Syntactical Analysis of the Class Calculus Gustav Bergmann Philosophy of Science, Vol. 9, No. 2. (Apr., 1942), pp. 227-232. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28194204%299%3A2%3C227%3ASAOTCC%3E2.0.CO%3B2-2 Philosophy of Science is currently published by The University of Chicago Press.
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SYNTACTICAL ANALYSIS OF T H E CLASS CALCULUS1 GUSTAV BERGMANN
In a paper read before the last Congress for the Unity of Science, Dr. Milton Singer2 distinguishes three main phases in the recent history of logic. The achievement he considers most characteristic of the first period is the development of the class calculus or so-called Boolean algebra. It begins with the work of Boole and DeMorgan and culminates in Schroeder's Algebra of Logic. In a minimum formulation, the results of this first stage can be summed up as, first, simplification and generalization of the traditional syllogistic machinery by means of a rather efficient new symbolism, and, second, adoption of the axiomatic method. To restate the last point: Boolean algebra was, since Leibniz, the first explicit attempt to construct what we call a formal system or calculus. Not realized a t that time, however, or a t least not adequately expressed, was the importance of the more recent distinction between scientific calculi on the one hand and purely linguistic or logical calculi on the other. As a consequence of this lack of explicit insight, the class calculus appeared to be on a par with scientific calculi, say, for instance, axiomatized geometry, Euclidean or otherwise. Its analytical character, even if it was seen, could not be formulated. Together with many others, this issue came to a head in the second period, marked by the creation of the sentential and functional calculus by Peirce, Frege, Peano and Russell. Certain technical difficulties, foremost among them the inability to eliminate in a satisfactory manner what is now known as the semantical paradoxes, brought home the epistemological impact and the full scope of the task undertaken; the tremendous task, that is, of a formalization of logic. Wittgenstein's criticism, incisive and ultimately fruitful as it was, offered no positive suggestion, it was even frankly nihilistic. The whole structure threatened t o fade into mathematical technicalities and epistemological insignificance. At this point help came, in a rather dramatic anaticlimax, from the mathematicians. I am here, of course, referring t o Hilbert's fundamental distinction between calculus and metacalculus (language and metalanguage). Here, so it seems, we have been provided with an approach which is both technically adequate and epistemologically significant. It certainly is the distinctive feature of what Dr. Singer very appropriately calls the third or metalogical period of modern logic. After this note was in the hands of the editor, I came across two recent publications i n which the point here raised is touched upon. See the formulation of the relation between the two calculi by A. Church in T h e Dictionary of Philosophy (ed. Runes), New York, 1942, p. 179, 2nd column; and the discussion of '0' and 'I' in Whitehead's Universal Algebra by W . V. Quine in T h e Philosophy of Alfred North Whitehead (ed. Schilpp), Northwestern University, 1941. See also Milton Singer "On Formal Method in Mathematical Logic." Unpublished Doctoral Dissertation, Univ. of Chicago, 1940. 227
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Syntax, as is very well known, is the name for a metalanguage which studies its object, the object language, formally, i.e., without reference to the possible referents of the object language. Formalizing logic, then, means defining the predicate 'analytic' as a purely syntactical term and without restricting its intuitive scope. The corresponding epistemological task is the characterization of 'true' and 'false' as semantical terms, i.e., as predicates of a metalanguage which speaks about both the object language and its referents. It is, however, not the purpose of this paper to discuss the very fascinating work which has been done along those lines. All I intend to do is to follow the history of the class-calculus through the three historical phases and to use it, as a wellknown and very elementary example, for the illustration of the syntactical approach. In particular I shall maintain that according to the more rigorous standards of logical syntax the so-called interpretation of the sentential calculus as a two-valued Boolean algebra can not even be formulated. The importance of this point, if any, is, of course, merely negative and clarificatory. Let me begin by elucidating the term 'scientific calculus'. Taking advantage of Carnap's terminology, I could simply say: a scientific calculus is a nonanalytical part of a descriptive object language, consisting of all the analytical consequences of a class of nonanalytical sentences, the so-called axioms. What Y = Y X', does that mean? Consider, for instance, the formula 'X which expresses part of the commutative law of the class calculus, and let us, for our very restricted purposes, neglect the difference between constants and variables. The following appears to be the case: The expressions (X', 'Y1,'X Y', 'Y X' refer to alleged extralinguistic objects, the so-called classes, the calculational symbols '+' and '=' to empirical operations and relationships. Therefore the axioms and theorems of the calculus appear to be what Carnap calls P-laws and any interpretation of them, which one may or may not find, would be hypothetical and synthetic a posteriori. Furthermore, the rules of tautological transformation, which are necessary for the derivation of the theorems from the axioms, are usually not formalized within a scientific calculus. To say the same thing differently: a scientific calculus is not a selfcontained calculus, but rather part of a more comprehensive object language. Consider, on the other hand, the "corresponding" formula of the sentential calculus (p v q = q V pl. Strictly speaking, the formulation of this so-called axiom of the sentential calculus or, as one better calls it, of this primitive sentence occurs in the metalanguage and has the form 'P((p V q = q V p')'. Here 'P9stands for the metalinguistic syntactical predicate (primitive sentence' and the quoted expression (including the quotation marks) is the name of all the object language sentences of this form. Thus one sees that syntax does indeed not speak about extralinguistic objects, but rather about the purely formal or linguistic aspects of its object language. Only the object language, if appropriately interpreted, refers to the extralinguistic material. This twofold remoteness is, to my mind, the most convincing nontechnical expression of the formal or, to use a more traditional terminology, thelanalytic a prioricharacter of logic.
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Even these few remarks are sufficient to understand the importance of what happened to the class calculus during the second period. At the end of the first period it still had, as we would call it today, the status of a scientific calculus; this because of its apparent extralinguistic reference. The corresponding epistemological thesis was, naturally enough, J. St. Mill's belief in the synthetic a posteriori character of arithmetic. This shows the importance of Russell's result when he obtained the class-calculus from a purely logical calculus, the so-called lower functional calculus, by some simple denotational changes. That this could be done, showed that the class calculus was not a scientific calculus, but itself a purely analytical structure. The use which has just been made of it in order to elucidate the nature of a scientific calculus was therefore, strictly speaking, illegitimate. This, however, was not all that happened. To understand the second half of the story, one must again remember that around 1920 the distinction between language and metalanguage and the present conception of analyticality had not yet fully emerged. As is well known, the argument centered around Russell's definition of material implication as a truthfunctional connective and, in connection with it, though usually on a slightly more technical level, around the so-called thesis of extensionality. Today we would put these two issues in the following way: (1) In an extensional object language the analyticality or, as Lewis would call it, the strict implication character of, let us say, the second horseshoe in modus ponens 'p. (p 3 q) 3 q' cannot be expressed within the calculus, but only as a metatheorem of the form 'An('p (p 3 q) 3 q')' (2) Is extensionality a purely formal regulative principle for the construction of languages or is it the expression of a genuine hypothesis of epistemological significance? An exhaustive answer t o this latter question is not quite simple; that much however can now be said with certainty: While the non-extensional calculi constructed by Lewis are not objectionable on formal grounds, it is still true that everything which they express and which cannot be expressed in the "corresponding" extensional calculi, can be expressed in a likewise extensional metalanguage of these calculi. From the epistemological standpoint of Scientific Empiricism, which is also shared by Lewis, such a relegation of the so-called modalities to the metalanguage is even an advantage. But it is only fair to say that a t that stage Lewis' preference was a t least understandable. This now is the point where the class calculus enters the picture. Russell's derivation of the class calculus makes essential use of the extensionality of his system. Thus Lewis was, from his standpoint, justified in questioning the value of such unification. For, while Boolean algebra and arithmetic are, as a matter of fact, extensional, Lewis thought of logic as an essentially nonextensional structure. Thus he was strongly motivated to relegate as much of Russell's extensional construction as possible t o the level of the class calculus which he, too, recognized as an earlier and no longer very significant stage in the development of logic. This, I believe, is the historical setting for his attempted interpretation of the sentential calculus as a two-valued Boolean algebra. During the second period this so-called interpretation attracted a certain amount of attention. There is therefore possibly some point in once
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stating explicitly that, from the syntactical standpoint, such an interpretation is fallacious and misleading.
I1 Let me begin with a general remark. If the translation of one logical calculus into another is to be of any significance a t all, one condition must certainly be fulfilled. Both calculi must be completely formalized as self-contained structures. In particular, the class calculus will have to be formalized as an object language, not in the fashion of a scientific calculus which, as we have seen, always silently presupposes a logical calculus. For what is the possible significance of a projection of the sentential calculus upon such a partial axiomatization of the class calculus? It is indeed not only possible but rather natural to formalize the class calculus in a self-contained fashion. This is the very elementary and instructive illustration of the syntactical method which I have mentioned a t the beginning. As our metalanguage we shall use English supplemented by one symbol ('F'), and an occasional use of the sentential connectives. What follows is of course not completely formal, but sufficient to indicate the idea of a more rigorous exposition. The construction of a calculus consists in the formulation of its formation and transformation rules. In our case both are exceedingly simple. Formation rules. We distinguish three kinds of signs. (1) Individual constants: 'Of, 'l', and, in case the calculus contains further elements, 'A', 'B', etc.; (2) Variables: 'X', 'Y', 'Z', etc.; (3) Calculational constants: 'f', i . ' '-) , , i = ' . The first three are the operators disjunction, conjunction and complement, the last one is the relator usually referred to as identity. Both individual constants and variables we call terms. There are, furthermore, some simple rules concerning the formation of compound terms, such as 'X Y', 'g',etc. Finally one has, as the core of any set of formation rules, the dejinition of a sentence. In this case: Any expression of the form 'X = Y' is a sentence. This is the only sentential design which exists in the calculus. The following remarks stress significant features: The sentential connectives do not occur in the object calculus. The inclusion sign is a defined relator, i.e., 'X C Y' stands for 'x.P = 0'. Operators, if appropriately applied to terms, yield terms and not sentences. Two terms, connected by one relator, form a sentence. Expressions which contain more than one relator are not sentences but meaningless strings of signs, e.g., 'X C Y = Z'. x = 1' are both sentences Transformation rules. 'X + Y = 1' and 'X of the class calculus, but only the second is what we call a formula of this calculus, that is, loosely and suggestively speaking, it holds for all substitution instances of the variable 'X'. TO give an illustration from a different calculus: 'p 3 q' and '(p 3 q) . (q 3 r) 3 (p 3 r)' are both sentences of the sentential calculus, but only the second one is a formula, for the formulae of the sentential calculus are the so-called tautologies; the one just mentioned the form of the hypothetical syllogism. The transformation rules of a calculus are the metalinguistic definition of what is considered to be a formula. To put it differently:
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they define the syntactical predicate 'being a formula' ('F'). Thus one would write ',- F('X Y = 1')' and 'F('X X = 1')'. Usually the definition of a formula is given in two steps. First, an initial set of sentential forms is singled out as the so-called primitive sentences (formulae) of the calculus. Second, rules are given how to obtain or, as one also says, deduce formulae from formulae. The formation rules of a self-contained formalization of the Boolean algebra, i.e., of a formalization which does not contain any other symbols than those of the calculus itself, have been sketched above. As primitive sentences one will conveniently choose one of Huntington's axiomatic sets which does not contain the inclusion sign as an undefined calculational symbol. All the axioms of such a set have the form of equations and can therefore immediately be stated in the form %('. . .')'; for instance, "(X-X = 0')'. If one proceeds in this manner, the second part of the definition, the deduction rules, becomes particularly simple. They consist of a substitution rule, saying that each variable in a formula can be uniformly replaced by any other term, and only one further rule, stating that the relator '=' is an identity. This is just the syntactical way of saying that this symbol can be handled exactly as the equality sign is handled in arithmetic: it is reflexive, symmetrical, transitive, and the calculus is extensional with respect t o it. Extensionality means: if F('A = B') and F(Y(A)'), then also F('f (B)'). Here 'f(A)' stands for a sentence which contains the partial expression 'A' and 'f(B)' is obtained from it by replacing one, several, or all occurrences of 'A' by 'B'. For curiosity's sake it might also be mentioned that this formalization offers an imtance of what one might call a negative primitive sentence or rule of refutation. This is the syntactical formula ',- F('1 = 0')', which vouchesafes the existence of two elements. Usually it is written in the form '1 # 0'. Let us now turn to the translation rules for what has been considered as a class interpretation of the sentential calculus. These are the rules: The constants and variables (with the exception of '1' and '0') are to be read as sentential constants and variables. The calculational constants are t o be read in the following way: 'v' ('or') for '+', '.' ('and') for '.', ',-' ('not') for 6-J , 'I' ('equivalent') for '=', '3' ('implies') for 'C'. Thus translated, a formula of the class-calculus which does not contain '1' or '0' yields a sentential formula; e.g., 'X Y =Y X' reads 'p V q = q V p'. Let, furthermore, 'f(X, Y, . . .)' be an expression wihch does not contain '0' or 'l', then the following is the case. The left side of a formula 'f(X, Y, . . .) = 1' yields a formula; e.g., 'X x = 1' gives 'p v ,- p' (excluded middle). The left side of a formula 'f(X, Y, . . .) = 0' yields the negation of formula; e.g., ' X . X = 0' gives 'p .,- p' (contradiction). The point is that this is not a significant translation of one calculus into another, but some kind of projection of both a calculus and its metacalculus into another calculus, for what is coordinated to the elements '1' and '0' respectively are not expressions of the sentential calculus but syntactical properties of the sentential formulae "corresponding" to the expressions 'f(X, Y, . . .)'
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a t the left. The same neglect of the fundamental distinction between a calculus and its syntax prevails in the other direction. Consider for instance the way the hypothetical syllogism is obtained. The corresponding class expression '(X C Y). (Y C 2 ) C (X C 2)' contains more than one relator and is therefore, as we have seen, a meaningless string of signs, let alone a formula. As a matter of fact, the hypothetical syllogism is not obtained from a formula within, but from a syntactical rule about the class calculus, namely from 'F('X C Y').F('Y C Z') 3 F('X C a;')'. The confusion is too obvious t o require further comment. Let me finish by pointing out myself how obvious all these remarks are. They are really just an exercise for students, to impress them with the analytical power of the syntactical method. The conception of the sentential calculus as a two-valued Boolean algebra has certainly no place in the third period of modern logic. How it came t o play a certain rdle in the second period, I have attempted t o indicate in the first part of this paper.
State University of Iowa.
