in that cosmology appears as the element of habit) can have no concrete being without action, as a separate object on which to work its government, just as action cannot exist without the immediate being of feeling on which to act. The truth is that pragmaticism is closely allied to the Hegelian absolute idealism, from which, however, it is sundered by its vigorous denial that the third category (which Hegel degrades to a mere stage of thinking) suffices to make the world, or is even so much as self-sufficient. Had Hegel, instead of regarding the first two stages with his smile of contempt, held on to them as independent or distinct elements of the triune Reality, pragmaticists might have looked up to him as the great vindicator of their truth. (Of course, the external trappings of his doctrine are only here and there of much significance.) For pragmaticism belongs essentially to the triadic class of philosophical doctrines, and is much more essentially so than Hegelianism is.†1 (Indeed, in one passage, at least, Hegel alludes to the triadic form of his exposition as to a mere fashion of dress.)

MILFORD, PA., September, 1904.

Peirce: CP 5.437 Cross-Ref:†† 437. POSTSCRIPT. During the last five months, I have met with references to several objections to the above opinions, but not having been able to obtain the text of

these objections, I do not think I ought to attempt to answer them. If gentlemen who attack either pragmatism in general or the variety of it which I entertain would only send me copies of what they write, more important readers they could easily find, but they could find none who would examine their arguments with a more grateful avidity for truth not yet apprehended, nor any who would be more sensible of their courtesy.

February 9, 1905.

Peirce: CP 5.438 Cross-Ref:†† VII

ISSUES OF PRAGMATICISM†1

§1 SIX CHARACTERS OF CRITICAL COMMON-SENSISME

438. Pragmaticism was originally enounced †P1 in the form of a maxim, as follows: Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object. Peirce: CP 5.438 Cross-Ref:†† I will restate this in other words, since ofttimes one can thus eliminate some unsuspected source of perplexity to the reader. This time it shall be in the indicative mood, as follows: The entire intellectual purport of any symbol consists in the total of all general modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol. Peirce: CP 5.439 Cross-Ref:†† 439. Two doctrines that were defended by the writer about nine years before the formulation of pragmaticism may be treated as consequences of the latter belief. One of these may be called Critical Common-sensism. It is a variety of the Philosophy of Common Sense, but is marked by six distinctive characters, which had better be enumerated at once. Peirce: CP 5.440 Cross-Ref:†† 440. Character I. Critical Common-sensism admits that there not only are indubitable propositions but also that there are indubitable inferences. In one sense, anything evident is indubitable; but the propositions and inferences which Critical Common-sensism holds to be original, in the sense one cannot "go behind" them (as the lawyers say), are indubitable in the sense of being acritical. The term "reasoning" ought to be confined to such fixation of one belief by another as is reasonable, deliberate, self-controlled. A reasoning must be conscious; and this consciousness is not mere "immediate consciousness," which (as I argued in 1868)†1 is simple Feeling viewed from another side, but is in its ultimate nature (meaning in that characteristic

element of it that is not reducible to anything simpler), a sense of taking a habit, or disposition to respond to a given kind of stimulus in a given kind of way. As to the nature of that, some éclaircissements will appear below and again in my third paper, on the Basis of Pragmaticism.†2 But the secret of rational consciousness is not so much to be sought in the study of this one peculiar nucleolus, as in the review of the process of self-control in its entirety. The machinery of logical self-control works on the same plan as does moral self-control, in multiform detail. The greatest difference, perhaps, is that the latter serves to inhibit mad puttings forth of energy, while the former most characteristically insures us against the quandary of Buridan's ass. The formation of habits under imaginary action (see the paper of January, 1878†3) is one of the most essential ingredients of both; but in the logical process the imagination takes far wider flights, proportioned to the generality of the field of inquiry, being bounded in pure mathematics solely by the limits of its own powers, while in the moral process we consider only situations that may be apprehended or anticipated. For in moral life we are chiefly solicitous about our conduct and its inner springs, and the approval of conscience, while in intellectual life there is a tendency to value existence as the vehicle of forms. Certain obvious features of the phenomena of self-control (and especially of habit) can be expressed compactly and without any hypothetical addition, except what we distinctly rate as imagery, by saying that we have an occult nature of which and of its contents we can only judge by the conduct that it determines, and by phenomena of that conduct. All will assent to that (or all but the extreme nominalist), but anti-synechistic thinkers wind themselves up in a factitious snarl by falsifying the phenomena in representing consciousness to be, as it were, a skin, a separate tissue, overlying an unconscious region of the occult nature, mind, soul, or physiological basis. It appears to me that in the present state of our knowledge a sound methodeutic prescribes that, in adhesion to the appearances, the difference is only relative and the demarcation not precise. Peirce: CP 5.441 Cross-Ref:†† 441. According to the maxim of Pragmaticism, to say that determination affects our occult nature is to say that it is capable of affecting deliberate conduct; and since we are conscious of what we do deliberately, we are conscious habitualiter of whatever hides in the depths of our nature; and it is presumable (and only presumable,†P1 although curious instances are on record), that a sufficiently energetic effort of attention would bring it out. Consequently, to say that an operation of the mind is controlled is to say that it is, in a special sense, a conscious operation; and this no doubt is the consciousness of reasoning. For this theory requires that in reasoning we should be conscious, not only of the conclusion, and of our deliberate approval of it, but also of its being the result of the premiss from which it does result, and furthermore that the inference is one of a possible class of inferences which conform to one guiding principle.†1 Now in fact we find a well-marked class of mental operations, clearly of a different nature from any others which do possess just these properties. They alone deserve to be called reasonings; and if the reasoner is conscious, even vaguely, of what his guiding principle is, his reasoning should be called a logical argumentation. There are, however, cases in which we are conscious that a belief has been determined by another given belief, but are not conscious that it proceeds on any general principle. Such is St. Augustine's "cogito, ergo sum." Such a process should be called, not a reasoning, but an acritical inference. Again, there are cases in which one belief is determined by another, without our being at all aware of it. These should be called associational suggestions of belief. Peirce: CP 5.442 Cross-Ref:††

442. Now the theory of Pragmaticism was originally based, as anybody will see who examines the papers of November 1877 and January 1878, upon a study of that experience of the phenomena of self-control which is common to all grown men and women; and it seems evident that to some extent, at least, it must always be so based. For it is to conceptions of deliberate conduct that Pragmaticism would trace the intellectual purport of symbols; and deliberate conduct is self-controlled conduct. Now control may itself be controlled, criticism itself subjected to criticism; and ideally there is no obvious definite limit to the sequence. But if one seriously inquires whether it is possible that a completed series of actual efforts should have been endless or beginningless (I will spare the reader the discussion), I think he can only conclude that (with some vagueness as to what constitutes an effort) this must be regarded as impossible.†1 It will be found to follow that there are, besides perceptual judgments, original (i.e., indubitable because uncriticized) beliefs of a general and recurrent kind, as well as indubitable acritical inferences. Peirce: CP 5.443 Cross-Ref:†† 443. It is important for the reader to satisfy himself that genuine doubt always has an external origin, usually from surprise; and that it is as impossible for a man to create in himself a genuine doubt by such an act of the will as would suffice to imagine the condition of a mathematical theorem, as it would be for him to give himself a genuine surprise by a simple act of the will. Peirce: CP 5.443 Cross-Ref:†† I beg my reader also to believe that it would be impossible for me to put into these articles over two per cent of the pertinent thought which would be necessary in order to present the subject as I have worked it out. I can only make a small selection of what it seems most desirable to submit to his judgment. Not only must all steps be omitted which he can be expected to supply for himself, but unfortunately much more that may cause him difficulty. Peirce: CP 5.444 Cross-Ref:†† 444. Character II. I do not remember that any of the old Scotch philosophers ever undertook to draw up a complete list of the original beliefs, but they certainly thought it a feasible thing, and that the list would hold good for the minds of all men from Adam down. For in those days Adam was an undoubted historical personage. Before any waft of the air of evolution had reached those coasts how could they think otherwise? When I first wrote, we were hardly orientated in the new ideas, and my impression was that the indubitable propositions changed with a thinking man from year to year. I made some studies preparatory to an investigation of the rapidity of these changes, but the matter was neglected, and it has been only during the last two years that I have completed a provisional inquiry which shows me that the changes are so slight from generation to generation, though not imperceptible even in that short period, that I thought to own my adhesion, under inevitable modification, to the opinion of that subtle but well-balanced intellect, Thomas Reid, in the matter of Common Sense (as well as in regard to immediate perception, along with Kant).†P1 Peirce: CP 5.445 Cross-Ref:†† 445. Character III. The Scotch philosophers recognized that the original beliefs, and the same thing is at least equally true of the acritical inferences, were of the general nature of instincts. But little as we know about instincts, even now, we are much better acquainted with them than were the men of the eighteenth century. We know, for example, that they can be somewhat modified in a very short time. The great facts have always been known; such as that instinct seldom errs, while reason

goes wrong nearly half the time, if not more frequently. But one thing the Scotch failed to recognize is that the original beliefs only remain indubitable in their application to affairs that resemble those of a primitive mode of life. It is, for example, quite open to reasonable doubt whether the motions of electrons are confined to three dimensions, although it is good methodeutic to presume that they are until some evidence to the contrary is forthcoming. On the other hand, as soon as we find that a belief shows symptoms of being instinctive, although it may seem to be dubitable, we must suspect that experiment would show that it is not really so; for in our artificial life, especially in that of a student, no mistake is more likely than that of taking a paper-doubt for the genuine metal. Take, for example, the belief in the criminality of incest. Biology will doubtless testify that the practice is inadvisable; but surely nothing that it has to say could warrant the intensity of our sentiment about it. When, however, we consider the thrill of horror which the idea excites in us, we find reason in that to consider it to be an instinct; and from that we may infer that if some rationalistic brother and sister were to marry, they would find that the conviction of horrible guilt could not be shaken off. Peirce: CP 5.445 Cross-Ref:†† In contrast to this may be placed the belief that suicide is to be classed as murder. There are two pretty sure signs that this is not an instinctive belief. One is that it is substantially confined to the Christian world. The other is that when it comes to the point of actual self-debate, this belief seems to be completely expunged and ex-sponged from the mind. In reply to these powerful arguments, the main points urged are the authority of the fathers of the church and the undoubtedly intense instinctive clinging to life. The latter phenomenon is, however, entirely irrelevant. For though it is a wrench to part with life, which has its charms at the very worst, just as it is to part with a tooth, yet there is no moral element in it whatever. As to the Christian tradition, it may be explained by the circumstances of the early Church. For Christianity, the most terribly earnest and most intolerant of religions (see The Book of Revelations of St. John the Divine) -- and it remained so until diluted with civilization -- recognized no morality as worthy of an instant's consideration except Christian morality. Now the early Church had need of martyrs, i.e., witnesses, and if any man had done with life, it was abominable infidelity to leave it otherwise than as a witness to its power. This belief, then, should be set down as dubitable; and it will no sooner have been pronounced dubitable, than Reason will stamp it as false. Peirce: CP 5.445 Cross-Ref:†† The Scotch School appears to have no such distinction concerning the limitations of indubitability and the consequent limitations of the jurisdiction of original belief. Peirce: CP 5.446 Cross-Ref:†† 446. Character IV. By all odds, the most distinctive character of the Critical Common-sensist, in contrast to the old Scotch philosopher, lies in his insistence that the acritically indubitable is invariably vague. Peirce: CP 5.446 Cross-Ref:†† Logicians have been at fault in giving Vagueness the go-by, so far as not even to analyze it. The present writer has done his best to work out the Stechiology (or Stoicheiology), Critic, and Methodeutic †1 of the subject, but can here only give a definition or two with some proposals respecting terminology. Peirce: CP 5.447 Cross-Ref:††

447. Accurate writers have apparently made a distinction between the definite and the determinate. A subject is determinate in respect to any character which inheres in it or is (universally and affirmatively) predicated of it, as well as in respect to the negative of such character, these being the very same respect. In all other respects it is indeterminate. The definite shall be defined presently. A sign (under which designation I place every kind of thought, and not alone external signs), that is in any respect objectively indeterminate (i.e., whose object is undetermined by the sign itself) is objectively general in so far as it extends to the interpreter the privilege of carrying its determination further.†P1 Example: "Man is mortal." To the question, What man? the reply is that the proposition explicitly leaves it to you to apply its assertion to what man or men you will.†2 A sign that is objectively indeterminate in any respect is objectively vague in so far as it reserves further determination to be made in some other conceivable sign, or at least does not appoint the interpreter as its deputy in this office.†3 Example: "A man whom I could mention seems to be a little conceited." The suggestion here is that the man in view is the person addressed; but the utterer does not authorize such an interpretation or any other application of what she says. She can still say, if she likes, that she does not mean the person addressed. Every utterance naturally leaves the right of further exposition in the utterer; and therefore, in so far as a sign is indeterminate, it is vague, unless it is expressly or by a well-understood convention rendered general. Usually, an affirmative predication covers generally every essential character of the predicate, while a negative predication vaguely denies some essential character. In another sense, honest people, when not joking, intend to make the meaning of their words determinate, so that there shall be no latitude of interpretation at all. That is to say, the character of their meaning consists in the implications and non-implications of their words; and they intend to fix what is implied and what is not implied. They believe that they succeed in doing so, and if their chat is about the theory of numbers, perhaps they may. But the further their topics are from such presciss, or "abstract," subjects, the less possibility is there of such precision of speech. In so far as the implication is not determinate, it is usually left vague; but there are cases where an unwillingness to dwell on disagreeable subjects causes the utterer to leave the determination of the implication to the interpreter; as if one says, "That creature is filthy, in every sense of the term." Peirce: CP 5.448 Cross-Ref:†† 448. Perhaps a more scientific pair of definitions would be that anything is general in so far as the principle of excluded middle does not apply to it and is vague in so far as the principle of contradiction does not apply to it.†1 Thus, although it is true that "Any proposition you please, once you have determined its identity, is either true or false"; yet so long as it remains indeterminate and so without identity, it need neither be true that any proposition you please is true, nor that any proposition you please is false. So likewise, while it is false that "A proposition whose identity I have determined is both true and false," yet until it is determinate, it may be true that a proposition is true and that a proposition is false.†P1 Peirce: CP 5.449 Cross-Ref:†† 449. In those respects in which a sign is not vague, it is said to be definite, and also with a slightly different mode of application, to be precise, a meaning probably due to præecisus having been applied to curt denials and refusals.†1 It has been the well-established, ordinary sense of precise since the Plantagenets; and it were much to be desired that this word, with its derivatives precision, precisive, etc., should, in the dialect of philosophy, be restricted to this sense. To express the act of rendering

precise (though usually only in reference to numbers, dates, and the like), the French have the verb préciser, which, after the analogy of décider, should have been précider. Would it not be a useful addition to our English terminology of logic, to adopt the verb to precide, to express the general sense, to render precise? Our older logicians with salutary boldness seem to have created for their service the verb to prescind, the corresponding Latin word meaning only to "cut off at the end," while the English word means to suppose without supposing some more or less determinately indicated accompaniment. In geometry, for example, we "prescind" shape from color, which is precisely the same thing as to "abstract" color from shape, although very many writers employ the verb "to abstract" so as to make it the equivalent of "prescind." But whether it was the invention or the courage of our philosophical ancestors which exhausted itself in the manufacture of the verb "prescind," the curious fact is that instead of forming from it the noun prescission, they took pattern from the French logicians in putting the word precision to this second use. About the same time †P1 (see Watts, Logick, 1725, I, vi, 9 ad fin.) the adjective precisive was introduced to signify what prescissive would have more unmistakably conveyed. If we desire to rescue the good ship Philosophy for the service of Science from the hands of lawless rovers of the sea of literature, we shall do well to keep prescind, presciss, prescission, and prescissive on the one hand, to refer to dissection in hypothesis, while precide, precise, precision, and precisive are used so as to refer exclusively to an expression of determination which is made either full or free for the interpreter. We shall thus do much to relieve the stem "abstract" from staggering under the double burden of conveying the idea of prescission as well as the unrelated and very important idea of the creation of ens rationis out of an {epos pteroen} -- to filch the phrase to furnish a name for an expression of non-substantive thought -- an operation that has been treated as a subject of ridicule -this hypostatic abstraction -- but which gives mathematics half its power. Peirce: CP 5.450 Cross-Ref:†† 450. The purely formal conception that the three affections of terms, determination, generality, and vagueness, form a group dividing a category of what Kant calls "functions of judgment" will be passed by as unimportant by those who have yet to learn how important a part purely formal conceptions may play in philosophy. Without stopping to discuss this, it may be pointed out that the "quantity" of propositions in logic, that is, the distribution of the first subject,†P1 is either singular (that is, determinate, which renders it substantially negligible in formal logic), or universal (that is, general), or particular (as the mediaeval logicians say, that is, vague or indefinite).†1 It is a curious fact that in the logic of relations it is the first and last quantifiers of a proposition that are of chief importance. To affirm of anything that it is a horse is to yield to it every essential character of a horse; to deny of anything that it is a horse is vaguely to refuse to it some one or more of those essential characters of the horse. There are, however, predicates that are unanalyzable in a given state of intelligence and experience. These are, therefore, determinately affirmed or denied. Thus, this same group of concepts reappears. Affirmation and denial are in themselves unaffected by these concepts, but it is to be remarked that there are cases in which we can have an apparently definite idea of a border line between affirmation and negation. Thus, a point of a surface may be in a region of that surface, or out of it, or on its boundary. This gives us an indirect and vague conception of an intermediary between affirmation and denial in general, and consequently of an intermediate, or nascent state, between determination and indetermination. There must be a similar intermediacy between generality and

vagueness. Indeed, in an article in the seventh volume of The Monist†1 there lies just beneath the surface of what is explicitly said, the idea of an endless series of such intermediacies. We shall find below some application for these reflections. Peirce: CP 5.451 Cross-Ref:†† 451. Character V. The Critical Common-sensist will be further distinguished from the old Scotch philosopher by the great value he attaches to doubt, provided only that it be the weighty and noble metal itself, and no counterfeit nor paper substitute. He is not content to ask himself whether he does doubt, but he invents a plan for attaining to doubt, elaborates it in detail, and then puts it into practice, although this may involve a solid month of hard work; and it is only after having gone through such an examination that he will pronounce a belief to be indubitable. Moreover, he fully acknowledges that even then it may be that some of his indubitable beliefs may be proved false. Peirce: CP 5.451 Cross-Ref:†† The Critical Common-sensist holds that there is less danger to heuretic science in believing too little than in believing too much. Yet for all that, the consequences to heuretics of believing too little may be no less than disaster. Peirce: CP 5.452 Cross-Ref:†† 452. Character VI. Critical Common-sensism may fairly lay claim to this title for two sorts of reasons; namely, that on the one hand it subjects four opinions to rigid criticism: its own; that of the Scotch school; that of those who would base logic or metaphysics on psychology or any other special science, the least tenable of all the philosophical opinions that have any vogue; and that of Kant; while on the other hand it has besides some claim to be called Critical from the fact that it is but a modification of Kantism. The present writer was a pure Kantist until he was forced by successive steps into Pragmaticism. The Kantist has only to abjure from the bottom of his heart the proposition that a thing-in-itself can, however indirectly, be conceived; and then correct the details of Kant's doctrine accordingly, and he will find himself to have become a Critical Common-sensist.

Peirce: CP 5.453 Cross-Ref:†† §2. SUBJECTIVE AND OBJECTIVE MODALITYE

453. Another doctrine which is involved in Pragmaticism as an essential consequence of it, but which the writer defended [306 ad fin] and North American Review, Vol. CXIII, pp. 449-472, 1871), [Vol. 9] before he had formulated, even in his own mind, the principle of pragmaticism, is the scholastic doctrine of realism. This is usually defined as the opinion that there are real objects that are general, among the number being the modes of determination of existent singulars, if, indeed, these be not the only such objects. But the belief in this can hardly escape being accompanied by the acknowledgment that there are, besides, real vagues, and especially real possibilities. For possibility being the denial of a necessity, which is a kind of generality, is vague like any other contradiction of a general. Indeed, it is the reality of some possibilities that pragmaticism is most concerned to insist upon. The article of January 1878†1 endeavored to gloze over this point as unsuited to the exoteric public addressed; or perhaps the writer wavered in his own mind. He said

that if a diamond were to be formed in a bed of cotton-wool, and were to be consumed there without ever having been pressed upon by any hard edge or point, it would be merely a question of nomenclature whether that diamond should be said to have been hard or not. No doubt this is true, except for the abominable falsehood in the word MERELY, implying that symbols are unreal. Nomenclature involves classification; and classification is true or false, and the generals to which it refers are either reals in the one case, or figments in the other. For if the reader will turn to the original maxim of pragmaticism at the beginning of this article, he will see that the question is, not what did happen, but whether it would have been well to engage in any line of conduct whose successful issue depended upon whether that diamond would resist an attempt to scratch it, or whether all other logical means of determining how it ought to be classed would lead to the conclusion which, to quote the very words of that article, would be "the belief which alone could be the result of investigation carried sufficiently far."†2 Pragmaticism makes the ultimate intellectual purport of what you please to consist in conceived conditional resolutions,†3 or their substance; and therefore, the conditional propositions, with their hypothetical antecedents, in which such resolutions consist, being of the ultimate nature of meaning, must be capable of being true, that is, of expressing whatever there be which is such as the proposition expresses, independently of being thought to be so in any judgment, or being represented to be so in any other symbol of any man or men. But that amounts to saying that possibility is sometimes of a real kind. Peirce: CP 5.454 Cross-Ref:†† 454. Fully to understand this, it will be needful to analyze modality, and ascertain in what it consists.†1 In the simplest case, the most subjective meaning, if a person does not know that a proposition is false, he calls it possible. If, however, he knows that it is true, it is much more than possible. Restricting the word to its characteristic applicability, a state of things has the Modality of the possible -- that is, of the merely possible -- only in case the contradictory state of things is likewise possible, which proves possibility to be the vague modality. One who knows that Harvard University has an office in State Street, Boston, and has impression that it is at No. 30, but yet suspects that 50 is the number, would say "I think it is at No. 30, but it may be at No. 50," or "it is possibly at No. 50." Thereupon, another, who does not doubt his recollection, might chime in, "It actually is at No. 50," or simply "it is at No. 50," or "it is at No. 50, de inesse." Thereupon, the person who had first asked, what the number was might say, "Since you are so positive, it must be at No. 50," for "I know the first figure is 5. So, since you are both certain the second is a 0, why 50 it necessarily is." That is to say, in this most subjective kind of Modality, that which is known by direct recollection is in the Mode of Actuality, the determinate mode. But when knowledge is indeterminate among alternatives, either there is one state of things which alone accords with them all, when this is in the Mode of Necessity, or there is more than one state of things that no knowledge excludes, when each of these is in the Mode of Possibility. Peirce: CP 5.455 Cross-Ref:†† 455. Other kinds of subjective Modality refer to a Sign or Representamen which is assumed to be true, but which does not include the Utterer's (i.e. the speaker's, writer's, thinker's or other symbolizer's) total knowledge, the different Modes being distinguished very much as above. There are other cases, however, in which, justifiably or not, we certainly think of Modality as objective. A man says, "I can go to the seashore if I like." Here is implied, to be sure, his ignorance of how he will decide to act. But this is not the point of the assertion. It is that the complete

determination of conduct in the act not yet having taken place, the further determination of it belongs to the subject of the action regardless of external circumstances. If he had said, "I must go where my employers may send me," it would imply that the function of such further determination lay elsewhere. In "You may do so and so," and "You must do so," the "may" has the same force as "can," except that in the one case freedom from particular circumstances is in question, and in the other freedom from a law or edict. Hence the phrase, "You may if you can." I must say that it is difficult for me to preserve my respect for the competence of a philosopher whose dull logic, not penetrating beneath the surface, leaves him to regard such phrases as misrepresentations of the truth. So an act of hypostatic abstraction which in itself is no violation of logic, however it may lend itself to a dress of superstition, may regard the collective tendencies to variableness in the world, under the name of Chance, as at one time having their way, and at another time overcome by the element of order; so that, for example, a superstitious cashier, impressed by a bad dream, may say to himself of a Monday morning, "May be, the bank has been robbed." No doubt, he recognizes his total ignorance in the matter. But besides that, he has in mind the absence of any particular cause which should protect his bank more than others that are robbed from time to time. He thinks of the variety in the universe as vaguely analogous to the indecision of a person, and borrows from that analogy the garb of his thought. At the other extreme stand those who declare as inspired (for they have no rational proof of what they allege), that an actuary's advice to an insurance company is based on nothing at all but ignorance. Peirce: CP 5.456 Cross-Ref:†† 456. There is another example of objective possibility: "A pair of intersecting rays, i.e., unlimited straight lines conceived as movable objects, can (or may) move, without ceasing to intersect, so that one and the same hyperboloid shall be completely covered by the track of each of them." How shall we interpret this, remembering that the object spoken of, the pair of rays, is a pure creation of the Utterer's imagination, although it is required (and, indeed, forced) to conform to the laws of space? Some minds will be better satisfied with a more subjective, or nominalistic, others with a more objective, realistic interpretation. But it must be confessed on all hands that whatever degree or kind of reality belongs to pure space belongs to the substance of that proposition, which merely expresses a property of space. Peirce: CP 5.457 Cross-Ref:†† 457. Let us now take up the case of that diamond which, having been crystallized upon a cushion of jeweler's cotton, was accidentally consumed by fire before the crystal of corundum that had been sent for had had time to arrive, and indeed without being subjected to any other pressure than that of the atmosphere and its own weight. The question is, was that diamond really hard? It is certain that no discernible actual fact determined it to be so. But is its hardness not, nevertheless, a real fact? To say, as the article of January 1878 seems to intend, that it is just as an arbitrary "usage of speech" chooses to arrange its thoughts, is as much as to decide against the reality of the property, since the real is that which is such as it is regardless of how it is, at any time, thought to be. Remember that this diamond's condition is not an isolated fact. There is no such thing; and an isolated fact could hardly be real. It is an unsevered, though presciss part of the unitary fact of nature. Being a diamond, it was a mass of pure carbon, in the form of a more or less transparent crystal (brittle, and of facile octahedral cleavage, unless it was of an unheard-of variety), which, if not trimmed after one of the fashions in which diamonds may be trimmed, took the shape of an octahedron, apparently regular (I

need not go into minutiæ), with grooved edges, and probably with some curved faces. Without being subjected to any considerable pressure, it could be found to be insoluble, very highly refractive, showing under radium rays (and perhaps under "dark light" and X-rays) a peculiar bluish phosphorescence, having as high a specific gravity as realgar or orpiment, and giving off during its combustion less heat than any other form of carbon would have done. From some of these properties hardness is believed to be inseparable. For like it they bespeak the high polemerization of the molecule. But however this may be, how can the hardness of all other diamonds fail to bespeak some real relation among the diamonds without which a piece of carbon would not be a diamond? Is it not a monstrous perversion of the word and concept real to say that the accident of the non-arrival of the corundum prevented the hardness of the diamond from having the reality which it otherwise, with little doubt, would have had? Peirce: CP 5.457 Cross-Ref:†† At the same time, we must dismiss the idea that the occult state of things (be it a relation among atoms or something else), which constitutes the reality of a diamond's hardness can possibly consist in anything but in the truth of a general conditional proposition. For to what else does the entire teaching of chemistry relate except to the "behavior" of different possible kinds of material substance? And in what does that behavior consist except that if a substance of a certain kind should be exposed to an agency of a certain kind, a certain kind of sensible result would ensue, according to our experiences hitherto. As for the pragmaticist, it is precisely his position that nothing else than this can be so much as meant by saying that an object possesses a character. He is therefore obliged to subscribe to the doctrine of a real Modality, including real Necessity and real Possibility. Peirce: CP 5.458 Cross-Ref:†† 458. A good question, for the purpose of illustrating the nature of Pragmaticism, is, What is Time? It is not proposed to attack those most difficult problems connected with the psychology, the epistemology, or the metaphysics of Time, although it will be taken for granted, as it must be according to what has been said, that Time is real.†1 The reader is only invited to the humbler question of what we mean by Time, and not of every kind of meaning attached to Past, Present, and Future either. Certain peculiar feelings are associated with the three general determinations of Time; but those are to be sedulously put out of view. That the reference of events to Time is irresistible will be recognized; but as to how it may differ from other kinds of irresistibility is a question not here to be considered. The question to be considered is simply, What is the intellectual purport of the Past, Present, and Future? It can only be treated with the utmost brevity. Peirce: CP 5.459 Cross-Ref:†† 459. That Time is a particular variety of objective Modality is too obvious for argumentation. The Past consists of the sum of faits accomplis, and this Accomplishment is the Existential Mode of Time. For the Past really acts upon us, and that it does, not at all in the way in which a Law or Principle influences us, but precisely as an Existent object acts. For instance, when a Nova Stella bursts out in the heavens, it acts upon one's eyes just as a light struck in the dark by one's own hands would; and yet it is an event which happened before the Pyramids were built. A neophyte may remark that its reaching the eyes, which is all we know, happens but a fraction of a second before we know it. But a moment's consideration will show him that he is losing sight of the question, which is not whether the distant Past can act

upon us immediately, but whether it acts upon us just as any Existent does. The instance adduced (certainly a commonplace enough fact), proves conclusively that the mode of the Past is that of Actuality. Nothing of the sort is true of the Future, to compass the understanding of which it is indispensable that the reader should divest himself of his Necessitarianism -- at best, but a scientific theory -- and return to the Common-sense State of Nature. Do you never say to yourself, "I can do this or that as well tomorrow as today"? Your Necessitarianism is a theoretical pseudo-belief -- a make-believe belief -- that such a sentence does not express the real truth. That is only to stick to proclaiming the unreality of that Time, of which you are invited, be it reality or figment, to consider the meaning. You need not fear to compromise your darling theory by looking out at its windows. Be it true in theory or not, the unsophisticated conception is that everything in the Future is either destined, i.e., necessitated already, or is undecided, the contingent future of Aristotle. In other words, it is not Actual, since it does not act except through the idea of it, that is, as a law acts; but is either Necessary or Possible, which are of the same mode since (as remarked above †1) Negation being outside the category of modality cannot produce a variation in Modality. As for the Present instant, it is so inscrutable that I wonder whether no sceptic has ever attacked its reality. I can fancy one of them dipping his pen in his blackest ink to commence the assault, and then suddenly reflecting that his entire life is in the Present -- the "living present," as we say, this instant when all hopes and fears concerning it come to their end, this Living Death in which we are born anew. It is plainly that Nascent State between the Determinate and the Indeterminate that was noticed above.†1 Peirce: CP 5.460 Cross-Ref:†† 460. Pragmaticism consists in holding that the purport of any concept is its conceived bearing upon our conduct. How, then, does the Past bear upon conduct? The answer is self-evident: whenever we set out to do anything, we "go upon," we base our conduct on facts already known, and for these we can only draw upon our memory. It is true that we may institute a new investigation for the purpose; but its discoveries will only become applicable to conduct after they have been made and reduced to a memorial maxim. In short, the Past is the storehouse of all our knowledge. Peirce: CP 5.460 Cross-Ref:†† When we say that we know that some state of things exists, we mean that it used to exist, whether just long enough for the news to reach the brain and be retransmitted to tongue or pen, or longer ago. Thus, from whatever point of view we contemplate the Past, it appears as the Existential Mode of Time. Peirce: CP 5.461 Cross-Ref:†† 461. How does the Future bear upon conduct? The answer is that future facts are the only facts that we can, in a measure, control; and whatever there may be in the Future that is not amenable to control are the things that we shall be able to infer, or should be able to infer under favorable circumstances. There may be questions concerning which the pendulum of opinion never would cease to oscillate, however favorable circumstances may be. But if so, those questions are ipso facto not real questions, that is to say, are questions to which there is no true answer to be given. It is natural to use the future tense (and the conditional mood is but a mollified future) in drawing a conclusion or in stating a consequence. "If two unlimited straight lines in one plane and crossed by a third making the sum . . . then these straight lines will meet on the side, etc." It cannot be denied that acritical inferences may refer to the

Past in its capacity as past; but according to Pragmaticism, the conclusion of a Reasoning power must refer to the Future. For its meaning refers to conduct, and since it is a reasoned conclusion must refer to deliberate conduct, which is controllable conduct. But the only controllable conduct is Future conduct. As for that part of the Past that lies beyond memory, the Pragmaticist doctrine is that the meaning of its being believed to be in connection with the Past consists in the acceptance as truth of the conception that we ought to conduct ourselves according to it (like the meaning of any other belief). Thus, a belief that Christopher Columbus discovered America really refers to the future. It is more difficult, it must be confessed, to account for beliefs that rest upon the double evidence of feeble but direct memory and upon rational inference. The difficulty does not seem insuperable; but it must be passed by. Peirce: CP 5.462 Cross-Ref:†† 462. What is the bearing of the Present instant upon conduct? Peirce: CP 5.462 Cross-Ref:†† Introspection is wholly a matter of inference.†1 One is immediately conscious of his Feelings, no doubt; but not that they are feelings of an ego. The self is only inferred. There is no time in the Present for any inference at all, least of all for inference concerning that very instant. Consequently the present object must be an external object, if there be any objective reference in it. The attitude of the Present is either conative or perceptive. Supposing it to be perceptive, the perception must be immediately known as external -- not indeed in the sense in which a hallucination is not external, but in the sense of being present regardless of the perceiver's will or wish. Now this kind of externality is conative externality. Consequently, the attitude of the present instant (according to the testimony of Common Sense, which is plainly adopted throughout) can only be a Conative attitude. The consciousness of the present is then that of a struggle over what shall be; and thus we emerge from the study with a confirmed belief that it is the Nascent State of the Actual. Peirce: CP 5.463 Cross-Ref:†† 463. But how is Temporal Modality distinguished from other Objective Modality? Not by any general character since Time is unique and sui generis. In other words there is only one Time. Sufficient attention has hardly been called to the surpassing truth of this for Time as compared with its truth for Space. Time, therefore, can only be identified by brute compulsion. But we must not go further.

Peirce: CP 5.464 Cross-Ref:†† BOOK III

UNPUBLISHED PAPERS

CHAPTER 1

A SURVEY OF PRAGMATICISM†1

§1. THE KERNEL OF PRAGMATISM

464. It is now high time to explain what pragmatism is. I must, however, preface the explanation by a statement of what it is not, since many writers, especially of the starry host of Kant's progeny, in spite of pragmatists' declarations, unanimous, reiterated, and most explicit, still remain unable to "catch on" to what we are driving at, and persist in twisting our purpose and purport all awry. I was long enough, myself, within the Kantian fold to comprehend their difficulty; but let it go. Suffice it to say once more that pragmatism is, in itself, no doctrine of metaphysics, no attempt to determine any truth of things. It is merely a method of ascertaining the meanings of hard words and of abstract concepts. All pragmatists of whatsoever stripe will cordially assent to that statement. As to the ulterior and indirect effects of practising the pragmatistic method, that is quite another affair. Peirce: CP 5.465 Cross-Ref:†† 465. All pragmatists will further agree that their method of ascertaining the meanings of words and concepts is no other than that experimental method by which all the successful sciences (in which number nobody in his senses would include metaphysics) have reached the degrees of certainty that are severally proper to them today; this experimental method being itself nothing but a particular application of an older logical rule, "By their fruits ye shall know them." Peirce: CP 5.466 Cross-Ref:†† 466. Beyond these two propositions to which pragmatists assent nem. con., we find such slight discrepancies between the views of one and another declared adherent as are to be found in every healthy and vigorous school of thought in every department of inquiry. The most prominent of all our school and the most respected, William James, defines pragmatism as the doctrine that the whole "meaning" of a concept expresses itself either in the shape of conduct to be recommended or of experience to be expected.†1 Between this definition and mine there certainly appears to be no slight theoretical divergence, which, for the most part, becomes evanescent in practice; and though we may differ on important questions of philosophy -- especially as regards the infinite and the absolute -- I am inclined to think that the discrepancies reside in other than the pragmatistic ingredients of our thought. If pragmatism had never been heard of, I believe the opinion of James on one side, of me on the other would have developed substantially as they have; notwithstanding our respective connecting them at present with our conception of that method. The brilliant and marvellously human thinker, Mr. F.C.S. Schiller, who extends to the philosophic world a cup of nectar stimulant in his beautiful Humanism, seems to occupy ground of his own, intermediate, as to this question, between those of James and mine. Peirce: CP 5.467 Cross-Ref:†† 467. I understand pragmatism to be a method of ascertaining the meanings, not of all ideas, but only of what I call "intellectual concepts," that is to say, of those upon the structure of which, arguments concerning objective fact may hinge. Had the light which, as things are, excites in us the sensation of blue, always excited the sensation of red, and vice versa, however great a difference that might have made in

our feelings, it could have made none in the force of any argument. In this respect, the qualities of hard and soft strikingly contrast with those of red and blue; because while red and blue name mere subjective feelings only, hard and soft express the factual behaviour of the thing under the pressure of a knife-edge. (I use the word "hard" in its strict mineralogical sense, "would resist a knife-edge.") My pragmatism, having nothing to do with qualities of feeling, permits me to hold that the predication of such a quality is just what it seems, and has nothing to do with anything else. Hence, could two qualities of feeling everywhere be interchanged, nothing but feelings could be affected. Those qualities have no intrinsic significations beyond themselves. Intellectual concepts, however -- the only sign-burdens that are properly denominated "concepts" -- essentially carry some implication concerning the general behaviour either of some conscious being or of some inanimate object, and so convey more, not merely than any feeling, but more, too, than any existential fact, namely, the "would-acts," "would-dos" of habitual behaviour; and no agglomeration of actual happenings can ever completely fill up the meaning of a "would-be." But [Pragmatism asserts], that the total meaning of the predication of an intellectual concept is contained in an affirmation that, under all conceivable circumstances of a given kind (or under this or that more or less indefinite part of the cases of their fulfillment, should the predication be modal) the subject of the predication would behave in a certain general way -- that is, it would be true under given experiential circumstances (or under a more or less definitely stated proportion of them, taken as they would occur, that is in the same order of succession, in experience).†P1 Peirce: CP 5.468 Cross-Ref:†† 468. A most pregnant principle, quite undeniably, will this "kernel of pragmatism" prove to be, that the whole meaning of an intellectual predicate is that certain kinds of events would happen, once in so often, in the course of experience, under certain kinds of existential conditions -- provided it can be proved to be true. But how is this to be done in the teeth of Messrs. Bradley, Taylor, and other high metaphysicians, on the one hand, and of the entire nominalistic nation, with its Wundts, its Haeckels, its Karl Pearsons, and many other regiments, in their divers uniforms, on the other? Peirce: CP 5.468 Cross-Ref:†† At this difficulty I have halted for weeks and weeks. It has not been that I could not furnish forth an ample supply of seductive persuasions to pragmatism, or even two or three scientific proofs of its truth. Without a recognition of the chief moments, or points, of these latter it is quite impossible that the power and heart's blood of any variety of doctrine or tendency that ought to be classed among the different species of pragmatism should be really comprehended. A man may very well feel advantages in applications of pragmatism without anything of that. He may even make new applications of the method, himself -- with much risk of blundering, however; but it appears very plain, both to reason and to observation of experience, that he cannot know in what interior eye, what pineal gland its soul and power reside, unless he clearly understands the chief conditions of its truth. Unfortunately, however, all the real proofs of pragmatism that I know -- and, I hardly doubt, all there are to be known -- require just as close and laborious exertion of attention as any but the very most difficult of mathematical theorems, while they add to that all those difficulties of logical analysis which force the mathematician to creep with exceeding caution, if not timorously. But mature consideration has brought me to see that, while those circumstances would render a task quite hopeless that I had never dreamed of undertaking, that of convincing the readers of a literary journal †1 by any honest

argument, of the truth of pragmatism, and consequently must prevent communicating to them quite the idea of this method that an accomplished pragmatist has, yet an idea perfectly fulfilling the reader's desire, that of enabling him to place pragmatism and its concepts in the area of his own thought, and of showing roughly how its concepts are related to familiar concepts [may be given].

Peirce: CP 5.469 Cross-Ref:†† §2. THE VALENCY OF CONCEPTS †2

469. I begin, then, with the first idea that it seems desirable to call to your attention. Everybody is familiar with the useful, though fluctuating and relative distinction of matter and form; and it is strikingly true that distinctions and classifications founded upon form are, with very rare exceptions, more important to the scientific comprehension of the behaviour of things than distinctions and classifications founded upon matter. Mendeléeff's classification of the chemical elements, with which all educated men are, by this time, familiar, affords neat illustrations of this, since the distinctions between what he calls "groups," that is to say, the different vertical columns of his table, consists in the elements of one such "group" entering into different forms of combination with hydrogen and with oxygen from those of another group; or as we usually say, their valencies differ; while the distinctions between what he calls the "series," that is, the different horizontal rows of the table, consist in the less formal, more material circumstance that their atoms have, the elements of one "series," greater masses than those of the other. Now everybody who has the least acquaintance with chemistry knows that, while elements in different horizontal rows but the same vertical column always exhibit certain marked physical differences, their chemical behaviours at corresponding temperatures are quite similar; and all the major distinctions of chemical behaviour between different elements are due to their belonging to different vertical columns of the table. Peirce: CP 5.469 Cross-Ref:†† This illustration has much more pertinence to pragmatism than appears at first sight; since my researches into the logic of relatives have shown beyond all sane doubt that in one respect combinations of concepts exhibit a remarkable analogy with chemical combinations; every concept having a strict valency. (This must be taken to mean that of several forms of expression that are logically equivalent, that one or ones whose analytical accuracy is least open to question, owing to the introduction of the relation of joint identity, follows the law of valency.) Thus, the predicate "is blue" is univalent, the predicate "kills" is bivalent (for the direct and indirect objects are, grammar aside, as much subjects as is the subject nominative); the predicate "gives" is trivalent, since A gives B to C, etc. Just as the valency of chemistry is an atomic character, so indecomposable concepts may be bivalent or trivalent. Indeed, definitions being scrupulously observed, it will be seen to be a truism to assert that no compound of univalent and bivalent concepts alone can be trivalent, although a compound of any concept with a trivalent concept can have at pleasure, a valency higher or lower by one than that of the former concept. Less obvious, yet demonstrable, is the fact that no indecomposable concept has a higher valency. Among my papers are actual analyses of a number greater than I care to state.†1 They are mostly more complex than would be supposed. Thus, the relation between the four bonds of an unsymmetrical carbon atom consists of twenty-four triadic relations.

Careful analysis shows that to the three grades of valency of indecomposable concepts correspond three classes of characters or predicates. Firstly come "firstnesses," or positive internal characters of the subject in itself; secondly come "secondnesses," or brute actions of one subject or substance on another, regardless of law or of any third subject; thirdly comes "thirdnesses," or the mental or quasi-mental influence of one subject on another relatively to a third. Since the demonstration of this proposition is too stiff for the infantile logic of our time (which is rapidly awakening, however), I have preferred to state it problematically, as a surmise to be verified by observation. The little that I have contributed to pragmatism (or, for that matter, to any other department of philosophy), has been entirely the fruit of this outgrowth from formal logic, and is worth much more than the small sum total of the rest of my work, as time will show.

Peirce: CP 5.470 Cross-Ref:†† §3. LOGICAL INTERPRETANTS

470. The next moment of the argument for pragmatism is the view that every thought is a sign. This is the doctrine of Leibniz, Berkeley, and the thinkers of the years about 1700. They were all extreme nominalists; but it is a great mistake to suppose that this doctrine is peculiarly nominalistic. I am myself a scholastic realist of a somewhat extreme stripe. Every realist must, as such, admit that a general is a term and therefore a sign. If, in addition, he holds that it is an absolute exemplar, this Platonism passes quite beyond the question of nominalism and realism; and indeed the doctrine of Platonic ideas has been held by the extremest nominalists. There is some reason to suspect that it was shared by Roscellinus himself. Peirce: CP 5.471 Cross-Ref:†† 471. The next point is still less novel; for not to mention references to it by the Greek commentators upon Aristotle, it was between six and seven centuries ago that John of Salisbury spoke of it as "fere in omnium ore celebre."†1 It is the distinction, to use that author's phrases, between that which a term nominat -- its logical breadth -- and that which it significat -- its logical depth.†2 In the case of a proposition, it is the distinction between that which its subject denotes and that which its predicate asserts. In the case of an argument, it is the distinction between the state of things in which its premisses are true and the state of things which is defined by the truth of its conclusion. Peirce: CP 5.472 Cross-Ref:†† 472. The action of a sign calls for a little closer attention. Let me remind you of the distinction referred to above between dynamical, or dyadic, action; and intelligent, or triadic action. An event, A, may, by brute force, produce an event, B; and then the event, B, may in its turn produce a third event, C. The fact that the event, C, is about to be produced by B has no influence at all upon the production of B by A. It is impossible that it should, since the action of B in producing C is a contingent future event at the time B is produced. Such is dyadic action, which is so called because each step of it concerns a pair of objects. Peirce: CP 5.473 Cross-Ref:†† 473. But now when a microscopist is in doubt whether a motion of an animalcule is guided by intelligence, of however low an order, the test he always used

to apply when I went to school, and I suppose he does so still, is to ascertain whether event, A, produces a second event, B, as a means to the production of a third event, C, or not. That is, he asks whether B will be produced if it will produce or is likely to produce C in its turn, but will not be produced if it will not produce C in its turn nor is likely to do so. Suppose, for example, an officer of a squad or company of infantry gives the word of command, "Ground arms!" This order is, of course, a sign. That thing which causes a sign as such is called the object (according to the usage of speech, the "real," but more accurately, the existent object) represented by the sign: the sign is determined to some species of correspondence with that object. In the present case, the object the command represents is the will of the officer that the butts of the muskets be brought down to the ground. Nevertheless, the action of his will upon the sign is not simply dyadic; for if he thought the soldiers were deaf mutes, or did not know a word of English, or were raw recruits utterly undrilled, or were indisposed to obedience, his will probably would not produce the word of command. However, although this condition is most usually fulfilled, it is not essential to the action of a sign. For the acceleration of the pulse is a probable symptom of fever and the rise of the mercury in an ordinary thermometer or the bending of the double strip of metal in a metallic thermometer is an indication, or, to use the technical term, is an index, of an increase of atmospheric temperature, which, nevertheless, acts upon it in a purely brute and dyadic way. In these cases, however, a mental representation of the index is produced, which mental representation is called the immediate object of the sign; and this object does triadically produce the intended, or proper, effect of the sign strictly by means of another mental sign; and that this triadic character of the action is regarded as essential is shown by the fact that if the thermometer is dynamically connected with the heating and cooling apparatus, so as to check either effect, we do not, in ordinary parlance speak of there being any semeiosy, or action of a sign, but, on the contrary, say that there is an "automatic regulation," an idea opposed, in our minds, to that of semeiosy. For the proper significate outcome of a sign, I propose the name, the interpretant of the sign. The example of the imperative command shows that it need not be of a mental mode of being. Whether the interpretant be necessarily a triadic result is a question of words, that is, of how we limit the extension of the term "sign"; but it seems to me convenient to make the triadic production of the interpretant essential to a "sign," calling the wider concept like a Jacquard loom, for example, a "quasi-sign." On these terms, it is very easy (not descending to niceties with which I will not annoy your readers) to see what the interpretant of a sign is: it is all that is explicit in the sign itself apart from its context and circumstances of utterance. Still, there is a possible doubt as to where the line should be drawn between the interpretant and the object. It will be convenient to give the mere glance, which is all that can be afforded, to this question as it applies to propositions. The interpretant of a proposition is its predicate; its object is the things denoted by its subject or subjects (including its grammatical objects, direct and indirect, etc.). Take the proposition "Burnt child shuns fire." Its predicate might be regarded as all that is expressed, or as "has either not been burned or shuns fire" or "has not been burned," or "shuns fire" or "shuns" or "is true"; nor is this enumeration exhaustive. But where shall the line be most truly drawn? I reply that the purpose of this sentence being understood to be to communicate information, anything belongs to the interpretant that describes the quality or character of the fact, anything to the object that, without doing that, distinguishes this fact from others like it; while a third part of the proposition, perhaps, must be appropriated to information about the manner in which the assertion is made, what warrant is offered for its truth, etc. But I rather incline to think that all this goes to the subject. On this view, the predicate is, "is either not a

child or has not been burned, or has no opportunity of shunning fire or does shun fire"; while the subject is "any individual object the interpreter may select from the universe of ordinary everyday experience." Peirce: CP 5.474 Cross-Ref:†† 474. I omit all I possibly can; but there is one fact extremely familiar in itself, that needs to be mentioned as being an indispensible point in the argument. It is that every man inhabits two worlds. These are directly distinguishable by their different appearances. But the greatest difference between them, by far, is that one of these two worlds, the Inner World, exerts a comparatively slight compulsion upon us, though we can by direct efforts so slight as to be hardly noticeable, change it greatly, creating and destroying existent objects in it; while the other world, the Outer World, is full of irresistible compulsions for us, and we cannot modify it in the least, except by one peculiar kind of effort, muscular effort, and but very slightly even in that way. Peirce: CP 5.475 Cross-Ref:†† 475.†1 Now the problem of what the "meaning" of an intellectual concept is can only be solved by the study of the interpretants, or proper significate effects, of signs. These we find to be of three general classes with some important subdivisions. The first proper significate effect of a sign is a feeling produced by it. There is almost always a feeling which we come to interpret as evidence that we comprehend the proper effect of the sign, although the foundation of truth in this is frequently very slight. This "emotional interpretant," as I call it, may amount to much more than that feeling of recognition; and in some cases, it is the only proper significate effect that the sign produces. Thus, the performance of a piece of concerted music is a sign. It conveys, and is intended to convey, the composer's musical ideas; but these usually consist merely in a series of feelings. If a sign produces any further proper significate effect, it will do so through the mediation of the emotional interpretant, and such further effect will always involve an effort. I call it the energetic interpretant. The effort may be a muscular one, as it is in the case of the command to ground arms; but it is much more usually an exertion upon the Inner World, a mental effort. It never can be the meaning of an intellectual concept, since it is a single act, [while] such a concept is of a general nature. But what further kind of effect can there be? Peirce: CP 5.476 Cross-Ref:†† 476. In advance of ascertaining the nature of this effect, it will be convenient to adopt a designation for it, and I will call it the logical interpretant, without as yet determining whether this term shall extend to anything beside the meaning of a general concept, though certainly closely related to that, or not. Shall we say that this effect may be a thought, that is to say, a mental sign? No doubt, it may be so; only, if this sign be of an intellectual kind -- as it would have to be -- it must itself have a logical interpretant; so that it cannot be the ultimate logical interpretant of the concept. It can be proved that the only mental effect that can be so produced and that is not a sign but is of a general application is a habit-change; meaning by a habit-change a modification of a person's tendencies toward action, resulting from previous experiences or from previous exertions of his will or acts, or from a complexus of both kinds of cause. It excludes natural dispositions, as the term "habit" does, when it is accurately used; but it includes beside associations, what may be called "transsociations," or alterations of association, and even includes dissociation, which has usually been looked upon by psychologists (I believe mistakenly), as of deeply contrary nature to association. Peirce: CP 5.477 Cross-Ref:††

477. Habits have grades of strength varying from complete dissociation to inseparable association. These grades are mixtures of promptitude of action, say excitability and other ingredients not calling for separate examination here. The habit-change often consists in raising or lowering the strength of a habit. Habits also differ in their endurance (which is likewise a composite quality). But generally speaking, it may be said that the effects of habit-change last until time or some more definite cause produces new habit-changes. It naturally follows that repetitions of the actions that produce the changes increase the changes. [It] is noticeable that the iteration of the action is often said to be indispensible to the formation of a habit; but a very moderate exercise of observation suffices to refute this error. A single reading yesterday of a casual statement that the "shtar chindis" means in Romany "four shillings," though it is unlikely to receive any reinforcement beyond the recalling of it, at this moment, is likely to produce the habit of thinking that "four" in the Gypsy tongue is "shtar," that will last for months, if not for years, though I should never call it to mind in the interval. To be sure, there has been some iteration just now, while I dwelt on the matter long enough to write these sentences; but I do not believe any reminiscence like this was needed to create the habit; for such instances have been extremely numerous in acquiring different languages. There are, of course, other means than repetition of intensifying habit-changes. In particular, there is a peculiar kind of effort, which may be likened to an imperative command addressed to the future self. I suppose the psychologists would call it an act of auto-suggestion. Peirce: CP 5.478 Cross-Ref:†† 478. We may distinguish three classes of events causative of habit-change. Such events may, in the first place, not be acts of the mind in which the habit-change is brought about, but experiences forced upon [it]. Thus, surprise is very efficient in breaking up associations of ideas. On the other hand, each new instance that is brought to the experience that supports an induction goes to strengthen that association of ideas -- that inward habit -- in which the tendency to believe in the inductive conclusion consists. But careful examination has pretty thoroughly satisfied me that no new association, no entirely new habit, can be created by involuntary experiences. Peirce: CP 5.479 Cross-Ref:†† 479. In the second place, the event that causes a habit-change may be a muscular effort, apparently. If I wish to acquire the habit of speaking of "speaking, writing, thinking," etc., instead of "speakin', writin', thinkin'," as I suspect I now do (though I am not sure) -- all I have to do is to make the desired enunciations a good many times; and to do this as thoughtlessly as possible, since it is an inattentive habit that I am trying to create. Everybody knows the facility with which habits may thus be acquired, even quite unintentionally. But I am persuaded that nothing like a concept can be acquired by muscular practice alone. When we seem to do that, it is not the muscular action but the accompanying inward efforts, the acts of imagination, that produce the habit. If a person who has never tried such a thing before undertakes to stand on one foot and to move the other round a horizontal circle, say, as being the easier way, clockwise if he is standing on the left foot, or counter-clockwise if he is standing on the right foot, and at the same time to move the fist of the same side as the moving foot round a horizontal circle in the opposite direction, that is, clockwise if the foot is moved counter-clockwise, and vice versa, he will, at first, find he cannot do it. The difficulty is that he lacks a unitary concept of the series of efforts that success requires. By practising the different parts of the movement, while attentively observing the kind of effort requisite in each part, he will, in a few minutes, catch the

idea, and will then be able to perform the movements with perfect facility. But the proof that it is in no degree the muscular efforts, but only the efforts of the imagination that have been his teachers, is that if he does not perform the actual motions, but only imagines them vividly, he will acquire the same trick with only so much additional practice as is accounted for by the difficulty of imagining all the efforts that will have to be made in a movement one has not actually executed. There is an obvious difficulty of determining just how much allowance should be made for this, in the fact [that] when the feat is learned in either way, it cannot be unlearned, so as to compare that way with the other. The only resort is to learn a considerable number of feats which depend upon acquiring a unitary conception of a series of efforts, learning some with actual muscular exercise and others by unaided imagination, and then forming one's judgment of whether the greater facility afforded by the actual muscular contractions is, or is not, greater than the support this gives the imagination. Saying the verse about "Peter Piper"; spelling without an instant's hesitation, in the old way, the name Aldibirontifoscoforniocrononhotontothologes (that is, thus: A-l, al, and here's my al; d-i, di, and here's my di, and here's my aldi; b-i, bi, and here's my bi, and here's my dibi, and here's my aldibi, etc.); making the pass with one hand upon a pack of cards, playing the thimbles and ball, and other turns of legerdemain all largely depend for their success upon a unitary conception of all that has to be done and just when it must be done. It is from such experiments that I have been led to estimate as nil the power of mere muscular effort in contributing to the acquisition of ideas.†P1 Peirce: CP 5.480 Cross-Ref:†† 480. Every concept, doubtless, first arises when upon a strong, but more or less vague, sense of need is superinduced some involuntary experience of a suggestive nature; that being suggestive which has a certain occult relation to the build of the mind. We may assume that it is the same with the instinctive ideas of animals; and man's ideas are quite as miraculous as those of the bird, the beaver, and the ant. For a not insignificant percentage of them have turned out to be the keys of great secrets. With beasts, however, conditions are comparatively unchanging, and there is no further progress. With man these first concepts (first in the order of development, but emerging at all stages of mental life) take the form of conjectures, though they are by no means always recognized as such. Every concept, every general proposition of the great edifice of science, first came to us as a conjecture. These ideas are the first logical interpretants of the phenomena that suggest them, and which, as suggesting them, are signs, of which they are the (really conjectural) interpretants. But that they are no more than that is evidently an after-thought, the dash of cold doubt that awakens the sane judgment of the muser. Meantime, do not forget that every conjecture is equivalent to, or is expressive of, such a habit that having a certain desire one might accomplish it if one could perform a certain act. Thus, the primitive man must have been sometimes asked by his son whether the sun that rose in the morning was the same as the one that set the previous evening; and he may have replied, "I do not know, my boy; but I think that if I could put my brand on the evening sun, I should be able to see it on the morning sun again; and I once knew an old man who could look at the sun though he could hardly see anything else; and he told me that he had once seen a peculiarly shaped spot on the sun; and that it was to be recognized quite unmistakably for several days." [Readiness] to act in a certain way under given circumstances and when actuated by a given motive is a habit; and a deliberate, or self-controlled, habit is precisely a belief. Peirce: CP 5.481 Cross-Ref:††

481. In the next step of thought, those first logical interpretants stimulate us to various voluntary performances in the inner world. We imagine ourselves in various situations and animated by various motives; and we proceed to trace out the alternative lines of conduct which the conjectures would leave open to us. We are, moreover, led, by the same inward activity, to remark different ways in which our conjectures could be slightly modified. The logical interpretant must, therefore, be in a relatively future tense. Peirce: CP 5.482 Cross-Ref:†† 482. To this may be added the consideration that it is not all signs that have logical interpretants, but only intellectual concepts and the like; and these are all either general or intimately connected with generals, as it seems to me. This shows that the species of future tense of the logical interpretant is that of the conditional mood, the "would-be." Peirce: CP 5.483 Cross-Ref:†† 483. At the time I was originally puzzling over the enigma of the nature of the logical interpretant, and had reached about the stage where the discussion now is, being in a quandary, it occurred to me that if I only could find a moderate number of concepts which should be at once highly abstract and abstruse, and yet the whole nature of whose meanings should be quite unquestionable, a study of them would go far toward showing me how and why the logical interpretant should in all cases be a conditional future. I had no sooner framed a definite wish for such concepts, than I perceived that in mathematics they are as plenty as blackberries. I at once began running through the explications of them, which I found all took the following form: Proceed according to such and such a general rule. Then, if such and such a concept is applicable to such and such an object, the operation will have such and such a general result; and conversely. Thus, to take an extremely simple case, if two geometrical figures of dimensionality N should be equal in all their parts, an easy rule of construction would determine, in a space of dimensionality N containing both figures, an axis of rotation, such that a rigid body that should fill not only that space but also a space of dimensionality N + 1, containing the former space, turning about that axis, and carrying one of the figures along with it while the other figure remained at rest, the rotation would bring the movable figure back into its original space of dimensionality, N, and when that event occurred, the movable figure would be in exact coincidence with the unmoved one, in all its parts; while if the two figures were not so equal, this would never happen. Peirce: CP 5.483 Cross-Ref:†† Here was certainly a stride toward the solution of the enigma. Peirce: CP 5.483 Cross-Ref:†† For the treatment of a score of intellectual concepts on that model, only a few of them being mathematical, seemed to me to be so refulgently successful as fully to convince me that to predicate any such concept of a real or imaginary object is equivalent to declaring that a certain operation, corresponding to the concept, if performed upon that object, would (certainly, or probably, or possibly, according to the mode of predication), be followed by a result of a definite general description. Peirce: CP 5.484 Cross-Ref:†† 484. Yet this does not quite tell us just what the nature is of the essential effect

upon the interpreter, brought about by the semio'sis [Click here to view] of the sign, which constitutes the logical interpretant. (It is important to understand what I mean by semiosis. All dynamical action, or action of brute force, physical or psychical, either takes place between two subjects [whether they react equally upon each other, or one is agent and the other patient, entirely or partially] or at any rate is a resultant of such actions between pairs. But by "semiosis" I mean, on the contrary, an action, or influence, which is, or involves, a coöperation of three subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs. {Sémeiösis} in Greek of the Roman period, as early as Cicero's time, if I remember rightly, meant the action of almost any kind of sign; and my definition confers on anything that so acts the title of a "sign.") Peirce: CP 5.485 Cross-Ref:†† 485. Although the definition does not require the logical interpretant (or, for that matter, either of the other two interpretants) to be a modification of consciousness, yet our lack of experience of any semiosis in which this is not the case, leaves us no alternative to beginning our inquiry into its general nature with a provisional assumption that the interpretant is, at least, in all cases, a sufficiently close analogue of a modification of consciousness to keep our conclusion pretty near to the general truth. We can only hope that, once that conclusion is reached, it may be susceptible of such a generalization as will eliminate any possible error due to the falsity of that assumption. The reader may well wonder why I do not simply confine my inquiry to psychical semiosis, since no other seems to be of much importance. My reason is that the too frequent practice, by those logicians who do not go to work [with] any method at all [or who follow] the method of basing propositions in the science of logic upon results of the science of psychology -- as contradistinguished from common-sense observations concerning the workings of the mind, observations well-known even if little noticed, to all grown men and women, that are of sound minds -- that practice is to my apprehension as unsound and insecure as was that bridge in the novel of "Kenilworth" that, being utterly without any sort of support, sent the poor Countess Amy to her destruction; seeing that, for the firm establishment of the truths of the science of psychology, almost incessant appeals to the results of the science of logic -- as contradistinguished from natural perceptions that one relation evidently involves another -- are peculiarly indispensable. Those logicians continually confound psychical truths with psychological truths, although the distinction between them is of that kind that takes precedence over all others as calling for the respect of anyone who would tread the strait and narrow road that leadeth unto exact truth. Peirce: CP 5.486 Cross-Ref:†† 486. Making that provisional assumption, then, I ask myself, since we have already seen that the logical interpretant is general in its possibilities of reference (i.e., refers or is related to whatever there may be of a certain description), what categories of mental facts there be that are of general reference. I can find only these four: conceptions, desires (including hopes, fears, etc.), expectations, and habits. I trust I have made no important omission. Now it is no explanation of the nature of the logical interpretant (which, we already know, is a concept) to say that it is a concept. This objection applies also to desire and expectation, as explanations of the same

interpretant; since neither of these is general otherwise than through connection with a concept. Besides, as to desire, it would be easy to show (were it worth the space), that the logical interpretant is an effect of the energetic interpretant, in the sense in which the latter is an effect of the emotional interpretant. Desire, however, is cause, not effect, of effort.†1 As to expectation, it is excluded by the fact that it is not conditional. For that which might be mistaken for a conditional expectation is nothing but a judgment that, under certain conditions, there would be an expectation: there is no conditionality in the expectation itself, such as there is in the logical interpretant after it is actually produced. Therefore, there remains only habit, as the essence of the logical interpretant. Peirce: CP 5.487 Cross-Ref:†† 487. Let us see, then, just how, according to the rule derived from mathematical concepts (and confirmed by others), this habit is produced; and what sort of a habit it is. In order that this deduction may be rightly made, the following remark will be needed. It is not a result of scientific psychology, but is simply a bit of the catholic and undeniable common sense of mankind, with no other modification than a slight accentuation of certain features. Peirce: CP 5.487 Cross-Ref:†† Every sane person lives in a double world, the outer and the inner world, the world of percepts and the world of fancies. What chiefly keeps these from being mixed up together is (besides certain marks they bear) everybody's well knowing that fancies can be greatly modified by a certain non-muscular effort, while it is muscular effort alone (whether this be "voluntary," that is, pre-intended, or whether all the intended endeavour is to inhibit muscular action, as when one blushes, or when peristaltic action is set up on experience of danger to one's person) that can to any noticeable degree modify percepts. A man can be durably affected by his percepts and by his fancies. The way in which they affect him will be apt to depend upon his personal inborn disposition and upon his habits. Habits differ from dispositions in having been acquired as consequences of the principle, virtually well-known even to those whose powers of reflexion are insufficient to its formulation, that multiple reiterated behaviour of the same kind, under similar combinations of percepts and fancies, produces a tendency -- the habit -- actually to behave in a similar way under similar circumstances in the future. Moreover -- here is the point -- every man exercises more or less control over himself by means of modifying his own habits; and the way in which he goes to work to bring this effect about in those cases in which circumstances will not permit him to practice reiterations of the desired kind of conduct in the outer world shows that he is virtually well-acquainted with the important principle that reiterations in the inner world -- fancied reiterations -- if well-intensified by direct effort, produce habits, just as do reiterations in the outer world; and these habits will have power to influence actual behaviour in the outer world; especially, if each reiteration be accompanied by a peculiar strong effort that is usually likened to issuing a command to one's future self.†P1 Peirce: CP 5.488 Cross-Ref:†† 488. I here owe my patient reader a confession. It is that when I said that those signs that have a logical interpretant are either general or closely connected with generals, this was not a scientific result, but only a strong impression due to a life-long study of the nature of signs. My excuse for not answering the question scientifically is that I am, as far as I know, a pioneer, or rather a backwoodsman, in the work of clearing and opening up what I call semiotic, that is, the doctrine of the

essential nature and fundamental varieties of possible semiosis; and I find the field too vast, the labor too great, for a first-comer. I am, accordingly, obliged to confine myself to the most important questions. The questions of the same particular type as the one I answer on the basis of an impression, which are of about the same importance, exceed four hundred in number; and they are all delicate and difficult, each requiring much search and much caution. At the same time, they are very far from being among the most important of the questions of semiotic. Even if my answer is not exactly correct, it can lead to no great misconception as to the nature of the logical interpretant. There is my apology, such as it may be deemed. Peirce: CP 5.489 Cross-Ref:†† 489. It is not to be supposed that upon every presentation of a sign capable of producing a logical interpretant, such interpretant is actually produced. The occasion may either be too early or too late. If it is too early, the semiosis will not be carried so far, the other interpretants sufficing for the rude functions for which the sign is used. On the other hand, the occasion will come too late if the interpreter be already familiar with the logical interpretant, since then it will be recalled to his mind by a process which affords no hint of how it was originally produced. Moreover, the great majority of instances in which formations of logical interpretants do take place are very unsuitable to serve as illustrations of the process, because in them the essentials of this semiosis are buried in masses of accidental and hardly relevant semioses that are mixed with the former. The best way that I have been able to hit upon for simplifying the illustrative example which is to serve as our matter upon which to experiment and observe is to suppose a man already skillful in handling a given sign (that has a logical interpretant) to begin now before our inner gaze for the first time, seriously to inquire what that interpretant is. It will be necessary to amplify this hypothesis by a specification of what his interest in the question is supposed to be. In doing this, I, by no means, follow Mr. Schiller's brilliant and seductive humanistic logic, according to which it is proper to take account of the whole personal situation in logical inquiries.†1 For I hold it to be very evil and harmful procedure to introduce into scientific investigation an unfounded hypothesis, without any definite prospect of its hastening our discovery of the truth. Now such a hypothesis Mr. Schiller's rule seems to me, with my present lights, to be. He has given a number of reasons for it; but, to my estimate, they seem to be of that quality that is well calculated to give rise to interesting discussions, and is consequently to be recommended to those who intend to pursue the study of philosophy as an entertaining exercise of the intellect, but is negligible [to] one whose earnest purpose is to do what in him lies toward bringing about a metamorphosis of philosophy into a genuine science. I cannot turn aside into Mr. Schiller's charming lane. When I ask what the interest is in seeking to discover a logical interpretant, it is not my fondness for strolling in paths where I can study the varieties of humanity that moves me, but the definite reflection that unless our hypothesis be rendered specific as to that interest, it will be impossible to trace out its logical consequences, since the way the interpreter will conduct the inquiry will greatly depend upon the nature of his interest in it. Peirce: CP 5.490 Cross-Ref:†† 490. I shall suppose, then, that the interpreter is not particularly interested in the theory of logic, which he may judge by examples to be profitless; but I shall suppose that he has embarked a great part of the treasures of his life in the enterprise of perfecting a certain invention; and that, for this end, it seems to him extremely desirable that he should acquire a demonstrative knowledge of the solution of a certain problem of reasoning. As to this problem itself, I shall suppose that it does not

fall within any class for which any general method of handling is known, and that indeed it is indefinite in every respect which might afford any familiar kind of handle by which any image fairly representing it could be held firmly before the mind and examined; so that, in short, it seems to elude reason's application or to slip from its grasp. Peirce: CP 5.490 Cross-Ref:†† Various problems answering this description might be instanced; but to fix our ideas, I will specify one of them, and will suppose that this is the very one which our imaginary inventor wishes to solve. It shall be the following "map-coloring problem": Let a globular body be bored through in two wide holes; and, though it is unnecessary, the edge at each end of each tunnel shall be smoothly rounded off. Then the problem is, supposing its utterer is free to divide the whole surface of this body -including the surfaces of the bores -- into regions in any way he likes (no region consisting of separated pieces), and supposing that it will then fall to the interpreter to color the whole area of each region in one color, but never giving to two regions that abut along a common boundary-line the same color; required to ascertain what will be the least number of different colors that will always suffice, no matter how the surface may have been divided. Peirce: CP 5.490 Cross-Ref:†† Under the high stimulus of his interest in this problem, and with that practical knack that we have supposed him to possess in coloring maps without too frequently being obliged to go back and alter the colors he had assigned to given regions, we need not doubt that our inquirer will be thrown into a state of high activity in the world of fancies, in experimenting upon coloring maps, while trying to make out what subconscious rule guides him, and renders him as successful as he usually is; and in trying, too, to discover what rule he had violated in each case where his first coloration has to be changed. This activity is, logically, an energetic interpretant of the interrogatory he puts to himself. Should he in this way succeed in working out a determinate rule for coloring every map on the two-tunnelled (or, what is the same thing, the two-bridged) everywhere unbounded surface with the fewest possible colors, there will be good hope that a demonstration may tread upon the heels of that rule, in which case, the problem will be solved in the most convenient form. Peirce: CP 5.490 Cross-Ref:†† But while he may very likely manage to formulate his own usually successful way of coloring the regions, it is very unlikely that he will obtain an unfailing rule for doing so. For after some of the first mathematicians in Europe had found themselves baffled by the far simpler problem, to prove that every map upon an ordinary sheet can be colored with four colors, one of the very first logico-mathematicians of our age, Mr. Alfred B. Kempe,†1 proposed a proof of it, somewhat, though not exactly, of the kind we are supposing our imaginary inventor to be aiming at. Yet I am informed that many years later a fatal flaw was discovered in Mr. Kempe's proof. I do not remember that I ever knew what the fallacy was. We may assume with confidence, then, that our imaginary interpreter will, at length, come to despair of solving the problem in that way. What way shall I imagine him to try next? Peirce: CP 5.490 Cross-Ref:†† It will be very natural for him to pass from endeavouring to define a uniformly successful rule of procedure, to endeavouring either; first, to define the topical conditions under which two different regions must be colored alike, if the colors are not to exceed a given number, whence he will deduce the conditions under which two

regions that do not abut must be colored differently; or else, first to define the conditions under which two regions cannot, by being stretched out, be brought into abuttal along a boundary, and thence to define the conditions under which two regions must be colored alike. Either of these methods is more promising than the one with which he began; and yet were either capable of being perfected without some very peculiar aperçu, the easier task of demonstrating that four colors suffice for every map on an ordinary limited sheet or globular surface must long ago have been brought to completion, which never has been accomplished, I believe, in print. We may assume, then, that he will, at length, come to abandon every such method. Meantime, he cannot fail to have noticed several obvious propositions that will be useful in his further inquiries. One of these will be that by minute alterations of the boundaries between regions, which alterations can neither diminish nor increase the number of colors that will in all cases just suffice, he can get rid of all points where four or more regions concur, and thus render the number of points of concurrence two-thirds as many as the number of boundaries, so that the latter number will be divisible by three, and the former by two, unless fewer colors are required than are generally necessary. He will also have remarked that there must for each color be at least one region of that color which abuts upon regions of all the other colors, that for each of these other colors there must be at least one region that besides abutting upon the first region abuts upon regions of all the remaining colors, etc. Peirce: CP 5.490 Cross-Ref:†† I shall suppose that it now occurs to him that it not only makes no difference what the proportionate dimensions either of the whole surface or of any of the regions are, but that it is equally indifferent whether any part of the whole surface be flat, convex, concave, curved, or broken by angles, or whether any boundaries are straight, curved, or broken by angles, and are convex or concave to either of the regions it bounds; whence it will follow that the problem belongs neither to Metrical, nor to Graphical (or Projective) Geometry, but to Topical Geometry, or Geometrical Topics. This is the most fundamental, and no doubt, in its own nature, much the easiest of the three departments of geometry. For just as Cayley showed,†1 metrics is but a special problem in the easier graphics; so quite obviously graphics is a special problem in the easier topics. For there is no other possible way of defining unlimited planes and rays, than by the topical statement (which does not fully define them) that the unbounded planes are a family of surfaces in 3-dimensional space of which any two contain one common line, only, which is a ray, and of which any three that do not all contain one common ray, have one point and only one in common; and further, any two points are both contained in one and in only one ray, while any three points not all in one ray are contained in one and only one unbounded plane.†P1 Peirce: CP 5.490 Cross-Ref:†† But though Topics must be the easiest kind of geometry, yet geometers were so accustomed to rely on considerations of measure and of flatness, that when they were deprived of these, they did not know how to handle problems; so that, apart from mere enumerations of forms, such as knots, we are still in possession of only one general theorem of Topics, Listing's census-theorem.†1 Consequently, our imagined investigator, as soon as he remarks that he has a problem in topical geometry before him, will infer that he must utilize that sole known theorem of topics; albeit it is sufficiently obvious that that theorem of itself is not adequate to furnishing a solution of his problem. I will state the census-theorem of Listing with some sacrifice of exactitude to perspicuity, insofar as it applies to the map-coloring problem. The surface which is divided into regions may be bounded by a line or unbounded. If it be

unbounded and separates [a] solid into two parts, I call it artiad; if it does not, I call it perissid. The Cyclosy, or ringiness, of the surface of a body unpierced by any tunnel (i.e., not bridged over by an unbounded bridge), is zero; and every tunnel through the body adds two to the cyclosy of its surface. The cyclosy of the simplest perissid surface, such as an unbounded plane, is one, and every tunnel connecting two parts of it in an additional way (or every cylindrical bridge, which will be a tunnel on the other side of the surface) adds two to the cyclosy. A region, or an uninterrupted boundary that does not return into itself (as I will assume is the case with all regions and boundaries between two regions), has zero cyclosy. I will further assume that there is more than one region on the surface. Under these circumstances, the Census theorem takes this form, supposing all points of concurrence of regions are points where three regions and no more run together; one third of the number of boundaries from one point of concurrence to the next diminished by the number of regions is equal to one less than the cyclosy of the whole surface, if this be bounded, or to two less than the cyclosy, if the surface be unbounded. In the case of the surface of the body pierced by two tunnels, the surface is unbounded, and its cyclosy is 4. The investigator will see at once that the number of colors must be at least 7, and is likely to be more. For were the body pierced by but one tunnel, let the number of regions each abutting upon all the rest be x. Then, the number of boundaries would be 1/2x (x-1); and the census-theorem applied to this case would be 1/6 x (x-1)-x = 2-2. That is x2 - 7x = 0, or x = 7. Since, then, even with but one tunnel seven colors might be required, at least that number will be required for the case of two tunnels. On the other hand, were 2 tunnels made in a projective plane, where the cyclosy would be 5, instead of 4, only 9 regions could touch one another; so that it is likely that for a surface of cyclosy 4, the requisite number of colors is less than 9. The investigator will, therefore, only have to ascertain whether 8, and if so whether 9, colors can be required. He is still not very near his solution, but he is not hopelessly removed from it. Peirce: CP 5.491 Cross-Ref:†† 491. In every case, after some preliminaries, the activity takes the form of experimentation in the inner world; and the conclusion (if it comes to a definite conclusion), is that under given conditions, the interpreter will have formed the habit of acting in a given way whenever he may desire a given kind of result. The real and living logical conclusion is that habit; the verbal formulation merely expresses it. I do not deny that a concept, proposition, or argument may be a logical interpretant. I only insist that it cannot be the final logical interpretant, for the reason that it is itself a sign of that very kind that has itself a logical interpretant. The habit alone, which though it may be a sign in some other way, is not a sign in that way in which that sign of which it is the logical interpretant is the sign. The habit conjoined with the motive and the conditions has the action for its energetic interpretant; but action cannot be a logical interpretant, because it lacks generality. The concept which is a logical interpretant is only imperfectly so. It somewhat partakes of the nature of a verbal definition, and is as inferior to the habit, and much in the same way, as a verbal definition is inferior to the real definition. The deliberately formed, self-analyzing habit -- self-analyzing because formed by the aid of analysis of the exercises that nourished it -- is the living definition, the veritable and final logical interpretant. Consequently, the most perfect account of a concept that words can convey will consist in a description of the habit which that concept is calculated to produce. But how otherwise can a habit be described than by a description of the kind of action to which it gives rise, with the specification of the conditions and of the motive?

Peirce: CP 5.492 Cross-Ref:†† 492. If we now revert to the psychological assumption originally made, we shall see that it is already largely eliminated by the consideration that habit is by no means exclusively a mental fact. Empirically, we find that some plants take habits. The stream of water that wears a bed for itself is forming a habit. Every ditcher so thinks of it. Turning to the rational side of the question, the excellent current definition of habit, due, I suppose, to some physiologist (if I can remember my bye-reading for nearly half a century unglanced at, Brown-Sequard †1 much insisted on it in his book on the spinal cord), says not one word about the mind. Why should it, when habits in themselves are entirely unconscious, though feelings may be symptoms of them, and when consciousness alone -- i.e., feeling -- is the only distinctive attribute of mind? Peirce: CP 5.492 Cross-Ref:†† What further is needed to clear the sign of its mental associations is furnished by generalizations too facile to arrest attention here, since nothing but feeling is exclusively mental. Peirce: CP 5.493 Cross-Ref:†† 493. But while I say this, it must not be inferred that I regard consciousness as a mere "epiphenomenon"; though I heartily grant that the hypothesis that it is so has done good service to science. To my apprehension, consciousness may be defined as that congeries of non-relative predicates, varying greatly in quality and in intensity, which are symptomatic of the interaction of the outer world -- the world of those causes that are exceedingly compulsive upon the modes of consciousness, with general disturbance sometimes amounting to shock, and are acted upon only slightly, and only by a special kind of effort, muscular effort -- and of the inner world, apparently derived from the outer, and amenable to direct effort of various kinds with feeble reactions; the interaction of these two worlds chiefly consisting of a direct action of the outer world upon the inner and an indirect action of the inner world upon the outer through the operation of habits. If this be a correct account of consciousness, i.e., of the congeries of feelings, it seems to me that it exercises a real function in self-control, since without it, or at least without that of which it is symptomatic, the resolves and exercises of the inner world could not affect the real determinations and habits of the outer world. I say that these belong to the outer world because they are not mere fantasies but are real agencies.

Peirce: CP 5.494 Cross-Ref:†† §4. OTHER VIEWS OF PRAGMATISM

494. I have now outlined my own form of pragmatism; but there are other slightly different ways of regarding what is practically the same method of attaining vitally distinct conceptions, from which I should protest from the depths of my soul against being separated. In the first place, there is the pragmatism of James, whose definition †1 differs from mine only in that he does not restrict the "meaning," that is, the ultimate logical interpretant, as I do, to a habit, but allows percepts, that is, complex feelings endowed with compulsiveness, to be such. If he is willing to do this, I do not quite see how he need give any room at all to habit. But practically, his view and mine must, I think, coincide, except where he allows considerations not at all

pragmatic to have weight. Then there is Schiller, who offers no less than seven alternative definitions of pragmatism.†2 The first is that pragmatism is the Doctrine that "truths are logical values." At first blush, this seems far too broad; for who, be he pragmatist or absolutist, can fail to prefer truth to fiction? But no doubt what is meant is that the objectivity of truth really consists in the fact that, in the end, every sincere inquirer will be led to embrace it -- and if he be not sincere, the irresistible effect of inquiry in the light of experience will be to make him so. This doctrine appears to me, after one subtraction, to be a corollary of pragmatism. I set it in a strong light in my original presentation of the method.†3 I call my form of it "conditional idealism." That is to say, I hold that truth's independence of individual opinions is due (so far as there is any "truth") to its being the predestined result to which sufficient inquiry would ultimately lead. I only object that, as Mr. Schiller himself seems sometimes to say, there is not the smallest scintilla of logical justification for any assertion that a given sort of result will, as a matter of fact, either always or never come to pass; and consequently we cannot know that there is any truth concerning any given question; and this, I believe, agrees with the opinion of M. Henri Poincaré,†1 except that he seems to insist upon the non-existence of any absolute truth for all questions, which is simply to fall into the very same error on the opposite side. But practically, we know that questions do generally get settled in time, when they come to be scientifically investigated; and that is practically and pragmatically enough. Mr. Schiller's second definition is Captain Bunsby's that "the 'truth' of an assertion depends on its application," which seems to me the result of a weak analysis. His third definition is that pragmatism is the doctrine that "the meaning of a rule lies in its application," which would make the "meaning" consist in the energetic interpretant and would ignore the logical interpretant; another feeble analysis. His fourth definition is that pragmatism is the doctrine that "all meaning depends on purpose." I think there is much to be said in favor of this, which would, however, make pragmatists of many thinkers who do not consider themselves as belonging to our school of thought. Their affiliations with us are, however, undeniable. His fifth definition is that pragmatism is the doctrine that "all mental life is purposive." His sixth definition is that pragmatism is "a systematic protest against all ignoring of the purposiveness of actual knowing." Mr. Schiller seems habitually to use the word "actual" in some peculiar sense. His seventh definition is that pragmatism is "a conscious application to epistemology (or logic) of a teleological psychology, which implies, ultimately, a voluntaristic metaphysics." Supposing by "psychology" he means not the science so called, but a critical acceptance of a sifted common-sense of mankind regarding mental phenomena, I might subscribe to this. I have myself called pragmatism "critical common-sensism"; but, of course, I do not mean this for a strict definition.

Peirce: CP 5.495 Cross-Ref:†† 495. Signor Giovanni Papini goes a step beyond Mr. Schiller in maintaining [that] pragmatism is indefinable.†1 But that seems to me to be a literary phrase. In the main, I much admire Papini's presentation of the subject. Peirce: CP 5.496 Cross-Ref:†† 496. There are certain questions commonly reckoned as metaphysical, and which certainly are so, if by metaphysics we mean ontology, which as soon as pragmatism is once sincerely accepted, cannot logically resist settlement. These are for example, What is reality? Are necessity and contingency real modes of being? Are the laws of nature real? Can they be assumed to be immutable or are they presumably

results of evolution? Is there any real chance, or departure from real law? But on examination, if by metaphysics we mean the broadest positive truths of the psycho-physical universe -- positive in the sense of not being reducible to logical formulæ -- then the very fact that these problems can be solved by a logical maxim is proof enough that they do not belong to metaphysics but to "epistemology," an atrocious translation of Erkenntnislehre. When we pass to consider the nature of Time, it seems that pragmatism is of aid, but does not of itself yield a solution. When we go on to the nature of Space, I boldly declare that Newton's view that it is a real entity is alone logically tenable; and that leaves such further questions as, Why should Space have three dimensions? quite unanswerable for the present. This, however, is a purely speculative question without much human interest. (It would, of course, be absurd to say that tridimensionality is without practical consequences.) For those metaphysical questions that have such interest, the question of a future life and especially that of One Incomprehensible but Personal God, not immanent in but creating the universe, I, for one, heartily admit that a Humanism, that does not pretend to be a science but only an instinct, like a bird's power of flight, but purified by meditation, is the most precious contribution that has been made to philosophy for ages.

Peirce: CP 5.497 Cross-Ref:†† CHAPTER 2

PRAGMATICISM AND CRITICAL COMMON-SENSISM†1

497. Jules.†P1 Your Pragmaticism, then, seems to be simply the theory that a portion of the "meaning" of thought, that is, I suppose, the substance of thought, which portion you term its "intellectual purport," lies in its reference to conditional resolves. But I fail to see what your vague Common-Sense has to do with it. For your Pragmaticism is too definite to fall under that head. Your doctrine of Common-Sense cannot be proved from your Pragmatism, either, since that certain given propositions are absolutely indubitable must itself be an indemonstrable axiom or must be false. So I don't see what relevancy to Pragmaticism, even if to any conditional resolve, there was supposed to be in your discourse about this critical acceptance of uncriticizable propositions. Peirce: CP 5.497 Cross-Ref:†† The Respondent. The Common-Sensism now so widely accepted is not critical of the substantial truth of uncriticizable propositions, but only as to whether a given proposition is of the number. Peirce: CP 5.498 Cross-Ref:†† 498. Jules. It seems to me they are the same, at any rate according to your style of dealing with such questions. For I can almost hear you argue that you must either believe a proposition or doubt or disbelieve it. If you believe it, you do not doubt it and cannot criticize it; if you doubt it, it cannot be indubitable.

Peirce: CP 5.498 Cross-Ref:†† Respondent. Just at this moment the question is whether those two propositions really are the same or distinct, and not what my general style of argumentation is; and I am happy to think that you do not yourself sincerely judge all the sages of human nature to have been conscious liars who from time immemorial have testified to their conviction that man possess no infallible introspective power into the secrets of his own heart, to know just what he believes and what he doubts. The denial of such a power is one of the clauses of critical common-sensism. The others are that there are indubitable beliefs which vary a little and but a little under varying circumstances and in distant ages; that they partake of the nature of instincts, this word being taken in a broad sense; that they concern matters within the purview of the primitive man; that they are very vague indeed (such as, that fire burns) without being perfectly so; that while it may be disastrous to science for those who pursue it to think they doubt what they really believe, and still more so really to doubt what they ought to believe, yet, on the whole, neither of these is so unfavorable to science as for men of science to believe what they ought to doubt, nor even for them to think they believe what they really doubt; that a philosopher ought not to regard an important proposition as indubitable without a systematic and arduous endeavour to attain to a doubt of it, remembering that genuine doubt cannot be created by a mere effort of will, but must be compassed through experience; that while it is possible that propositions that really are indubitable, for the time being, should nevertheless be false, yet in so far as we do not doubt a proposition we cannot but regard it as perfectly true and perfectly certain; that while holding certain propositions to be each individually perfectly certain, we may and ought to think it likely that some one of them, if not more, is false.†P1 Peirce: CP 5.499 Cross-Ref:†† 499. This is the doctrine of Critical Common-sensism, and the present pertinency of it is that a pragmaticist, to be consistent, is obliged to embrace it. . . . For brevity's sake, I shall confine myself to the easiest stated but far from the best of several reasons. Because the pragmaticist, then, recognizes that the substance of what he thinks lies in a conditional resolve, he cannot fail also to recognize that to learn the very truth is the way to satisfy the wishes of his heart. For this reason he will be of all men the man whose mind is most open to conviction, and will be keen up the scent of whatever can go toward teaching him to distinguish accurately between truth and falsity, probability and improbability. This will suffice to make the pragmaticist attentive to all those matters of every-day facts which critical common-sensism takes into account. It remains to say why he, more than another man, should be inclined to draw these inferences from those facts. His doctrine essentially insists upon the close affinity between thinking in particular and endeavour in general. Since, therefore, action in general is largely a matter of instinct, he will be pretty sure to ask himself whether it be not the same with belief. That this question once asked admits of but one answer is shown by the fact that even John Locke was obliged to fall into line here, notwithstanding the nominalistic metaphysics, the most blinding of all systems, as metaphysics generally is the most powerful of all causes of mental cecity, because it deprives the mind of the power to ask itself certain questions, as the habit of wearing a confining dress deprives one's joints of their suppleness. Now Locke was a man of strong prejudices, while the pragmaticist -- it cannot be said too often -- will be the most open-minded of all men. But once it is settled that belief may be a matter of instinct and of desire, the inquiry [will arise] whether almost every man will not have his quite irresistible beliefs spring up ineluctably, together with the question

whether these will not be pretty uniform. Now irresistible instinctive desires are such familiar and such almost invarying phenomena -- so few men, let us say, are able to hold their breath for five minutes, even among strong thinkers who are apt to be great breath-holders -- that there can be no doubt how this will get answered. . . . You see for yourself that pragmaticism will be sure to carry critical common-sensism in its arms, do you not? Peirce: CP 5.500 Cross-Ref:†† 500. Jules. Perhaps so, but how do you know that your pragmaticistic doctrine is true? Peirce: CP 5.500 Cross-Ref:†† Respondent. Why just place the two doctrines -- pragmaticism and its alternative -- side by side in your mind. Peirce: CP 5.500 Cross-Ref:†† Jules. Gracious! Have the gracious gods confined us to two alternatives? I don't know what power of two would suffice to count up the varieties of pragmatism itself. Peirce: CP 5.500 Cross-Ref:†† Respondent. Yes; but the choice between them shall be considered later. Without disrespect to those who only differ from me about the mint and cummin of the law, I may speak of the traditional logic as the principal alternative, as presenting itself at the first and only great parting of the ways. For the old logicians, thought has no meaning except itself, any more than a fugue of Bach has. If you start with any concept, any twig of the tree of Porphyry, say canary-birds, and wish to include the similar Palestine bird, you can think of a serin-finch. To be sure, this includes a dozen other species; but that cannot be helped; it is the nearest concept there is. Suppose you wish to include sparrows besides. Then the only distinct concept there is, is coracomorphs (or crow-like birds) which, in the hazy light of the non-ornithological mind, would be pretty much the same as "ordinary birds." Rising through birds, vertebrates, animals, living creatures, natural objects, things, we come, in the ninth remove from canary-birds, to substances. Now that substance is a category is the general opinion of all schools of logicians -- indeed, it is, with the unbroken agreement of all, except some who really do thinking on their own account, an irregular pluralism of functions. Peirce: CP 5.500 Cross-Ref:†† Jules. Say, with the exception of one curious specimen, the Axi-vectis persicus.†P1 Peirce: CP 5.501 Cross-Ref:†† 501. Respondent. There have been a number of different lists of categories; but all the logicians, to whom I have been referring, agree that the concepts, which are categories, are all simple and are the only simple concepts. That means that while something may be true of one category that is not true of another, yet such differences are not such as to constitute the characters of the concepts. Each is other than each of the rest but this difference is unspecifiable and thus indefinite. At the same time there is nothing indefinite in the concepts themselves. It sounds paradoxical, yet it is precisely like what is to be remarked about different qualities of feeling. The perfume of the orange flower is perfectly definite and no complexity is to be detected in it. The same thing is true of the rose, of peppermint, and of sandal-wood. They are all very

different -- and one can predicate various qualities of them.†1 The odour of sandalwood is heavy, orange flower is cool, rose has an exquisite purity, peppermint has a clean smell; but these qualities do not at all constitute the odors, nor are they any part of the smells themselves. Of their relations it is inconceivable that anything should ever be predicate except that each is other than each. Those relations therefore are as indefinite as they possibly could be while there is not the slightest indefiniteness [in the feelings related]. Indeed I am prepared to assert that the bigwigs of logic make concepts to be nothing else than another kind of qualities of feeling. Inquire of one of these gentlemen whether or not concepts are qualities of feeling. "Oh mi!" he will reply, "nothing could be more different." But ask him what the difference is, and pursue the inquiry sharply, and it will turn out that it is impossible to say what it is. He will begin by saying that concepts are general, feelings not so. But that position cannot be maintained for thirty seconds. They are different no doubt; but the difference is altogether indefinite. It is precisely like the difference between smells and colours. It must be so, because at the very outset they defined concepts as qualities of feeling, not in these very words, of course, but in the very meaning of these words, when they said that concepts possess, as immediate objects, all the characters †P1 that they possess at all, each in itself, regardless of anything else. Peirce: CP 5.501 Cross-Ref:†† Jules. Yes, you have asserted that with admirable distinctness. Now I invite you to prove it.

Peirce: CP 5.502 Cross-Ref:†† CHAPTER 3

CONSEQUENCES OF CRITICAL COMMON-SENSISM†1

§1. INDIVIDUALISM 502. Doctor X. The best of your screed in the April Monist was about the singleness of symbols.†2 It is in truth bad morals to use words in other than their original senses. But apply this rule to your own use of the verb "is." When the child first uses this verb, he applies it to some sensory reality. How, then, in the face of your code of terminological ethics, do you ever dare to use the verb "to be" for indicating anything not sensory? I, for my part, stick to the one meaning of "is" rigidly. When you talk of "general objects being real," and the like, you seem not to be aware that the verbs "to be," "to be real," and "to exist" have ever precisely one and the same significance. Peirce: CP 5.502 Cross-Ref:†† Pragmaticist. My statements today are not designed to answer objections, but merely to correct misapprehensions. However, as your objection seems motived by a grave misapprehension, I had better set that right. Only in order to guard against my misapprehending you in my turn, let me ask you whether you mean to say that throughout that vast philosophical "treatise" of yours, the signification of the verb "is"

is really and truly in every instance the same. Peirce: CP 5.502 Cross-Ref:†† Doctor X. It is. Peirce: CP 5.502 Cross-Ref:†† Pragmaticist. And everywhere that meaning is "is real"? Peirce: CP 5.502 Cross-Ref:†† Doctor X. Yes. Peirce: CP 5.502 Cross-Ref:†† Pragmaticist. Thanks. Then the misapprehension that motives your objection is that you understand me to be more of a scholastic realist than you are, while in fact it is the other way. For this signification of "is," which occurs in hundreds of instances in your book, and is everywhere one and the same, by virtue of that fact satisfies the definition of a general, or, as the scholastics more accurately said, of a universal; and you tell me that when you say that that universal "is" is always the same, you mean that it "is real" everywhere, as well as the same. So that makes you a scholastic realist. But you go much further than that; for when you say that the signification is the same, you of course make this signification the subject of the verb "is," which according to your statement is equivalent to a declaration that this signification which is one and the same in so many places of your book is a sensory object. Not Bernardus Carnotensis himself, who seems like you to have combined scholastic realism with individualism, went to the length of making any general a sensory object. Peirce: CP 5.502 Cross-Ref:†† Doctor X. Now you are quibbling, you know. Peirce: CP 5.503 Cross-Ref:†† 503. Pragmaticist. Do not answer me, I beg. For remember that you, Doctor X, like Y, Z, and W are not an existent individual, even if you are a sensory object, but only a general type -- unless indeed you are a mere man of straw. A reply could only vindicate you by showing that you were not a real type, when you would prove yourself to be a man of straw; in which case I might be provoked into the personality of burning you up, in our good old mediaeval fashion; though I believe you limit yourself to roasting your antagonists. For my part, I have found the combustion of a man of straw one of the best means of stopping my logical chimney from smoking; while your doctrine would seem to debar you from the employment of that useful device. It is perhaps true that the sectators of individualism, the essence of whose doctrine is that reality and existence are coextensive, i.e., are either alike true or alike false of every subject, must, to be logical, go along with you in holding that "real" and "existent" have the same meaning, or Inhalt. But many a logician, as soon as he is convinced that that party is under that obligation (individualism furnishing the principle of the consequence as well as furnishing the antecedent) would regard that circumstance as creating a reductio ad absurdum of individualism, inasmuch as reality means a certain kind of non-dependence upon thought, and so is a cognitionary character, while existence means reaction with the environment, and so is a dynamic character; and accordingly the two meanings, he would say, are clearly not the same. Individualists are apt to fall into the almost incredible misunderstanding that all other men are individualists, too -- even the scholastic realists, who, they suppose, thought that "universals exist." It is true that there are indications of there having been some

who thought so in that greater darkness before the dawn of Aristotle's Analytics and Topics, when such grotesque weldings of doctrine as that of nominalistic Platonism are heard of, and when Roscellin may possibly have said that universals were flatus vocis. But I ask, can anybody who has seen Westminster Abbey, who had read the Prologue to the Canterbury Tales, and who stops to consider that the metaphysics of the Plantagenet age must have more adequately represented the general intellectual standing of that age, when metaphysics absorbed its greatest heuristic minds, than the metaphysics of our day can represent our general intellectual condition, can any such person believe that the great doctors of that time believed that generals exist? They certainly did not so opine, but regarded generals as modes of determination of individuals; and such modes were recognized as being of the nature of thought. Now whoever cares to know what pragmaticism is should understand that on its metaphysical side it is an attempt to solve the problem: In what way can a general be unaffected by any thought about it? Hence, before we treat of the evidences of pragmaticism, it will be needful to weigh the pros and cons of scholastic realism. For pragmaticism could hardly have entered a head that was not already convinced that there are real generals. Peirce: CP 5.504 Cross-Ref:†† 504. . . . [Sic] Another misapprehension: You seem to imagine that your argument from the talk of the child will be as convincing to me as it is to you. It is not so, because (aside from "is" not being one of the technical terms to which common-sense limited my maxim), I do not think that the import of any word (except perhaps a pronoun) is limited to what is in the utterer's mind actualiter, so that when I mention the Greek language my meaning should be limited to such Greek words as I happen to be thinking of at the moment. It is, on the contrary, according to me, what is in the mind, perhaps not even habitualiter, but only virtualiter, which constitutes the import.†P1 To say that I hold that the import, or adequate ultimate interpretation, of a concept is contained, not in any deed or deeds that will ever be done, but in a habit of conduct, or general moral determination of whatever procedure there may come to be, is no more than to say that I am a pragmaticist. Now every animal must have habits. Consequently, it must have innate habits. In so far as it has cognitive powers, it must have in posse innate cognitive habits, which is all that anybody but John Locke ever meant by innate ideas.†P1 To say that I hold this for true is implied in my confession of the doctrine of Common-Sense -- not quite that of the old Scotch School, but a critical philosophy of common-sense. It is impossible rightly to apprehend the pragmaticist's position without fully understanding that nowhere would he be less at home than in the ranks of individualists, whether metaphysical (and so denying scholastic realism), or epistemological (and so denying innate ideas).

Peirce: CP 5.505 Cross-Ref:†† §2. CRITICAL PHILOSOPHY AND THE PHILOSOPHY OF COMMON-SENSE

505. Doctor Y. Allow me. You speak of holding a Critical Philosophy of Common-Sense. What meaning would you have me attach to that phrase, seeing that Critical Philosophy and the Philosophy of Common-Sense, the two rival and opposed ways of answering Hume, are at internecine war, impacificable. The Common-Sense philosopher opines that, be Criticism never so indefatigable, it will have to come to a halt somewhere, and leave some belief uncriticized; namely, wherever no stimulus to

doubt has ever been experienced. An uncriticized belief must, says the Common-sensist, ipso facto be regarded as the very truth. That sounds conclusive. Yet it does not satisfy any of the Criticists, at all -- be they Kantians or be they of one of the modern kinds that do not usually go by the name of Criticists. That a belief should be accepted as the bed-rock of truth simply and solely because it has not been criticized -- oh, this is to their minds too monstrous! They insist that first principles be scientifically established. To think otherwise, says the great Wundt,†1 is to ask that philosophy should come into being by equivocal generation. He so resuscitates the phrase which, in the days when men believed in armary unguents, in mummial philtres, and in sigillary medicines, denoted the manner in which they thought that Satan's flies and the vilest of crawling things could be produced, in order to hint how much out of good odor common-sense is in his estimation. Of a principle proposed for the foundation of philosophy, think the Criticists, it must either be proved that the very circumstances and form of human knowledge require its acceptance, or better, that scientific psychology should show that its truth is unavoidable, or still better, that physiology should support it as it supports parallelism, or best of all, that histology should almost bring it within the field of the microscope, as caryocinesis is supposed almost to give ocular demonstration of some high proposition. Now, without asking whether it be Common-Sense or one of the Critical methods that is right, one cannot help seeing that Criticism and Common-sense are so immiscible that to plunge into either is to lose all touch with the other. The Criticist believes in criticizing first principles, while the Common-sensist thinks such criticism is all nonsense. So I can find no meaning in your straddling phrase. Peirce: CP 5.505 Cross-Ref:†† Pragmaticist. The phrase denotes a particular stripe of Common-sensism, which is separated from the old Scotch kind by four distinguishing marks. The mark that I find it convenient to describe first is that the Critical Common-sensist holds that all the veritably indubitable beliefs are vague -- often in some directions highly so. Logicians have too much neglected the study of vagueness, not suspecting the important part it plays in mathematical thought. It is the antithetical analogue of generality. A sign is objectively general, in so far as, leaving its effective interpretation indeterminate, it surrenders to the interpreter the right of completing the determination for himself. "Man is mortal." "What man?" "Any man you like." A sign is objectively vague, in so far as, leaving its interpretation more or less indeterminate, it reserves for some other possible sign or experience the function of completing the determination. "This month," says the almanac-oracle, "a great event is to happen." "What event?" "Oh, we shall see. The almanac doesn't tell that."†P1 The general might be defined as that to which the principle of excluded middle does not apply. A triangle in general is not isosceles nor equilateral; nor is a triangle in general scalene. The vague might be defined as that to which the principle of contradiction does not apply. For it is false neither that an animal (in a vague sense) is male, nor that an animal is female. Mr. Kempe's great memoir in the Philosophical Transactions for 1886,†1 the most solid piece of work upon any branch of the stecheology of relations that has ever been done, in addition to its intrinsic value, has that of taking us out of the logician's rut, and showing us how the mathematician conceives of logical objects. Thus, in Section 4, he says that the four angular points of a square "are not distinguishable from" one another. On first reading this, a person [Peirce] who was preoccupied with conceptions derived from logicians, was moved to write to the author and ask whether he did not mean "do not differ" from one another. For if the angles are undistinguished, how do we know there are more than one of them? But

though some suggestions of the letter were adopted in a supplement to the memoir, Mr. Kempe stood to "undistinguished" and "undistinguishable";†2 and in Section 29 he is more explicit, saying of the units of a singulary †P2 system of units (i.e., a system all whose units are undistinguished from one another), "that no definition can be given of, or remark made about, one which is not equally applicable to each of the others." In Section 73, he goes further in using the expression "undistinguished in dress or other circumstance," showing that he means to exclude distinction by means of relations. All this is utterly paradoxical to the logician, who will say that two vertices of a square are distinguished from each other in not being opposite the same vertex, and in various other ways. But the difficulty disappears as soon as he recognizes that Kempe's units are not supposed to be real objects, but are only vague ideas, to which nobody ever supposed the principle of contradiction to apply. Peirce: CP 5.506 Cross-Ref:†† 506. Notwithstanding their contrariety, generality and vagueness are, from a formal point of view, seen to be on a par. Evidently no sign can be at once vague and general in the same respect, since insofar as the right of determination is not distinctly extended to the interpreter it remains the right of the utterer. Hence also, a sign can only escape from being either vague or general by not being indeterminate. But that no sign can be absolutely and completely indeterminate †1 is proved in 3.93 where Plutarch's anecdote about appealing from Phillip drunk to Phillip sober is put to use. Yet every proposition actually asserted must refer to some non-general subject; for the doctrine that a proposition has but a single subject has to be given up in the light of the Logic of Relations. (See The Open Court, pp. 3416 et seq.) [3.417ff.] Indeed, all propositions refer to one and the same determinately singular subject, well-understood between all utterers and interpreters; namely, to The Truth, which is the universe of all universes, and is assumed on all hands to be real.†2 But besides that, there is some lesser environment of the utterer and interpreter of each proposition that actually gets conveyed, to which that proposition more particularly refers and which is not general. The Open Court paper referred to [above] made this plain, but left unnoticed some truths of the first importance about vagueness. No communication of one person to another can be entirely definite, i.e., non-vague. We may reasonably hope that physiologists will some day find some means of comparing the qualities of one person's feelings with those of another, so that it would not be fair to insist upon their present incomparability as an inevitable source of misunderstanding. Besides, it does not affect the intellectual purport of communications. But wherever degree or any other possibility of continuous variation subsists, absolute precision is impossible. Much else must be vague, because no man's interpretation of words is based on exactly the same experience as any other man's. Even in our most intellectual conceptions, the more we strive to be precise, the more unattainable precision seems. It should never be forgotten that our own thinking is carried on as a dialogue, and though mostly in a lesser degree, is subject to almost every imperfection of language. I have worked out the logic of vagueness with something like completeness,†1 but need not inflict more of it upon you, at present. Peirce: CP 5.507 Cross-Ref:†† 507. That veritably indubitable beliefs are especially vague could be proved a priori. But proof not being aimed at today, it will be simpler to say that the Critical Common-sensist's personal experience is that a suitable line of reflexion, accompanied by imaginary experimentation, always excites doubt of any very broad proposition if it be defined with precision. Yet there are beliefs of which such a critical sifting invariably leaves a certain vague residuum unaffected.

Peirce: CP 5.508 Cross-Ref:†† 508. One ought then to ask oneself, whether, since much of the original belief has disappeared under an attentive dissection, perseverance might not affect the destruction of what remains of it. This question always appears reasonable as long as one stands far enough away from the facts of the case, and views them as one would a painting of Monet. Peirce: CP 5.508 Cross-Ref:†† But the answer that a closer scrutiny dictates in some cases is that it is not because insufficient pains have been taken to precide †2 the residuum, that it is vague: it is that it is vague intrinsically. Take, for example, our belief in the Order of Nature. The criticisms of it in 342; 6.395ff; 2.749ff; 6.35ff; 6.613, as well as by various other writers, of whom may be mentioned as long antecedent to the writer, Renouvier,†3 Delboeuf,†4 Fouillée,†5 Blood,†6 and James,†7 and no doubt there were others, and since that time Dewey †8 and I know not who else, appear to me to have stripped it of all rational precision. As precisely defined it can hardly be said to be absolutely indubitable considering how many thinkers there are who do not believe it. But who can think that there is no order in nature? Peirce: CP 5.509 Cross-Ref:†† 509. Could I be assured that other men candidly and with sufficient deliberation doubt any proposition which I regard as indubitable, that fact would inevitably cause me to doubt it, too. I ought not, however, lightly to admit that they do so doubt a proposition after the most thorough criticism by myself and anxious consideration of any other criticisms which I have been able to find and understand has left it quite indubitable by me, since there are other states of mind that can easily be mistaken for doubt. If, indeed, the phenomenon in question were at all a common one, instead of being among the rarest of experiences, I should return to a variety of Common-sensism which has always strongly attracted me, namely, that there is no definite and fixed collection of opinions that are indubitable, but that criticism gradually pushes back each individual's indubitables, modifying the list, yet still leaving him beliefs indubitable at the time being. The reason I have of late given up that opinion, attractive as I find it, is that the facts of my experience accord better with the theory of a fixed list, the same for all men. I do not suppose that it is absolutely fixed, (for my synechism would revolt at that) but that it is so nearly so, that for ordinary purposes it may be taken as quite so. Peirce: CP 5.510 Cross-Ref:†† 510. Doubt is a state of mind marked by a feeling of uneasiness; but we cannot, from a logical, least of all from a pragmaticistic point of view, regard the doubt as consisting in the feeling. A man in doubt is usually trying to imagine how he shall, or should, act when or if he finds himself in the imagined situation. He supposes himself to have an end in view, and two different and inconsistent lines of action offer themselves. His action is in imagination (or perhaps really) brought to a stop because he does not know whether (so to speak) the right hand road or the left hand road is the one that will bring him to his destination; and (to continue the figure of speech) he waits at the fork for an indication, and kicks his heels. His pent up activity finds vent in feeling, which becomes the more prominent from his attention being no longer absorbed in action. A true doubt is accordingly a doubt which really interferes with the smooth working of the belief-habit. Every natural or inbred belief manifests itself in natural or inbred ways of acting, which in fact constitute it a belief-habit. (I need not repeat that I do not say that it is the single deeds that constitute the habit. It is the

single "ways," which are conditional propositions, each general). A true doubt of such a belief must interfere with this natural mode of acting. If a philosophist, reflecting upon the belief from an extraneous or unnatural point of view, develops new modes of manifestation of that belief (as, for example, by associating it with certain phrases), these new habits must not be regarded as expressions of the natural belief simply; for they inevitably involve something more. Consequently, if subsequent reflexion results in doubt of them, it is not necessarily doubt of the original belief, although it may be mistaken for such doubt. Peirce: CP 5.511 Cross-Ref:†† 511. These considerations lead me, quite naturally, to mention another mark of the Critical Common-sensist that separates him from the old school. Namely, he opines that the indubitable beliefs refer to a somewhat primitive mode of life, and that, while they never become dubitable in so far as our mode of life remains that of somewhat primitive man, yet as we develop degrees of self-control unknown to that man, occasions of action arise in relation to which the original beliefs, if stretched to cover them, have no sufficient authority. In other words, we outgrow the applicability of instinct -- not altogether, by any manner of means, but in our highest activities. The famous Scotch philosophers lived and died out before this could be duly appreciated. Peirce: CP 5.512 Cross-Ref:†† 512. Doctor Y. What do you mean by "somewhat primitive"? And by what sort of reasoning can a dubitable proposition about experience become indubitable? Peirce: CP 5.512 Cross-Ref:†† Pragmaticist. A searching question, because some of our beliefs, which seem as indubitable as any, are of such a character that they can hardly have entered the minds, say, of Neanderthal men, and in any case, cannot possibly have been transmitted to us from the first conscious animals. Consequently, Common-sensism has to grapple with the difficulty that if there are any indubitable beliefs, these beliefs must have grown up; and during the process, cannot have been indubitable beliefs. Still, I see no reason for thinking that beliefs that were dubitable became indubitable. Every decent house dog has been taught beliefs that appear to have no application to the wild state of the dog; and yet your trained dog has not, I guess, been observed to have passed through a period of scepticism on the subject. There is every reason to suppose that belief came first, and the power of doubting long after. Doubt, usually, perhaps always, takes its rise from surprise, which supposes previous belief; and surprises come with novel environment. I will only add that though precise reasoning about precise experiential doubt could not entirely destroy doubt, any more than the action of finite conservative forces could leave a body in a continuous state of rest, yet vagueness, which is no more to be done away with in the world of logic than friction in mechanics, can have that effect. Peirce: CP 5.513 Cross-Ref:†† 513. As I was saying, a modern recognition of evolution must distinguish the Critical Common-sensist from the old school. Modern science, with its microscopes and telescopes, with its chemistry and electricity, and with its entirely new appliances of life, has put us into quite another world; almost as much so as if it had transported our race to another planet. Some of the old beliefs have no application except in extended senses, and in such extended senses they are sometimes dubitable and subject to just criticism. It is above all the normative sciences, esthetics, ethics, and logic, that men are in dire need of having severely criticized, in their relation to the new world created by science. Unfortunately, this need is as unconscious as it is great.

The evils are in some superficial way recognized; but it never occurs to anybody that the study of esthetics, ethics, and logic can be seriously important, because these sciences are conceived by all, but their deepest students, in the old way. It only concerns my present purpose to glance at this state of things. The needed new criticism must know whereon it stands; namely, on the beliefs that remain indubitable; and young Critical Common-sensists of intellectual force who burn for a task in which they can worthily sacrifice their lives without encouragement, reward, recognition, or a hearing (and I trust such young men still live) can find in this field their heart's desire. Peirce: CP 5.514 Cross-Ref:†† 514. Yet a third mark of the Critical Common-sensist is that he has a high esteem for doubt. He may almost be said to have a sacra fames for it. Only, his hunger is not to be appeased with paper doubts: he must have the heavy and noble metal, or else belief. Peirce: CP 5.514 Cross-Ref:†† He quite acknowledges that what has been indubitable one day has often been proved on the morrow to be false. He grants the presciss proposition that it may be so with any of the beliefs he holds. He really cannot admit that it may be so with all of them; but here he loses himself in vague unmeaning contradictions. Peirce: CP 5.515 Cross-Ref:†† 515. Doctor Y. Can indubitable propositions be demonstrable? Peirce: CP 5.515 Cross-Ref:†† Pragmaticist. Indubitable propositions must be ultimate premisses, or at least, must be held without reference to precise proofs. For what one cannot doubt one cannot argue about; and no precise empirical argument can free its conclusion altogether from rational doubt. Peirce: CP 5.516 Cross-Ref:†† 516. Yet it is true that whenever one turns a critical glance upon one of our original beliefs -- say, the belief in the order of nature -- the mind at once seems vaguely to pretend to have reasons for believing it. One dreams of an inductive proof. One surmises that the belief results from something like an inductive proof that has been forgotten. Very likely it did, in a sense of the term "inductive process" that is so generalized as to include uncontrolled thought. But this admission must be accompanied by the emphatic denial that the indubitable belief is inferential, or is "accepted." It simply remains unshaken as it always was. That does not at all interfere with the theory that in the psychological process of its development, the occurrence of single experiences, such as might have been predictively deduced from it, were an indispensable factor, while an original potentiality of the belief-habit must have been a correlative factor. All this is perfectly consistent, too, with the necessity of criticising the ordinary axioms of reasoning and of morals, as well as ordinarily developed ideals, as soon as they are extended so as to become applicable to the new world created by science. Peirce: CP 5.517 Cross-Ref:†† 517. I was saying that the Critical Common-sensist feels that the danger -- the scientific danger, at any rate; and Philosophy is a department of pure Heuretic Science even less concerned, for example, about practical religion, if possible, than religion ought to be about it -- does not lie in believing too little but in believing too much.

The indolent university student, no matter at what pains his professor of philosophy may be to set him upon his own legs, yet having a well-grounded respect for that professor's superior acquirements and force of intellect, finds it much easier to accept all he says, as true because he says it, than to submit said professor's arguments to searching criticism; and he thus becomes, on the average, quite as much a slave to authority as was the average scholar of the medieval schools. Only, instead of bowing to Aristotle and the universal voice of the church sounding semper eadem, he submits to the yoke of some young doctrinaire with whom every other like him disagrees. With such sentiments, the Critical Common-sensist sets himself in serious earnest to the systematic business of endeavoring to bring all his very general first premisses to recognition, and of developing every suspicion of doubt of their truth, by the use of logical analysis, and by experimenting in imagination. If, besides being a Critical Common-sensist, he is also a pragmaticist, he will further hold that everything in the substance of his beliefs can be represented in the schemata of his imagination; that is to say, in what may be compared to composite photographs of continuous series of modifications of images; these composites being accompanied by conditional resolutions †P1 as to conduct. Peirce: CP 5.517 Cross-Ref:†† These resolutions should cover all classes of circumstances, in the sense that they would produce (or, perhaps†P2 more strictly, manifestations of whatever it may be in our occult nature that produces) determinations of habit corresponding to every possible pragmaticistic application of the propositions believed. Peirce: CP 5.518 Cross-Ref:†† 518. Pragmaticism is, of course, in its developed fullness too recent a phenomenon in the history of philosophy to have disclosed anything to direct experience, concerning its tendencies, that is particularly trustworthy. Thus far, however, it would appear that, as a matter of fact, pragmaticists press their peculiar doubts about first principles a good deal further and with a more straightforward earnestness than Kantians do. For when a Kant expresses a doubt, one has still to learn whether it is the substance of the proposition that he doubts or merely its attachment to one faculty or to another. One has even known of a pragmaticist being called by a Kantian "David Hume Redivivus";†1 but I fancy it was more like the David Hume of some mediumistic séance. Peirce: CP 5.519 Cross-Ref:†† 519. Doctor X. I should think that so passionate a lover of doubt would make a clean sweep of his beliefs. Peirce: CP 5.519 Cross-Ref:†† Pragmaticist. You naturally would, holding the infant's mind to be a tabula rasa and the adult's a school slate, on which doubts are written with a soapstone pencil to be cleaned off with the dab of a wet sponge; but if they are marked with talc on man's "glassy essence," they may disappear for a long time only to be revived by a breath. Peirce: CP 5.520 Cross-Ref:†† 520. Doctor X. Yours seemed marked with T-A-L-K. Doubtless your pragmaticist dotes too much on doubt to risk subjecting it to scientific experimentation. Peirce: CP 5.520 Cross-Ref:††

Pragmaticist. Bah! I may as well capitulate first as last. I have been betrayed by that execrable banality, auri sacra fames, which is not even good poetry, since it is inaccurate. For what it expresses is the hoarding passion of the miser, a sort of collector's rage for gold as gold, while all it means in the familiar passages is no more than simple cupidity; and that is what I meant in applying it to the Critical Common-sensist's eager pursuit of doubt. He is none of those overcultivated Oxford dons -- I hope their day is over -- whom any discovery that brought quietus to a vexed question would evidently vex because it would end the fun of arguing around it and about it and over it. On the contrary what he adores, if he is a good pragmaticist, is power; not the sham power of brute force, which, even in its own specialty of spoiling things, secures such slight results; but the creative power of reasonableness, which subdues all other powers, and rules over them with its sceptre, knowledge, and its globe, love. It is as one of the chief lieutenants of reasonableness that he highly esteems doubt, although it is not amiable. Peirce: CP 5.521 Cross-Ref:†† 521. As for allaying doubts concerning the first principles of logic and philosophy by means of scientific experiments, he does not attempt that for two reasons, one a fine little reason that insinuates itself between the joints of the Wundtian armour, the other more like wholesome country air. Peirce: CP 5.521 Cross-Ref:†† The small reason is sufficient. It is that any such idioscopic †1 inquiry must proceed upon the virtual assumption of sundry logical and metaphysical beliefs; and it is rational to settle the validity of those before undertaking an operation that supposes their truth. Now whether the truth of them be explicitly laid down on critical grounds, or the doctrine of Common-Sense prevent our pretending to doubt it, along with all these other sound first principles will be admitted, and so the whole inquiry will be concluded before the first outward experiment is made. But this preliminary inquiry is long and arduous. Peirce: CP 5.522 Cross-Ref:†† 522. Doctor X. A sort of Panama Commission business, apparently. I should say, Pitch in like Lesseps, unless your second objection is more serious. What is that? Peirce: CP 5.522 Cross-Ref:†† Pragmaticist. There is nothing novel in it. It is that nothing is so unerring as instinct within its proper field, while reason goes wrong about as often as right -perhaps oftener. Now those vague beliefs that appear to be indubitable have the same sort of basis as scientific results have. That is to say, they rest on experience -- on the total everyday experience of many generations of multitudinous populations. Such experience is worthless for distinctively scientific purposes, because it does not make the minute distinctions with which science is chiefly concerned; nor does it relate to the recondite subjects of science, although all science, without being aware of it, virtually supposes the truth of the vague results of uncontrolled thought upon such experiences, cannot help doing so, and would have to shut up shop if she should manage to escape accepting them. No "wisdom" could ever have discovered argon; yet within its proper sphere, which embraces objects of universal concern, the instinctive result of human experience ought to have so vastly more weight than any scientific result, that to make laboratory experiments to ascertain, for example, whether there be any uniformity in nature or no, would vie with adding a teaspoonful of saccharine to the ocean in order to sweeten it.

Peirce: CP 5.523 Cross-Ref:†† 523. Doctor Y. Is there any further peculiarity which distinguishes Critical Common-sensism from that of Reid and Dugald Stewart? Peirce: CP 5.523 Cross-Ref:†† Pragmaticist. Yes; for it criticizes the critical method, follows its footsteps, tracks it to its lair. To the accusation that Common-Sense accepts a proposition as indubitable because it has not been criticized, the answer is that this confounds two uses of the word "because." Neither the philosophy of Common-Sense nor the man who holds it accepts any belief on the ground that it has not been criticized. For, as already said, such beliefs are not "accepted." What happens is that one comes to recognize that one has had the belief-habit as long as one can remember; and to say that no doubt of it has ever arisen is only another way of saying the same thing. But it is quite true that the Common-sensist like everybody else, the Criticist included, believes propositions because they have not been criticized in the sense that he does not doubt certain propositions that he would have doubted if he had criticized them. For in the first place, to criticize is ipso facto to doubt, and in the second place criticism can only attack a proposition after it has given it some precise sense in which it is impossible entirely to remove the doubt. It is probably true, too, that the Common-sensist believes unquestioningly some propositions that might have been criticized and that are not true. We are all liable to do that; but perhaps he is more in danger of it than other men. Still, as a fact, it is difficult to find a Criticist who does not hold to more fundamental beliefs than any Critical Common-sensist does. Peirce: CP 5.524 Cross-Ref:†† 524. The Critical Philosopher seems to opine that the fact that he has not hitherto doubted a proposition is no reason why he should not henceforth doubt it. (At which Common-Sense whispers that, whether it be "reason" or no, it will be a well-nigh insuperable obstacle to doubt.) Accordingly, he will not stop to ask whether he actually does doubt it or not, but at once proceeds to examine it. Now if it happens that he does actually doubt the proposition, he does quite right in starting a critical inquiry. But in case he does not doubt, he virtually falls into the Cartesian error of supposing that one can doubt at will. A proposition that could be doubted at will is certainly not believed. For belief, while it lasts, is a strong habit, and as such, forces the man to believe until some surprise breaks up the habit. The breaking of a belief can only be due to some novel experience, whether external or internal. Now experience which could be summoned up at pleasure would not be experience. Peirce: CP 5.525 Cross-Ref:†† 525. Kant (whom I more than admire) is nothing but a somewhat confused pragmatist. A real is anything that is not affected by men's cognitions about it; which is a verbal definition, not a doctrine. An external object is anything that is not affected by any cognitions, whether about it or not, of the man to whom it is external. Exaggerate this, in the usual philosopher fashion, and you have the conception of what is not affected by any cognitions at all. Take the converse of this definition and you have the notion of what does not affect cognition, and in this indirect manner you get a hypostatically abstract notion of what the Ding an sich would be. In this sense, we also have a notion of a sky-blue demonstration; but in half a dozen ways the Ding an sich has been proved to be nonsensical; and here is another way. It has been shown [3.417ff] that in the formal analysis of a proposition, after all that words can convey has been thrown into the predicate, there remains a subject that is

indescribable and that can only be pointed at or otherwise indicated, unless a way, of finding what is referred to, be prescribed. The Ding an sich, however, can neither be indicated nor found. Consequently, no proposition can refer to it, and nothing true or false can be predicated of it. Therefore, all references to it must be thrown out as meaningless surplusage. But when that is done, we see clearly that Kant regards Space, Time, and his Categories just as everybody else does, and never doubts or has doubted their objectivity. His limitation of them to possible experience is pragmatism in the general sense; and the pragmaticist, as fully as Kant, recognizes the mental ingredient in these concepts. Only (trained by Kant to define), he defines more definitely, and somewhat otherwise, than Kant did, just how much of this ingredient comes from the mind of the individual in whose experience the cognition occurs. The kind of Common-sensism which thus criticizes the Critical Philosophy and recognizes its own affiliation to Kant has surely a certain claim to call itself Critical Common-sensism.

Peirce: CP 5.526 Cross-Ref:†† §3. THE GENERALITY OF THE POSSIBLE

526. Doctor Z. You say that no collection of individuals could ever be adequate to the extension of a concept in general, which is, of course, the old peripatetic doctrine. But really I do not quite see how you propose to reconcile that to the proposition that the meaning extends no further than to future embodiments of it. Peirce: CP 5.526 Cross-Ref:†† Pragmaticist. The original paper on pragmaticism was completed in September 1877†P1 and appeared in Popular Science Monthly for January 1878. At that time, modern investigation of the doctrines of multitude had not begun.†P2 Indeed, there are indications in that paper of an endless series not being regarded as a collection.†1 Yet the philosophical importance of the new studies was fully recognized by the pragmaticist from the first.†P3 Peirce: CP 5.527 Cross-Ref:†† 527. In 3.527ff the objectivity of possibility was asserted; and the hypothesis defended in vol. 6, Bk. I, chs. 1 and 2 supposes possibility to be real.†2 It was, indeed, implied in the scholastic realism maintained in the N.A. Rev., Vol. CXIII (pp. 454 et seq.) [vol. 9]. But the paper of January 1878 evidently endeavors to avoid asking the reader to admit a real possibility. The theory of modality is far too great a question to be treated incidentally to any other.†1 But the distinct recognition of real possibility is certainly indispensable to pragmaticism. Peirce: CP 5.528 Cross-Ref:†† 528. The pragmaticist has always explicitly stated that the intellectual purport of a concept consists in the truth of certain conditional propositions asserting that if the concept be applicable, and the utterer of the proposition or his fellow have a certain purpose in view, he would act in a certain way. A purpose is essentially general, and so is a way of acting; and a conditional proposition is a proposition about a universe of possibility. At the same time, the conditional proposition refers only to possible individual actions. If there be any paradox here, it is partially resolved in the important paper in The Monist, Vol. VII [3.526ff], where it is shown that an endless series of experiences, each entirely consistent with those that precede it, cannot itself

be experienced (as such endless series), but involves a first dose of ideality, or generality. It is not a perfect general, it is true; but the whole endless series of steps from this to true continuity (which is perfect generality elevated to the mode of conception of the Logic of Relations) are there described. Whoever wishes to complete the theory of modality should set out from the results there demonstrated. Peirce: CP 5.529 Cross-Ref:†† 529. The conditional proposition which the pragmaticist holds to constitute the purport of a concept is in the article of January 1878 spoken of as expressing a "fact"; and it does indeed express the actual state of mind of a person who has made, or is ready to make, a resolve -- not a categorical resolve, but a resolve conditional upon having a certain purpose. But nobody would make any difficulty in admitting that every resolve is limited to future acts, and acts are the most perfectly individual objects there are. Peirce: CP 5.530 Cross-Ref:†† 530. I apprehend that I have thus substantially answered your question; but in order to make my answer a little clearer, I will illustrate it by the consideration of the continuity of Space. In this illustration I shall adopt the Leibnizian conception of Space in place of the Newtonian, which I believe to be the true one. In that Leibnizian view, Space is merely a possibility limited by an impossibility; a possibility of no matter what affections of bodies (determining their relative positions), together with the impossibility of those affections being actualized otherwise than under certain limitations, expressed in the postulates of topical, graphical, and metrical geometry. No collection of points, though it be abnumerable to the billionth degree, could fill a line so that there would be room for no more points; and in that respect the line is truly general; no possible multitude of singulars is adequate to it. Space is thus truly general; and yet it is, so to say, nothing but the way in which actual bodies conduct themselves. Peirce: CP 5.531 Cross-Ref:†† 531. Doctor Z. But the idea of Leibnizian Space, if there were such a thing, would not be a concept. It would be a Vorstellung, or composite of images. Kant might perhaps have called it a Schema, since he defines a schema as a determination of intuition by a concept through the reproductive imagination.†1 Of course, it would not be one of those transcendental schemata, which he talks of in the Critik; but it possesses much the same sort of bastard generality. Peirce: CP 5.531 Cross-Ref:†† Pragmaticist. The breakneck hurry in which the C.d.r.V. was written is its only defence against a charge of slovenly workmanship. Every detail is left in the rough; and there is no more unfinished apartment in the whole glorious edifice than that devoted to the Schematization of the Categories. Kant says that no image, and consequently we may add, no collection of images, is adequate to representing what a schema represents.†2 If that be the case, I should like to know how a schema is not as general as a concept. If I ask him, all he seems to answer is that it is the product of a different "faculty." Peirce: CP 5.532 Cross-Ref:†† 532. But what you would seem to mean by a concept is the meaning of some general symbol, this meaning being conceived as referring to the symbol. Now it is precisely the pragmatist's contention that symbols, owing their origin (on one side) to human conventions, cannot transcend conceivable human occasions. At any rate, it is

plain that no possible collection of single occasions of conduct can be, or adequately represent all conceivable occasions. For there is no collection of individuals of any general description which we could not conceive to receive the addition of other individuals of the same description aggregated to it. The generality of the possible, the only true generality, is distributive, not collective. You perhaps do not see how this remark bears upon your question.

Peirce: CP 5.533 Cross-Ref:†† §4. VALUATION

533. Doctor W. I should like to know what the attitude of your pragmaticism is to the question of whether or no valuation is a factor of all intellectual meaning. Peirce: CP 5.533 Cross-Ref:†† Pragmaticist. Well, collective and distributive universality can bide their time. Considering how it stood in the mid-channel of pragmatistic thought to join ethics to logic, it seems to me strange that we had to wait until 1903 for any pragmatist to assert that logic ought to be based upon ethics. Perhaps some one of us had said it before; but I only know that it was then said in a course of lectures before the Lowell Institute in Boston,†1 and was maintained on the ground that reasoning is thought subjected to self-control, and that the whole operation of logical self-control takes precisely the same quite complicated course which everybody ought to acknowledge is that of effective ethical self-control. Mr. Schiller in the same year published an essay entitled "The Ethical Basis of Metaphysics."†2 The title is promising, but the essay is, for me, reduced to gibberish by the author's talking about the real, without the slightest hint of what he means by this word except that it is something the character of which is affected (and it would seem very greatly) by anybody's thinking that it possesses or does not possess that character. In short he treats a verbal definition as a doctrine, and stoutly denying it, leaves the word a mystery. To meet some such fatal blank has more than once been my ill-luck in trying to read Schiller. It is my stupidity no doubt. Peirce: CP 5.533 Cross-Ref:†† To return to self-control, which I can but slightly sketch, at this time, of course there are inhibitions and coördinations that entirely escape consciousness. There are, in the next place, modes of self-control which seem quite instinctive. Next, there is a kind of self-control which results from training. Next, a man can be his own training-master and thus control his self-control. When this point is reached much or all the training may be conducted in imagination. When a man trains himself, thus controlling control, he must have some moral rule in view, however special and irrational it may be. But next he may undertake to improve this rule; that is, to exercise a control over his control of control. To do this he must have in view something higher than an irrational rule. He must have some sort of moral principle. This, in turn, may be controlled by reference to an esthetic ideal of what is fine. There are certainly more grades than I have enumerated. Perhaps their number is indefinite. The brutes are certainly capable of more than one grade of control; but it seems to me that our superiority to them is more due to our greater number of grades of self-control than it is to our versatility. Peirce: CP 5.534 Cross-Ref:††

534. Doctor Y. Is it not due to our faculty of language? Peirce: CP 5.534 Cross-Ref:†† Pragmaticist. To my thinking that faculty is itself a phenomenon of self-control. For thinking is a kind of conduct, and is itself controllable, as everybody knows. Now the intellectual control of thinking takes place by thinking about thought. All thinking is by signs; and the brutes use signs. But they perhaps rarely think of them as signs. To do so is manifestly a second step in the use of language. Brutes use language, and seem to exercise some little control over it. But they certainly do not carry this control to anything like the same grade that we do. They do not criticize their thought logically. One extremely important grade of thinking about thought, which my logical analyses have shown to be one of chief, if not the chief, explanation of the power of mathematical reasoning, is a stock topic of ridicule among the wits. This operation is performed when something, that one has thought about any subject, is itself made a subject of thought. You remember how in the last Intermède to the Malade Imaginaire, the doctor puts a question to the candidate for the medical degree?

Si mihi licentiam dat Dominus Praeses, Et tanti docti Doctores, Et assistantes illustres, Très sçavanti Bacheliero, Quem estimo et honoro, Domandabo causam et rationem quare Opium facit dormire.

To which the candidate replies,

Mihi a docto Doctore Domandatur causam et rationem quare Opium facit dormire: A quoi respondeo, Quia est in eo Virtus dormitiva, Cujus est natura Sensus assoupire.

Whereupon the chorus bursts out,

Bene, bene, bene, bene respondere, Dignus, dignus est entrare In nostro docto corpore. (Bene, bene respondere.)

Even in this burlesque instance, this operation of hypostatic abstraction is not quite utterly futile. For it does say that there is some peculiarity in the opium to which the sleep must be due; and this is not suggested in merely saying that opium puts people to sleep. By the way, John Locke's account †1 of a real function of this sort at Montpellier three years after the play was first performed, with such tragic effect upon Molière, shows that there was more truth than caricature in the Intermède. In order to get an inkling -- though a very slight one -- of the importance of this operation in mathematics, it will suffice to remember that a collection is an hypostatic abstraction, or ens rationis,†2 that multitude is the hypostatic abstraction derived from a predicate of a collection,†3 and that a cardinal number is an abstraction attached to a multitude.†4 So an ordinal number is an abstraction attached to a place,†5 which in its turn is a hypostatic abstraction from a relative character of a unit of a series, itself an abstraction again. Now, Doctor Z, as well as I can make out, what you mean by a concept is a predicate considered by itself, except for its connection with the word or other symbol expressing it, and now regarded as denotative of the concept. Such a concept is not merely prescissively abstracted, but, as being made a subject of thought, is hypostatically abstract. So understood, it is true that it is more removed from the perceptual objects than is the Vorstellung, or composite of images. But for all that, its intellectual purport is just the same. It is only the grammatico-logical form that is transmuted. Peirce: CP 5.535 Cross-Ref:†† 535. And you, Doctor W., will see that since pragmaticism makes the purport to consist in a conditional proposition concerning conduct, a sufficiently deliberate consideration of that purport will reflect that the conditional conduct ought to be regulated by an ethical principle, which by further self-criticism may be made to accord with an esthetical ideal. For I cannot admit that any ideal can be too high for a duly transfigured esthetics. So, although I do not think that an esthetic valuation is essentially involved, actualiter (so to speak) in every intellectual purport, I do think that it is a virtual factor of a duly rationalized purport. That is to say, it really does belong to the purport, since conduct may depend upon its being appealed to. Yet in ordinary cases, it will not be needful that this should be done. Such seem to me to be the facts, phrase them how you may. Peirce: CP 5.536 Cross-Ref:†† 536. Doctor W. I am glad to hear you say so. And what do you think of Humanism? Peirce: CP 5.536 Cross-Ref:†† Pragmaticist. Why if you had said Anthropomorphism, I should have replied that I heartily embrace most of the clauses of that doctrine, if some right of private

interpretation be allowed me. I hold, for instance, that man is so completely hemmed in by the bounds of his possible practical experience, his mind is so restricted to being the instrument of his needs, that he cannot, in the least, mean anything that transcends those limits. The strict consequence of this is, that it is all nonsense to tell him that he must not think in this or that way because to do so would be to transcend the limits of a possible experience. For let him try ever so hard to think anything about what is beyond that limit, it simply cannot be done. You might as well pass a law that no man shall jump over the moon; it wouldn't forbid him to jump just as high as he possibly could. Peirce: CP 5.536 Cross-Ref:†† For much the same reason, I do not believe that man can have the idea of any cause or agency so stupendous that there is any more adequate way of conceiving it than as vaguely like a man. Therefore, whoever cannot look at the starry heaven without thinking that all this universe must have had an adequate cause, can in my opinion not otherwise think of that cause half so justly than by thinking it is God. Peirce: CP 5.537 Cross-Ref:†† 537. But when you talk of Humanism, I am utterly perplexed to know what it means. One of its clauses seems borrowed from Hegel, on whom how greatly its author [Schiller] dotes is well known. Namely, he apparently does not wish to have phenomena torn to pieces; or at any rate not if that introduces any falsity; and he does not wish us to devote any attention to the effects of conditions that do not occur, or at any rate not to substitute the solution of such a problem for the true problems of nature. For my part, I think such talk shows great ignorance of the conditions of science. Then again, as I understand it, this Humanism is to be a philosophy not purely intellectual because every department of man's nature must be voiced in it. For my part, I beg to be excused from having any dealings with such a philosophy. I wish philosophy to be a strict science, passionless and severely fair. I know very well that science is not the whole of life, but I believe in the division of labor among intellectual agencies. The apostle of Humanism says that professional philosophists "have rendered philosophy like unto themselves, abstruse, arid, abstract, and abhorrent."†1 But I conceive that some branches of science are not in a healthy state if they are not abstruse, arid, and abstract, in which case, like the Aristotelianism which is this gentleman's particular bête noire, it will be as Shakespeare said (of it, remember)

"Not harsh and crabbed, as dull fools suppose, But musical as is Apollo's lute," etc.†2

Peirce: CP 5.538 Cross-Ref:†† CHAPTER 4

BELIEF AND JUDGMENT

§1. PRACTICAL AND THEORETICAL BELIEFS †1

538. Let us begin by considering practical belief, such as that anthracite is a convenient fuel, leaving purely theoretical belief, such as that the pole of the earth describes an oval of a few rods' diameter, or that there is an imaginary circle which is twice cut by every real circle, for a supplementary study. Let us use the word "habit," throughout this book, not in its narrower, and more proper sense, in which it is opposed to a natural disposition (for the term acquired habit will perfectly express that narrower sense), but in its wider and perhaps still more usual sense, in which it denotes such a specialization, original or acquired, of the nature of a man, or an animal, or a vine, or a crystallizable chemical substance, or anything else, that he or it will behave, or always tend to behave, in a way describable in general terms upon every occasion (or upon a considerable proportion of the occasions) that may present itself of a generally describable character. Now to say that a man believes anthracite to be a convenient fuel is to say no more nor less than that if he needs fuel, and no other seems particularly preferable, then, if he acts deliberately, bearing in mind his experiences, considering what he is doing, and exercizing self-control, he will often use anthracite. A practical belief may, therefore, be described as a habit of deliberate behavior. The word "deliberate" is hardly completely defined by saying that it implies attention to memories of past experience and to one's present purpose, together with self-control. The acquisition of habits of the nervous system and of the mind is governed by the principle that any special character of a reaction to a given kind of stimulus is (unless fatigue intervenes) more likely to belong to a subsequent reaction to a second stimulus of that kind, than it would be if it had not happened to belong to the former reaction. But habits are sometimes acquired without any previous reactions that are externally manifest. A mere imagination of reacting in a particular way seems to be capable after numerous repetitions of causing the imagined kind of reaction really to take place upon subsequent occurrences of the stimulus. In the formation of habits of deliberate action, we may imagine the occurrence of the stimulus, and think out what the results of different actions will be. One of these will appear particularly satisfactory; and then an action of the soul takes place which is well described by saying that that mode of reaction "receives a deliberate stamp of approval." The result will be that when a similar occasion actually arises for the first time it will be found that the habit of really reacting in that way is already established. I remember that one day at my father's table, my mother spilled some burning spirits on her skirt. Instantly, before the rest of us had had time to think what to do, my brother, Herbert, who was a small boy, had snatched up the rug and smothered the fire. We were astonished at his promptitude, which, as he grew up, proved to be characteristic. I asked him how he came to think of it so quickly. He said, "I had considered on a previous day what I would do in case such an accident should occur."†1 This act of stamping with approval, "endorsing" as one's own, an imaginary line of conduct so that it shall give a general shape to our actual future conduct is what we call a resolve. It is not at all essential to the practical belief, but only a somewhat frequent attachment. Peirce: CP 5.539 Cross-Ref:†† 539. Let us now pass to the consideration of purely theoretical belief. If an opinion can eventually go to the determination of a practical belief, it, in so far, becomes itself a practical belief; and every proposition that is not pure metaphysical jargon and chatter must have some possible bearing upon practice. The diagonal of a square is incommensurable with its side. It is difficult to see what experiential

difference there can be between commensurable and incommensurable magnitudes; but there is this, that it is useless to try to find the exact expression of the diagonal as a rational fraction of the side. Still, it does not follow that because every theoretical belief is, at least indirectly, a practical belief, this is the whole meaning of the theoretical belief. Of theoretical beliefs, in so far as they are not practical, we may distinguish between those which are expectations, and those which are not even that. One of the simplest, and for that reason one of the most difficult, of the ideas which it is incumbent upon the author of this book to endeavor to cause the reader to conceive, is that a sense of effort and the experience of any sensation are phenomena of the same kind, equally involving direct experience of the duality of the Without and the Within.†1 The psychology of the sense of effort is not yet satisfactorily made out. It seems to be a sensation which somehow arises when striped muscles are under tension. But though this is the only way of stimulating it, yet an imagination of it is by association called up, upon the occasion of other slight sensations, even when muscles are uncontracted; and this imagination may sometimes be interpreted as a sign of effort. But though the sense of effort is thus merely a sensation, like any other, it is one in which the duality which appears in every sensation is specially prominent. A sense of exertion is at the same time a sense of being resisted. Exertion cannot be experienced without resistance, nor resistance without exertion. It is all one sense, but a sense of duality. Every sensation involves the same sense of duality, though less prominently. This is the direct perception of the external world of Reid and Hamilton.†2 This is the probatio ambulandi, which Diogenes Laertius perhaps gets mislocated. An idealist need not deny the reality of the external world, any more than Berkeley did. For the reality of the external world means nothing except that real experience of duality. Still, many of them do deny it -- or think they do. Very well; an idealist of that stamp is lounging down Regent Street, thinking of the utter nonsense of the opinion of Reid, and especially of the foolish probatio ambulandi, when some drunken fellow who is staggering up the street unexpectedly lets fly his fist and knocks him in the eye. What has become of his philosophical reflections now? Will he be so unable to free himself from prepossessions that no experience can show him the force of that argument? There may be some underlying unity beneath the sudden transition from meditation to astonishment. Grant that: does it follow that that transition did not take place? Is not the transition a direct experience of the duality of the inward past and outward present? A poor analyst is he who cannot see that the Unexpected is a direct experience of duality, that just as there can be no effort without resistance, so there can be no subjectivity of the unexpected without the objectivity of the unexpected, that they are merely two aspects of one experience given together and beyond all criticism. If the idealist should pick himself up and proceed to argue to the striker, saying "you could not have struck me, because you have no independent existence, you know," the striker might answer, "I dare say I have not separate existence enough for that; but I have separate existence enough to make you feel differently from what you were expecting to feel." Whatever strikes the eye or the touch, whatever strikes upon the ear, whatever affects nose or palate, contains something unexpected. Experience of the unexpected forces upon us the idea of duality. Will you say, "Yes, the idea is forced upon us, but it is not directly experienced, because only what is within is directly experienced"? The reply is that experience means nothing but just that of a cognitive nature which the history of our lives has forced upon us. It is indirect, if the medium of some other experience or thought is required to bring it out. Duality, thought abstractly, no doubt requires the intervention of reflection; but that upon which this reflection is based, the concrete duality, is there in the very experience itself.

Peirce: CP 5.540 Cross-Ref:†† 540. In the light of these remarks, we perceive that there is just this difference between a practical belief and an expectation so far as it involves no purpose [or] effort; namely that the former is expectant of muscular sensation, the latter of sensation not muscular. The expectancy consists in the stamp of approval, the act of recognition as one's own, being placed by a deed of the soul upon an imaginary anticipation of experience; so that, if it be fulfilled, though the actual experience will, at all events, contain enough of the unexpected to be recognized as external, yet the person who stands in expectancy will almost claim the event as his due, his triumphant "I told you so" implying a right to expect as much from a justly-regulated world. A man who goes among a barbarous tribe and announces a total eclipse of the sun next day, will expect, not only "his" eclipse from Nature, but due credit for it from that People. In all this, I am endeavoring so to shape what I have to say as to exhibit, besides, the close alliance, the family identity, of the ideas of externality and unexpectedness. Peirce: CP 5.541 Cross-Ref:†† 541. As to purely theoretical beliefs not expectacious, if they are to mean anything, they must be somehow expectative. The word "expect" is now and then applied by careless and ignorant speakers, especially the English, to what is surmised in regard to the past. It is not illogical language: it is only elliptical. "I expect that Adam must have felt a little sore over the extraction of his rib," may be interpreted as meaning that the expectation is, that so it will be found when the secrets of all hearts are laid bare. History would not have the character of a true science if it were not permissible to hope that further evidences may be forthcoming in the future by which the hypotheses of the critics may be tested. A theory which should be capable of being absolutely demonstrated in its entirety by future events, would be no scientific theory but a mere piece of fortune telling. On the other hand, a theory, which goes beyond what may be verified to any degree of approximation by future discoveries is, in so far, metaphysical gabble. To say that a quadratic equation which has no real root has two different imaginary roots does not sound as if it could have any relation to experience. Yet it is strictly expectative. It states what would be expectable if we had to deal with quantities expressing the relations between objects, related to one another like the points of the plane of imaginary quantity. So a belief about the incommensurability of the diagonal relates to what is expectable for a person dealing with fractions; although it means nothing at all in regard to what could be expected in physical measurements, which are, of their very nature, approximate only. Let us examine a highly abstract belief; and see whether there is any expectancy in it. Riemann †1 declared that infinity has nothing to do with the absence of a limit but relates solely to measure. This means that if a bounded surface be measured in a suitable way it will be found infinite, and that if an unbounded surface be measured in a suitable way, it will be found finite. It relates to what is expectable for a person dealing with different systems of measurement. The Roman church requires the faithful to believe that the elements of the eucharist are really transformed into flesh and blood, although all their "sensible accidents," that is, all that could be expected from physical experience, remain those of bread and wine. The Protestant episcopal church requires its ministers to teach that the elements remain really bread and wine, although they have miraculous spiritual effects different from those of ordinary bread and wine. "No indeed," say the Romanists, "they not only have those spiritual effects but they really are transmuted." But the layman declares that he cannot understand the difference. "That is not necessary," says the priest, "you can believe it implicitly."

What does that mean? It means that the layman is to trust that if he could understand the matter and know the truth, he would find that the priest was right.†1 But trust -and the word belief means trust primarily -- essentially refers to the future, or to a contingent future. The implication is that the layman may sometime know, presumably will, in another world; and that he may expect that if he ever does come to know, he will find the priest to be right. Thus, analysis shows that even in regard to so excessively metaphysical a matter, the belief, if there can be any belief, has to involve expectation as its very essence. Peirce: CP 5.542 Cross-Ref:†† 542. It now begins to look strongly as if perhaps all belief might involve expectation as its essence. That is as much as can justly be said. We have as yet no assurance that this is true of every kind of belief. One class of accepted truths which we have neglected is that of direct perceptual facts. I lay down a wafer, before me. I look at it, and say to myself, "That wafer looks red." What element of expectation is there in the belief that the wafer looks red at this moment? Peirce: CP 5.542 Cross-Ref:†† In order to handle this question, it is necessary to draw a distinction. Every belief is belief in a proposition. Now every proposition has its predicate which expresses what is believed, and its subjects which express of what it is believed. The grammarians of today prefer to say that a sentence has but one subject, which is put in the nominative. But from a logical point of view the terminology of the older grammarians was better, who spoke of the subject nominative and the subject accusative. I do not know that they spoke of the subject dative; but in the proposition, "Anthony gave a ring to Cleopatra," Cleopatra is as much a subject of what is meant and expressed as is the ring or Anthony. A proposition, then, has one predicate and any number of subjects. The subjects are either names of objects well known to the utterer and to the interpreter of the proposition (otherwise he could not interpret it) or they are virtually almost directions how to proceed to gain acquaintance with what is referred to. Thus, in the sentence "Every man dies," "Every man" implies that the interpreter is at liberty to pick out a man and consider the proposition as applying to him. In the proposition "Anthony gave a ring to Cleopatra," if the interpreter asks, What ring? the answer is that the indefinite article shows that it is a ring which might have been pointed out to the interpreter if he had been on the spot; and that the proposition is only asserted of the suitably chosen ring. The predicate on the other hand is a word or phrase which will call up in the memory or imagination of the interpreter images of things such as he has seen or imagined and may see again. Thus, "gave" is the predicate of the last proposition; and it conveys its meaning because the interpreter has had many experiences in which gifts were made; and a sort of composite photograph of them appears in his imagination. I am told that "Saccharin is 500 times as sweet as cane-sugar." But I never heard of saccharin. On inquiry, I find it is the sulphimide of orthosulphobenzoic acid; that is, it is phthalimide in which one CO group is replaced by SO[2]. I can see on paper that there might be such a body. That it is "500 times sweeter than sugar" produces a rather confused idea of a very familiar general kind. What I am to expect is expressed by the predicate, while the subjects inform me on what occasion I am to expect it. Diogenes Laertius, Suidas, Plutarch, and an anonymous biographer tell us that Aristotle was unable to pronounce the letter R.†1 I place Aristotle perfectly, of course. He is the author of works I often read and profoundly admire and whose fame far surpasses that of any other logician -The Prince of Philosophers. I have also met people who could not pronounce R; but in other respects they did not seem to be much like Aristotle -- not even Dundreary.

Should I meet him in the Elysian Fields, I shall know what to expect. That is an impossible supposition; but should I ever meet a great logician, spindle-shanked and pig-eyed, who cannot pronounce R, I shall be interested to see whether he has other characteristics of Aristotle. This example has been selected as one which should seem to a superficial eye to involve no gleam of expectation; and if this testimony of four respectable witnesses, as independent as under the circumstances they could be, is destined never to receive confirmation nor contradiction, nor in any other way to have its probable consequences confronted by future experience, then in truth no expectation does it carry. In that case, it is an idle tale that might, for any practical purpose, have been as well the creation of some ironical poet. In that case, it is, properly speaking, no contribution to knowledge, for at least it is only probability, and probability cannot be reckoned as knowledge, unless it is destined to be indefinitely heightened in the future. Knowledge which should have no possible bearing upon any future experience -- bring no expectation whatever -- would be information concerning a dream. But in truth no such thing can be presumed of any knowledge. We expect that in time it will produce, or reinforce, or weaken some definite expectation. Give science only a hundred more centuries of increase in geometrical progression, and she may be expected to find that the sound waves of Aristotle's voice have somehow recorded themselves. If not, it were better to hand the reports over to the poets to make something pretty of, and thus turn them to some human use. But the right thing to do is to expect the verification. It is the degenerate pronunciation that is to be expected; the occasion is when Aristotle's voice shall become virtually heard again or when we shall have some other information which shall confirm or refute those reports. Peirce: CP 5.543 Cross-Ref:†† 543. Now if the reader should say, "Talk as you please, the assertion that Aristotle was {praulos} simply brings to the mental ear the voice of a man unable to pronounce the letter R, and labels that image with an indication of Aristotle, a man who lived three hundred years before Christ," the author may surprise him and grieve any whom he may have convinced, by declaring "I agree with you entirely"; only this assertion, which is identical with the previous one, though translated into other language, means nothing unless it be that Aristotle having been brought, directly or indirectly, to our experience, will be found, if found at all, to be incapable of pronouncing the R. Let us distinguish between the proposition and the assertion of that proposition. We will grant, if you please, that the proposition itself merely represents an image with a label or pointer attached to it. But to assert that proposition is to make oneself responsible for it, without any definite forfeit, it is true, but with a forfeit no smaller for being unnamed.†1 Now an ex post facto law is forbidden by the Constitution of the United States of America, but an ex post facto contract is forbidden by the constitution of things. A man cannot promise what the past shall have been, if he tries. It is evident that to guarantee that, if a piece of work has not already been done right, one will pay for it, and to guarantee that, if it shall be found not to have already been done right, one will pay for it, have one and the same meaning. One or other of them therefore must be an elliptical or otherwise unliteral expression, or else both are so. But nobody will maintain that to promise to pay for the work, if it shall be ascertained not to have been already done right, really means to promise to so pay, if it shall in fact not have been already done right, whether it be ascertained or not. It would be equally absurd to say that there was any third meaning which should have reference to an unascertained past. It follows, then, that to contract to pay money if something in the past has been done or not done can only mean that

the money shall be paid if it is ascertained that the event has happened or has not happened. But there would be no reason why the literal sense should not be understood if it made any sense. Hence there can be no meaning in making oneself responsible for a past event independent of its future ascertainment. But to assert a proposition is to make oneself responsible for its truth. Consequently, the only meaning which an assertion of a past fact can have is that, if in the future the truth be ascertained, so it shall be ascertained to be. There seems to be no rational escape from this. Peirce: CP 5.544 Cross-Ref:†† 544. Now let us take up the perceptual judgment "This wafer looks red." It takes some time to write this sentence, to utter it, or even to think it. It must refer to the state of the percept at the time that it, the judgment, began to be made. But the judgment does not exist until it is completely made. It thus only refers to a memory of the past; and all memory is possibly fallible and subject to criticism and control. The judgment, then, can only mean that so far as the character of the percept can ever be ascertained, it will be ascertained that the wafer looked red. Peirce: CP 5.545 Cross-Ref:†† 545. Perhaps the matter may be stated less paradoxically. Everybody will agree that it would be perfectly meaningless to say that sulphur had the singular property of turning pink when nobody was looking at it, instantly returning to yellowness before the most rapid glance could catch its pink color, or to say that copper was subject to the law that as long as there was no pressure upon it, it was perfectly yielding, becoming hard in proportion as it was pressed; and generally, a law which never should operate would be an empty formula. Indeed, something not very far from the assertion about copper is contained in all treatises on dynamics, although not limited to any particular substance. Namely, it is set down that no tangential force can be exerted upon a perfect fluid. But no writer puts it forth as a statement of fact; it is given as a definition merely. A law, then, which never will operate has no positive existence. Consequently, a law which has operated for the last time has ceased to exist as a law, except as a mere empty formula which it may be convenient to allow to remain. Hence to assert that a law positively exists is to assert that it will operate, and therefore to refer to the future, even though only conditionally. But to say that a body is hard, or red, or heavy, or of a given weight, or has any other property, is to say that it is subject to law and therefore is a statement referring to the future.

Peirce: CP 5.546 Cross-Ref:†† §2. JUDGMENT AND ASSERTION †1

546. Every new concept first comes to the mind in a judgment. This argument evades the consideration of the difficult question of the logical nature of the judgment, but draws attention to a fact that ordinary speech recognizes; namely, that a judgment is something that ripens in the mind, and further that there is a vernacular phrase which betrays a feature of the ripe judgment, the phrase "I says to myself, says I." The phrase indicates the easily verified fact that the ripe judgment, at least, involves an element closely analogous to assertion. But what is that? What is the nature of assertion? We have no magnifying-glass that can enlarge its features, and

render them more discernible; but in default of such an instrument we can select for examination a very formal assertion, the features of which have purposely been rendered very prominent, in order to emphasize its solemnity. If a man desires to assert anything very solemnly, he takes such steps as will enable him to go before a magistrate or notary and take a binding oath to it. Taking an oath is not mainly an event of the nature of a setting forth, Vorstellung, or representing. It is not mere saying, but is doing. The law, I believe, calls it an "act." At any rate, it would be followed by very real effects, in case the substance of what is asserted should be proved untrue. This ingredient, the assuming of responsibility, which is so prominent in solemn assertion, must be present in every genuine assertion. For clearly, every assertion involves an effort to make the intended interpreter believe what is asserted, to which end a reason for believing it must be furnished. But if a lie would not endanger the esteem in which the utterer was held, nor otherwise be apt to entail such real effects as he would avoid, the interpreter would have no reason to believe the assertion. Nobody takes any positive stock in those conventional utterances, such as "I am perfectly delighted to see you," upon whose falsehood no punishment at all is visited. At this point, the reader should call to mind, or, if he does not know it, should make the observations requisite to convince himself, that even in solitary meditation every judgment is an effort to press home, upon the self of the immediate future and of the general future, some truth. It is a genuine assertion, just as the vernacular phrase represents it; and solitary dialectic is still of the nature of dialogue. Consequently it must be equally true that here too there is contained an element of assuming responsibility, of "taking the consequences." Peirce: CP 5.547 Cross-Ref:†† 547. That is the first point of this argument; namely, that the judgment, which is the sole vehicle in which a concept can be conveyed to a person's cognizance or acquaintance, is not a purely representitious event, but involves an act, an exertion of energy, and is liable to real consequences, or effects. To this an eager adversary of pragmaticism might make answer to the effect that if there be an assumption of responsibility in a judgment, it can only be in a ripe judgment; whereas the concept makes its appearance before the judgment is ripe, when it is still in the problematic or interrogatory mood; and that this shows that the volitional element is quite extraneous to the substance, or "meaning," of the concept. But the reply will be that this answer quite mistakes the aim of the argument. For it is no pragmaticistic doctrine that responsibility attaches to a concept; but the argument is that the predication of a concept is capable of becoming the subject of responsibility, since it actually does become so in the act of asserting that predication. Peirce: CP 5.548 Cross-Ref:†† 548. Thereupon it follows that the concept has a capability of having a bearing upon conduct; and this fact will lend it intellectual purport. For it cannot be denied that one, at least, of the functions of intelligence is to adapt conduct to circumstances, so as to subserve desire. If the argument is correct, this applies to any concept whatsoever, unless there be a concept that cannot be predicated.

Peirce: CP 5.549 Cross-Ref:†† CHAPTER 5

TRUTH

§1. TRUTH AS CORRESPONDENCE †1

549. A state of things is an abstract constituent part of reality, of such a nature that a proposition is needed to represent it. There is but one individual, or completely determinate, state of things, namely, the all of reality. A fact is so highly a prescissively abstract state of things, that it can be wholly represented in a simple proposition, and the term "simple," here, has no absolute meaning, but is merely a comparative expression. Peirce: CP 5.550 Cross-Ref:†† 550. A mathematical form of a state of things is such a representation of that state of things as represents only the samenesses and diversities involved in that state of things, without definitely qualifying the subjects of the samenesses and diversities. It represents not necessarily all of these; but if it does represent all, it is the complete mathematical form. Every mathematical form of a state of things is the complete mathematical form of some state of things. The complete mathematical form of any state of things, real or fictitious, represents every ingredient of that state of things except the qualities of feeling connected with it. It represents whatever importance or significance those qualities may have; but the qualities themselves it does not represent. Peirce: CP 5.551 Cross-Ref:†† 551. Before any conclusion shall be made to rest upon this almost self-evident proposition, a way of setting it quite beyond doubt shall be explained. As at present enunciated, it is merely put forward as a private opinion of the writer's which will serve to explain the great interest he attaches to the emphatic dualism of the three normative sciences, which may be regarded as being the sciences of the conditions of truth and falsity, of wise and foolish conduct, of attractive and repulsive ideas. Should the reader become convinced that the importance of everything resides entirely in its mathematical form, he, too, will come to regard this dualism as worthy of close attention. Meantime that it exists, and is more marked in these sciences than in any others, is an indisputable fact. To what is this circumstance to be attributed? Skipping the easy reasoning by which it can be shown that this dualism cannot be due to any peculiar quality of feeling that may be connected with these sciences, nor to any intellectual peculiarity of them, which negative propositions will become obtrusively plain at a later stage of our reasoning, we may turn at once to the affirmative reason for attributing the dualism to the reference of the normative sciences to action. It is curious how this reason seems to seek to escape detection, by putting forward an apparent indication that it is not there. For it is evident that it is in esthetics that we ought to seek for the deepest characteristics of normative science, since esthetics, in dealing with the very ideal itself whose mere materialization engrosses the attention of practics and of logic, must contain the heart, soul, and spirit of normative science. But that dualism which is so much marked in the True and False, logic's object of study, and in the Useful and Pernicious of the confessional of Practics, is softened almost to obliteration in esthetics. Nevertheless, it would be the height of stupidity to

say that esthetics knows no good and bad. It must never be forgotten that evil of any kind is none the less bad though the occurrence of it be a good. Because in every case the ultimate in some measure abrogates, and ought to abrogate, the penultimate, it does not follow that the penultimate ought not to have abrogated the antepenultimate in due measure. On the contrary, just the opposite follows. Peirce: CP 5.552 Cross-Ref:†† 552. Esthetic good and evil are closely akin to pleasure and pain. They are what would be pleasure or pain to the fully developed superman. What, then, are pleasure and pain? The question has been sufficiently discussed, and the answer ought by this time to be ready. They are secondary feelings or generalizations of such feelings; that is, of feelings attaching themselves to, and excited by, other feelings.†1 A toothache is painful. It is not pain, but pain accompanies it; and if you choose to say that pain is an ingredient of it, that is not far wrong. However, the quality of the feeling of toothache is a simple, positive feeling, distinct from pain; though pain accompanies it. To use the old consecrated terms, pleasure is the feeling that a feeling is "sympathetical," pain that it is "antipathetical." The feeling of pain is a symptom of a feeling which repels us; the feeling of pleasure is the symptom of an attractive feeling. Attraction and repulsion are kinds of action. Feelings are pleasurable or painful according to the kind of action which they stimulate. In general, the good is the attractive -- not to everybody, but to the sufficiently matured agent; and the evil is the repulsive to the same. Mr. Ferdinand C.S. Schiller †1 informs us that he and James have made up their minds that the true is simply the satisfactory. No doubt; but to say "satisfactory" is not to complete any predicate whatever. Satisfactory to what end? Peirce: CP 5.553 Cross-Ref:†† 553. That truth is the correspondence of a representation with its object is, as Kant †2 says, merely the nominal definition of it. Truth belongs exclusively to propositions. A proposition has a subject (or set of subjects) and a predicate. The subject is a sign; the predicate is a sign; and the proposition is a sign that the predicate is a sign of that of which the subject is a sign.†3 If it be so, it is true. But what does this correspondence or reference of the sign, to its object, consist in? The pragmaticist answers this question as follows. Suppose, he says, that the angel Gabriel were to descend and communicate to me the answer to this riddle from the breast of omniscience. Is this supposable; or does it involve an essential absurdity to suppose the answer to be brought to human intelligence? In the latter case, "truth," in this sense, is a useless word, which never can express a human thought. It is real, if you will; it belongs to that universe entirely disconnected from human intelligence which we know as the world of utter nonsense. Having no use for this meaning of the word "truth," we had better use the word in another sense presently to be described. But if, on the other hand, it be conceivable that the secret should be disclosed to human intelligence, it will be something that thought can compass. Now thought is of the nature of a sign. In that case, then, if we can find out the right method of thinking and can follow it out -- the right method of transforming signs -- then truth can be nothing more nor less than the last result to which the following out of this method would ultimately carry us. In that case, that to which the representation should conform, is itself something in the nature of a representation, or sign -- something noumenal, intelligible, conceivable, and utterly unlike a thing-in-itself. Peirce: CP 5.554 Cross-Ref:†† 554. Truth is the conformity of a representamen to its object, its object, ITS

object, mind you. The International Dictionary at the writer's elbow, the Century Dictionary which he daily studies, the Standard which he would be glad sometimes to consult, all contain the word yes; but that word is not true simply because he is going to ask on this eighth of January 1906, in Pike County, Pennsylvania, whether it is snowing. There must be an action of the object upon the sign to render the latter true. Without that, the object is not the representamen's object. If a colonel hands a paper to an orderly and says, "You will go immediately and deliver this to Captain Hanno," and if the orderly does so, we do not say the colonel told the truth; we say the orderly was obedient, since it was not the orderly's conduct which determined the colonel to say what he did, but the colonel's speech which determined the orderly's action. Here is a view of the writer's house: what makes that house to be the object of the view? Surely not the similarity of appearance. There are ten thousand others in the country just like it. No, but the photographer set up the film in such a way that according to the laws of optics, the film was forced to receive an image of this house. What the sign virtually has to do in order to indicate its object -- and make it its -- all it has to do is just to seize its interpreter's eyes and forcibly turn them upon the object meant: it is what a knock at the door does, or an alarm or other bell, or a whistle, a cannon-shot, etc. It is pure physiological compulsion; nothing else. Peirce: CP 5.554 Cross-Ref:†† So, then, a sign, in order to fulfill its office, to actualize its potency, must be compelled by its object. This is evidently the reason of the dichotomy of the true and the false. For it takes two to make a quarrel, and a compulsion involves as large a dose of quarrel as is requisite to make it quite impossible that there should be compulsion without resistance.

Peirce: CP 5.555 Cross-Ref:†† §2. TRUTH AND SATISFACTION †1

555. It appears that there are certain mummified pedants who have never waked to the truth that the act of knowing a real object alters it. They are curious specimens of humanity, and as I am one of them, it may be amusing to see how I think. It seems that our oblivion to this truth is due to our not having made the acquaintance of a new analysis that the True is simply that in cognition which is Satisfactory. As to this doctrine, if it is meant that True and Satisfactory are synonyms, it strikes me that it is not so much a doctrine of philosophy as it is a new contribution to English lexicography. Peirce: CP 5.556 Cross-Ref:†† 556. But it seems plain that the formula does express a doctrine of philosophy, although quite vaguely; so that the assertion does not concern two words of our language but, attaching some other meaning to the True, makes it to be coextensive with the Satisfactory in cognition. Peirce: CP 5.557 Cross-Ref:†† 557. In that case, it is indispensable to say what is meant by the True: until this is done the statement has no meaning. I suppose that by the True is meant that at which inquiry aims. Peirce: CP 5.558 Cross-Ref:††

558. It is equally indispensable to ascertain what is meant by Satisfactory; but this is by no means so easy. Whatever be meant, however, if the doctrine is true at all, it must be necessarily true. For it is the very object, conceived in entertaining the purpose of the inquiry, that is asserted to have the character of satisfactoriness. Peirce: CP 5.559 Cross-Ref:†† 559. Is the Satisfactory meant to be whatever excites a certain peculiar feeling of satisfaction? In that case, the doctrine is simply hedonism in so far as it affects the field of cognition. For when hedonists talk of "pleasure," they do not mean what is so-called in ordinary speech, but what excites a feeling of satisfaction. Peirce: CP 5.560 Cross-Ref:†† 560. But to say that an action or the result of an action is Satisfactory is simply to say that it is congruous to the aim of that action. Consequently, the aim must be determined before it can be determined, either in thought or in fact, to be satisfactory. An action that had no other aim than to be congruous to its aim would have no aim at all, and would not be a deliberate action. Peirce: CP 5.561 Cross-Ref:†† 561. The hedonists do not offer their doctrine as an induction from experience but insist that, in the nature of things, that is, from the very essence of the conceptions, an action can have no other aim than "pleasure." Now it is conceivable that an action should be disconnected from every other in its aim. Such an action, then, according to hedonistic doctrine, can have no other aim than that of satisfying its own aim, which is absurd. Peirce: CP 5.562 Cross-Ref:†† 562. But if the hedonist replies that his position does not relate to satisfaction, but to a feeling that only arises upon satisfaction, the rejoinder will be that feeling is incomprehensible; so that no necessary truth can be discovered about it. But as a matter of observation we do, now and then, meet with persons who very largely behave with a view of experiencing this or that feeling. These people, however, are exceptional, and are wretched beings sharply marked off from the mass of busy and happy mankind. Peirce: CP 5.563 Cross-Ref:†† 563. It is, however, no doubt true that men act, especially in the action of inquiry, as if their sole purpose were to produce a certain state of feeling, in the sense that when that state of feeling is attained, there is no further effort. It was upon that proposition that I originally based pragmaticism, laying it down in the article that in November 1877†1 prepared the ground for my argument for the pragmaticistic doctrine (Pop. Sci. Monthly for January, 1878†2). In the case of inquiry, I called that state of feeling "firm belief," and said, "As soon as a firm belief is reached we are entirely satisfied, whether the belief be true or false,"†3 and went on to show how the action of experience consequently was to create the conception of real truth. Early in 1880, in the opening paragraphs of my memoir in Vol. III of the American Journal of Mathematics,†4 I referred the matter to the fundamental properties of protoplasm, showing that purposive action must be action virtually directed toward the removal of stimulation. Peirce: CP 5.564 Cross-Ref:†† 564. My paper of November 1877, setting out from the proposition that the agitation of a question ceases when satisfaction is attained with the settlement of

belief, and then only, goes on to consider how the conception of truth gradually develops from that principle under the action of experience; beginning with willful belief, or self-mendacity, the most degraded of all intellectual conditions; thence rising to the imposition of beliefs by the authority of organized society; then to the idea of a settlement of opinion as the result of a fermentation of ideas; and finally reaching the idea of truth as overwhelmingly forced upon the mind in experience as the effect of an independent reality.

Peirce: CP 5.565 Cross-Ref:†† §3. DEFINITIONS OF TRUTH †1

565. Logical. (1) Truth is a character which attaches to an abstract proposition, such as a person might utter. It essentially depends upon that proposition's not professing to be exactly true. But we hope that in the progress of science its error will indefinitely diminish, just as the error of 3.14159, the value given for π, will indefinitely diminish as the calculation is carried to more and more places of decimals. What we call π is an ideal limit to which no numerical expression can be perfectly true. If our hope is vain; if in respect to some question -- say that of the freedom of the will -- no matter how long the discussion goes on, no matter how scientific our methods may become, there never will be a time when we can fully satisfy ourselves either that the question has no meaning, or that one answer or the other explains the facts, then in regard to that question there certainly is no truth. But whether or not there would be perhaps any reality is a question for the metaphysician, not the logician. Even if the metaphysician decides that where there is no truth there is no reality, still the distinction between the character of truth and the character of reality is plain and definable. Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth. A further explanation of what this concordance consists in will be given below. Reality is that mode of being by virtue of which the real thing is as it is, irrespectively of what any mind or any definite collection of minds may represent it to be. The truth of the proposition that Caesar crossed the Rubicon consists in the fact that the further we push our archaeological and other studies, the more strongly will that conclusion force itself on our minds forever -- or would do so, if study were to go on forever. An idealist metaphysician may hold that therein also lies the whole reality behind the proposition; for though men may for a time persuade themselves that Caesar did not cross the Rubicon, and may contrive to render this belief universal for any number of generations, yet ultimately research -- if it be persisted in -- must bring back the contrary belief. But in holding that doctrine, the idealist necessarily draws the distinction between truth and reality. Peirce: CP 5.566 Cross-Ref:†† 566. In the above we have considered positive scientific truth. But the same definitions equally hold in the normative sciences. If a moralist describes an ideal as the summum bonum, in the first place, the perfect truth of his statement requires that it should involve the confession that the perfect doctrine can neither be stated nor conceived. If, with that allowance, the future development of man's moral nature will only lead to a firmer satisfaction with the described ideal, the doctrine is true. A

metaphysician may hold that the fact that the ideal thus forces itself upon the mind, so that minds in their development cannot fail to come to accept it, argues that the ideal is real: he may even hold that that fact (if it be one) constitutes a reality. But the two ideas, truth and reality, are distinguished here by the same characters given in the above definitions. Peirce: CP 5.567 Cross-Ref:†† 567. These characters equally apply to pure mathematics. Projective geometry is not pure mathematics, unless it be recognized that whatever is said of rays holds good of every family of curves of which there is one and one only through any two points, and any two of which have a point in common. But even then it is not pure mathematics until for points we put any complete determinations of any two-dimensional continuum. Nor will that be enough. A proposition is not a statement of perfectly pure mathematics until it is devoid of all definite meaning, and comes to this -- that a property of a certain icon is pointed out and is declared to belong to anything like it, of which instances are given. The perfect truth cannot be stated, except in the sense that it confesses its imperfection. The pure mathematician deals exclusively with hypotheses. Whether or not there is any corresponding real thing, he does not care. His hypotheses are creatures of his own imagination; but he discovers in them relations which surprise him sometimes. A metaphysician may hold that this very forcing upon the mathematician's acceptance of propositions for which he was not prepared, proves, or even constitutes, a mode of being independent of the mathematician's thought, and so a reality. But whether there is any reality or not, the truth of the pure mathematical proposition is constituted by the impossibility of ever finding a case in which it fails. This, however, is only possible if we confess the impossibility of precisely defining it. Peirce: CP 5.568 Cross-Ref:†† 568. The same definitions hold for the propositions of practical life. A man buys a bay horse, under a warranty that he is sound and free from vice. He brings him home and finds he is dyed, his real colour being undesirable. He complains of false representations; but the seller replies, "I never pretended to state every fact about the horse; what I said was true, so far as it professed to be true." In ordinary life all our statements, it is well understood, are, in the main, rough approximations to what we mean to convey. A tone or gesture is often the most definite part of what is said. Even with regard to perceptual facts, or the immediate judgments we make concerning our single percepts, the same distinction is plain. The percept is the reality. It is not in propositional form. But the most immediate judgment concerning it is abstract. It is therefore essentially unlike the reality, although it must be accepted as true to that reality. Its truth consists in the fact that it is impossible to correct it, and in the fact that it only professes to consider one aspect of the percept.†1 Peirce: CP 5.569 Cross-Ref:†† 569. But even if it were impossible to distinguish between truth and reality, that would not in the least prevent our defining what it is that truth consists in. Truth and falsity are characters confined to propositions. A proposition is a sign which separately indicates its object. Thus, a portrait with the name of the original below it is a proposition. It asserts that if anybody looks at it, he can form a reasonably correct idea of how the original looked. A sign is only a sign in actu by virtue of its receiving an interpretation, that is, by virtue of its determining another sign of the same object. This is as true of mental judgments as it is of external signs. To say that a proposition is true is to say that every interpretation of it is true. Two propositions are equivalent

when either might have been an interpretant of the other. This equivalence, like others, is by an act of abstraction (in the sense in which forming an abstract noun is abstraction) conceived as identity. And we speak of believing in a proposition, having in mind an entire collection of equivalent propositions with their partial interpretants. Thus, two persons are said to have the same proposition in mind. The interpretant of a proposition is itself a proposition. Any necessary inference from a proposition is an interpretant of it. When we speak of truth and falsity, we refer to the possibility of the proposition being refuted; and this refutation (roughly speaking) takes place in but one way. Namely, an interpretant of the proposition would, if believed, produce the expectation of a certain description of percept on a certain occasion. The occasion arrives: the percept forced upon us is different. This constitutes the falsity of every proposition of which the disappointing prediction was the interpretant. Peirce: CP 5.569 Cross-Ref:†† Thus, a false proposition is a proposition of which some interpretant represents that, on an occasion which it indicates, a percept will have a certain character, while the immediate perceptual judgment on that occasion is that the percept has not that character. A true proposition is a proposition belief in which would never lead to such disappointment so long as the proposition is not understood otherwise than it was intended.†1 Peirce: CP 5.570 Cross-Ref:†† 570. All the above relates to complex truth, or the truth of propositions. This is divided into many varieties, among which may be mentioned ethical truth, or the conformity of an assertion to the speaker's or writer's belief, otherwise called veracity, and logical truth, that is, the concordance of a proposition with reality, in such way as is above defined. Peirce: CP 5.571 Cross-Ref:†† 571. (2) The word truth has also had great importance in philosophy in widely different senses, in which it is distinguished as simple truth, which is that truth which inheres in other subjects than propositions. Peirce: CP 5.571 Cross-Ref:†† Plato in the Cratylus (385B) maintains that words have truth; and some of the scholastics admitted that an incomplex sign, such as a picture, may have truth. Peirce: CP 5.572 Cross-Ref:†† 572. But truth is also used in senses in which it is not an affection of a sign, but of things as things. Such truth is called transcendental truth. The scholastic maxim was Ens est unum, verum, bonum. Among the senses in which transcendental truth was spoken of was that in which it was said that all science has for its object the investigation of truth, that is to say, of the real characters of things. It was, in other senses, regarded as a subject of metaphysics exclusively. It is sometimes defined so as to be indistinguishable from reality, or real existence. Another common definition is that truth is the conformity, or conformability, of things to reason. Another definition is that truth is the conformity of things to their essential principles. Peirce: CP 5.573 Cross-Ref:†† 573. (3) Truth is also used in logic in a sense in which it inheres only in subjects more complex than propositions. Such is formal truth, which belongs to an argumentation which conforms to logical laws.

Peirce: CP 5.574 Cross-Ref:†† CHAPTER 6

METHODS FOR ATTAINING TRUTH

§1. THE FIRST RULE OF LOGICP†1

574. Certain methods of mathematical computation correct themselves; so that if an error be committed, it is only necessary to keep right on, and it will be corrected in the end. For instance, I want to extract the cube root of 2. The true answer is 1.25992105. . . . The rule is as follows: Peirce: CP 5.574 Cross-Ref:†† Form a column of numbers, which for the sake of brevity we may call the A's. The first 3 A's are any 3 numbers taken at will. To form a new A, add the last two A's, triple the sum, add to this sum the last A but two, and set down the result as the next A. Now any A, the lower in the column the better, divided by the following A gives a

fraction which increased by 1 is approximately [cube root of 2] [Click here to view]

Correct

Sum of

Computation

Erroneous

Sum of

Two Triple Computation

1

1

0

0

1

1

3

1

1

3

4

5

15

4

5

15

15

19

57 Error!16

58

73

219

223

281

843

858

1081

3243

61 235 904

20 77

Two

60 231

296 1139

888 3417

Triple

3301

4159 12477

12700

3478

4382 13146

13381

1 3301/12700 1.2599213

Error +.0000002

1 3478/13381 1.2599208

Error -.0000002

Peirce: CP 5.574 Cross-Ref:†† You see the error committed in the second computation, though it seemed to multiply itself greatly, became substantially corrected in the end. Peirce: CP 5.574 Cross-Ref:†† If you sit down to solve ten ordinary linear equations between ten unknown quantities, you will receive materials for a commentary upon the infallibility of mathematical processes. For you will almost infallibly get a wrong solution. I take it as a matter of course that you are not an expert professional computer. He will proceed according to a method which will correct his errors if he makes any. Peirce: CP 5.575 Cross-Ref:†† 575. This calls to mind one of the most wonderful features of reasoning and one of the most important philosophemes in the doctrine of science, of which, however, you will search in vain for any mention in any book I can think of; namely, that reasoning tends to correct itself, and the more so, the more wisely its plan is laid. Nay, it not only corrects its conclusions, it even corrects its premisses. The theory of Aristotle is that a necessary conclusion is just equally as certain as its premisses, while a probable conclusion is somewhat less so. Hence, he was driven to his strange distinction between what is better known to Nature and what is better known to us. But were every probable inference less certain than its premisses, science, which piles inference upon inference, often quite deeply, would soon be in a bad way. Every astronomer, however, is familiar with the fact that the catalogue place of a fundamental star, which is the result of elaborate reasoning, is far more accurate than any of the observations from which it was deduced. Peirce: CP 5.576 Cross-Ref:†† 576. That Induction tends to correct itself, is obvious enough. When a man undertakes to construct a table of mortality upon the basis of the Census, he is engaged in an inductive inquiry. And lo, the very first thing that he will discover from the figures, if he did not know it before, is that those figures are very seriously vitiated by their falsity. The young find it to their advantage to be thought older than they are, and the old to be thought younger than they are. The number of young men who are just 21 is altogether in excess of those who are 20, although in all other cases the ages expressed in round numbers are in great excess. Now the operation of inferring a law in a succession of observed numbers is, broadly speaking, inductive; and therefore we see that a properly conducted Inductive research corrects its own premisses. Peirce: CP 5.577 Cross-Ref:††

577. That the same thing may be true of a Deductive inquiry our arithmetical example has shown. Theoretically, I grant you, there is no possibility of error in necessary reasoning. But to speak thus "theoretically," is to use language in a Pickwickian sense. In practice, and in fact, mathematics is not exempt from that liability to error that affects everything that man does. Strictly speaking, it is not certain that twice two is four. If on an average in every thousand figures obtained by addition by the average man there be one error, and if a thousand million men have each added 2 to 2 ten thousand times, there is still a possibility that they have all committed the same error of addition every time. If everything were fairly taken into account, I do not suppose that twice two is four is more certain than Edmund Gurney †1 held the existence of veridical phantasms of the dying or dead to be. Deductive inquiry, then, has its errors; and it corrects them, too. But it is by no means so sure, or at least so swift to do this as is Inductive science. A celebrated error in the Mécanique Céleste concerning the amount of theoretical acceleration of the moon's mean motion deceived the whole world of astronomy for more than half a century.†2 Errors of reasoning in the first book of Euclid's Elements, the logic of which book was for two thousand years subjected to more careful criticism than any other piece of reasoning without exception ever was or probably ever will be, only became known after the non-Euclidean geometry had been developed. The certainty of mathematical reasoning, however, lies in this, that once an error is suspected, the whole world is speedily in accord about it. Peirce: CP 5.578 Cross-Ref:†† 578. As for Retroductive Inquiries, or the Explanatory Sciences, such as Geology, Evolution, and the like, they always have been and always must be theatres of controversy. These controversies do get settled, after a time, in the minds of candid inquirers; though it does not always happen that the protagonists themselves are able to assent to the justice of the decision. Nor is the general verdict always logical or just. Peirce: CP 5.579 Cross-Ref:†† 579. So it appears that this marvellous self-correcting property of Reason, which Hegel made so much of, belongs to every sort of science, although it appears as essential, intrinsic, and inevitable only in the highest type of reasoning, which is induction. But the logic of relatives shows that the other types of reasoning, Deduction and Retroduction, are not so thoroughly unlike Induction as they might be thought, and as Deduction, at least, always has been thought to be. Stuart Mill alone among the older logicians in his analysis of the Pons Asinorum came very near to the view which the logic of relatives forces us to take.†1 Namely, in the logic of relatives, treated let us say, in order to fix our ideas, by means of those existential graphs of which I gave a slight sketch in the last lecture,†2 [we] begin a Deduction by writing down all the premisses. Those different premisses are then brought into one field of assertion, that is, are colligated, as Whewell †3 would say, or joined into one copulative proposition. Thereupon, we proceed attentively to observe the graph. It is just as much an operation of Observation as is the observation of bees. This observation leads us to make an experiment upon the Graph. Namely, we first duplicate portions of it; and then we erase portions of it, that is, we put out of sight part of the assertion in order to see what the rest of it is. We observe the result of this experiment, and that is our deductive conclusion. Precisely those three things are all that enter into the experiment of any Deduction -- Colligation, Iteration, Erasure.†4 The rest of the process consists of observing the result. It is not, however, in every Deduction that all the three possible elements of the Experiment take place. In

particular, in ordinary syllogism the iteration may be said to be absent. And that is the reason that ordinary syllogism can be worked by a machine.†5 There is but one conclusion of any consequence to be drawn by ordinary syllogism from given premisses. Hence, it is that we fall into the habit of talking of the conclusion. But in the logic of relatives there are conclusions of different orders, depending upon how much iteration takes place.†6 What is the conclusion deducible from the very simple first principles of number? It is ridiculous to speak of the conclusion. The conclusion is no less than the aggregate of all the theorems of higher arithmetic that have been discovered or that ever will be discovered. Now let us turn to Induction. This mode of reasoning also begins by a colligation. In fact, it is precisely the colligation that gave induction its name, {epagein} with Socrates,†1 {synagögé} with Plato,†2 {epagögé} with Aristotle.†3 It must, by the rule of predesignation,†4 be a deliberate experiment. In ordinary induction we proceed to observe something about each instance. Relative induction is illustrated by the process of making out the law of the arrangement of the scales of a pine-cone. It is necessary to mark a scale taken as an instance, and counting in certain directions to come back to that marked scale. This double observation of the same instance corresponds to Iteration in deduction. Finally, we erase the particular instances and leave the class or system sampled directly connected with the characters, relative or otherwise, which have been found in the sample of it. Peirce: CP 5.580 Cross-Ref:†† 580. We see, then, that Induction and Deduction are after all not so very unlike. It is true that in Induction we commonly make many experiments and in Deduction only one. Yet this is not always the case. The chemist contents himself with a single experiment to establish any qualitative fact. True, he does this because he knows that there is such a uniformity in the behaviour of chemical bodies that another experiment would be a mere repetition of the first in every respect. But it is precisely such a knowledge of a uniformity that leads the mathematician to content himself with one experiment. The inexperienced student in mathematics will mentally perform a number of geometrical experiments, which the veteran would regard as superfluous, before he will permit himself to come to a general conclusion. For example, if the question is, how many rays can cut four rays fixed in space, the experienced mathematician will content himself with imagining that two of the fixed rays intersect and that the other two likewise intersect. He will see, then, that there is one ray through the two intersections and another along the intersection of the two planes of pairs of intersecting fixed rays, and will unhesitatingly declare thereupon that but two rays can cut four fixed rays, unless the fixed rays are so situated that an infinite multitude of rays will cut them all. But I dare say many of you would want to experiment with other arrangements of the four fixed rays, before making any confident pronouncement. A friend of mine who seemed to have difficulties in adding up her accounts was once counselled to add each column five times and adopt the mean of the different results. It is evident that when we run a column of figures down as well as up, as a check, or when we review a demonstration in order to look out for any possible flaw in the reasoning, we are acting precisely as when in an induction we enlarge our sample for the sake of the self-correcting effect of induction. Peirce: CP 5.581 Cross-Ref:†† 581. As for retroduction, it is itself an experiment. A retroductive research is an experimental research; and when we look upon Induction and Deduction from the point of view of Experiment and Observation, we are merely tracing in those types of reasoning their affinity to Retroduction. It begins always with colligation, of course,

of a variety of separately observed facts about the subject of the hypothesis. How remarkable it is, by the way, that the entire army of logicians from Zeno to Whateley should have left it to this mineralogist [Whewell] to point out colligation as a generally essential step in reasoning. To return to Retroduction, then, it begins with colligation. Something corresponding to iteration may or may not take place. And then comes an Observation. Not, however, an External observation of the objects as in Induction, nor yet an observation made upon the parts of a diagram, as in Deduction; but for all that just as truly an observation. For what is observation? What is experience? It is the enforced element in the history of our lives. It is that which we are constrained to be conscious of by an occult force residing in an object which we contemplate. The act of observation is the deliberate yielding of ourselves to that force majeure -- an early surrender at discretion, due to our foreseeing that we must, whatever we do, be borne down by that power, at last. Now the surrender which we make in Retroduction, is a surrender to the Insistence of an Idea. The hypothesis, as the Frenchman says, c'est plus fort que moi. It is irresistible; it is imperative. We must throw open our gates and admit it at any rate for the time being. Peirce: CP 5.582 Cross-Ref:†† 582. Thus it is that inquiry of every type, fully carried out, has the vital power of self-correction and of growth. This is a property so deeply saturating its inmost nature that it may truly be said that there is but one thing needful for learning the truth, and that is a hearty and active desire to learn what is true. If you really want to learn the truth, you will, by however devious a path, be surely led into the way of truth, at last. No matter how erroneous your ideas of the method may be at first, you will be forced at length to correct them so long as your activity is moved by that sincere desire. Nay, no matter if you only half desire it, at first, that desire would at length conquer all others, could experience continue long enough. But the more veraciously truth is described at the outset, the shorter by centuries will the road to it be. Peirce: CP 5.583 Cross-Ref:†† 583. In order to demonstrate that this is so, it is necessary to note what is essentially involved in the Will to Learn. The first thing that the Will to Learn supposes is a dissatisfaction with one's present state of opinion. There lies the secret of why it is that our American universities are so miserably insignificant. What have they done for the advance of civilization? What is the great idea or where is [the] single great man who can truly be said to be the product of an American university? The English universities, rotting with sloth as they always have, have nevertheless in the past given birth to Locke and to Newton, and in our time to Cayley, Sylvester, and Clifford. The German universities have been the light of the whole world. The medieval University of Bologna gave Europe its system of law. The University of Paris and that despised scholasticism took Abelard and made him into Descartes. The reason was that they were institutions of learning while ours are institutions for teaching. In order that a man's whole heart may be in teaching he must be thoroughly imbued with the vital importance and absolute truth of what he has to teach; while in order that he may have any measure of success in learning he must be penetrated with a sense of the unsatisfactoriness of his present condition of knowledge. The two attitudes are almost irreconcilable. But just as it is not the self-righteous man who brings multitudes to a sense of sin, but the man who is most deeply conscious that he is himself a sinner, and it is only by a sense of sin that men can escape its thraldom; so it is not the man, who thinks he knows it all, that can bring other men to feel their need of learning, and it is only a deep sense that one is miserably ignorant that can

spur one on in the toilsome path of learning. That is why, to my very humble apprehension, it cannot but seem that those admirable pedagogical methods, for which the American teacher is distinguished, are of little more consequence than the cut of his coat, that they surely are as nothing compared with that fever for learning that must consume the soul of the man who is to infect others with the same apparent malady. Let me say that of the present condition of Harvard I really know nothing at all except that I know the leaders of the department of philosophy to be all true scholars, particularly marked by eagerness to learn and freedom from dogmatism. And in every age, it can only be the philosophy of that age, such as it may be, which can animate the special sciences to any work that shall really carry forward the human mind to some new and valuable truth. Because the valuable truth is not the detached one, but the one that goes toward enlarging the system of what is already known. Peirce: CP 5.584 Cross-Ref:†† 584. The Inductive Method springs directly out of dissatisfaction with existing knowledge. The great rule of predesignation, which must guide it, is as much as to say that an induction to be valid must be prompted by a definite doubt or at least an interrogation; and what is such an interrogation but first, a sense that we do not know something; second, a desire to know it; and third, an effort -- implying a willingness to labor -- for the sake of seeing how the truth may really be. If that interrogation inspires you, you will be sure to examine the instances; while if it does not, you will pass them by without attention. Peirce: CP 5.585 Cross-Ref:†† 585. I repeat that I know nothing about the Harvard of today, but one of the things which I hope to learn during my stay in Cambridge is the answer to this question, whether the Commonwealth of Massachusetts has set up this university to the end that such young men as can come here may receive a fine education and may thus be able to earn handsome incomes, and have a canvas-back and a bottle of Clos de Vougeot for dinner -- whether this is what she is driving at -- or whether it is that, knowing that all America looks largely to sons of Massachusetts for the solutions of the most urgent problems of each generation, she hopes that in this place something may be studied out which shall be of service in the solutions of those problems. In short, I hope to find out whether Harvard is an educational establishment or whether it is an institution for learning what is not yet thoroughly known, whether it is for the benefit of the individual students or whether it is for the good of the country and for the speedier elevation of man into that rational animal of [which] he is the embryonic form. Peirce: CP 5.585 Cross-Ref:†† There is one thing that I am sure a Harvard education cannot fail to do, because it did that much even in my time, and for a very insouciant student; I mean that it cannot fail to disabuse the student of the popular notion that modern science is so very great a thing as to be commensurate with Nature and indeed to constitute of itself some account of the universe, and to show him that it is yet, what it appeared to Isaac Newton to be, a child's collection of pebbles gathered upon the beach -- the vast ocean of Being lying there unsounded. Peirce: CP 5.586 Cross-Ref:†† 586. It is not merely that in all our gropings we bump up against problems which we cannot imagine how to attack, why space should have but three dimensions, if it really has but three, why the Listing numbers which define its shape should all equal one, if they really do, or why some of them should be zero, as Listing himself

and many geometers think they are, if that be the truth, of why forces should determine the second derivative of the space rather than the third or fourth, of why matter should consist of about seventy distinct kinds, and all those of each kind apparently exactly alike, and these different kinds having masses nearly in arithmetical progression and yet not exactly so, of why atoms should attract one another at a distance in peculiar ways, if they really do, or if not what produced such vortices, and what gave the vortices such peculiar laws of attraction, of how or by what kind of influence matter came to be sifted out, so that the different kinds occur in considerable aggregations, of why certain motions of the atoms of certain kinds of protoplasm are accompanied by sensation, and so on through the whole list.†1 These things do indeed show us how superficial our science still is; but its littleness is made even more manifest when we consider within how narrow a range all our inquiries have hitherto lain. The instincts connected with the need of nutrition have furnished all animals with some virtual knowledge of space and of force, and made them applied physicists. The instincts connected with sexual reproduction have furnished all animals at all like ourselves with some virtual comprehension of the minds of other animals of their kind, so that they are applied psychists. Now not only our accomplished science, but even our scientific questions have been pretty exclusively limited to the development of those two branches of natural knowledge.†1 There may for aught we know be a thousand other kinds of relationship which have as much to do with connecting phenomena and leading from one to another, as dynamical and social relationships have. Astrology, magic, ghosts, prophecies, serve as suggestions of what such relationships might be. Peirce: CP 5.587 Cross-Ref:†† 587. Not only is our knowledge thus limited in scope, but it is even more important that we should thoroughly realize that the very best of what we, humanly speaking, know [we know] only in an uncertain and inexact way. Peirce: CP 5.587 Cross-Ref:†† Nobody would dream of contending that because the sun has risen and set every day so far, that afforded any reason at all for supposing that it would go on doing so to all eternity. But when I say that there is not the very slightest reason for thinking that no material atoms ever go out of existence or come into existence, there I fail to carry the average man with me; and I suppose the reason is, that he dimly conceives that there is some reason, other than the pure and simple induction, for holding matter to be ingenerable and indestructible. For it is plain that if it be a mere question of our weighings or other experiences, all that appears is that not more than one atom in a million or ten million becomes annihilated before the deficiency of mass is pretty certain to be balanced by another atom's being created. Now when we are speaking of atoms, a million or ten million is an excessively minute quantity. So that as far as purely inductive evidence is concerned we are very very far from being entitled to think that matter is absolutely permanent. If you put the question to a physicist his reply will probably be, as it certainly ought to be, that physicists only deal with such phenomena as they can either directly or indirectly observe, or are likely to become able to observe until there is some great revolution in science, and to that he will very likely add that any limitation upon the permanence of matter would be a purely gratuitous hypothesis without anything whatever to support it. Now this last part of the physicist's reply is, in regard to the order of considerations which he has in mind, excellent good sense. But from an absolute point of view, I think it leaves something out of account. Do you believe that the fortune of the Rothschilds will endure forever? Certainly not; because although they may be safe enough as far

as the ordinary causes go which engulf fortunes, yet there is always a chance of some revolution or catastrophe which may destroy all property. And no matter how little that chance may be, as far as this decade or this generation goes, yet in limitless decades and generations, it is pretty sure that the pitcher will get broken, at last. There is no danger, however slight, which in an indefinite multitude of occasions does not come as near to absolute certainty as probability can come. The existence of the human race, we may be as good as sure, will come to an end at last. For not to speak of the gradual operation of causes of which we know, the action of the tides, the resisting medium, the dissipation of energy, there is all the time a certain danger that the earth may be struck by a meteor or wandering star so large as to ruin it, or by some poisonous gas. That a purely gratuitous hypothesis should turn out to be true is, indeed, something so exceedingly improbable that we cannot be appreciably wrong in calling it zero. Still, the chance that out of an infinite multitude of gratuitous hypotheses an infinitesimal proportion, which may itself be an infinite multitude, should turn out to be true, is zero multiplied by infinity, which is absolutely indeterminate. That is to say we simply know nothing whatever about it. Now that any single atom should be annihilated is a gratuitous hypothesis. But there are, we may suppose, an infinite multitude of atoms, and a similar hypothesis may be made for each. And thus we return to my original statement that as to whether any finite number or even an infinite number of atoms are annihilated per year, that is something of which we are simply in a state of blank ignorance, unless we have found out some method of reasoning altogether superior to induction. If, therefore, we should detect any general phenomenon of nature which could very well be explained, not by supposing any definite breach of the laws of nature, for that would be no explanation at all, but by supposing that a continual breach of all the laws of nature, every day and every second, was itself one of the laws or habitudes of nature, there would be no power in induction to offer the slightest logical objection to that theory. But as long as we are aware of no such general phenomena tending to show such continual inexactitude in law, then we must remain absolutely without any rational opinion upon the matter pro or con. Peirce: CP 5.588 Cross-Ref:†† 588. There are various ways in which the natural cocksuredness and conceit of man struggles to escape such confession of total ignorance. But they seem to be all quite futile. One of the commonest, and at the same time the silliest, is the argument that God would for this or that excellent reason never act in such an irregular manner. I think all the men who talk like that must be near-sighted. For to suppose that any man who could see the moving clouds and survey a wide expanse of landscape and note its wonderful complexity, and consider how unimaginably small it all was in comparison to the whole face of the globe, not to speak of the millions of orbs in space, and who would not presume to predict what move Morphy or Steinitz might make in so simple a thing as a game of chess, should undertake to say what God would do, would seem to impeach his sanity. But if instead of its being a God, after whose image we are made, and whom we can, therefore, begin to understand, it were some metaphysical principle of Being, even more incomprehensible, whose action the man pretended to compute, that would seem to be a pitch of absurdity one degree higher yet. Peirce: CP 5.589 Cross-Ref:†† 589. People talk of a hypothesis where there is a vera causa. But in such cases the inference is not hypothetic but inductive. A vera causa is a state of things known to be present and known partially at least to explain the phenomena, but not known to

explain them with quantitative precision. Thus, when seeing ordinary bodies round us accelerated toward the earth's centre and seeing also the moon, which both in its albedo and its volcanic appearance altogether resembles stone, to be likewise accelerated toward the earth, and when finding these two accelerations are in the inverse duplicate ratios of their distances from that centre, we conclude that their nature, whatever it may be, is the same, we are inferring an analogy, which is a type of inference having all the strength of induction and more, besides.†1 For the sake of simplicity, I have said nothing about it in these lectures; but I am here forced to make that remark. Moreover, when we consider that all that we infer about the gravitation of the moon is a continuity between the terrestrial and lunar phenomena, a continuity which is found throughout physics, and when we add to that, the analogies of electrical and magnetical attractions, both of which vary inversely as the square of the distance, we plainly recognize here one of the strongest arguments of which science affords any example. Newton was entirely in the right when he said, Hypotheses non fingo.†2 It is they who have criticized the dictum whose logic is at fault. They are attributing an obscure psychological signification to force, or vis insita, which in physics only connotes a regularity among accelerations. Thus inferences concerning veræ causæ are inductions not retroductions, and of course have only such uncertainty and inexactitude as belong to induction. When I say that a reductive inference is not a matter for belief at all, I encounter the difficulty that there are certain inferences which, scientifically considered, are undoubtedly hypotheses and yet which practically are perfectly certain. Such for instance is the inference that Napoleon Bonaparte really lived at about the beginning of this century, a hypothesis which we adopt for the purpose of explaining the concordant testimony of a hundred memoirs, the public records of history, tradition, and numberless monuments and relics. It would surely be downright insanity to entertain a doubt about Napoleon's existence. A still better example is that of the translations of the cuneiform inscriptions which began in mere guesses, in which their authors could have had no real confidence. Yet by piling new conjectures upon former conjectures apparently verified, this science has gone on to produce under our very eyes a result so bound together by the agreement of the readings with one another, with other history, and with known facts of linguistics, that we are unwilling any longer to apply the word theory to it. You will ask me how I can reconcile such facts as these with my dictum that hypothesis is not a matter for belief. In order to answer this question I must first examine such inferences in their scientific aspect and afterwards in their practical aspect. The only end of science, as such, is to learn the lesson that the universe has to teach it. In Induction it simply surrenders itself to the force of facts. But it finds, at once -- I am partially inverting the historical order, in order to state the process in its logical order -- it finds I say that this is not enough. It is driven in desperation to call upon its inward sympathy with nature, its instinct for aid, just as we find Galileo at the dawn of modern science making his appeal to il lume naturale. But in so far as it does this, the solid ground of fact fails it. It feels from that moment that its position is only provisional. It must then find confirmations or else shift its footing. Even if it does find confirmations, they are only partial. It still is not standing upon the bedrock of fact. It is walking upon a bog, and can only say, this ground seems to hold for the present. Here I will stay till it begins to give way. Moreover, in all its progress, science vaguely feels that it is only learning a lesson. The value of Facts to it, lies only in this, that they belong to Nature; and Nature is something great, and beautiful, and sacred, and eternal, and real -- the object of its worship and its aspiration. It therein takes an entirely different attitude toward facts from that which Practice takes. For Practice, facts are the arbitrary forces with which it has to reckon and to wrestle.

Science, when it comes to understand itself, regards facts as merely the vehicle of eternal truth, while for Practice they remain the obstacles which it has to turn, the enemy of which it is determined to get the better. Science feeling that there is an arbitrary element in its theories, still continues its studies, confident that so it will gradually become more and more purified from the dross of subjectivity; but practice requires something to go upon, and it will be no consolation to it to know that it is on the path to objective truth -- the actual truth it must have, or when it cannot attain certainty must at least have high probability, that is, must know that, though a few of its ventures may fail, the bulk of them will succeed. Hence the hypothesis which answers the purpose of theory may be perfectly worthless for art. After a while, as Science progresses, it comes upon more solid ground. It is now entitled to reflect: this ground has held a long time without showing signs of yielding. I may hope that it will continue to hold for a great while longer. This reflection, however, is quite aside from the purpose of science. It does not modify its procedure in the least degree. It is extra-scientific. For Practice, however, it is vitally important, quite altering the situation. As Practice apprehends it, the conclusion no longer rests upon mere retroduction, it is inductively supported. For a large sample has now been drawn from the entire collection of occasions in which the theory comes into comparison with fact, and an overwhelming proportion, in fact all the cases that have presented themselves, have been found to bear out the theory. And so, says Practice, I can safely presume that so it will be with the great bulk of the cases in which I shall go upon the theory; especially as they will closely resemble those which have been well tried. In other words there is now reason to believe in the theory, for belief is the willingness to risk a great deal upon a proposition. But this belief is no concern of science, which has nothing at stake on any temporal venture but is in pursuit of eternal verities (not semblances to truth) and looks upon this pursuit, not as the work of one man's life, but as that of generation after generation, indefinitely. Thus those retroductive inferences which at length acquire such high degrees of certainty, so far as they are so probable, are not pure retroductions and do not belong to science, as such; while, so far as they are scientific and are pure retroductions, have no true probability and are not matters for belief. We call them in science established truths, that is, they are propositions into which the economy of endeavor prescribes that, for the time being, further inquiry shall cease.

Peirce: CP 5.590 Cross-Ref:†† §2. ON SELECTING HYPOTHESES †1

590. If we are to give the names of Deduction, Induction, and Abduction to the three grand classes of inference, then Deduction must include every attempt at mathematical demonstration, whether it relate to single occurrences or to "probabilities," that is, to statistical ratios; Induction must mean the operation that induces an assent, with or without quantitative modification, to a proposition already put forward, this assent or modified assent being regarded as the provisional result of a method that must ultimately bring the truth to light; while Abduction must cover all the operations by which theories and conceptions are engendered. Peirce: CP 5.591 Cross-Ref:†† 591. How is it that man ever came by any correct theories about nature? We know by Induction that man has correct theories; for they produce predictions that are

fulfilled. But by what process of thought were they ever brought to his mind? A chemist notices a surprising phenomenon. Now if he has a high admiration of Mill's Logic, as many chemists have, he will remember that Mill tells him that he must work on the principle that, under precisely the same circumstances, like phenomena are produced. Why does he then not note that this phenomenon was produced on such a day of the week, the planets presenting a certain configuration, his daughter having on a blue dress, he having dreamed of a white horse the night before, the milkman having been late that morning, and so on? The answer will be that in early days chemists did use to attend to some such circumstances, but that they have learned better. How have they learned this? By an induction. Very well, that induction must have been based upon a theory which the induction verified. How was it that man was ever led to entertain that true theory? You cannot say that it happened by chance, because the possible theories, if not strictly innumerable, at any rate exceed a trillion -- or the third power of a million; and therefore the chances are too overwhelmingly against the single true theory in the twenty or thirty thousand years during which man has been a thinking animal, ever having come into any man's head. Besides, you cannot seriously think that every little chicken, that is hatched, has to rummage through all possible theories until it lights upon the good idea of picking up something and eating it. On the contrary, you think the chicken has an innate idea of doing this; that is to say, that it can think of this, but has no faculty of thinking anything else. The chicken you say pecks by instinct. But if you are going to think every poor chicken endowed with an innate tendency toward a positive truth, why should you think that to man alone this gift is denied? If you carefully consider with an unbiassed mind all the circumstances of the early history of science and all the other facts bearing on the question, which are far too various to be specifically alluded to in this lecture, I am quite sure that you must be brought to acknowledge that man's mind has a natural adaptation to imagining correct theories of some kinds, and in particular to correct theories about forces, without some glimmer of which he could not form social ties and consequently could not reproduce his kind. In short, the instincts conducive to assimilation of food, and the instincts conducive to reproduction, must have involved from the beginning certain tendencies to think truly about physics, on the one hand, and about psychics, on the other. It is somehow more than a mere figure of speech to say that nature fecundates the mind of man with ideas which, when those ideas grow up, will resemble their father, Nature. Peirce: CP 5.592 Cross-Ref:†† 592. But if that be so, it must be good reasoning to say that a given hypothesis is good, as a hypothesis, because it is a natural one, or one readily embraced by the human mind. It must concern logic in the highest degree to ascertain precisely how far and under what limitations this maxim may be held. For of all beliefs, none is more natural than the belief that it is natural for man to err. The logician ought to find out what the relation is between these two tendencies. Peirce: CP 5.593 Cross-Ref:†† 593. It behooves a man first of all to free his mind of those four idols of which Francis Bacon speaks in the first book of the Novum Organum. So much is the dictate of Ethics, itself. But after that, what? Descartes, as you know, maintained that if a man could only get a perfectly clear and distinct idea †1 -- to which Leibniz added the third requirement that it should be adequate †2 -- then that idea must be true. But this is far too severe. For never yet has any man attained to an apprehension perfectly clear and distinct, let alone its being adequate; and yet I suppose that true ideas have been entertained. Ordinary ideas of perception, which Descartes thought

were most horribly confused, have nevertheless something in them that very nearly warrants their truth, if it does not quite so. "Seeing is believing," says the instinct of man. Peirce: CP 5.594 Cross-Ref:†† 594. The question is what theories and conceptions we ought to entertain. Now the word "ought" has no meaning except relatively to an end. That ought to be done which is conducive to a certain end. The inquiry therefore should begin with searching for the end of thinking. What do we think for? What is the physiological function of thought? If we say it is action, we must mean the government of action to some end. To what end? It must be something, good or admirable, regardless of any ulterior reason. This can only be the esthetically good. But what is esthetically good? Perhaps we may say the full expression of an idea?†1 Thought, however, is in itself essentially of the nature of a sign. But a sign is not a sign unless it translates itself into another sign in which it is more fully developed. Thought requires achievement for its own development, and without this development it is nothing. Thought must live and grow in incessant new and higher translations, or it proves itself not to be genuine thought. Peirce: CP 5.595 Cross-Ref:†† 595. But the mind loses itself in such general questions and seems to be floating in a limitless vacuity. It is of the very essence of thought and purpose that it should be special, just as truly as it is of the essence of either that it should be general. Yet it illustrates the point that the valuable idea must be eminently fruitful in special applications, while at the same time it is always growing to wider and wider alliances. Peirce: CP 5.596 Cross-Ref:†† 596. Classical antiquity was far too favorable to the sort of concept that was

fortis, et in se ipso totus, teres atque rotundus.†2

I often meet with such theories in philosophical books, especially in the works of theological students and of others who draw their ideas from antiquity. Such is the circular theory, which assumes itself and returns into itself -- the aristocratical theory which holds itself aloof from vulgar facts. Logic has not the least objection to such a view, so long as it maintains its self-sufficiency, keeps itself strictly to itself, as its nobility obliges it to do, makes no pretension of meddling with the world of experience, and does not ask anybody to assent to it. Peirce: CP 5.597 Cross-Ref:†† 597.†1 Auguste Comte, at the other extreme, would condemn every theory that was not "verifiable." Like the majority of Comte's ideas, this is a bad interpretation of a truth. An explanatory hypothesis, that is to say, a conception which does not limit its purpose to enabling the mind to grasp into one a variety of facts, but which seeks to connect those facts with our general conceptions of the universe, ought, in one sense, to be verifiable; that is to say, it ought to be little more than a ligament of numberless possible predictions concerning future experience, so that if they fail, it fails. Thus, when Schliemann entertained the hypothesis that there really had been a city of Troy and a Trojan War, this meant to his mind among other things that when he should come to make excavations at Hissarlik he would probably find

remains of a city with evidences of a civilization more or less answering to the descriptions of the Iliad, and which would correspond with other probable finds at Mycenae, Ithaca, and elsewhere. So understood, Comte's maxim is sound. Nothing but that is an explanatory hypothesis. But Comte's own notion of a verifiable hypothesis was that it must not suppose anything that you are not able directly to observe.†2 From such a rule it would be fair to infer that he would permit Mr. Schliemann to suppose he was going to find arms and utensils at Hissarlik, but would forbid him to suppose that they were either made or used by any human being, since no such beings could ever be detected by direct percept. He ought on the same principle to forbid us to suppose that a fossil skeleton had ever belonged to a living ichthyosaurus. This seems to be substantially the opinion of M. Poincaré at this day. The same doctrine would forbid us to believe in our memory of what happened at dinnertime today. I have for many years been an adherent of what is technically called Common Sense in philosophy, myself; and do not think that my Tychistic opinions conflict with that position; but I nevertheless think that such theories as that of Comte and Poincaré about verifiable hypotheses frequently deserve the most serious consideration; and the examination of them is never lost time; for it brings lessons not otherwise so easily learned. Of course with memory would have to go all opinions about everything not at this moment before our senses. You must not believe that you hear me speaking to you, but only that you hear certain sounds while you see before you a spot of black, white, and flesh color; and those sounds somehow seem to suggest certain ideas which you must not connect at all with the black and white spot. A man would have to devote years to training his mind to such habits of thought, and even then it is doubtful whether it would be possible. And what would be gained? If it would alter our beliefs as to what our sensuous experience is going to be, it would certainly be a change for the worse, since we do not find ourselves disappointed in any expectations due to common sense beliefs. If on the other hand it would not make any such difference, as I suppose it would not, why not allow us the harmless convenience of believing in these fictions, if they be fictions? Decidedly we must be allowed these ideas, if only as cement for the matter of our sensations. At the same time, I protest that such permission would not be at all enough. Comte, Poincaré, and Karl Pearson take what they consider to be the first impressions of sense, but which are really nothing of the sort, but are percepts that are products of psychical operations, and they separate these from all the intellectual part of our knowledge, and arbitrarily call the first real and the second fictions. These two words real and fictive bear no significations whatever except as marks of good and bad. But the truth is that what they call bad or fictitious, or subjective, the intellectual part of our knowledge, comprises all that is valuable on its own account, while what they mark good, or real, or objective, is nothing but the pretty vessel that carries the precious thought. Peirce: CP 5.598 Cross-Ref:†† 598. I can excuse a person who has lost a dear companion and whose reason is in danger of giving way under the grief, for trying, on that account, to believe in a future life. I can more than excuse him because his usefulness is at stake, although I myself would not adopt a hypothesis, and would not even take it on probation, simply because the idea was pleasing to me. Without judging others, I should feel, for my own part, that that would be a crime against the integrity of the reason that God has lent to me. But if I had the choice between two hypotheses, the one more ideal and the other more materialistic, I should prefer to take the ideal one upon probation, simply because ideas are fruitful of consequences, while mere sensations are not so; so that the idealistic hypothesis would be the more verifiable, that is to say, would predict

more, and could be put the more thoroughly to the test. Peirce: CP 5.598 Cross-Ref:†† Upon this same principle, if two hypotheses present themselves, one of which can be satisfactorily tested in two or three days, while the testing of the other might occupy a month, the former should be tried first, even if its apparent likelihood is a good deal less. Peirce: CP 5.599 Cross-Ref:†† 599. It is a very grave mistake to attach much importance to the antecedent likelihood †1 of hypotheses, except in extreme cases; because likelihoods are mostly merely subjective, and have so little real value, that considering the remarkable opportunities which they will cause us to miss, in the long run attention to them does not pay. Every hypothesis should be put to the test by forcing it to make verifiable predictions. A hypothesis on which no verifiable predictions can be based should never be accepted, except with some mark attached to it to show that it is regarded as a mere convenient vehicle of thought -- a mere matter of form. Peirce: CP 5.600 Cross-Ref:†† 600. In an extreme case, where the likelihood is of an unmistakably objective character, and is strongly supported by good inductions, I would allow it to cause the postponement of the testing of a hypothesis. For example, if a man came to me and pretended to be able to turn lead into gold, I should say to him, "My dear sir, I haven't time to make gold." But even then the likelihood would not weigh with me directly, as such, but because it would become a factor in what really is in all cases the leading consideration in Abduction, which is the question of Economy -- Economy of money, time, thought, and energy.†2 Peirce: CP 5.601 Cross-Ref:†† 601. It is Prof. Ernst Mach †3 who has done the most to show the importance in logic of the consideration of Economy although I had written a paper on the subject as early as 1878.†4 But Mach goes altogether too far. For he allows thought no other value than that of economizing experiences. This cannot for an instant be admitted. Sensation, to my thinking, has no value whatever except as a vehicle of thought. Peirce: CP 5.602 Cross-Ref:†† 602. Proposals for hypotheses inundate us in an overwhelming flood, while the process of verification to which each one must be subjected before it can count as at all an item, even of likely knowledge, is so very costly in time, energy, and money -- and consequently in ideas which might have been had for that time, energy, and money, that Economy would override every other consideration even if there were any other serious considerations. In fact there are no others. For abduction commits us to nothing. It merely causes a hypothesis to be set down upon our docket of cases to be tried. Peirce: CP 5.603 Cross-Ref:†† 603. I shall be asked, Do you really mean to say that we ought not to adopt any opinion whatever as an opinion until it has sustained the ordeal of furnishing a prediction that has been verified? Peirce: CP 5.603 Cross-Ref:†† In order to answer that question, it will be requisite to inquire how an abduction can be justified, here understanding by abduction any mode or degree of acceptance of a proposition as a truth, because a fact or facts have been ascertained

whose occurrence would necessarily or probably result in case that proposition were true. The abduction so defined amounts, you will remark, to observing a fact and then professing to say what idea it was that gave rise to that fact. One would think a man must be privy to the counsels of the Most High so to presume. The only justification possible, other than some such positive fact which would put quite another color upon the matter, is the justification of desperation. That is to say, that if he is not to say such things, he will be quite unable to know anything of positive fact. Peirce: CP 5.603 Cross-Ref:†† In a general way, this justification certainly holds. If man had not had the gift, which every other animal has, of a mind adapted to his requirements, he not only could not have acquired any knowledge, but he could not have maintained his existence for a single generation. But he is provided with certain instincts, that is, with certain natural beliefs that are true. They relate in part to forces, in part to the action of minds. The manner in which he comes to have this knowledge seems to me tolerably clear. Certain uniformities, that is to say certain general ideas of action, prevail throughout the universe, and the reasoning mind is [it]self a product of this universe. These same laws are thus, by logical necessity, incorporated in his own being. For example, what we call straight lines are nothing but one out of an innumerable multitude of families of nonsingular lines such that through any two points there is one and one only. The particular family of lines called straight has no geometrical properties that distinguish it from any other of the innumerable families of lines of which there is one and one only through any two points. It is a law of dynamics that every dynamical relation between two points, no third point being concerned, except by combinations of such pairs, is altogether similar, except in quantity, to every such dynamical relation between any other two points on the same ray, or straight line. It is a consequence of this that a ray or straight line is the shortest distance between two points; whence, light appears to move along such lines; and that being the case, we recognize them by the eye, and call them straight. Thus, the faculty of sight naturally causes us to assign great prominence to such lines; and thus when we come to form a hypothesis about the motion of a particle left uninfluenced by any other, it becomes natural for us to suppose that it moves in a straight line. The reason this turns out true is, therefore, that this first law of motion is a corollary from a more general law which, governing all dynamics, governs light, and causes the idea of straightness to be a predominant one in our minds. Peirce: CP 5.604 Cross-Ref:†† 604. In this way, general considerations concerning the universe, strictly philosophical considerations, all but demonstrate that if the universe conforms, with any approach to accuracy, to certain highly pervasive laws, and if man's mind has been developed under the influence of those laws, it is to be expected that he should have a natural light, or light of nature, or instinctive insight, or genius, tending to make him guess those laws aright, or nearly aright. This conclusion is confirmed when we find that every species of animal is endowed with a similar genius. For they not only one and all have some correct notions of force, that is to say, some correct notions, though excessively narrow, of phenomena which we, with our broader conceptions, should call phenomena of force, and some similarly correct notions about the minds of their own kind and of other kinds, which are the two sufficient cotyledons of all our science, but they all have, furthermore, wonderful endowments of genius in other directions. Look at the little birds, of which all species are so nearly identical in their physique, and yet what various forms of genius do they not display in modelling their nests? This would be impossible unless the ideas that are naturally

predominant in their minds were true. It would be too contrary to analogy to suppose that similar gifts were wanting to man. Nor does the proof stop here. The history of science, especially the early history of modern science, on which I had the honor of giving some lectures in this hall some years ago,†1 completes the proof by showing how few were the guesses that men of surpassing genius had to make before they rightly guessed the laws of nature. . . .

Peirce: CP 5.605 Cross-Ref:†† APPENDIX

§ 1. KNOWLEDGE †1

605. This word is used in logic in two senses: (1) as a synonym for Cognition, and (2), and more usefully, to signify a perfect cognition, that is, a cognition fulfilling three conditions: first, that it holds for true a proposition that really is true; second, that it is perfectly self-satisfied and free from the uneasiness of doubt; third, that some character of this satisfaction is such that it would be logically impossible that this character should ever belong to satisfaction in a proposition not true. Peirce: CP 5.606 Cross-Ref:†† 606. Knowledge is divided, firstly, according to whatever classification of the sciences is adopted. Thus, Kantians distinguish formal and material knowledge. Secondly, knowledge is divided according to the different ways in which it is attained, as into immediate and mediate knowledge. Immediate knowledge is a cognition, or objective modification of consciousness, which is borne in upon a man with such resistless force as to constitute a guarantee that it (or a representation of it) will remain permanent in the development of human cognition. Such knowledge is, if its existence be granted, either borne in through an avenue of sense, external or internal, as a percept of an individual, or springs up within the mind as a first principle of reason or as a mystical revelation. Mediate knowledge is that for which there is some guarantee behind itself, although, no matter how far criticism be carried, simple evidency, or direct insistency, of something has to be relied upon. The external guarantee rests ultimately either upon authority, i.e., testimony, or upon observation. In either case mediate knowledge is attained by Reasoning, which see for further divisions.†2 It is only necessary to mention here that the Aristotelians distinguished knowledge {hoti}, or of the facts themselves, and knowledge {dioti}, or of the rational connection of facts, the knowledge of the how and why (cf. the preceding topic). They did not distinguish between the how and the why, because they held that knowledge {dioti} is solely produced by Syllogism †1 in its greatest perfection, as demonstration. The term empirical knowledge is applied to knowledge, mediate or immediate, which rests upon percepts; while the terms philosophical and rational knowledge are applied to knowledge, mediate or immediate, which rests chiefly or wholly upon conclusions or revelations of reason. Thirdly, knowledge is divided, according to the character of the immediate object, into apprehensive and judicative knowledge, the former being of a percept, image, or Vorstellung, the latter of the

existence or non-existence of a fact. Fourthly, knowledge is divided, according to the manner in which it is in the mind, into actual, virtual, and habitual knowledge. See Scotus, Opus Oxoniense, lib. I, dist. iii. quest. 2, paragraph beginning "Loquendo igitur." Fifthly, knowledge is divided according to its end, into speculative and practical.

Peirce: CP 5.607 Cross-Ref:†† §2. REPRESENTATIONISM †2

607. The doctrine that percepts stand for something behind them. Peirce: CP 5.607 Cross-Ref:†† In a certain sense it must be admitted, even by presentationists, that percepts only perform the function of conveying knowledge of something else. That is to say, they have to be combined and generalized to become useful knowledge; so that they may be said to represent their own generalizations. In this, representationists and presentationists may agree. But the dispute between them consists in this, that the representationist regards the percept in the light of testimony or a picture, from which by inference, or a mental act analogous to inference, the hidden cause of the percept may become known; while the presentationist holds that perception is a two-sided consciousness in which the percept appears as forcibly acting upon us, so that in perception the consciousness of an active object and of a subject acted on are as indivisible as, in making a muscular effort, the sense of exertion is one with and inseparable from the sense of resistance. The representationist would not allow that there is any bilateral consciousness even in the latter sense, regarding the bilaterality as a quasi-inference, or product of the mind's action; while the presentationist insists that there is nothing intellectual or intelligible in this duality. It is, he says, a hard fact experienced but never understood. A representationist will naturally regard the theory that everything in the outward world is atoms, their masses, motions, and energy, as a statement of the real fact which percepts represent. The presentationist, on the other hand, will more naturally regard it as a formula which is fitted to sum up and reconcile the percepts as the only ultimate facts. These are, however, merely different points of view in which neither ought to find anything absolutely contrary to his own doctrine.

Peirce: CP 5.608 Cross-Ref:†† §3. ULTIMATE †1

608. (1) Last in a series; especially in a series of purposes each, except the last, subsidiary to an ulterior one following it in the arrangement considered, or of actions each of which, except the last, leads to the performance of another. Peirce: CP 5.608 Cross-Ref:†† Thus, the phrase ultimate signification implies that a sign determines another sign of the same object, and this another; and so on until something is reached which is a sign only for itself. Ultimate fact implies that there is a series of facts each explicable by the one following it, until a fact is reached utterly inexplicable. (Cf.

Hamilton's Reid, Note A, V, ii 6, et seq.) Peirce: CP 5.609 Cross-Ref:†† 609. (2) Applied also to the limiting state of an endless series of states which approach indefinitely near to the limiting state, and on the whole nearer and nearer, without necessarily ever reaching it; although the word ultimate does not imply a denial of actual attainment. Peirce: CP 5.609 Cross-Ref:†† Thus, it has been held that a real object is that which will be represented in the ultimate opinion about it. This implies that a series of opinions succeed one another, and that it is hoped that they may ultimately tend more and more towards some limiting opinion, even if they do not reach and rest in a last opinion. Cf. Truth and Error, Logical [Bk. III, ch. 5, §3].

Peirce: CP 5.610 Cross-Ref:†† §4. MR. PETERSON'S PROPOSED DISCUSSION †1

610. Very valuable ideas ofttimes appear so obvious, when once set forth, that high laudation of their inventors would invite ridicule. Such, we are told, was the notion that obseded C. Colombo, and such is Mr. Peterson's proposal †2 to start in The Monist a discussion of philosophical terminology. It may be a very simple proposal, but nobody, as far as one careful reader of The Monist remembers, had made it before; and its utility to students of phenomenology, normative science, and metaphysics will have a high coefficient in its proportionality to the advantage they take of it. Duty calls upon us to contribute, each one what he can that will be useful, whether in the way of question or in that of answer. It seems likely that in my lifetime of study I may have learned something of the way to investigate questions such as Mr. Peterson puts; and if so, here is an opportunity to be of aid to other students. Peirce: CP 5.611 Cross-Ref:†† 611. Experience, the first term concerning which Mr. Peterson asks for light, is somewhat remarkable for having been employed as nearly as possible in the same sense from Polus the Acragentine (i.e. native of Girgenti) sophist down to Avenarius and Haeckel. As my first step in investigating its meaning, I should look out its equivalent empeiria,†P1 in Bonitz's Index Aristotelicus. For every serious student of philosophy ought to be able to read the common dialect of Greek at sight, and needs on his shelves the Berlin Aristotle, in the fifth volume of which is that index. On looking out empeiria there, what first strikes one is that it is not a very common word with Aristotle, nor yet an unusual one, since Bonitz cites something over a dozen passages in which it occurs. The first (Post. Anal. II, xix) runs: "From sense are engendered memories, and from multiplied memory of the same thing is engendered experience; for many memories make up a single experience." Waitz (Organon, II, 429) has a minute note on this passage. Another passage to which the Index refers (Nic. Ethics, VI, viii) is thus translated by Stewart in his valuable "Notes" on the work: "If we . . . ask why a boy may be a mathematician, but cannot understand philosophy or natural science, we find that it is because the truths of mathematics are abstract" [a bad explanation but that does not affect the evidence as to the meaning of empeiria], "whereas the principles of philosophy and natural science are reached

through long experience. A boy does not realize the meaning of the principles of philosophy and natural science, but merely repeats by rote the formulæ used to express them." In the Politics (A, xi) Aristotle remarks that theorizing is free, while experience is necessitated, and goes on to speak of experience with live stock, etc. In another place in the Politics (E, ix) he says that the military commander of greatest experience in strategy is to be preferred, even though his habit of peculation be known; while for the chief of police, or for a treasurer, experience is of no account in comparison with integrity. But the cynosural passage is the first chapter of Book A of the Metaphysics; and here he remarks (as he likewise does in the Ethics), that experience is a knowledge (gnosis)†P1 of singulars. Therein Aristotle's language differs from that of the Socrates of Plato, with whom empeiria is the skill that results from long dealings with any matter. Aristotle never intended to say that there is no other cognition of singulars than in experience; for that would directly contradict his doctrine that experience is a mass of memories relating to the same subject. His remark was, however, understood in the Middle Ages to be a definition of experience, and was repeated as such, a blunder that was not so unnatural as it would have been if the scholastic doctors had dealt with direct experience. The teachings of the Aristotelic Index having been exhausted, I turn to Harper's Latin Lexicon, which informs me that no writer of the Golden Age used experientia in the general sense, though that acception became common in the Silver Age, especially with Tacitus. The next work that I personally should consult would be my own notes collected during more than forty years. I always carry a pad of the size of a Post Card, of thick papers (50 in a pad, enough to last for two days, at least); and on these I note whatever elements of experience may reach me.†1 I keep these in drawers and boxes like the card catalogue of a library. I arrange and rearrange them from time to time. It is a treasure more valuable than a policy of insurance. I probably have near two hundred thousand such notes. But in order to bring what I have to say to a close, I will quote from the definition of experience given by the father of modern experiential philosophy, Dr. John Locke. In the Essay concerning Humane †P1 Understanding, (II, i, 2) we read (and the italics are in the original): "Whence has [the mind] all the materials of reason and knowledge? To this I answer, in one word, from experience: in that all our knowledge is founded, and from that it ultimately derives itself. Our observation employed either about external sensible objects, or about the internal operations of our minds, perceived and reflected on by ourselves, is that which supplies our understanding with all the materials of thinking." This definition so formally stated, by such an authority, quite peerless for our present purpose, should be accepted as definitive and as a landmark that it would be a crime to displace or disturb. For in order that philosophy should become a successful science, it must, like biology, have its own vocabulary; and as in biology, it must be the rule that whoever wishes to introduce a new concept is to invent a new word to express it. This is no suggestion of the moment.†2 I am, for my humble part, maturely convinced that philosophy will never be upon the road to sound results until we dismiss our affection for old words and our dislike of newfangled words, and make its vocabulary over after the fashion of taxonomic zoölogy and botany. I limit my recommendation to technical terms; for I can pretend to no competence to give advice about belles-lettres. Yet even there I perceive that people read old authors, and admire them for saying what they never meant to say; because the modern readers forget that two or three centuries ago words still familiar suggested quite different ideas from those the same words now suggest. Peirce: CP 5.612 Cross-Ref:††

612. But somebody may object that Locke's definition is vague, being founded on a misconception of the nature of perception. Suppose, the objector will say, that a newborn male infant were to be brought up among a colony of men on a desert island, without ever having seen a woman and barely having heard of such a creature. Suppose that, arrived at the age of twenty, he were to meet on the beach a Pacific Island woman who had swum over from another island. Would not the irresistible, the only possible cognition he could have of this creature be strongly colored by his own instincts? It would be the ineluctable result of "observation employed concerning an external sensible object." The word "experience," however, is employed by Locke chiefly to enable him to say that human cognitions are inscribed by the individual's life-history upon a tabula rasa, and are not, like those of the lower animals, gifts of inborn instinct. His definition is vague for the reason that he never realized how important the innate element of our directest perceptions really is. Peirce: CP 5.613 Cross-Ref:†† 613. To such an objector I might say, My dear fellow, you must be joking; for under the guise of an objection you reinforce what I was saying with a new argument for restricting the use of the word "experience" to the expression of that vague idea which Locke so well defines. You make it plain that a distinct word is wanted, or rather two distinct words, to express the two more precise concepts which you suggest. The idea of the word "experience" was to refer to that which is forced upon a man's recognition, will-he nill-he, and shapes his thoughts to something quite different from what they naturally would have been. But the philosophers of experience, like many of other schools, forget to how great a degree it is true that the universe is all of a piece, and that we are all of us natural products, naturally partaking of the characteristics that are found everywhere through nature. It is in some measure nonsensical to talk of a man's nature as opposed to what perceptions force him to think. True, man continually finds himself resisted, both in his active desires and in that passive inertia of thought which causes any new phenomenon to give him a shock of surprise. You may think of an element of knowledge which thus resists his superficial tendencies; but to express precisely that idea you must have a new word: it will not answer the purpose to call it experience. You may also reflect that every man's environment is in some measure unfavorable to his development; and so far as this affects his cognitive development, you have there an element that is opposed to the man's nature. But surely the word experience would be ill-chosen to express that. Peirce: CP 5.614 Cross-Ref:†† 614. But I am encroaching far too much upon the space of this number, and am taking too much advantage of our good editor's indulgence. I did wish to consider what element of his philosophy Comte had specially in mind in christening it Positive. He plainly meant that it should be unlike the metaphysical thought which kneads over and over what we know already, and would be like the sort of material which is furnished by a microscope or by an archæologist's spade. I hope Mr. Peterson's suggestion may bring a whole crop of fruit.

CHARLES SANTIAGO †1 SANDERS PEIRCE.

Peirce: CP 5.1 Fn 1 p 1 †1 From "Pragmatic and Pragmatism," Dictionary of Philosophy and Psychology, ed. by J.M. Baldwin, The Macmillan Co., New York; vol. 2, pp. 321-322 (1902). Peirce: CP 5.1 Fn 2 p 1 †2 Anthropologie in pragmatischer Hinsicht, Vorrede. Peirce: CP 5.2 Fn 3 p 1 †3 This paragraph was contributed by William James. Peirce: CP 5.3 Fn 4 p 1 †4 See 402; 526n. Peirce: CP 5.3 Fn 1 p 2 †1 New World, pp. 327-47; reprinted in 1897 in The Will to Believe, and Other Essays in Popular Philosophy. Peirce: CP 5.3 Fn 2 p 2 †2 In 1898; University of California Chronicle; reprinted in 1920 in Collected Essays and Reviews, ed. by R.B. Perry. Peirce: CP 5.3 Fn 3 p 2 †3 See 433 and vol. 1, bk. IV, ch. 4. Peirce: CP 5.3 Fn 4 p 2 †4 See 402n3, 429. Peirce: CP 5.4 Fn 1 p 3 †1 See vol. 6, bk. I, B. Peirce: CP 5.5 Fn 2 p 3 †2 From "Pragmatism" [1], c. 1905. Peirce: CP 5.5 Fn 3 p 3 †3 See Kritik der reinen Vernunft, A832, B860. Peirce: CP 5.5 Fn 4 p 3 †4 See 1.294ff. Peirce: CP 5.5 Fn 5 p 3 †5 Cf. 27, 469ff. Peirce: CP 5.7 Fn 1 p 4 †1 Cf. 1.303ff; 41ff. Peirce: CP 5.7 Fn 2 p 4 †2 Cf. 1.322ff; 45ff. Peirce: CP 5.7 Fn 3 p 4 †3 Cf. 1.337ff; 59ff. Peirce: CP 5.7 Fn 4 p 4 †4 Cf. 3.491. Peirce: CP 5.8 Fn 1 p 5

†1 "Linear Associative Algebra," §1, American Journal of Mathematics, vol. 4, pp. 97-229 (1881). Peirce: CP 5.8 Fn 2 p 5 †2 See also 4.233ff. Peirce: CP 5.11 Fn 1 p 6 †1 From "Pragmatism (Editor [3])," c. 1906. For the remainder, see bk. III, ch. 1. Peirce: CP 5.12 Fn 1 p 7 †1 To judge from the Letters of William James, vol. 2, p. 233, there was a meeting of this club in the autumn of 1874. Peirce: CP 5.12 Fn 2 p 7 †2 A friend of William James, who worked with Mr. Justice Holmes on Kent's Commentaries. Peirce: CP 5.12 Fn 3 p 7 †3 Cf. The Emotions and the Will, ch. 11, p. 505, 3d ed. (1875). Peirce: CP 5.12 Fn 1 p 8 †1 See 64. Peirce: CP 5.13 Fn 2 p 8 †2 Author of a textbook on rhetoric in use at Harvard College in Peirce's day. Peirce: CP 5.13 Fn P1 Para 1/6 p 9 Cross-Ref:†† †P1 Pragmatism. It is a singular instance of that over-modesty and unyielding self-underestimate on my part of which I am so justly proud as my principal claim to distinction that I should have omitted pragmatism, my own offspring, with which the world resounds. See Baldwin's Dictionary where is my original definition of 1878 and an exegesis, not very deep, of William James. Pragmatism is a method in philosophy. Philosophy is that branch of positive science (i.e., an investigating theoretical science which inquires what is the fact, in contradistinction to pure mathematics which merely seeks to know what follows from certain hypotheses) which makes no observations but contents itself with so much of experience as pours in upon every man during every hour of his waking life. The study of philosophy consists, therefore, in reflexion, and pragmatism is that method of reflexion which is guided by constantly holding in view its purpose and the purpose of the ideas it analyzes, whether these ends be of the nature and uses of action or of thought. Peirce: CP 5.13 Fn P1 Para 2/6 p 9 Cross-Ref:†† ". . . the whole subsequent argument has already had its main lines mapped out by our introductory discussion of that Weltanschauung which Professor James has called pragmatism." -- F.C.S. Schiller (in Personal Idealism, edited by Henry Cecil Sturt, 1902, p. 63). Peirce: CP 5.13 Fn P1 Para 3/6 p 9 Cross-Ref:†† The passage of Professor James here alluded to is as follows: ". . . Mr. Charles Sanders Peirce has rendered thought a service by disentangling from the particulars of its application the principle by which these men were instinctively guided, and by singling it out as fundamental and giving to it a Greek name. He calls it the principle of pragmatism." -- William James, The Varieties of Religious Experience, 1902, p.

444. Peirce: CP 5.13 Fn P1 Para 4/6 p 9 Cross-Ref:†† It will be seen [from the original statement] that pragmatism is not a Weltanschauung but is a method of reflexion having for its purpose to render ideas clear. Peirce: CP 5.13 Fn P1 Para 5/6 p 9 Cross-Ref:†† Pragmatistic, a., Having the character of pragmatism, as a method in philosophy. Peirce: CP 5.13 Fn P1 Para 6/6 p 9 Cross-Ref:†† Pragmatist, n., in philosophy, one who professes to practice pragmatism. Thus Schiller of Oxford, author of Riddles of the Sphinx, is a pragmatist, although he does not very thoroughly understand the nature of pragmatism. -- From Peirce's personal interleaved copy of the Century Dictionary, c. 1902. Peirce: CP 5.14 Fn 1 p 11 †1 Delivered at Cambridge, Massachusetts, March 26 to May 17, 1903; James described them in his Pragmatism, p. 5, as "flashes of brilliant light relieved against Cimmerian darkness." He states that they were delivered at the Lowell Institute; the available records, however, show that they were given in Sever Hall, Harvard, under the auspices of the Harvard department of philosophy. Peirce: CP 5.21 Fn 1 p 18 †1 Cf. 2.661f. Peirce: CP 5.22 Fn P1 p 19 Cross-Ref:†† †P1 I.e., he receives (-2)n2 cents if n tails intervene between two successive heads. Peirce: CP 5.23 Fn 1 p 20 †1 See 19. Peirce: CP 5.24 Fn 2 p 20 †2 Cf. 1.88, 2.647. Peirce: CP 5.27 Fn 1 p 21 †1 See 394ff. Peirce: CP 5.29 Fn 1 p 22 †1 Cf. 2.334, 2.435ff. Peirce: CP 5.30 Fn 2 p 22 †2 Cf. 2.332ff, 3.432f. Peirce: CP 5.32 Fn 1 p 23 †1 Cf. 3.203f, 539, 541. Peirce: CP 5.34 Fn 1 p 24 †1 Cf. 108f., 1.191, 1.574, 1.611f. Peirce: CP 5.36 Fn 1 p 25 †1 See his Grammar of Science, Introduction, pp. 26-27, where he seems to say that society must not allow a bad stock to perpetuate itself.

Peirce: CP 5.36 Fn 2 p 25 †2 Henry Rutgers Marshall's Instinct and Reason, p. 569, Macmillan Co. (1898). Peirce: CP 5.37 Fn 1 p 27 †1 See vol. 1, bk. III for a detailed study of phenomenology. Peirce: CP 5.38 Fn 2 p 27 †2 Cf. 43, 1.525. Peirce: CP 5.40 Fn 1 p 28 †1 See 4.232f. Peirce: CP 5.41 Fn 1 p 29 †1 Second draught. On the first page Peirce wrote "This won't do; it will have to be rewritten"; but no later draught of this part has been found. The third draught is given in 59-65. Peirce: CP 5.41 Fn 2 p 29 †2 Cf. 1.300ff. Peirce: CP 5.42 Fn 3 p 29 †3 Cf. 1.43. Peirce: CP 5.45 Fn 1 p 32 †1 Cf. 1.322ff. Peirce: CP 5.45 Fn 2 p 32 †2 See his Philosophiæ Naturalis Principia Mathematica, liber I, def. IV. Peirce: CP 5.45 Fn 1 p 33 †1 Cf. 1.321, 3.363, 3.527, 4.157. Peirce: CP 5.47 Fn 1 p 34 †1 Cf. 1.121, 1.316, 2.750. Peirce: CP 5.47 Fn P1 p 34 Cross-Ref:†† †P1 I would not have anybody accept any doctrine of logic simply because minute and thorough criticism has resulted in making me perfectly confident of its truth. But I will not allow this scruple to prevent my saying that for my part -- who am characterized in some of the books as a sceptic in philosophy and have even been called a modern Hume ["David Hume Redivivus," Pt. I of "Mr. Charles S. Peirce's Onslaught on the Doctrine of Necessity" by Paul Carus in The Monist, vol. 2, pp. 560ff.] -- I have after long years of the severest examination become fully satisfied that, other things being equal, an anthropomorphic conception, whether it makes the best nucleus for a scientific working hypothesis or not, is far more likely to be approximately true than one that is not anthropomorphic. Suppose, for example, it is a question between accepting Telepathy or Spiritualism. The former I dare say is the preferable working hypothesis because it can be more readily subjected to experimental investigation. But as long as there is no reason for believing it except phenomena that Spiritualism is equally competent to explain, I think Spiritualism is much the more likely to be approximately true, as being the more anthropomorphic and natural idea; and in like manner, as between an old-fashioned God and a modern patent Absolute, recommend me to the anthropomorphic conception if it is a question

of which is the more likely to be about the truth. [See vol. 6, bk. II, chs. 4 and 7.] Peirce: CP 5.52 Fn 1 p 38 †1 Cf. 1.334. Peirce: CP 5.54 Fn 2 p 38 †2 Cf. 115ff, 151ff, 4.539, 4.541. Peirce: CP 5.57 Fn 1 p 39 †1 57 and 58 occur as part of a digression at the end of "Lecture IV." That part of the notebook repeats much of the foregoing, and, with the exception of what follows, is not being published. Peirce: CP 5.59 Fn 1 p 40 †1 Third draught. Cf. 1.337ff. Peirce: CP 5.60 Fn 1 p 41 †1 Cf. 1.635. Peirce: CP 5.64 Fn 1 p 43 †1 But see Wright's Philosophical Discussions, edited by C.E. Norton (1877). Peirce: CP 5.64 Fn 1 p 44 †1 Asa Gray, the famous Harvard botanist. Peirce: CP 5.66 Fn 1 p 47 †1 There were two draughts of this lecture. It is difficult to determine which is the final one. The following is from version "b." Peirce: CP 5.66 Fn 2 p 47 †2 Cf. 1.527ff. Peirce: CP 5.67 Fn 3 p 47 †3 Cf. 4.218ff. Peirce: CP 5.71 Fn P1 p 49 Cross-Ref:†† †P1 This gives an idea of the second degree of degenerate Thirdness. Those of you who have read Professor Royce's Supplementary Essay [in The World and the Individual, vol. 1, p. 505, n. 1] will have remarked that he avoids this result, which does not suit his philosophy, by not allowing his map to be continuous. But to exclude continuity is to exclude what is best and most living in Hegel -- from the alternative "a" version. Peirce: CP 5.71 Fn 1 p 50 †1 See vol. 7. Peirce: CP 5.73 Fn 2 p 50 †2 Cf. vol 2, bk. II, ch. 2 and ch. 3. Peirce: CP 5.75 Fn 1 p 52 †1 See "Aesthetics in Washington," in A Memorial of Horatio Greenough, by Henry T. Tuckerman, p. 82 (1853). Peirce: CP 5.77 Fn P1 Para 1/3 p 52 Cross-Ref:†† †P1 Grant me that the three categories of Firstness, Secondness, and Thirdness, or Quality, Reaction, and Representation, have in truth the enormous

importance for thought that I attribute to them, and it would seem that no division of theories of metaphysics could surpass in importance a division based upon the consideration of what ones of the three categories each of different metaphysical systems have fully admitted as real constituents of nature. Peirce: CP 5.77 Fn P1 Para 2/3 p 52 Cross-Ref:†† It is, at any rate, a hypothesis easy to try; and the exact logic of hypothesis allots great weight to that consideration. There will be then these seven possible classes: i. Nihilism, so-called, and idealistic sensualism. ii. The doctrine of [Wincenty] Lutoslawski and his unpronounceable master [Mickiewicz]. iii. Hegelianism of all shades. ii iii. Cartesianism of all kinds, Leibnizianism, Spinozism, and the metaphysics of the physicists of today. i iii. Berkeleyanism. i ii. Ordinary Nominalism. ii iii. The metaphysics that recognizes all the categories. It ought to be subdivided, but I shall not stop to consider its subdivisions. It embraces Kantism, Reid's Philosophy, and the Platonic philosophy of which Aristotelianism is a special development. Peirce: CP 5.77 Fn P1 Para 3/3 p 53 Cross-Ref:†† A great variety of thinkers call themselves Aristotelians, even the Hegelians, on the strength of special agreements. No modern philosophy, or very little, has any real right to the title. I should call myself an Aristotelian of the scholastic wing, approaching Scotism, but going much further in the direction of scholastic realism. -From the beginning of "Lecture IV." Peirce: CP 5.82 Fn 1 p 54 †1 82-87 are from the "a" version; 88-93 follow 81 after an unpublished section which is a duplication of most of 82-87, 2.283f. and 3.423f. Peirce: CP 5.84 Fn 1 p 58 †1 See 3.641, 4.51f, 4.85. Peirce: CP 5.85 Fn 2 p 58 †2 See 3.63, 3.421f, 3.468ff. Peirce: CP 5.85 Fn 3 p 58 †3 Logik, §3, 1; see also 2.19-20, 2.151ff. Peirce: CP 5.93 Fn 1 p 64 †1 In the manuscript, what is here published follows shortly after the note to 77 in Lecture III. Peirce: CP 5.102 Fn 1 p 67 †1 Cf. 151ff. Peirce: CP 5.102 Fn 2 p 67

†2 Cf. 2.367. Peirce: CP 5.103 Fn 3 p 67 †3 See 3.547f. Peirce: CP 5.106 Fn 1 p 69 †1 See 119. Peirce: CP 5.108 Fn 1 p 70 †1 See 2.186f. Peirce: CP 5.108 Fn 2 p 70 †2 Cf. 34ff, 440. Peirce: CP 5.110 Fn 3 p 70 †3 See 2.39ff. Peirce: CP 5.111 Fn 1 p 71 †1 Introduction to Ethics by T.S. Jouffroy, trans. by William H. Channing. Peirce: CP 5.111 Fn 2 p 71 †2 Dr. James Walker, president of Harvard University, and professor of moral and intellectual philosophy. Peirce: CP 5.111 Fn 3 p 71 †3 Probably The Elements of Morality, including Polity. Peirce: CP 5.112 Fn 1 p 72 †1 Cf. 1.333. Peirce: CP 5.115 Fn 1 p 73 †1 Cf. 151ff, 568, 4.539f. Peirce: CP 5.118 Fn 1 p 75 †1 §9 of ch. 2, bk. III, vol. 1 follows here in the ms., but apparently was not read. Peirce: CP 5.119 Fn 1 p 76 †1 Cf. vol. 6, bk. I, A. Peirce: CP 5.120 Fn 1 p 77 †1 The third and final draught; cf. vol. 1, bk. IV. Peirce: CP 5.120 Fn 2 p 77 †2 Cf. vol. 1, bk. II, ch. 2, §5. Peirce: CP 5.120 Fn 3 p 77 †3 See 61, and 1.126ff. Peirce: CP 5.125 Fn 1 p 79 †1 Théorie mathématique des effets du jeu de billard, G.G. Coriolis, Paris (1835). Peirce: CP 5.126 Fn 1 p 80 †1 See 1.247f, 4.239ff. Peirce: CP 5.128 Fn 1 p 81

†1 See Oeuvres de Descartes, t. III, lettre 183, A. et P. Tannery, Paris (1897-1910). Peirce: CP 5.129 Fn 1 p 82 †1 Cf. 1.573ff, 2.196f. Peirce: CP 5.130 Fn 2 p 82 †2 See 2.186f. Peirce: CP 5.138 Fn 1 p 87 †1 See 2.317n, 2.393. Peirce: CP 5.144 Fn 1 p 89 †1 Vol. 2, bk. III, ch. 2, Part III. Peirce: CP 5.144 Fn 1 p 90 †1 Chapter 25, bk. II. Peirce: CP 5.147 Fn 1 p 92 †1 See 4.571. Peirce: CP 5.149 Fn 2 p 92 †2 See his Logic, bk. II, ch. 4, §4. Peirce: CP 5.151 Fn 1 p 94 †1 Cf. 2.367f. Peirce: CP 5.151 Fn 2 p 94 †2 Cf. 2.440. Peirce: CP 5.153 Fn 1 p 95 †1 See 448n, 4.539. Peirce: CP 5.153 Fn 1 p 96 †1 Cf. 2.287n. Peirce: CP 5.154 Fn 2 p 96 †2 Cf. 2.324, 2.357. Peirce: CP 5.155 Fn 3 p 96 †3 Cf. 3.532, where a bar is to be inserted over the second 1. Peirce: CP 5.156 Fn 1 p 97 †1 See 3.532, where the above is interpreted as an instance of subalternation. Peirce: CP 5.157 Fn 1 p 98 †1 Cf. 3.562B. Peirce: CP 5.160 Fn 1 p 99 †1 See 2.152ff. Peirce: CP 5.160 Fn 2 p 99 †2 Cf. 2.654ff. Peirce: CP 5.161 Fn 3 p 99 †3 Cf. 2.100ff; 2.266ff; 2.619ff. Peirce: CP 5.162 Fn 1 p 100

†1 See vol. 4, bk. II, for a detailed study of diagrams. Peirce: CP 5.162 Fn 2 p 100 †2 See 3.363f; 3.559, 4.233. Peirce: CP 5.163 Fn 3 p 100 †3 See 579, 2.442ff, 4.505f, 4.565f. Peirce: CP 5.164 Fn 1 p 101 †1 See vol. 4, bk. II, ch. 2. Peirce: CP 5.167 Fn 1 p 102 †1 Cf. vol. 2, bk. III, B. Peirce: CP 5.168 Fn 1 p 103 †1 See Die Schule der Chemie, Julius A. Stöckhardt, Part I, §6. Peirce: CP 5.169 Fn 1 p 104 †1 See Lettres sur la théorie des probabilités, 3me lettre. Peirce: CP 5.171 Fn 1 p 105 †1 Cf. 1.118, 2.623ff, 2.753f. Peirce: CP 5.172 Fn 1 p 107 †1 See 6.307ff. Peirce: CP 5.175 Fn 1 p 108 †1 See 166. Peirce: CP 5.176 Fn 2 p 108 †2 See e.g., Kritik der Reinen Vernunft, A7, B10, 11. Peirce: CP 5.177 Fn 1 p 109 †1 Ibid, A 656, B 684. Peirce: CP 5.178 Fn 2 p 109 †2 Cf. 4.427. Peirce: CP 5.178 Fn 1 p 110 †1 See 4.353. Peirce: CP 5.178 Fn 2 p 110 †2 See his Neues Organon, Bd. I., S. 111ff. Peirce: CP 5.178 Fn 3 p 110 †3 Vorlesungen über die Algebra der Logik (Exakte Logik), Bd. III, 12. Peirce: CP 5.178 Fn 4 p 110 †4 See vol. 3, No. VII. Peirce: CP 5.180 Fn 1 p 112 †1 Peirce was scheduled to deliver six lectures; he seems, however, to have given all seven. Peirce: CP 5.181 Fn 2 p 112 †2 See de Anima, bk. III, ch. 8.

Peirce: CP 5.181 Fn 3 p 112 †3 See The Principles of Human Knowledge, §13. Peirce: CP 5.187 Fn 1 p 117 †1 Ch. XXI. Peirce: CP 5.192 Fn 1 p 119 †1 Cf. 280ff. Peirce: CP 5.194 Fn 1 p 121 †1 See 3.63. Peirce: CP 5.198 Fn 1 p 123 †1 See Cours de philosophie positive, 28me leçon. Peirce: CP 5.199 Fn 2 p 123 †2 See 170. Peirce: CP 5.201 Fn 1 p 124 †1 There is a record of the fifth of the Lowell Lectures, "The Doctrine of Multitude, Infinity and Continuity," being delivered on December 7, 1903. It does not seem possible, due to the discrepancy of dates, that this is the lecture meant, but no other has been uncovered. See, however, vol. 4, bk. I, No. VI. Peirce: CP 5.207 Fn 1 p 127 †1 See 400ff. Peirce: CP 5.207 Fn 1 p 128 †1 See e.g., 3.217ff. Peirce: CP 5.207 Fn 2 p 128 †2 See Kirchhoff's Vorlesungen ü. math. Physik, Bd. I, Vorrede. Leipzig (1874-6). Peirce: CP 5 Book 2 Question 1 Fn 1 p 135 †1 Journal of Speculative Philosophy, vol. 2, pp. 103-114 (1868); intended as Essay IV of the "Search for a Method," 1893. Peirce: CP 5.213 Fn P1 p 135 Cross-Ref:†† †P1 The word intuitus first occurs as a technical term in St. Anselm's Monologium. [Monologium, LXVI; Cf. Prantl, III, S. 332, 746n.] He wished to distinguish between our knowledge of God and our knowledge of finite things (and in the next world, of God, also); and thinking of the saying of St. Paul, Videmus nunc per speculum in ænigmate: tunc autem facie ad faciem, [LXX], he called the former speculation and the latter intuition. This use of "speculation" did not take root, because that word already had another exact and widely different meaning. In the middle ages, the term "intuitive cognition" had two principal senses; 1st, as opposed to abstractive cognition, it meant the knowledge of the present as present, and this is its meaning in Anselm; but 2d, as no intuitive cognition was allowed to be determined by a previous cognition, it came to be used as the opposite of discursive cognition (see Scotus, In sentent., lib. 2, dist. 3, qu. 9), and this is nearly the sense in which I employ it. This is also nearly the sense in which Kant uses it, the former distinction being expressed by his sensuous and non-sensuous. (See Werke, herausg. Rosenkranz, Thl. 2, S. 713, 31, 41, 100, u.s.w.) An enumeration of six meanings of

intuition may be found in Hamilton's Reid, p. 759. Peirce: CP 5.215 Fn 1 p 137 †1 See Prantl, II, 73ff. Peirce: CP 5.215 Fn P1 p 137 Cross-Ref:†† †P1 The proposition of Berengarius is contained in the following quotation from his De Sacra Cæna: "Maximi plane cordis est, per omnia ad dialecticam confugere, quia confugere ad eam ad rationem est confugere, quo qui non confugit, cum secundum rationem sit factus ad imaginem dei, suum honorem reliquit, nec potest renovari de die in diem ad imaginem dei." The most striking characteristic of medieval reasoning, in general, is the perpetual resort to authority. When Fredigisus and others wish to prove that darkness is a thing, although they have evidently derived the opinion from nominalistic-Platonistic meditations, they argue the matter thus: "God called the darkness, night;" then, certainly, it is a thing, for otherwise before it had a name, there would have been nothing, not even a fiction to name. [See Prantl, II, 19f.] Abelard [Ouvrages, p. 179] thinks it worth while to cite Boëthius, when he says that space has three dimensions, and when he says that an individual cannot be in two places at once. The author of De Generibus et Speciebus [ibid., p. 517], a work of a superior order, in arguing against a Platonic doctrine, says that if whatever is universal is eternal, the form and matter of Socrates, being severally universal, are both eternal, and that, therefore, Socrates was not created by God, but only put together, "quod quantum a vero deviet, palam est." The authority is the final court of appeal. The same author, where in one place he doubts a statement of Boëthius [ibid., p. 535f], finds it necessary to assign a special reason why in this case it is not absurd to do so. Exceptio probat regulam in casibus non exceptis. Recognized authorities were certainly sometimes disputed in the twelfth century; their mutual contradictions insured that; and the authority of philosophers was regarded as inferior to that of theologians. Still, it would be impossible to find a passage where the authority of Aristotle is directly denied upon any logical question. "Sunt et multi errores eius," says John of Salisbury [Metalogicon, Lib. IV, cap. XXVIII], "qui in scripturis tam ethnicis, quam fidelibus poterunt inveniri; verum in logica parem habuisse non legitur." "Sed nihil adversus Aristotelem," says Abelard, and in another place, "Sed si Aristotelem Peripateticorum principem culpare possumus, quam amplius in hacarte recepimus?" The idea of going without an authority, or of subordinating authority to reason, does not occur to him. Peirce: CP 5.219 Fn 1 p 139 †1 An Essay Towards a New Theory of Vision, 1709. Peirce: CP 5.223 Fn P1 p 142 Cross-Ref:†† †P1 Proceedings of the American Academy, May 14, 1867. [1.549] Peirce: CP 5.223 Fn P2 Para 1/3 p 142 Cross-Ref:†† †P2 The above theory of space and time does not conflict with that of Kant so much as it appears to do. They are in fact the solutions of different questions. Kant, it is true, makes space and time intuitions, or rather forms of intuition, but it is not essential to his theory that intuition should mean more than "individual representation." The apprehension of space and time results, according to him, from a mental process -- the "Synthesis der Apprehension in der Anschauung." (See Critik d. reinen Vernunft. Ed. 1781, pp. 98 et seq.) My theory is merely an account of this synthesis.

Peirce: CP 5.223 Fn P2 Para 2/3 p 142 Cross-Ref:†† The gist of Kant's Transcendental Æsthetic is contained in two principles. First, that universal and necessary propositions are not given in experience. Second, that universal and necessary facts are determined by the conditions of experience in general. By a universal proposition is meant merely, one which asserts something of all of a sphere -- not necessarily one which all men believe. By a necessary proposition, is meant one which asserts what it does, not merely of the actual condition of things, but of every possible state of things; it is not meant that the proposition is one which we cannot help believing. Experience, in Kant's first principle, cannot be used for a product of the objective understanding, but must be taken for the first impressions of sense with consciousness conjoined and worked up by the imagination into images, together with all which is logically deducible therefrom. In this sense, it may be admitted that universal and necessary propositions are not given in experience. But, in that case, neither are any inductive conclusions which might be drawn from experience, given in it. In fact, it is the peculiar function of induction to produce universal and necessary propositions. Kant points out, indeed, that the universality and necessity of scientific inductions are but the analogues of philosophic universality and necessity; and this is true, in so far as it is never allowable to accept a scientific conclusion without a certain indefinite drawback. But this is owing to the insufficiency in the number of the instances; and whenever instances may be had in as large numbers as we please, ad infinitum, a truly universal and necessary proposition is inferable. As for Kant's second principle, that the truth of universal and necessary propositions is dependent upon the conditions of the general experience, it is no more nor less than the principle of Induction. I go to a fair and draw from the "grab-bag" twelve packages. Upon opening them, I find that every one contains a red ball. Here is a universal fact. It depends, then, on the condition of the experience. What is the condition of the experience? It is solely that the balls are the contents of packages drawn from that bag, that is, the only thing which determined the experience, was the drawing from the bag. I infer, then, according to the principle of Kant, that what is drawn from the bag will contain a red ball. This is induction. Apply induction not to any limited experience but to all human experience and you have the Kantian philosophy, so far as it is correctly developed. Peirce: CP 5.223 Fn P2 Para 3/3 p 143 Cross-Ref:†† Kant's successors, however, have not been content with his doctrine. Nor ought they to have been. For, there is this third principle: "Absolutely universal propositions must be analytic." For whatever is absolutely universal is devoid of all content or determination, for all determination is by negation. The problem, therefore, is not how universal propositions can be synthetical, but how universal propositions appearing to be synthetical can be evolved by thought alone from the purely indeterminate. Peirce: CP 5.227 Fn P1 p 144 Cross-Ref:†† †P1 Werke, vii. (2), 11. Peirce: CP 5.257 Fn 1 p 152 †1 Cf. A Treatise Concerning Human Knowledge, §§1-6. Peirce: CP 5.262 Fn P1 p 153 Cross-Ref:†† †P1 This argument, however, only covers a part of the question. It does not go to show that there is no cognition undetermined except by another like it.

Peirce: CP 5.264 Fn 1 p 156 †1 Journal of Speculative Philosophy, vol. 2, pp. 140-157 (1868); intended as Essay V of the "Search for a Method," 1893. Peirce: CP 5.269 Fn 1 p 159 †1 Cf. 2.466. Peirce: CP 5.269 Fn 2 p 159 †2 Cf. 2.470. Peirce: CP 5.270 Fn 1 p 160 †1 Cf. 2.508ff. Peirce: CP 5.275 Fn 1 p 163 †1 Cf. 2.623f. Peirce: CP 5.275 Fn 2 p 163 †2 Cf. his Analytica Priora, Bk. III, ch. 23. Peirce: CP 5.276 Fn P1 Para 1/11 p 164 Cross-Ref:†† †P1 Several persons versed in logic have objected that I have here quite misapplied the term hypothesis, and that what I so designate is an argument from analogy. It is a sufficient reply to say that the example of the cipher has been given as an apt illustration of hypothesis by Descartes (Rule 10 Oeuvres choisies: Paris, 1865, page 334), by Leibniz (Nouv. Ess., lib. 4, ch. 12, §13, Ed. Erdmann, p. 383 b), and (as I learn from D. Stewart: Works, vol. 3, pp. 305 et seq.) by Gravesande, Boscovich, Hartley, and G.L. Le Sage. The term Hypothesis has been used in the following senses: 1. For the theme or proposition forming the subject of discourse. 2. For an assumption. Aristotle divides theses or propositions adopted without any reason into definitions and hypotheses. The latter are propositions stating the existence of something. Thus the geometer says, "Let there be a triangle." 3. For a condition in a general sense. We are said to seek other things than happiness {ex hypotheseös}, conditionally. The best republic is the ideally perfect, the second the best on earth, the third the best {ex hypotheseös}, under the circumstances. Freedom is the {hypothesis} or condition of democracy. 4. For the antecedent of a hypothetical proposition. 5. For an oratorical question which assumes facts. 6. In the Synopsis of Psellus, for the reference of a subject to the things it denotes. 7. Most commonly in modern times, for the conclusion of an argument from consequence and consequent to antecedent. This is my use of the term. 8. For such a conclusion when too weak to be a theory accepted into the body of a science. [Cf. 2.511n, 2.707.] Peirce: CP 5.276 Fn P1 Para 2/11 p 164 Cross-Ref:†† I give a few authorities to support the seventh use: Chauvin. -- Lexicon Rationale, 1st Ed. -- "Hypothesis est propositio, quæ assumitur ad probandum aliam veritatem incognitam. Requirunt multi, ut hæc hypothesis vera esse cognoscatur, etiam antequam appareat, an alia ex ea deduci possint. Verum aiunt alii, hoc unum desiderari, ut hypothesis pro vera admittatur, quod nempe ex hac talia deducitur, quæ respondent phænomenis, et satisfaciunt omnibus difficultatibus, quæ hac parte in re, et in iis quæ de ea apparent, occurrebant." Peirce: CP 5.276 Fn P1 Para 3/11 p 164 Cross-Ref:†† Newton. -- "Hactenus phænomena coelorum et maris nostri per vim gravitatis exposui, sed causam gravitatis nondum assignavi . . . Rationem vero harum gravitatis

proprietatum ex phænomenis nondum potui deducere, et hypotheses non fingo. Quicquid enim ex phænomenis non deducitur, hypothesis vocanda est . . . In hac Philosophiâ Propositiones deducuntur ex phænomenis, et redduntur generales per inductionem." Principia. Ad fin. Peirce: CP 5.276 Fn P1 Para 4/11 p 165 Cross-Ref:†† Sir Wm. Hamilton. -- "Hypotheses, that is, propositions which are assumed with probability, in order to explain or prove something else which cannot otherwise be explained or proved." -- Lectures on Logic (Am. Ed.), p. 188. Peirce: CP 5.276 Fn P1 Para 5/11 p 165 Cross-Ref:†† "The name of hypothesis is more emphatically given to provisory suppositions, which serve to explain the phenomena in so far as observed, but which are only asserted to be true, if ultimately confirmed by a complete induction." -- Ibid., p. 364. Peirce: CP 5.276 Fn P1 Para 6/11 p 165 Cross-Ref:†† "When a phenomenon is presented which can be explained by no principle afforded through experience, we feel discontented and uneasy; and there arises an effort to discover some cause which may, at least provisionally, account for the outstanding phenomenon; and this cause is finally recognized as valid and true, if, through it, the given phenomenon is found to obtain a full and perfect explanation. The judgment in which a phenomenon is referred to such a problematic cause, is called a Hypothesis." -- Ibid., pp. 449, 450. See also Lectures on Metaphysics, p. 117. Peirce: CP 5.276 Fn P1 Para 7/11 p 165 Cross-Ref:†† J.S. Mill. -- "An hypothesis is any supposition which we make (either without actual evidence, or on evidence avowedly insufficient), in order to endeavor to deduce from it conclusions in accordance with facts which are known to be real; under the idea that if the conclusions to which the hypothesis leads are known truths, the hypothesis itself either must be, or at least is likely to be true." -- Logic (6th Ed.), vol. 2, p. 8. [Book III, ch. XIV, §4.] Peirce: CP 5.276 Fn P1 Para 8/11 p 165 Cross-Ref:†† Kant. -- "If all the consequents of a cognition are true, the cognition itself is true. . . . It is allowable, therefore, to conclude from consequent to a reason, but without being able to determine this reason. From the complexus of all consequents alone can we conclude the truth of a determinate reason . . . The difficulty with this positive and direct mode of inference (modus ponens) is that the totality of the consequents cannot be apodeictically recognized, and that we are therefore led by this mode of inference only to a probable and hypothetically true cognition (Hypotheses)." -- Logik by Jäsche; Werke, Ed. Rosenk. and Sch., vol. 3, p. 221. Peirce: CP 5.276 Fn P1 Para 9/11 p 165 Cross-Ref:†† "A hypothesis is the judgment of the truth of a reason on account of the sufficiency of the consequents." -- Ibid., p. 262. Peirce: CP 5.276 Fn P1 Para 10/11 p 165 Cross-Ref:†† Herbart. -- "We can make hypotheses, thence deduce consequents, and afterwards see whether the latter accord with experience. Such suppositions are termed hypotheses." -- Einleitung; Werke, vol. 1, p. 53. Peirce: CP 5.276 Fn P1 Para 11/11 p 165 Cross-Ref:††

Beneke. -- "Affirmative inferences from consequent to antecedent, or hypotheses." -- System der Logik, vol. 2, p. 103. There would be no difficulty in greatly multiplying these citations. Peirce: CP 5.277 Fn 1 p 165 †1 See J.S. Mill, Logic, bk. II, ch. 3, § 3. Peirce: CP 5.277 Fn 1 p 166 †1 See 2.513. Peirce: CP 5.283 Fn 1 p 169 †1 See 233f. Peirce: CP 5.288 Fn P1 p 172 Cross-Ref:†† †P1 A judgment concerning a minimum of information, for the theory of which see my paper on Comprehension and Extension [2.409ff]. Peirce: CP 5.289 Fn P2 p 172 Cross-Ref:†† †P2 Observe that I say in itself. I am not so wild as to deny that my sensation of red today is like my sensation of red yesterday. I only say that the similarity can consist only in the physiological force behind consciousness -- which leads me to say, I recognize this feeling the same as the former one, and so does not consist in a community of sensation. [Cf. 1.313, 1.383, 1.388; 3.419, 4.157.] Peirce: CP 5.289 Fn 1 p 173 †1 See 504n. Peirce: CP 5.289 Fn P1 p 173 Cross-Ref:†† †P1 Accordingly, just as we say that a body is in motion, and not that motion is in a body we ought to say that we are in thought and not that thoughts are in us. Peirce: CP 5.290 Fn P1 p 174 Cross-Ref:†† †P1 On quality, relation, and representation, see 1.553f. Peirce: CP 5.292 Fn 1 p 176 †1 Cf. 2.643. Peirce: CP 5.294 Fn 1 p 178 †1 Cf. 1.550ff; 3.7, 3.44. Peirce: CP 5.297 Fn 1 p 179 †1 Cf. 372ff, 394ff, 1.351, 1.390ff, 2.711, 3.155ff. Peirce: CP 5.299 Fn 1 p 180 †1 Cf. 3.93. Peirce: CP 5.299 Fn 2 p 180 †2 Berkeley, Principles of Human Knowledge, §10 of the Introduction. Peirce: CP 5.300 Fn 1 p 181 †1 Cf. his Treatise of Human Nature, Pt. I, § 3 and Pt. III, § 5. Peirce: CP 5.300 Fn P1 p 181 Cross-Ref:†† †P1 No person whose native tongue is English will need to be informed that contemplation is essentially (1) protracted, (2) voluntary, and (3) an action, and that it is never used for that which is set forth to the mind in this act. A foreigner can

convince himself of this by the proper study of English writers. Thus, Locke (Essay concerning Human Understanding, Book II, chap. 19, § 1) says, "If it [an idea] be held there [in view] long under attentive consideration, 'tis Contemplation; and again (ibid., Book II, chap. 10, § 1) "keeping the Idea which is brought into it [the mind] for some time actually in view, which is called Contemplation." This term is therefore unfitted to translate Anschauung; for this latter does not imply an act which is necessarily protracted or voluntary, and denotes most usually a mental presentation, sometimes a faculty, less often the reception of an impression in the mind, and seldom, if ever, an action. To the translation of Anschauung by intuition, there is, at least, no such insufferable objection. Etymologically, the two words precisely correspond. The original philosophical meaning of intuition was a cognition of the present manifold in that character; and it is now commonly used, as a modern writer says, "to include all the products of the perceptive (external or internal) and imaginative faculties; every act of consciousness, in short, of which the immediate object is an individual, thing, act, or state of mind, presented under the condition of distinct existence in space and time." Finally, we have the authority of Kant's own example for translating his Anschauung by Intuitus; and indeed this is the common usage of Germans writing Latin. Moreover, intuitiv frequently replaces anschauend or anschaulich. If this constitutes a misunderstanding of Kant, it is one which is shared by himself and nearly all his countrymen. [See an anonymous comment on this note in the Journal of Speculative Philosophy, vol. II, p. 191.] Peirce: CP 5.302 Fn 1 p 182 †1 See 238ff. Peirce: CP 5.311 Fn P1 p 186 Cross-Ref:†† †P1 By an ideal, I mean the limit which the possible cannot attain. Peirce: CP 5.311 Fn 1 p 187 †1 Cf. 354f, 2.654f. Peirce: CP 5.312 Fn P1 p 187 Cross-Ref:†† †P1 Eadem natura est, quæ in existentia per gradum singularitatis est determinata, et in intellectu, hoc est ut habet relationem ad intellectum ut cognitum ad cognoscens, est indeterminata. -- Quaest. Subtillissimae, lib. 7, qu. 18. Peirce: CP 5.312 Fn P1 p 188 Cross-Ref:†† †P1 See his argument Summa logices, part. 1, cap. 16. Peirce: CP 5.313 Fn 1 p 188 †1 Cf. 6.270. Peirce: CP 5.318 Fn 1 p 190 †1 Journal of Speculative Philosophy, vol. 2, pp. 193-208 (1868); with corrections of 1893; intended as Essay VI of the "Search for a Method," 1893. Peirce: CP 5.318 Fn 2 p 190 †2 See 295ff. Peirce: CP 5.318 Fn 3 p 190 †3 Originally "are." Peirce: CP 5.318 Fn 1 p 191

†1 See 254, 265. Peirce: CP 5.319 Fn 2 p 191 †2 Cf. 2.186ff. Peirce: CP 5.320 Fn P1 p 191 Cross-Ref:†† †P1 The word suppositio is one of the useful technical terms of the middle ages which was condemned by the purists of the renaissance as incorrect. The early logicians made a distinction between significatio and suppositio. [Cf. Prantl, II, 286ff; III, 51f.] Significatio is defined as "rei per vocem secundum placitum representatio." [Ibid., footnote 199.] It is a mere affair of lexicography, and depends on a special convention (secundum placitum), and not on a general principle. Suppositio belongs, not directly to the vox, but to the vox as having this or that significatio. "Unde significatio prior est suppositione et differunt in hoc, quia significatio est vocis, suppositio vrto est termini jam compositi ex voce et significatione." [Ibid., footnote 201.] The various suppositiones which may belong to one word with one significatio are the different senses in which the word may be taken, according to the general principles of the language or of logic. Thus, the word table has different significationes in the expressions "table of logarithms" and "writing-table"; but the word man has one and the same significatio, and only different suppositiones, in the following sentences: "A man is an animal," "a butcher is a man," "man cooks his food," "man appeared upon the earth at such a date," &c. Some later writers have endeavored to make "acceptio" do service for "suppositio"; but it seems to me better, now that scientific terminology is no longer forbidden, to revive supposition. I should add that as the principles of logic and language for the different uses of the different parts of speech are different, supposition must be restricted to the acceptation of a substantive. The term copulatio was used for the acceptation of an adjective or verb. Peirce: CP 5.320 Fn 1 p 192 †1 See 311. Peirce: CP 5.322 Fn P1 p 193 Cross-Ref:†† †P1 "If any one will by ordinary syllogism prove that because every man is an animal, therefore every head of a man is a head of an animal, I shall be ready to -- set him another question." -- De Morgan: "On the Syllogism No. IV. and on the Logic of Relations." [Transactions, Cambridge Philosophical Society, X, pt. II, p. 337. Cf. Principia Mathematica, *37.62.] Peirce: CP 5.323 Fn 1 p 194 †1 See 311. Peirce: CP 5.324 Fn 2 p 194 †2 Originally "and." Peirce: CP 5.326 Fn 1 p 195 †1 See 2.480f, 2.500ff. Peirce: CP 5.327 Fn 2 p 195 †2 Not italicized in the original. Peirce: CP 5.327 Fn P1 p 195 Cross-Ref:†† †P1 That is, in the Kantian sense. -- 1893.

Peirce: CP 5.327 Fn 3 p 195 †3 See 263. Peirce: CP 5.328 Fn 1 p 196 †1 An Essay Concerning Human Understanding, Bk. IV, ch. xvii, §4. Peirce: CP 5.328 Fn 2 p 196 †2 E.g. J.S. Mill; see his System of Logic, Bk. II, ch. iii. Peirce: CP 5.328 Fn 3 p 196 †3 Cf. 2.614. Peirce: CP 5.328 Fn P1 p 197 Cross-Ref:†† †P1 Mr. Mill thinks the syllogism is merely a formula for recalling forgotten facts. Whether he means to deny, what all logicians since Kant have held, that the syllogism serves to render confused thoughts distinct, or whether he does not know that this is the usual doctrine, does not appear. Peirce: CP 5.330 Fn 1 p 199 †1 Added, 1893. Peirce: CP 5.330 Fn 2 p 199 †2 Added, 1893. Peirce: CP 5.330 Fn 3 p 199 †3 The phrase "that . . . known" was originally, "that everything will be known at some time some number of years into the future." Peirce: CP 5.330 Fn P1 p 199 Cross-Ref:†† †P1 The difference between the two statements is like that between "Every man is the son of some woman," and "Some woman is the mother of every man." -1893. Peirce: CP 5.331 Fn P1 p 200 Cross-Ref:†† †P1 So far as there is any validity in this conception. -- 1893. Peirce: CP 5.332 Fn P1 p 201 Cross-Ref:†† †P1 "So zeigt sich jener Schlusssatz dadurch als falsch, obgleich für sich dessen Prämissen und ebenso dessen Consequenz ganz richtig sind." -- Hegel's Werke, vol. v., p. 124. [Berlin, 1841.] Peirce: CP 5.333 Fn 1 p 204 †1 "is . . . undoubted" originally "is dependent on the." Peirce: CP 5.333 Fn P1 p 204 Cross-Ref:†† †P1 Again the distinction is analogous to that between "Every man is the son of some woman or other," and "Some one woman is the mother of all men." -- 1893. Peirce: CP 5.335 Fn 1 p 205 †1 Cf. 2.646, 4.121ff, 4.219ff, 6.174ff. Peirce: CP 5.337 Fn P1 p 206 Cross-Ref:†† †P1 The usage of ordinary language has no relevancy in the matter. Peirce: CP 5.339 Fn 1 p 209 †1 Cf. 403.

Peirce: CP 5.339 Fn P1 Para 1/2 p 210 Cross-Ref:†† †P1 This seems to me to be the main difficulty of freedom and fate. But the question is overlaid with many others. The Necessitarians seem now to maintain less that every physical event is completely determined by physical causes (which seems to me {in 1869 seemed -- 1893} irrefragable) than that every act of will is determined by the strongest motive. This has never been proved. Its advocates seem to think that it follows from universal causation, but why need the cause of an act lie within the consciousness at all? If I act from a reason at all, I act voluntarily; but which of two reasons shall appear strongest to me on a particular occasion may be owing to what I have eaten for dinner. Unless there is a perfect regularity as to what is the strongest motive with me, to say that I act from the strongest motive is mere tautology. If there is no calculating how a man will act except by taking into account external facts, the character of his motives does not determine how he acts. Mill and others have, therefore, not shown that a man always acts from the strongest motive. Hobbes [Leviathan, ch. VI] maintained that a man always acts from a reflection upon what will please him most. This is a very crude opinion. Men are not always thinking of themselves. Peirce: CP 5.339 Fn P1 Para 2/2 p 210 Cross-Ref:†† Self-control seems to be the capacity for rising to an extended view of a practical subject instead of seeing only temporary urgency. This is the only freedom of which man has any reason to be proud; and it is because love of what is good for all on the whole, which is the widest possible consideration, is the essence of Christianity, that it is said that the service of Christ is perfect freedom. Peirce: CP 5.340 Fn 1 p 210 †1 Cf. 2.618. Peirce: CP 5.340 Fn P1 p 212 Cross-Ref:†† †P1 This is the principle which was most usually made the basis of the resolution of the Insolubilia. See, for example, Pauli Veneti, Sophismata Aurea. Soph. 50. The authority of Aristotle is claimed for this mode of solution. Sophist. Elench., cap. 25. The principal objection which was made to this mode of solution, viz., that the principle that every proposition implies its own truth, cannot be proved, I believe that I have removed. The only arguments against the truth of this principle were based on the imperfect doctrines of moda