An Essay In Modal Logic

PRINTED IN THE NETHERLANDS DRUKKERIJ HOLLA?SD N.V.. AMSTERDAM

AN ESSAY IN

MODAL L O G I C

G E O R G H. V O N W R I G H T Professor of Philosophy University of Cambridge

1951

N O R T H - H O L L A N D P U B L I S H I N G COMPANY AMSTERDAM

PREFACE The present essay in modal logic is the result of investigations which originated from the following observation : There is an obvious formal analogy between so-called quantifiers on the one hand and a variety of concepts, including the traditional modalities, on the other hand. It might be thought convenient to call the family of concepts, to which the resemblance in question applies, modal concepts. This observation suggested to me that the use of truth-tables and normal forms as decision methods in quantification theory - a problem on which I had been working before - might, with due modifications, be transferred to modal logic. The suggestion is here followed out in the construction of a hierarchy and variety of modal systems. Some of them resemble in their formal structure the lower functional calculus with only one-place predicates ; others resemble the lower functional calculus with two-place predicates and not more than two overlapping quantifiers. It is characteristic of all these systems that their decision problem has an effective solution. I n Appendix I is shown, how a classical but notoriously obscure chapter of modal logic, viz. the theory of the modal syllogism, can be cleared up with the logical instruments of one of the modal systems previously outlined. A mechanical method for testing the validity of modal syllogisms is obtained through a simple modification of one of the methods for testing “ordinary” syllogisms. I n Appendix I1 a main branch of the logic of modal concepts is developed in an axiomatic form. Three axiomatic systems, called the Systems M , M’, and M “ , are presented. It is shown that the System M contains Lewis’s 52, and that the Systems M’ and MI’ are equivalent to Lewis’s 54 and 55 respectively. I am very much indebted to Mr. P. Geach for numerous suggestions and observations on various aspects of my work and for invaluable assistance in preparing the h a 1 draft of the manuscript.

VI

PREFACE

To Mr. A. R. Anderson I am indebted for an important correction in one of the truth-tables. I am also indebted to the editors of “Studies in Logic” for the opportunity of acquainting myself with the essay by R. Feys and J. C. C . McKinsey called Modal Logics I before its publication in this series. The work of Feys and McKinsey has been useful to me in comparing my Systems M , M‘, and M“ with the “classical” systems Sl-S5 of C. I. Lewis.

GEORG HENRIK VON WRIGHT Cambridge, England May, 1951

I TRUTH-LOGIC AND MODAL LOGIC

We shall distinguish between truth-concepts or truth-categories and modal concepts or modal categories. The logic of truth-concepts we shall call truth-logic, and the logic of modal concepts we shall call modal logic. The basic truth-categories are the two so-called truth-values, viz. truth and falsehood. Further examples of truth-categories are the concept of a truth-function and the instances of such functions : negation, conjunction, disjunction, (material)implication, (material) equivalence, tautology, and contradiction. It is of some importance to observe that the words “tautology” and “contradiction” are used in this essay as names of truth-functions exclusively. The words in question are sometimes used as synonyms for certain modal words. The modal categories we shall understand in a somewhat wider sense than is usually the case. We shall distinguish between four kinds of m d i . First, there are the alethic modes or modes of truth. These are the modalities with which so-called modal logic traditionally has been concerned. They can conveniently be divided into two subkinds. Sometimes we consider the modes in which a proposition is (or is not) true. A proposition is pronounced necessarily, possibly, or contingently true. Sometimes we consider the modes in which a property is present (or absent) in a thing. A property is pronounced necessarily, possibly, or contingently present in a certain thing. Aquinas made this distinction, when he said that the modal assertion could be de dicto or de re. We shall employ his terminology. Secondly, there are the epistemic modes or modes of knowing. They have to some extent, but not very systematically, been treated by logicians. The basic epistemic modalities are : verified (known to be true), falsified (known to be false), and undecided

2

TRUTH-LOGIC AND MODAL LOQIC

(neither known to be true nor known to be false). Like the alethic modalities, the epistemic modalities can also be taken either de dicto or de re. Thirdly, there are the deontic modes or modes of obligation. They have hardly at all been treated by logicians in modern times. The basic deontic modalities are: obligatory (“ought to”), permitted (“mayyy),and forbidden (“must not”). Fourthly, there are the existential modes or modes of existence. Their treatment is sometimes called quantification theory and is usually not regarded as a branch of modal logic. Whether universality, existence, and emptiness should be counted as modal attributes or not, is largely a matter of terminological convenience. One should, however, not fail to observe that there are essential similarities between alethic, epistemic, and deontic modalities on the one hand and quantifiers on the other hand. These similarities can be schematically exhibited in a table: 1 alethic

necessary possible contingent impossible

1

epistemic

verified undecided falsified

1

deontic

obligatory permitted indifferent forbidden

I

existential

universal existing empty

There are other groups, in addition to the above four, of concepts which could be called modal. We shall not, however, continue the list. Though modal concepts are different from truth-concepts, the two realms of categories are not logically totally disconnected. If a proposition is true, it is possible. This cannot be converted: not all possible propositions are true. Similarly, if a proposition is verified, it is true. This cannot be converted either: not all true propositions are verified. If a property is true of a thing, the property Positively obnoxious, however, seems to me the classification of truth and falsehood as modalities. The similarities, listed in the table, between the four kinds of modal categories cannot be extended to truth-concepts.

TRUTH-LOQIC AND MODAL LOGIC

3

exists. - The deontic categories, however, appear to be wholly disconnected from truth-categories. (Cf. below p. 41.) There are not only important similarities but also significant differences between the various kinds of modalities. (If there were no such differences, modal logic would be trivial relatively to quantification theory.) Some of these differences consist in the different ways in which modalities of the various kinds are related to truth-concepts. Another noteworthy difference is this : Nothing can be at the same time necessary and impossible, verified and falsified, obligatory and forbidden. But a property can be at the same time universal and empty, uiz. if the Universe of Discourse happens to be empty. These and other differences will have to be studied in detail later. There are some interesting mixed forms of modal categories. One is a combination of epistemic and existential modalities. The property of being known to have or to lack a certain property can be universal, i.e. belong to all things, or existent, i.e. belong to some thing(s), or empty, i.e. belong to no thing. Another mixed form is a combination of alethic and epistemic modalities. It may be possible for a certain proposition to be(come) verified. In this case we call the proposition in question verifiable. Or it may be possible for ti certain proposition to be(come) falsified. I n this case we call the proposition in question falsifiable. I n other words: a proposition is verifiable, if it is possible to come to know that it is true, and falsifiable if it is possible to come to know that it is false. Related to the problems of mixed modalities are the problems of auper-imposed or higher order modalities. The proposition that a certain proposition is necessary, possible, or impossible may in its turn be pronounced necessary, possible, or impossible. The question then arises, whether these modalities of second and higher order can be “reduced” to modalities of the first order. (The reduction problem.) For instance: are the necessarily necessary and the (simply) necessary the same; are the possibly necessary and the (simply) necessary the same? These and similar questions are notoriously obscure and the answers given to them by logicians and philosophers vary considerably.

4

TRUTH-LOUIC AND MODAL LOU10

The existential modalities will not be treated in this essay. (Some elements only of their theory will be mentioned in the next section.) As mentioned in the Preface I have dealt with the topic in previous publications, to which I shall sometimes make references. 1 The deontic modalities will only be considered briefly. I have dealt with them a t some length elsewhere. The study of modality is relevant to the study of logical proof and hence also to the foundations of mathematics. This relevance is particularly clear in the case of the intuitionist approach to the foundation problems. Unfortunately, there will not be room for a discussion of intuitionist logic within the scope of the present essay. 1 On the Idea of Logical Truth 1-11. Societas Scientiarum Fennics, Commentationes Physico-Mathematicae XIV 4 and XV 10. Helsingfors 1948 and 1950. 2 Deontic Logic. Mind 60, 1951.

I1

SOME ELEMENTS OF TRUTH-LOGIC. QUANTIFICATION A proposition is sometimes a truth-function of other propositions. The concept of a truth-function and the various truth-functions are assumed to be familiar to the reader. For truth-functions we shall use the following symbols: for negation, & for conjunction, v for disjunction, + for (material) implication, and ++ for (material) equivalence. It is of some importance to observe that the terms “negation”, “conjunction”, “disjunction”, “implication”, “equivalence”, “tautology”, and “contradiction” will be consistently used to designate truth-functions. They thus refer to propositions (and not to sentences or expressions). Let a and b be sentences. N a will be called the negationsentence of a. It expresses the negation of the proposition expressed by a. Similarly, a & b will be called the conjunction-sentence, a v b the disjunction-sentence, a + b the implication-sentence, and a tf b the equivalence-sentence of a and b. A sentence, which is taken as an “unanalyzed whole” is called an atomic sentence. By a molecular complex of n sentences we understand: i. Any one of the n sentences themselves. ii. The negation-sentence of any molecular complex of the n sentences, and the conjunction-, disjunction-, implication-, and equivalence-sentence of any two molecular complexes of the n sentences. The n sentences are called constituents of their molecular com-

-

1 Strictly speaking, a and b cannot be called sentences. They are sentencevariables or “schematic letters”, for which sentences can be substituted. For the sake of brevity, however, I shall speak about the schematic letters m sentences (and not as sentence-variables).

6

SOME ELEMENTS OF TRUTH-LO QIC. QUANTIFICATION

plexes. If they are atomic sentences they are called atomic constituents. Any molecular complex of n sentences expresses a truth-function of the propositions expressed by the n sentences themselves. Which truth-function it expresses can be investigated and decided in truth-tables. The truth-table technique is assumed to be familiar to the reader. Any molecular complex of n sentences has certain so-called normal forms (in terms of the n sentences). This means that, given any molecular complex of n sentences, another molecular complex can be found, which has a characteristic structure and expresses the same truth-function of the same n sentences as the given complex. There are the conjunctive and the disjunctive normal forms and the perfect (ausgezeichnete) conjunctive and disjunctive normal forms. The definitions of the normal forms and the technique used for finding them are assumed to be familiar to the reader. As to the use of brackets we adopt the convention that the symbol & has a stronger combining force than the symbols v, +, and tt; the symbol v than -+ and ++; and the symbol + than tf. Thus, e.g., we can for ( ( ( a& b ) v c) + d ) ++e write simply a & b v c +d e e . If the symbol is prefixed to a molecular complex (other than a negation-sentence), the complex should be within brackets. One should note the difference between, e.g., a & b and -( -a & b ) . N

--

If the proposition that a certain thing has a certain property is true, then we say that the property is present in the thing and that the thing is a positive case or a positive instance of the property. If the proposition that a certain thing has a certain property is false, then we say that the property is absent in the thing and that the thing is a negative case or a negative instance of the property. It is convenient to call presence and absence (of a property in a thing) presence-values and to introduce the concept of a presencefunction in strict analogy to the concept of a truth-function. A property is a presence-function of some other properties, we may say, if the presence-value of the former in a thing is uniquely

I

S O M E ELEhfENTS OF TRUTH-LOGIC. QUANTIFICATION

determined by the presence-values of the latter in the same thing. Having introduced the concept of a presence-function, we can d e b e the negation(-property) of a given property, and the conjunction-, disjunction-, implication-, and equivalence(-property) of n given properties. For these presence-functions we shall use the same symbols as for the corresponding truth-functions. As (variable) names of properties we shall use big letters A , B, . . . Names of properties will also be called predicates. We can define the concepts of an atomic predicate and of a molecular complex of n predicates in strict analogy to the concepts of an atomic sentence and a molecular complex of n sentences. Molecular complexes of predicates have normal forms, which are strictly analogous to the normal forms of molecular complexes of sentences. If the letter E is prefixed either to an atomic predicate or to a molecular complex of atomic predicates, we get an atomic Esentence. It expresses the proposition that at least one thing has the property designated by the atomic predicate or the molecular complex of predicates in question. For instance:' E ( A & B ) . If the letter U is prefixed either to an atomic predicate or to a molecular complex of atomic predicates, we get an atomic Usentence. It expresses the proposition that everything has the property designated by the atomic predicate or the molecular complex of predicates in question. For instance: U N A . The letters E and U are called quantifiers (also operators). The use of U w i l l be regarded as an abbreviation for the use of E N. By an E-sentence we shall understand an atomic E-sentence or an atomic U-sentence or a molecular complex of atomic E- and/or U-sentences. For instance: E ( A & B ) + U A. As to the use of brackets we adopt the convention that a molecular complex (other than a negation-predicate) should be enclosed within brackets, when preceded by E or U . As regards brackets E and U are thus used in the same way as N. The Quantified Logic of Properties or the System E studies E-sentences. N

N

For a detailed treatment cf. On the Idea of Logical Truth I, pp. 11-20.

I11

ALETHIC MODALITIES A.

DE

DICTO

The alethic modalities are said to be de dicto when they are about the mode or way in which a proposition is or is not true. The modalities are used de dicto in phrases such as “it is necessary that . . .”, “it is impossible that . . .”, etc. I n this chapter two systems of alethic modalities de dicto will be developed, viz. the System 2Ml and the System Ml + M,. The first could be called the Logic of Pure First Order Alethic Modalities, and the second could be called the Logic of Mixed First Order Alethic Modalities. Systems (pure and mixed) of higher order modalities w ill be considered in a later chapter. 1. The System Ml

As an undefined alethic modality we introduce the concept of possibility. It is the only undefined alethic modality we need. If a proposition is not possible, it is called impossible. If the negation of a proposition is impossible, the proposition is called necessary. If a proposition and its negation are both possible, the proposition is called contingent. Contingency is thus a narrower concept than possibility. Every contingent proposition is possible, but not every possible proposition is contingent. The above alethic modalities are attributes of a single proposition. The following alethic modalities are attributes of a pair of propositions : Two propositions are incompatible, if their conjunction is impossible (and compatible if it is possible). Two propositions are strictly equivalent, if their (material) equivalence is necessary.

DE DICTO. TEE SYSTEM MI

9

One proposition strictly implies another proposition, if the (material) implication of the second by the first is necessary. The concepts of (im)possibility, necessity, contingency, and (in)compatibility are ‘hatural” concepts in the sense that they are used in discourse outside logic. The concepts of strict implication and strict equivalence have no such extra-logical use and are, therefore, to be regarded as “technical” or “artificial” concepts. They are, however, related to two important ‘hatural” ideas, viz. the ideas of entailment (“follows from”, logical consequence) and of identity. If a proposition entails another, it strictly implies it. The converse, it seems, is not universally true. This has to do with the so-called paradoxes of strict implication. (Cf. below p. 18.) If two sentences express the same proposition, their equivalencesentence expresses a necessary proposition. But the equivalence of two propositions may be necessary without the propositions being the same. We shall not in this essay attempt to define the difference between entailment and identity on the one hand and strict implication and strict equivalence on the other hand. But we shall sometimes have occasion to point out the difference. It is perhaps of some interest to note that entailment, contrary to what might have been expected, is not a purely modal idea. The proposition that the proposition expressed by a is possible, will be expressed by Ma. The proposition that the proposition expressed by a is impossible, is the negation of the proposition that it is possible. It can thus be expressed by Ma. The proposition that the proposition expressed by a is necessary, is the negation of the proposition that its negation is possible. It can thus be expressed by M a. We shall also use the shorter expression Na. The proposition that the proposition expressed by a is contingent can be expressed by Ma & M a. The proposition that the propositions expressed by a and by b are compatible can be expressed by M ( a & b). The proposition that the propositions expressed by a and by b

-

- -

-

10

ALETHIC WODALITIES

are strictly equivalent can be expressed by N ( a tf b ) or by "(a & b v - a & b). The proposition that the proposition expressed by a strictly implies the proposition expressed by b can be expressed by N(a -+b ) or by M(a & b). M and N are called alethic operators. As to the use of brackets be it remarked that M and N are used (and E and U ) . One should note the difin the same way as ference between, e.g., M a & b and M ( a & b). If the operator M is prefixed to an atomic sentence or a molecular complex of atomic sentences, we get an atomic M,-sentence. Similarly, we define atomic Nl-sentences. Molecular complexes of atomic Ml- and/or N,-sentences we shall call M,-sentences. The System Ml studies M,-sentences. N

-

-

N

A task of particular importance is to develop a technique for deciding, whether Ml-sentences express truths of logic, or not. (The decision problem.) Sometimes Ml-sentences express truths of logic for reasons which have nothing to do with the specific character of modal concepts. For instance: If one proposition is possible, if another proposition is possible, then the second proposition is impossible, if the first is impossible. In symbols: ( M b - + M a ) + ( ~ M a + ~ 2 M b ) . This is a truth of logic. It is an instance of a variant of the inference scheme called modus tollens, which is valid for a n y propositions, whether modal or not. It is, therefore, a trivial truth from the point of view of modal logic. Sometimes, however, M,-sentences express truths of logic for reasons which depend upon the specific logical nature of modal concepts. For instance : A proposition which is strictly implied by a possible proposition, is itself a possible proposition. In symbols : M a & N(a --f b ) + Mb. This obviously is a truth of logic. It is not, however, an instance of any law of logic, which is valid for just a n y propositions, whether modal or not. It is, therefore, an interesting truth from the point of view of modal logic.

DE DICTO. THE SYSTEM MI

11

If a MI-sentence expresses a truth of logic for reasons which do not depend on the specific nature of modal concepts, then this truth can be established or proved by means of a truth-table of propositional logic. If, however, a MI-sentence expresses a truth of logic for reasons which depend on the specific nature of modal concepts, then its truth can never be established by means of propositional logic alone. A peculiar decision method, therefore, has to be found for MI-sentences. Possibility and impossibility we shall call M-values. A proposition is called an alethic modal function or a M-function of n propositions, if the M-value of the former is uniquely determined by the M-values of the latter. It is intuitively clear that not any proposition which is a truthfunction of some other propositions is also a M-function of them. (Otherwise modal logic would be trivial.) E.g., the conjunction of two propositions is not a M-function of them. From the separate possibilities of two propositions nothing can be concluded as t o the possibility of their conjunction. Sometimes .the propositions are, and sometimes they are not, conjunctively possible. It is possible that it will be raining to-morrow and possible that there will be thunder to-morrow and also possible that there will be both rain and thunder to-morrow. On the other hand, it is possible that my coat will be cut in half and possible that it will not, but it is not possible that it will be both cut and not cut (Aristotle). Further, it is possible that somebody is teaching and possible that nobody is being taught, but not possible that somebody is teaching but nobody being taught. It is also intuitively clear that any proposition which is the disjunction of two propositions is a M-function of them. The proposition is possible, if and only if at least one of the propositions, of which it is a disjunction, is possible. It is possible that there will be rain or thunder to-morrow, if and only if it is possible that there will be rain or possible that there will be thunder to-morrow. This we lay down as a Principle of Distribution for Alethic Modalities or a Principle of M-Distribution :

12

ALETRIC MODALITIES

If a proposition i s the disjunction of two propositions, then the proposition that the proposition is possible is the disjunction of the proposition that the first proposition i s possible and the proposition that the second proposition i s possible. (This principle can, naturally, be extended to disjuntions of any number n of propositions.) As we know (cf. above p. 6), any molecular complex of n sentences has what we propose to call a perfect disjunctive normal form. This is a 0-, 1-, or more-than-1-termed disjunction-sentence of n-termed conjunction-sentences. Each of the n Sentences or its negation-sentence occurs in every one of the conjunction-sentences. We have the following Principle of M-Extensionality : If one proposition necessarily has the same truth-value as another proposition, then the proposition that the first proposition i s possible necessarily has the same truth-value as the proposition that the second proposition i s possible. It follows from the Principles of M-Distribution and M-Extensionality that any molecular complex of n sentences expresses a M-function of the propositions expressed by the conjunctionsentences in its perfect disjunctive normal form. Consider now an atomic Ml-sentence. It consists of the operator M followed by (an atomic sentence which is not itself a Ml-sentence or) a molecular complex of sentences, none of which are themselves Ml-sentences. Let the molecular complex of sentences be in the perfect disjunctive normal form. Let the number of conjunctionsentences in the normal form be m. Consider next the m atomic Ml-sentences which consist of the operator M followed by one of the m conjunction-sentences. We shall call these m atomic Mlsentences the Ml-constituents of the initially given atomic Mlsentence. Example: M ( a ++ b ) is an atomic Ml-sentence. The perfect disjunctive normal form of a + b is a & b v - a & b. Hence b ) are the two Ml-constituents of M(a & b ) and M(- a & M ( a ++ b). Since, in virtue of the Principles of M-Distribution and MExtensionality, a e b expresses a M-function of the propositions

-

-

13

DE DICTO. THE SYSTEM MI

b, it follows that M(a ++ b ) expressed by a & b and - a & expresses a truth-function (disjunction) of the propositions expressed by M(a & b ) and M ( N a & b). Generally speaking: an atomic Ml-sentence expresses a truth-function (disjunction) of the propositions expressed by its Ml-constituents. Are the Ml-constituents of any atomic Ml-sentence logically independent of one another, i.e. can the propositions which they express be true and false in any combination of truth-values? The answer is that they can, though subject to one important restriction. If the molecular complex of sentences which follows after the operator M in the Ml-sentence in question expresses the tautology of the propositions expressed by its atomic constituents, then not all the Ml-constituents of the Ml-sentence in question can express false propositions. Example : The Ml-constituents of M(a v a ) are Ma and M a. Since a v a expresses the tautology of the proposition expressed by a, it cannot be the case that both Ma and M a express false propositions. The above restriction can be laid down as a (Special) Principle of Possibility : A n y given proposition either i s itself possible or has a negation that is possible. We shall here construct a truth-table for the following atomic Ml-sentences: Ma and M a and M(a & b ) and M(a v 6 ) and M(a -+ b ) and M(a tf b ) and M(a v a). The perfect disjunctive normal form of a (in terms of a and b ) is a & b v a & b. The normal form of a & b is a & b. The normal form of a v b is a & b v a & N b v - a & b. The normal form of a + b is a & b v N C L & b v - a & N b . The normal form of a t t b is a & b v - a & N b . The normal form of a v N a, finally, is a & b v a & bv a&b v -a & b. Thus the seven atomic MI-sentences have altogether four Ml-constituents, viz. M(a & b ) and M(a & b ) and M(N a & b ) and M ( w a & ~ b ) . In distributing truth-values on the MI-constituents we have to observe the restriction imposed by the Principle of Possibility. The subsequent calculation of truth-values on the basis of the N

N

N

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N

N

N

N

N

N

N

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14

ALETHIC MODALITIES

initial distribution is governed only by the Principles of M-Distribution and M-Extensionality and principles of the truth-logic of propositions. The table looks as follows :

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What is the truth-table €or M(a & a )T The perfect disjunctive normal form of a & a is “empty”, i.e, is a 0-termed disjunctionsentence. Thus M(a & a ) too is a 0-termed disjunction-sentence of MI-constituents. It might be argued that a disjunction is true, if and only if at least one of its members is true, and that a 0-termed disjunction, since it has no members, is never true (always false). From this argument would follow that M ( a & - a ) expresses the contradiction of the propositions expressed by its MI-constituents. If M(a & - a ) expresses the contradiction, then its negationsentence M(a & a ) expresses the tautology of the propositions expressed by its MI-constituents. But M ( a & a ) means the same as N ( a v a). Thus, on the above criterion for the truth of a 0-termed disjunction, it follows that the proposition that a tautology is necessary and a contradiction impossible are truths of

- -

-

-

DE DICTO. THE SYSTEM MI

15

logic. This certainly agrees with our logical intuitions. We cannot, however, take the above criterion for the truth of a 0-termed disjunction for granted, for which reason we lay down the result as a Principle of M-Tautology : If a proposition is a tautology, then the proposition that it is necessary is a tautology too. Consider a Ml-sentence. It is a molecular complex of atomic 2Ml- and/or Nl-sentences. (Cf. above p. 10.) Atomic N,-sentences can be regarded as abbreviations for negation-sentences of certain atomic Ml-sentences. (Cf. above p. 9.) If the Ml-sentence happened to contain atomic Nl-sentences, we replace them by negation-sentences of atomic M,-sentences. Thus we get a new molecular complex, all the constituents of which are atomic Ml-sentences. We now turn our attention to the (molecular complexes of) sentences which follow after the operators M in this new molecular complex of atomic Ml-sentences. We make a complete list of all atomic sentences which are constituents of at least one of the (molecular complexes of) sentences in question. Thereupon we transform these (molecular complexes of) sentences into their perfect disjunctive normal forms in terms of all sentences which occur in the list of constituents. The various conjunction-sentences in these normal forms preceded by the operator M we shall call the Ml-constituents of the original Ml-sentence. (Cf. the example below. ) We know already that any atomic Ml-sentence expresses a truth-function of the propositions expressed by its Ml-constituents. Since any molecular complex of atomic Ml- and/or Nl-sentences expresses a truth-function of the propositions expressed by the atomic Ml- and/or Nl-sentences themselves, it follows that any Ml-sentence expresses a truth-function of the propositions expressed by its Ml-constituents. Which truth-function of the propositions expressed by its Mlconstituents a given Ml-sentence expresses, can be investigated and decided in truth-tables. This fact constitutes a solution of the decision problem for the System Ml.

16

'

ALETHIC MODALITIES

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DE DICTO. THE SYSTEM M i

17

expresses the tautology of the propositions expressed by its Mlconstituents.

A Ml-sentence which expresses the tautology of the propositions expressed by its Ml-constituents, will be said to express a Mltautology or a truth of logic in the System Ml. A true proposition to the effect that a certain Ml-sentence expresses a MI-tautology, will be called a law of logic in the System Ml. We mention below some examples of such truths and laws in the System M I . The Ml-sentences are easily shown to express tautologies by means of truth-tables. i. Two laws on the relation of possibility to necessity, and vice versa: N a. A proposition is possible, if and only if 1. Afa e its negation is not necessary. 2. N a --f Ma. If a proposition is necessary, it is also possible. ( A necesse esse ad posse valet consequentia.) ii. Four laws for the distribution of operators: 1. N ( a & 6 ) tf N a & Nb. The conjunction of two propositions is necessary, if and only if the two propositions are themselves necessary. A l a v Mb. The disjunction of two propositions 2. M(a v b) is possible, if and only if at least one of the propositions is itself possible. 3 . N a v Nb -+ N ( a v b ) . If a t least one of two propositions is necessary, then their disjunction is necessary. 4. M(a & b) + M a & Mb. If the conjunction of two propositions is possible, then both the propositions are themselves possible. The second of these laws can be said to “reflect” the Principle of M-Distribution but should not be confused with it. I n the proof of ii . 2 this principle is already assumed. iii. Six laws on strict implication: 1. Na & N ( a --f b) --f Nb. A proposition which is strictly implied by a necessary proposition is itself necessary. Remembering that N

-

N

18

ALETHIC MODALITIES

strict implication is a weaker form of entailment we have the following corollary : From a necessary proposition only necessary propositions can follow. 2. Ma & N(a --f 6 ) --f Mb. A proposition which is strictly implied by a possible proposition is itself possible. Corollary: From a possible proposition only possible propositions can follow. 3. Nu & N(u & b -+ c) + N(b -+ c). A necessary premiss may be omitted. 4. Na --f N ( b -+ a). A necessary proposition is strictly implied by any proposition. There is no corollary for entailment. Ma + N(a -+ b). An impossible proposition strictly im5. plies any proposition. There is again no corollary for entailment. 6. N(N a --f a ) --f Na. A proposition which is strictly implied or entailed by its own negation is necessary. (The consequentia mirabilis.) To 4. and 5. we can refer as the paradoxes of strict implication. Every Ml-sentence has what we propose to call an absolutely perfect disjunctive normal form. This we get by replacing any one of the atomic Ml- and/or Nl-sentences, of which the given Mlsentence is a molecular complex, by a disj unction-sentence of M,-constituents of the Ml-sentence and transforming the molecular complex of MI-constituents, thus obtained, into its perfect disjunctive normal form. If this perfect disjunctive normal form contains conjunction-sentences which express contradictions in virtue of the Principle of Possibility, we omit them. What remains after we make these omissions is the absolutely perfect disjunctive normal form of the original Ml-sentence. (If some of the atomic Ml- and/or Nl-sentences would have to be replaced by a 0-termed disjunction-sentence of MI-constituents, we replace it by the letter 0. 0 is treated as a sentence. I n the perfect disjunctive normal form that we get, every conjunctionsentence will contain either 0 or 0. If it contains 0, we omit the conjunction-sentence from the normal form, and if it contains 0, we omit 0 from the conjunction-sentence.) The absolutely perfect disjunctive normal form shows with which ones of a finite number of mutually exclusive and jointly N

N

N

N

DE DICTO. THE S Y S T E M MI

f Mo

19

exhaustive possibilities the Ml-sentence in question expresses agreement and with which ones it expresses disagreement. If it agrees with all possibilities it expresses a truth of logic. Note. - Readers, who are familiar with my paper On the Idea of Logical Truth I , will easily recognize that the System M I or the Logic of Pure First Order Alethic Modalities presents a close analogy to the Quantified Logic of Properties (System E) or to that part of the so-called predicate calculus which contains only one-place predicates and no sentence-variables and no free individual variables. The logic of the words “possible”, “impossible”, and “necessary”, in other words, is very much similar to the logic of the words “some”, “no”, and “all”. It is indeed not surprising that this should be the case. For, popularly speaking, the possible is that which is true under s o m e circumstances, the impossible that which is true under n o circumstances, and the necessary that which is true under a l l circumstances. The Principle of M-Distribution corresponds to the principle which I called in the paper mentioned the Principle of Existence and which might also be called the Principle of E-Distribution. The only relevant difference in formal structure between the two systems is the absence of an analogue to the Special Principle of Possibility in the Quantified Logic of Properties. For, considering the possibility of an empty Universe of Discourse, it cannot truly be regarded ~ E Ia principle of logic that either a property or its negationproperty should exist (nor, therefore, that universdity should entail existence). But it must be regarded as a principle of logic that either a proposition or its negation-proposition should be possible (and therefore also that necessity should entail possibility). (Cf. above p. 3.)

The System M I + Mo For sentences taken as “unanalyzed wholes” we could introduce the term atomic M,-sentences. By Mo-sentences we could then mean molecular complexes of atomic M,-sentences. By the System N o we might understand the truth-logic of propositions (“ordinary” propositional logic). It studies sentences as “unanalyzed wholes” and their molecular complexes, i.e. M,sentences. By Ml + Mo-sentenceswe shall understand molecular complexes of Ml- and/or Mo-sentences. (The “and/or” indicates that the definition should be understood so as to include also H0-sentences and Ml-sentences as “degenerate” cases of Ml + Mo-sentences.) 2.

Consider a Ml

+ Mo-sentence.

20

ALETHIC MODALITIES

The sentence contains a certain number of atomic M,-sentences. (We shall here disregard atomic N,-sentences, since they can be replaced by negation-sentences of atomic Ml-sentences.) Each one of the atomic M,-sentences consists of the operator M followed by a molecular complex of atomic Mo-sentences. We make a complete list of all the atomic M,-sentences which occur either in the molecular complexes after the operators M in the atomic M,-sentences or elsewhere in the M , + Mo-sentence in question. The atomic Mo-sentences, thus listed, we shall call the Mo-constituents of the M, + M,-sentence. Example : Ma & M(a --f b ) tf a v c is a M, + M,-sentence. It contains two atomic Ml-sentences, wiz. Ma and M(a -t b ) . The atomic Mo-sentences which occur in the molecular complexes after the operators M in the atomic M,-sentences are a and b. Tho atomic Mo-sentences which occur elsewhere in the Ml + Mo-sentence in question are a and c. Hence the Mo-constituents of the M , + Mosentence are the three sentences a and b and c. We replace the molecular complexes of atomic M,-sentences which follow after the operators M in the atomic &?,-sentences by their perfect disjunctive normal forms in terms of all the Moconstituents of the M , + Mo-sentence in question. Thereupon we distribute the operators M . Thus we get a new MI + No-sentence, in which all the atomic M,-sentences which occur consist of the operator M followed by a conjunction-sentence of M,-constituents and/or negation-sentences of Mo-constituents. These atomic M,sentences we shall call the Ml-constituents of the initially given M , + Mo-sentence. Example: The perfect disjunctive normal form of a in terms of the three Mo-constituents a and b and c is a & b & c v a & b & cv a&-b&cvc;:&-b&-c. The normal form of a + b is a&b&c v a&b&-c v -a&b&c v -a&b&-c v -a&-b&cv-a&-b&-c. Replacing a and a - t b by their normal forms in M a & M(a -+ b ) + a v c and distributing the operators M we get a new M , + Mo-sentence which contains c ) and the atomic M,-sentences M(a & b & c ) and M(a & b & M ( a & - b & c ) and M ( a & - b & - c ) and N ( - a & b & - c )

-

-

21

DE DICTO. THE SYSTEM MI f Mo

and M ( - a & b & c ) and M ( - a & N b & c ) and M ( N a & b& c). These are the eight M,-constituents of our M, + Mosentence. Thus every M, + Mo-sentence has I&-constituents and Mlconstituents. If the number of Mo-constituents is n, the number of M,-constituents is (at most) 2". Example : The M, + Mo-sentence Ma & M(a --f b) a v c has 3 Mo-constituents and 8 or 23 M,-constituents. It has the maximum number of M,-constituents. From the way in which they are derived it is plain, however, that a M, + Mo-sentence with 3 Moconstituents could have less than 8 M,-constituents. E.g., the M, + Mo-sentence Ma & M (a --f b) tf a v c has 3 M,-constituents but only 4 M,-constituents. The distribution of truth-values over the propositions expressed by the Mo- and MI-constituents of a given M, + Mo-sentence is subject to one and only one restriction. Any given Ml-constituent will consist of the operator M followed by a conjunction-sentence of Mo-constituents and/or negation-sentences of Mo-constituents. If a distribution of truth-values over the propositions expressed by the Mo-constituentsis such as to make this conjunction-sentence express a true proposition, then the corresponding Ml-constituent will also express a true proposition. Example: If the Mo-constituents of a certain M , + Mo-sentence are a and b and c and if one of the M,-constituents is M(a & b & c), then, if a and b express true propositions and c a! false proposition, M(a & b & c) will necessarily express a true proposition. The above restriction we shall lay down as a (General) Principle of Possibility : If a proposition i s true, then it i s also possible. It is easy to see that the Special Principle of Possibility, which we introduced in the System M,, is but a consequence of the General Principle of Possibility. For, since any given proposition is either true or false, it will necessarily be the case that either the proposition itself or its negation is possible. And this is what the Special Principle asserts. Thus, having introduced the General N

N

-

N

N

N

22

ALETHIC MODALITIES

Principle, we can dispense with the Special Principle. Henceforth, in speaking about the Principle of Possibility, we shall always mean the General Principle. It might also be called the ab-esse-ad-psseprinciple. It follows from the above that, if a Ml + Mo-sentencehas 1 Moconstituent and 2 Ml-constituents, then there are 4 or 2(2'+1--1) combinations of truth-values. If it has 2 Mo-constituents and 4 MIconstituents, there are 32 or 2(2'+2-1)combinations of truth-values. Generally speaking, if there are n Mo-constituents and 2n Mlconstituents, there are 2(2n+n-1) combinations of truth-values. Any Ml + Mo-sentence expresses a truth-function of the propositions expressed by its Mo- and Ml-constituents. Which truthfunction it expresses, can be investigated and decided in a truthtable. This fact constitutes a solution of the decision problem of the System Ml + M,. The technique of constructing truth-tables in the System Ml + M, will be illustrated by an example. Consider the Ml + Mo-sentence ( a -+b ) + ( N M b + Ma). It says that, if a proposition materially implies an impossible proposition, then it is itself impossible. Is this a truth of logic, or not? The Mo-constituents are a and b. Ma means the same as M(a & b v a & b ) which means the same as M ( a & b ) v M ( a & b). And Mb means the same as M(a & b v a & 6 ) which means the same as M(a & b ) v M(a & b). Thus the Ml-constituents are M ( a & b ) and M ( a & ~ b ) and M ( - a & b). In distributing truth-values over the Mo- and Ml-constituents we have to observe the limitation imposed by the Principle of Possibility. The subsequent calculation of truth-values depends solely upon principles of the truth-logic of propositions (the System M,) and the Principle of M-Distribution and M-Extensionality. The table looks as follows: N

-

N

N

It is seen that the M, iMo-sentence which we are investigating does not express the tautology of the propositions expressed by its Mo- and M,-constituents. It does not, therefore, express a truth of modal logic. The proposition which it expresses, however, might be called “almost a tautology”, since there is only one case out of twenty possible cases, in which it is false.

+

A M, Mo-sentence which expresses the tautology of the propositions expressed by its Mo- and MI-constituents, will be said to express a M I + Mo-tautology or a truth of logic in the System

MI

+ dl,.

We mention below some examples of such tautologies. They are easily verified from truth-tables.

24

ALETHIC MODALITIES

i. Four laws on the relation of truth to modality: 1. N a + a. A necesse esse ad esse valet consequentia. 2. a -+ Ma. Ab esse ad posse valet consequentia. M a -+ a. What is impossible is false. 3. 4. a -+ M N a. What is false is also possibly false. N

N

N

The four laws might be said to “reflect” (alternative formulations of) the General Principle of Possibility but should not be confused with it. I n the proof of i . 1-i.4 this principle is already assumed. The four laws might be said to establish an order of strength between truth-values and modalities. Necessity is stronger than truth and truth is stronger than possibility. Impossibility is stronger than falsehood, and falsehood than possible falsehood.

ii. Three laws on implication: 1. 2. 3.

(a -+ b) + (a --f Mb). ( a -+ b ) -+ ( N u -+ b). ( a + b ) + ( N u -+Mb).

These three laws might be said to concern the relative modal strength of premisses and conclusions in arguments. The premiss of an argument can be (modally) strengthened and the conclusion (modally) weakened and the argument remains valid. Every Ml + Mo-sentence has what we propose to call an absolutely perfect disjunctive normal form. This we get by replacing the Nl + Mo-sentence by a molecular complex of its No-and Nlconstituents and transforming the new Jfl + N,-sentence, thus obtained, into its perfect disjunctive normal form. If this perfect disjunctive normal form contains conjunction-sentences which express contradictions in virtue of the Principle of Possibility, we omit them. What remains, these omissions having been made, is the absolutely perfect disjunctive normal form of the initially given Nl + No-sentence. (For possible 0-termed disjunctions cf. above p. 18.) The absolutely perfect disjunctive normal form shows with which ones of a finite number of mutually exclusive and jointly exhaustive possibilities the Ml + Mo-sentencein question expresses

DE RE

26

agreement and with which ones it expresses disagreement. If it agrees with all possibilities it expresses a truth of logic. Note. - As we have seen (p. 19), there is no analogue to the Principle of Possibility in the QuantSed Logic of Properties or in that part of the predicate calculus which contains only one-place predicates and no sentencevariables or free individual variables. If sentence-variablesand free individual variables are permitted, we get a more comprehensive fragment of the predicate calculus, which answers to the System M I + M,, of modal logic. I n this more comprehensive fragment there ia an analogue to the (General) Principle of Possibility, viz. the well-known “axiom” that if a property is true of a thing, then this property exists. We might call it the Principle of Existence.

B. DE RE The alethic modalities are said to be de re when they are about the mode or way in which an individual thing has or has not a certain property. The modalities are used de re in phrases such as “Jones is possibly (not possibly, necessarily) dead”, etc. There is an unproblematic use of the alethic modalities de re which is merely a terminological and frequently convenient alternative to their use de dicto. To say that Jones is possibly dead is a shorter way of saying that it is possible that Jones is dead, etc. I n the sentence “Jones is possibly (not possibly, necessarily) dead” we can regard “possibly (not possibly, necessarily) dead” as a predicate or name of a property. Properties which are thus named by alethic modal words prefixed to ordinary predicates we shall call “modalized”. Let A be a predicate. For the property of possibly having the property called A we can introduce the predicate MA, and for the property of necessarily having the property called A the predicate N A . These composite predicates can be treated as atomic predicates are treated. Thus we can, e.g., use the signs N, &, v, +, and * to form molecular complexes of them. The molecular complexes denote properties which are presence-functions of the properties denoted by the constituents MA, NA, etc. themselves. For example: M A & N B is the name of the property which a thing has, if and N

20

ALETHIC MODALITIES

only if it possibly has the property called A but not necessarily the property called B. We could develop a calculus or System Aflr which is “isomorphous” with the System M I and only differs from it in the feature that its expressions are predicates instead of sentences. The decision problem of the System M I , deals with the question whether a given expression names a property which is tautologically present in all things, or not. We could also develop a System MI, + No, which deals with mixed expressions of names of modalized and not modalized properties. This System Mlr + M,,, is “isomorphous” with the System M I + M,,. (The System M w means the truth-logic of properties. Cf. above p. 6f.) The existence of “isomorphous” systems de dicto and de re is trivial and need not concern us longer. What is interesting, however, is the following question: Is there, in addition to the use of modalities de re as a terminological alternative to their use de dicto, another autonomous use of them which cannot be translated into their use de dicto? We shall not here discuss this question in all its width. But we shall draw attention to a point which appears to be worth observing. Let us compare the modalized properties called N A and M A with the property called A from the point of view of their respective extensions. It is clear that whatever necessarily has a certain property also has the property itself, and that whatever has the property also possibly has the property. These relations cannot be converted. If something has a certain property it need not necessarily have it, and from the mere fact that something possibly has a certain property does not follow that it has this property. This might suggest that N A names a property which is (normally) of smaller extension than the property named by A , and that A names a property which is (normally) of smaller extension than the property named by M A . The use of the modalities de re to name properties of different though related extensions would constitute an autonomous use of

DE RE

21

them, with interesting consequences when combined with the use of existential modalities (quantification). However, the above suggestion about the extensions of modalized and not modalized properties turns out to be misleading, if - as is at least not unplausible - we accept the following Principle of Predication : If a property can be significantly predicated of the individuals of a certain Universe of Discourse, then either the property i s necessarily present in s m e or all individuals and necessarily absent in the rest, or else the property i s possibly but not necessarily (i.e. contingently) present in some or all individuals and possibly but not necessarily (i.e. contingently) absent in the rest. This principle, or some modification of it, can be said to underlie the classification of properties into “formal” or “1ogica.l” and “material” or “descriptive” properties, which is sometimes made. Arithmetical properties, e.g., are formal properties of numbers : any given natural number is either necessarily or impossibly a prime number, etc. Colours, e.g., are material properties of physical bodies: any given piece of solid matter is either Contingently red or contingently not red, etc. (One can think of exceptions to the principle, e.g. among higher order properties, but I am not convinced that they are not apparent exceptions only. The question will not be discussed here.) If the above Principle of Predication is accepted, it cannot at the same time be the case that some individuals are possibly but not actually, some actually but not necessarily, and some necessarily instances of the property called A . It would have to be the case either that the property called M A is co-extensive with the Universe of Discourse and the property called N A empty, or that the property called N A is co-extensive with the property called A and also with the property called M A . This would make a combination of alethic modalities de re and quantification uninteresting, since nothing would ensue from a combined use which would not follow from quantification alone in combination with a statement on the formal or material nature of the properties involved. Thus, e.g., if A is the name of a logical

28

ALETHIC MODALITIES

property, then ENA (“nothing is necessarily A”) would express a proposition of the same truth-value as the one expressed by N EA (“nothing is A ” ) , and E M A (“something is possibly A ” ) a proposition of the same truth-value as the one expressed by EA. If again A is the name of a descriptive property, then N E N A trivially expresses a true proposition and E M A expresses the same proposition as E ( A v A). For the above reasons we shall in this essay disregard the use of the alethic modalities de re in combination with quantification. N

N

Note. - The alethic modalities, as understood in this essay, cover the ground of that which is also called logical possibility, impossibility, necessity, etc. It should, however, be observed that the same modal words are used in ordinary laguage in other senses as well. An important use of them is connected with the notions of an ability and of a disposition and with the verb can. For example : “Jones can speak German” (= “it is possible for Jones to make himself understood in German”) ;“Jones cannot speak German” (= “it is impossible for Jones to make himself understood in German”). We shall call the modal concepts ,which refer to abilities and dispositions, dynamic modalities. (Iam indebted for the term to Mr. GEACH.) The dynamic modalities, it appears, are (genuinely) used de re only. It is important to note that the combination of these modalities with quantifiers is not trivialized by our Principle of Predication. - If Jones is speaking German, Jones can speak German; but Jones may be able t o speak German though he is not now speaking it. There is nobody who cannot not speak German, i.e. cannot stop speaking and always speaks German. Further, some men can speak German, others cannot. The question whether the dynamic modalities, Le. the logic of abilities and dispositions, is subject to exactly the same formal rules aa the alethic modalities will have to be investigated separately.

IV

EPISTEMIC MODALITIES

A. DE

DICTO

The epistemic modalities are said to be de dicto when they are about the mode or way in which a proposition is or is not known (to be true). The epistemic modalities are used de dicto in phrases such as “it is known that . . .”, “it is unknown whether . . .”, or “it is known that not . . .”. It is important to distinguish between two interpretations of the phrase “it is known (verified) that a”, wiz. i. “the proposition expressed by a is known to be true (verified)”, and ii. “it is known (verified) that a expresses a true proposition”. (a is a sentence-variable. Cf. above p. 5.) I n this essay we shall throughout understand the phrase “it is known (verified) that a” in the interpretation i cbove. As an undefined epistemic modality we introduce the concept known to be true or verified. It is the only undefined epistemic modality we need. If the negation of a proposition is verified, the proposition is called falsified. If neither a proposition nor its negation is verified (falsified),the proposition is called undecided. The proposition that the proposition expressed by a is verified, we shall express by Va. The proposition that the proposition expressed by a is falsified, is the same as the proposition that its negation is verified. It can thus be expressed by V -a. We shall also use the expression Fa. The proposition that the proposition expressed by a is undecided can be expressed by Va & V a or, alternatively, by

-

N

N

-Fu&-F-u. V and F are called epistemic operators. As regards brackets, V and P resemble the other modal operators.

30

EPISTEMIC MODALITIES

By an atomic Vl-sentence we understand the operator V prefixed to an atomic sentence or a molecular complex of atomic sentences. Similarly, we define atomic Fl-sentences. Molecular complexes of atomic Vl- and/or F,-sentences we call Vl-sentences. For the sake of convenience, we can by Vo-sentencesunderstand the same as above by M,-sentences. (Cf. p. 19.) We can then define V, + V,-sentences as molecular complexes of Vl- and/or V,sentences. The governing principles of the Systems M, and Ml + M, were those of M-Distribution, Possibility, M-Extensionality, and MTautology. It is important to find out, to what extent the Systems V, and V, + V, are governed by analogous principles. To the Principle of Distribution for Alethic Modalities there corresponds a Principle of Distribution for Epistemic Modalities. We can also call it the Principle of J’-Distribution. It states that if a proposition is the disjunction of two propositions, then the proposition that the proposition is not falsified is the disjunction of the proposition that the first proposition is not falsified and the proposition that the second proposition is not falsified. To the Special Principle of Possibility there corresponds a Special Principle of Non-Falsification. It states that any given proposition is either itself not falsified or has a negation that is not falsified. To the General Principle of Possibility there corresponds a General Principle of Non-Falsification. It states that if a proposition is true, then it is not falsified. To the Principle of M-Extensionality there corresponds the following Principle of V-Intensionality : If one proposition i s known necessarily to have the same truthvalue as another proposition, then the proposition that the first proposition i s not falsified necessarily has the same truth-value as the proposition that the second proposition i s not falsified. If in the above formulation we substitute for “is known necessarily to have” the phrase “necessarily has”, we obtain the formulation of what might be called a Principle of V-Extensionality. Its N

DE DICTO

31

truth is debatable. If we reject it, this would constitute a reason for calling the system of epistemic modalities “intensional” as opposed to the system of alethic modalities which is “extensional”. It should be observed that from the point of view of proving propositions in the respective systems it does not make any ddference whether we assume the systems to be “extensional” or “intensional”. For the mere fact that the strict equivalence of two propositions entails the strict equivalence of two further propositions cannot be used in a proof unless the strict equivalence of the two first propositions is also lcnown. Proof, one might say, is itself an “intensional” process. To the Principle of M-Tautology there corresponds the following Principle of V-Tautology : If a proposition i s lcnown to be a tautology, then the proposition that it i s verified i s a tautology too. (This principle is of technical consequence only. Iis acceptance means that the criterion of truth of a 0-termed disjunction, which we adopted in the logic of the alethic modalities, can be used for the epistemic modalities as well.) With the above modifications in the governing principles the Systems Vl and Vl + Vo can be developed and studied in strict analogy to the Systems Jl, and Nl + No. The truths of logic of the systems are tautologies of propositions expressed by certain Vland Vl + Vo-constituents( Vl- and Vl+ Vo-tautologies). The decision problem can be effectively solved using truth-tables and/or normal forms. If in the sentences expressing M-tautologies mentioned above on pp. 17-18 we replace the operator N by the operator V and the F , we obtain sentences expressing Vl-tautologies. operator M by N

From the point of view of “formal behaviour”, the verified corresponds to the necessary, the undecided to the contingent, and the falsified to the impossible. The not falsified corresponds to the (alethic) possible. It is indeed a remarkable fact of language that there is no unique epistemic word which corresponds to the alethic word “possible”.

32

EPISTEMIC YODALI!CIES

There is, in other words, no specific term to cover the ground of both the words “verified” and “undecided”. It is further a remarkable fact of language that the word “possible” is frequently used also in an epistemic sense. The meaning of the epistemic “possible” is somewhat vague. The word is sometimes used epistemically as a strict formal equivalent to its alethic use. Then “possible” means the same as “verified or undecided” or “not known to be false”. This use, however, appears to be less natural and therefore probably also less common than the use of the word “possible” to mean the same as “undecided” or “neither known to be true nor known to be false”. Under this second use an alethically impossible proposition can be (truly) pronounced epistemically possible. If, e.g., Goldbach’s conjecture is actually false but nobody ever manages to refute it, then the conjecture would be epistemically possible and yet alethically impossible. It should be observed that the ab-esse-ad-posse-principle(Principle of Possibility), which is one of the cardinal truths about the alethically possible, is false for the epistemic use of “possible” as equivalent to “undecided”. This agrees very well with ordinary language : The prisoner of war, who has for several years been unable to communicate with his family, might say “my mother is possibly dead by now”. But if, on his return home from captivity, he finds that his mother is dead, he would no longer say that she is possibly dead. The idea that there is an alethic sense of the word, in which his mother continues to be possibly dead though already actually dead, would not even enter his head. This is not uninteresting to point out, because it shows that logicians, who without qualification assert or assume the validity of the ab-esse-ad-posseprinciple, are apt to ignore relevant facts about the use of language. The ambiguous use of “possible”, sometimes to denote a state of knowledge and sometimes to denote a mode of truth, is probably relevant in connexion with some traditional philosophic puzzles concerning scepticism and certainty. We shall not, however, discuss these connexions here.

DE RE

33

B. DE RE The epistemic modalities are said to be de re when they are about the mode or way in which an individual thing is known to possess or to lack a certain property. The modalities are used de re in phrases such as “Jones is (not) known (not) to be dead”, etc. The epistemic modalities are sometimes used de re merely to provide a convenient terminological alternative to their use de dicto. It is shorter and, on the whole, more convenient to say “Jones is known to be dead” than to say “it is known that Jones is dead”. It should, however, be observed that the equivalence of “Jones is known to be dead” and “it is known that Jones is dead” presupposes that the latter sentence is interpreted as under i on p. 29. Under this interpretation “it is known that Jones is dead” means the same as “the proposition expressed by ‘Jones is dead’ is known to be true”. If “Jones is known to be dead” is equivalent to this, it must mean “the individual, whose name is ‘Jones’, is known to be dead”. The knowledge is thus of the individual or the thing (de re). Hence the truth of the proposition that Jones is known to be dead is independent of whether we know or not that‘the person in question is called “Jones” just as, under interpretation i on p. 29, the truth of the proposition that it is known that Jones is dead is independent of whether we know or not what the sentence “Jones is dead” means. I n the sentence “Jones is (not) known (not) to be dead” we can regard “(not) known (not) to be dead” as a predicate or name of a property. Properties which are thus named by epistemic modaI words prefixed to ordinary names of properties we shall call “epistemically modalized”. For the property of being known to possess the property named by A we can introduce the composite name V A , and for the property of being known not to possess the property named by A we can introduce the composite name PA. Such composite predicates can be handled as atomic predicates are handled. (Cf. above p. 25 and below p. 46.) I am indebted to Mr. P. Geach for certain observations in this context.

34

EPISTEMIC MODALITIES

It is a trivial fact that there exists a System V17)which is “isomorphous” with the System V,, and a System V,, + V,, which is “isomorphous” with the System V, + V,, and only differ from the systems de dicto in that their expressions are predicates instead of sentences. There is, however, undoubtedly a,lso a non-trivial use of the epistemic modalities de re which cannot be translated into their use de dicto. This will be plain from the following considerations: It is clear that, normally, not all individuals which are actually positive instances of a certain property are also known to be positive instances of it (though the converse holds), and that not all individuals which are not known not to be positive instances of a certain property are actually positive instances of it (though again the converse holds). This means that V A , normally, names a property which is included in but not co-extensive with the property named by A and that A in its turn, normally, names a property which is included in but not co-extensive with the property named by V A. I n other words: The Principle of Predication (p. 27), and the logical or descriptive nature of the originally given property, do not affect the extensions of the epistemically modalized versions of the original property in the same peculiar way as they affect the extensions of its alethically modalized versions. (Cf. above p. 27.) This fact makes it possible to quantify epistemically modalized properties with effects which are not obtainable by means of quantification without modalization. Thus, e.9.) the proposition that something is known to have the property called A does not depend for its truth-value on the formal or material nature of the property called A , nor is it (generally) entailed by the proposition that something has the property called A . It is easy to see that not all propositions which are obtained through a quantification of epistemically modalized properties can be expressed in terms of epistemic modalities de dicto (and quantifiers). E.g., the proposition that something is known t o have the property called A is different from the proposition that it is known that something has the property called A and from any other N

N

DE RE

35

proposition which can be expressed by prefixing an epistemic operator to an E-sentence containing A . (Cf. below p. 49.) It thus appears that the epistemic modalities de re have a non-trivial and autonomous use in combination with the existential modalities (quantification concepts). This combined use w ill be the object of closer study in a later chapter. Note. - Throughout this essay the epistemic modalities are treated as “absolute”, i.e. mention is not being made of a person who possesses or does not possess knowledge. We could develop an alternative system in which the epistemic modalities are treated as “relative” to persons. I n this system “unknown to 9, we should have to deal with expressions like “known to d’, etc. Introducing quantifiers we should get a combined system dealing with expressions like ‘‘known to somebody”, “unknown to everybody”, etc. T h i s combination of epistemic and existential modalities will not be studied in the present essay.

V

DEONTIC MODALITIES The deontic modalities are about the mode or way in which we are permitted or not to perform an act. They are used in phrases such as “it is obligatory to . . .”, “it is permitted to . . . ”, or “it is forbidden to . . .”. I n ordinary language the word “act” is ambiguously applied to properties of a certain kind and to individual instances of such properties. We call, e.g., theft as such, without regard to concrete cases, an act. But we also call the theft committed by Jones on a certain occasion an act. It should be observed that deontic concepts are here regarded as attributes of act-properties and not of actindividuals. “Act” thus henceforth mea,ns a kind of property. It is convenient to call an act, which is a presence-function (cf. above p. 6) of other acts, a performance-function. For whether this act is or is not performed by an agent on a certain occasion uniquely depends upon which of these other acts are performed and are not performed by that agent on that occasion. The concepts of negation, conjunction, disjunction, implication, equivalence, tautology, and contradiction apply to acts in the same way as they apply to all properties. The same symbols as before will be used for those concepts. As variable names of acts we shall use variable names of properties A , B, . . . .

As an undefined deontic modality we introduce the concept of There is a more elaborate account of this topic in my article Deontic Logic in Mind 60, 1951. The concept of an act-individual presents some complications. The description of the individual involves the mention both of an agent by which and of an occasion on which the act is performed. The occasion, moreover, need not be spatio-temporally continuous. These complications, however. need not concern us here.

DEONTIC MODALITIES

37

permission. It is the only undefined deontic modality which we need. If an act is not permitted, it is called forbidden. We m u s t n o t do that which we are n o t allowed t o do. If the negation of an act is forbidden, the act itself is called obligatory. We ought to do that which we are n o t allowed n o t t o do. If an act and its negation are both permitted, the act is called (morally) indifferent. Two acts are (morally) incompatible, if their conjunction is forbidden (and compatible if it is permitted). Doing one act commits us to do another act, if the implication of the two acts is obligatory. 1 The proposition that the act named by A is permitted will be expressed by P A . The proposition that the act named by A is forbidden can be expressed by N PA. The proposition that the act named by A is obligatory can be expressed by P A. We shall also use the shorter expression OA. The proposition that the act named by A is (morally) indifferent can be expressed by P A & P A. The proposition that the acts named by A and by B are (morally) incompatible can be expressed by N P ( A & B ) . The proposition that doing the act named by A commits us to do the act named by B can be expressed by O ( A -+B). P and 0 are called deontic operators. As regards brackets, P and 0 resemble the other modal operators. By an atomic P-sentence we understand the operator P prefixed to an atomic name of an act or a molecular complex of atomic names of acts. Similarly, we define atomic 0-sentences. N

-

N

By the implication (-act) of two acts we understand the act which is performed by an agent, if and only if it is not the case that he performs the first act but omits the second act. That the implication of two acts is obligatory thus means that we are not allowed to perform the first act without performing the second act as well. I n other words: performing the first act commits us to perform the second act.

38

DEONTIC MODALITIES

Molecular complexes of atomic P- and/or 0-sentences we call P-sentences. The System P studies P-sentences. To the Principles of Distribution for Alethic and Epistemic Modalities there corresponds a Principle of Distribution for Deontic Modalities. (Principle of P-Distribution.) It states that if an act is the disjunction of two acts, then the proposition that the act is permitted is the disjunction of the propositions that the first act is permitted and the proposition that the second act is permitted. To the (Special) Principles of Possibility and Non-Falsification there corresponds a Principle of Permission. It states that either any given act is itself permitted, or its negation is permitted. The self-evident character of the Principle of Permission should be apparent from the following considerations : If the principle were not true, it would be possible for an act and its negation both to be forbidden. But that the negation of an act is forbidden means that the act itself is obligatory. Thus to say that the act and its negation are both forbidden means the same as to say that the act itself is both forbidden and obligatory. This surely conflicts with ordinary use of language and our common sense intuitions concerning obligation concepts. The deontic modalities are extensional. If two acts necessarily have the same performance-value in one and the same individual, then the two propositions that the two acts are permitted necessarily have the same truth-value. (Principle of P-Extensionality.) Ordinary language and our common sense logical intuitions apparently do not provide a clear answer to the question whether a tautologous act must be obligatory or not. From the point of view of formal logic, therefore, the safest course seems to be to regard O(Av A ) and correspondingly P(A & - A ) as expressing contingently truth or falsehood depending upon material circumstances. It should, however, be observed that if there really existed an act, say A , which is such that P(A & - A ) expresses a true proposition, then every act would be permitted. For, since A &-A

-

39

DEONTIC MODALITIES

and A & A & B are names of acts which necessarily have the same performance-value, we should, in virtue of the Principle of P-Extensionality, have P ( A & A & B ) and from this PB follows. The assumption that there existed an act, the tautology-act of which is not obligatory, would thus lead to “moral anarchy” or “moral nihilism”. 1 This may be considered a n argument in favour of accepting a Principle of P-Tautology as true, even if not as logically true. Using the Principles of Permission, P-Distribution, and PExtensionality, the System P can be developed in close analogy to the previous systems for alethic and epistemic modalities. The truths of logic of the system are tautologies of the propositions expressed by certain P-constituents (P-tautologies). The decision problem can be solved using truth-tables and/or normal forms. We shall here mention some P-tautologies which concern the idea of commitment: 1. OA & O(A --f B) + OB. If doing what we ought to do commits us to do something else, then this new act is also something which we ought to do. 2. P A & O(A -+ B) --f PB. If doing what we are free to do commits us to do something else, then this new act is also something which we are free to do. I n other words : doing the permitted can never commit us to do the forbidden. PB & O(A -+B) + PA. If doing something commits 3. us to do the forbidden, then we are forbidden to do the first thing. Following our conscience, we might say, is not a sufficient criterion that we are doing the right.2 N

N

-

N

I am indebted for this observation to Mr. J. Hintikka. Philosophers have sometimes argued in the following way: To keep our promises cannot be (unconditionally) obligatory, since we may promise t o do something which is in fact forbidden. If, however, the rule of promisekeeping says: it is forbidden t o give a promise ( A )and not keep it ( B ) ,then this piece of reasoning is invalid. For, P B does not in combination with O ( A + B)give - O ( A + B ) - which means the same as P ( A & B) but only PA. I n other words : if what we promise is forbidden, then we are forbidden to give the promise, and not :if what we promise is forbidden, then we are permitted to give the promise without keeping it. This is one

-

-

-

40

DEONTIC MODALITIES

4. O ( A + B V C ) & N P B $ N P C - + N P A . An act which commits us to a choice between forbidden alternatives is forbidden. 5. (O(Av B ) & P A & P B ) . It is logically impossible to be obliged to choose between forbidden alternatives. 6. OA & O(A & B + C ) + O(B -+C ) . If doing two things, the first of which we ought to do, commits us to do a third thing, then doing the second thing alone commits us to do the third thing. Our commitments, we might say, are not affected by our (other) obligations. 7. O(N A -+ A ) -+ OA. If failure to perform an act commits us to perform it, then this act is obligatory. N

N

N

The following differences and resemblances are noteworthy : The operators M and N , when prefixed to sentences yield new sentences, and when prefixed to names of properties yield new names of properties. The same is true of the operators V and F . The operators P and 0, however, when prefixed to names of properties (acts) yield sentences. M A denotes a property, viz. the property of possibly being A . But P A expresses a proposition, viz. the proposition that it is permitted to do A . The quantifiers E and U resemble in this respect the deontic operators. When prefixed to predicates they yield sentences. If A names the property of being red, EA expresses the proposition that there are red things. It follows from the above that the deontic (and the existential) unlike the alethic and the epistemic modalities cannot be taken alternatively de dicto and de re. Further, it follows from the above that the deontic (and the of many examples which show that moral arguments may involve considerable logical subtleties and that, therefore, a logical study of moral concepts is philosophically relevant. The two last tautologies bring out the distinction which Aquinas made between a n agent’s being perplexus simpliciter and perplexus secundum quid. It is logically impossible t o be perplexus simpliciter but possible t o be perplexus secundum quid. I a m indebted for this and some other observations concerning the deontic modalities to M i . P. Geach.

DEONTIC MODALITIES

41

existential) unlike the alethic and the epistemic modalities are not capable of higher orders. Since M u is a sentence, M M a is also a sentence, and since M A is a predicate, M M A is also a predicate. P P A has no meaning. Similarly, E E A has no meaning. There is an interesting respect, in which the deontic modalities differ drom the alethic, the epistemic, and the existential alike. As we know, a proposition is possible, if it is true (General Principle of Possibility), and a proposition is not falsified, if it is true (General Principle of Non-Falsification), and a property exists, if it is true of a thing (Principle of Existence). The deontic modalities, however, exhibit no analogous connexion with truth and falsehood (matters of fact). If an act is performed or omitted by an agent, nothing whatever follows as regards its deontic nature. This observation is well known to ethical philosophers. N o t e . - I n this essay the deontic modalities are treated as “absolute”. Alternative systems could be developed, in which the deontic modalities are in various ways made “relative”. Instead of dealing with propositions of the type “ A is permitted”, we might consider propositions of the types “ A is permitted according to the moral code C”, or “ A is permitted to x”, or “x permits y to do A”, etc. These types of proposition can in their turn be subject to quantification. Then we get new propositions of the types “ A is permitted within some moral code”, or “ A is permitted to everybody”, or “somebody permits everybody to do A”, etc. Deontic modalities can also be combined with alethic and with epistemic modalities. An act may be necessarily permitted, or known to be forbidden, etc. Extensions of the System P will not, however, be studied in this essay.

~~

1 There are other senses of “higher order” in which we may speak about deontic and existential modalities of higher order. We shall not, however, deal with them here.

VI COMBINED MODALITIES I n this chapter we shall study the combination of epistemic and existential modalities. It is the only combination of modalities which we study, but not the only one that exists. The study can conveniently be pursued in three stages. We shall successively develop a System V E , a System E V , and a System

V E + EV.

The System V E deals with expressions, in which an epistemic modal phrase is prefixed to a quantification statement. For example : “it is known that something is red”. I n these expressions the modalities are used exclusively de dicto. The System E V deals with expressions, in which a quantifier is prefixed to the name of an “epistemically modalized” property. For example : “something is known to be red”. I n these expressions the modalities are used exclusively de re. The System V E + E V , finally, deals with molecular complexes of expressions belonging to the Systems V E and E V respectively. For example: “if something is known to be red, then it is known that something is red”. I n these expressions the modalities occur both de dicto and de re. There is a System M E which is closely analogous in formal structure to the System VE. It deals with expressions, in which an alethic modal phrase is prefixed to a quantification statement. For example: “it is necessary that something is red”. The System M E covers an important part of the ground of inquiry in traditional modal logic, viz. the study of the so-called modal syllogism. We shall deal with the modal syllogism in an appendix. The reason, why we prefer here to deal with the System V E rather than with the traditionally more important System M E lies in the fact that the System V E can be supplemented with the Systems E V and V E E V , whereas a corresponding supplement-

+

43

THE SYSTEM VE

+

ation of the System M E with Systems E M and M E E M appears dubious. This is due to the characteristic difference between the uses de re of the alethic and the epistemic modalities, which we have noted above. 1.

The System VE

The concepts of an atomic E-sentence, an atomic U-sentence, and an E-sentence have been defined in a previous chapter. (Cf. above p. 7.) By an atomic VE-sentence we understand the epistemic operator V followed by an E-sentence. For instance: V(EA v UB). By an atomic FE-sentence we understand the epistemic operator F followed by an E-sentence. For instance: FE(A & B). By a VE-sentence, finally, we shall understand an atomic VEsentence or an atomic FE-sentence or a molecular complex of (atomic) VE- and/or PE-sentences. For instance: VEA & V ( U A v EB). The previous rules for the use of brackets are adopted. The System V E studies VE-sentences. The governing principles of the system are those of the System V and the System E (the Quantified Logic of Properties). Consider a VE-sentence. If the operators U and/or V occur in the sentence we replace them by the operators E and/or F in and “V” = “ F N”. accordance with the rules “U” = ‘‘- E We thus obtain a new VE-sentence which expresses the same proposition as the original one but does not contain the operators U and/or V. We next make a complete list of all atomic predicates which occur in the VE-sentence. We thereupon replace every one of the molecular complexes of predicates which occur in the VE-sentence by its perfect disjunctive normal form in terms of all predicates in our list. After doing this, we distribute the operator E in front of every one of the normal forms. The atomic E-sentences which occur in the VE-sentence after the distribution we shall call the E-constituents of the (original) VE-sentence. N”

44

COMBINED MODALITIES

We now make a complete list of all the E-constituents. We thereupon replace every one of the E-sentences which occur in the VE-sentence by its perfect disjunctive normal form in terms of all E-constituents in our list. After doing this, we replace the operator F, wherever it stands without a negation-sign in front of itself, by F and distribute the N F in front of every one of the last mentioned normal forms. The atomic VE-sentences which occur in the VE-sentence after the distribution we shall call the VE-constituents of the (original) VE-sentence. A simple example will illustrate the above train of thought. Let the VE-sentence be VEA v F U A (“it is known t h a t something is A or it is not known that not everything is A”). After elimination of the operators U and V we get F EA v F E -A. The sole atomic predicate, which occurs in the VE-sentence, is A. The perfect disjunctive normal form of A in terms of A is A,and the normal form of -A is A. There is thus no distribution of the operator E in this case. The E-constituents are EA and E -A. The perfect disjunctive normal form of EA in terms of the EA & E Av EA & E -A. The two E-constituents is E -A is EA & E -A v EA & E -A. normal form of We thus obtain the new VE-sentence F(NEA & E A v EA & E -A) v F(EA & E -A v EA & E - A ) . If we replace the first F by N F and distribute the two N F’s which occur, we get (- F(- EA & E -A) v F(- EA & E A)) v F(EA & -E -A) v F(-EA & -E -A). The VE-constituents of the original VE-sentence are thus F(EA & -E -A) and F(-EA & E -A) and F(-EA & WE-A). If A is the sole atomic predicate which occurs in a VE-sentence, there are never more than two E-constituents, viz. EA and E -A, and never more than four (= 22) VE-constituents, viz. F(EA & E A) and F(EA & E -A) and F(- EA & E A ) and F(- E A & N E A). The maximum number of constituents need not occur; the expression discussed above turned out to have two E-constituents, but only three VE-constituents.

--

-

-

--

N

--

N

--

N

-

-

N

-- -- - - N

N

N

45

THE SYSTEM V E

If A and B are the only atomic predicates which occur in a VEsentence, there is a maximum number of four (= 22) E-constituents and sixteen (= z4) VE-constituents. We leave their formation as an exercise to the reader. Generally speaking, if there are n atomic predicates in a VEsentence, the sentence may have 2" E-constituents and 2@) VEconstituents. The distribution of truth-values over the VE-constituents is subject to one restriction and one only. It is the restriction imposed by the (Special) Principle of Non-Falsification. It states that, if a VE-sentence has the maximum number of VE-constituents, then not all its VE-constituents can express true propositions. What this restriction means in practice is readily seen from a simple example. The disjunction E A & E A v E A & E Av N EA & E A v EA & N E A expresses a (E-) tautology, i.e. one of its four members must express a true proposition. Hence it is an impossibility that all the propositions which the members express should be known to be false. If a VE-sentence has not got the maximum number of VEconstituents, there are no restrictions on the distribution of truthvalues. Every VE-sentence expresses a truth-function of the proposition expressed by its VE-constituents. If it expresses the tautology of the propositions expressed by its VE-constituents, it is said to express a VE-tautology or a truth of logic in the System VE. Which truth-function of the propositions expressed by its VEconstituents a given VE-sentence expresses can always be investigated and decided in a truth-table. This fact constitutes a solution of the decision problem of the System VE. Every VE-sentence has what we propose to call an absolutely perfect disjunctive normal form. This we obtain, if we replace the VE-sentence by a molecular complex of its VE-constituents and transform this molecular complex into the perfect disjunctive normal form. If the conjunction-sentence of the maximum number of VE-constituents occurs in the normal form, we omit it. The absolutely perfect disjunctive normal form shows, with which N

N

N

N

N

N

46

COMBINED MODALITIES

ones of a finite number of mutually exclusive and jointly exhaustive possibilities the VE-sentence expresses agreement and with which ones it expresses disagreement. Agreement with all possibilities is a necessary and sufficient criterion of logical truth. 2. The System E V

By an atomic V-predicate we understand the epistemic operator V followed by either an atomic predicate or a molecular complex of predicates. For instances: V(A & B ) . By an atomic F-predicate we understand the epistemic operator F followed by either an atomic predicate or a molecular complex of predicates. For instance: F A. By a V-predicate we shall understand an atomic V-predicate or an atomic F-predicate or a molecular complex of (atomic) Vand/or F-predicates. For instance: V(A & B) -+F A . By an atomic EV-sentence we understand the operator E followed by a V-predicate. For instance E(VA v FB). By an atomic UV-sentence we understand the operator U followed by a V-predicate. For instance: UV(A & B). By a E V-sentence, finally, we understand an atomic E V-sentence or an atomic UV-sentence or a molecular complex of atomic E V and/or UV-sentences. For instance : EVA & U (FA v VB). The same rules for brackets apply as in the System VE. The System E V studies E V-sentences. The governing principles of the System E V are the same as the governing principles of the System VE. Every E V-sentence has V- and E V-constituents. The procedure by means of which they are derived is mutatis mutundis the same as the one by means of which we derive the E- and VE-constituents of VE-sentences. We need not, therefore, describe this procedure in detail nor illustrate its working. There is, however, one characteristic difference between the Systems V E and E V with regard to the constituents. It will be seen from the following considerations : If A is the sole atomic predicate which occurs in a E V-sentence, there are not more than two V-constituents. viz. FA and F -A,

-

-

47

THE SYSTEN EV

and not more than four (= 27 E V-constituents, viz. E(F A & F A ) andE(PA&NF-A)andE(-FA&F-A) andE(NFA& F A ) . Of the four EV-constituents, however, the first expresses a proposition which is always false in virtue of the Principle of Non-Falsification. For, E ( P A & F - A ) means that there exists a thing - in the appropriate Universe of Discourse which is known to be neither A nor not-A, and this is an impossiN

N

N

bility. Generally speaking, if there are n atomic predicates in a E V sentence, the sentence may have 2" V-constituents and 2(2n)E V constituents. Of the 2(2n)E 8-constituents, however, one expresses an impossible proposition. The existence of an EV-constituent which expresses an always false proposition ought to be taken into account, when truth-values are distributed over the E V-constituents. This is the onlyrestriction upon the distribution of truth-values. Every E V-sentence expresses a truth-function of the propositions expressed by its E V-constituents. If it expresses the tautology of the propositions expressed by its EV-constitueqts, it is said to express a EV-tautology or a truth of logic in the System E V . Which truth-function of the propositions expressed by its E V constituents a given E V-sentence expresses can always be investigated and decided in a truth-table. This fact constitutes a solution of the decision problem of the System E V . Every EV-sentence has what we propose to call an absolutely perfect disjunctive normal form. This we obtain, if we replace the E V-sentence by a molecular complex of its E V-constituents and transform this molecular complex into the perfect disjunctive normal form. If the constituent which expresses an always false proposition occurs in a member of the disjunction, we omit this member, and if its negation-sentence occurs we omit the negationsentence. The absolutely perfect disjunctive normal form shows, with which ones of a finite number of mutually exclusive and jointly exhaustive possibilities the E V-sentence expresses agreement and with which ones it expresses disagreement. Agreement with all possibilities is a necessary and sufficient criterion of logical truth.

48

COMBINED MODALITIES

3. The System V E + E V By a V E + EV-sentence we understand a VE-sentence or an E V-sentence or a molecular complex of VE- and E V-sentences. The System V E + E V studies V E + EV-sentences. Every V E + EV-sentence has E-, V-, VE- and EV-constituents. The E-constituents are the E-constituents of the VE-sentences which occur as constituents in the V E + EV-sentence. The Vconstituents are the V-constituents of the E V-sentences which occur in the V E + EV-sentence. The VE-constituents are the VE-constituents of the VE-sentences, and the E V-constituents are the EV-constituents of the EV-sentences, which occur in the BE + E V-sentence. In the derivation of constituents care should be taken that they axe all “in terms of” the same set of atomic predicates. Thus a V E + EV-sentence, in which there are n atomic predicates, has (at most) 2” E-, 2n V-, 2(2”) VE-, and 2(2n) EV-constituents. Between the VE- and EV-constituents there are logical interdependencies which restrict the distribution of truth-values. The decision problem of the System V E + E V, therefore, is not trivial relative to the decision problem of the Systems V E and E V which was trivial relative to the decision problem of the “underlying” or “basic” Systems V and E themselves. As we have pointed out before, there is a close resemblance in formal structure between the Systems M I , V,, P,and E. This means that the various modal systems of the first order bear a certain resemblance to the most elementary branch of what is also known as quantification theory or to that which we have called the Quantified Logic of Properties. It is characteristic of this branch of quantification theory that it only deals with expressions which in the traditional notation of the so-called predicate calculus - can be brought into a form where no two quantifiers “overlap”. We might, therefore, also call the branch in question the theory of simple quantification. It is characteristic of the systems of combined epistemic and existential modalities that they deal with expressions which - in

THE SYSTEM V E

+

EV

49

our notation - sometimes contain two but never more than two of the operators V, F , E , and U in succession. This fact suggests a formal analogy between these systems and that branch of quantification theory in which one quantifier -in the traditional notation - may “overlap” other quantifiers provided it is not itself “overlapped” by a quantifier. We shall call this branch the theory of double quantification. I have elsewhere tried to show, how the decision problem is to be solved for the theory of double quantification. The solution essentially depends upon the working of two principles concerning the commutability in the order of operators. I have called them the Principles of Existential Symmetry and Asymmetry respectively. The first states that any formula which -in the traditional notation - contains two successive universal or existential operators is equivalent to a formula containing the same operators in the opposite order. The second states that any formula which contains an existential operator immediately succeeded by an universd operator entails a formula in which the universal operator immediately precedes the existential operator. To these two principles there correspond two principles concerning the commutability of existential and epistemic operators. There is the Principle of V-U-Commutation : If it is known that everything possesses a certain property, then everything is known to possess this property. It should be observed that this principle cannot be converted. For, I may know of each thing in a certain Universe of Discourse that i t has a certain property and yet not know that there is not some further thing in the universe that lacks the property. There is further the Principle of E- V-Commutation : If there is a thing which is known to possess a certain property, then it is known that some thing possesses this property. It is obvious that this principle cannot be converted either. For, it may be the case that a certain property is known not to be empty and yet not be the case that it is known of any single thing that it has the property in question. On the Idea of Logical Truth I I .

50

COMBINED MODALITIES

I n order to show the working of the principles for the purpose of solving the decision problem of the System V E + E V we shall start from considerations about the simplest possible case, i.e. from V E + EV-sentences containing one single atomic name (A) of a property. A sentence of this simple structure has (at most) the following four VE-constituents :

-

F(EA & E A ) , for which we introduce the abbreviation fl. 2. P(EA & E - A ) , abbreviated f2. 3. P(- EA & E A), abbreviated f,. 4. P(- E A & E A ) , abbreviated f,. 1.

N

-N

It has further (at most) four EV-constituents: E(PA & P A), for which we introduce the abbreviation el. 2. E ( F A & F A), abbreviated e2. 3. E(- P A & P - A ) , abbreviated e,. 4. E(- F A & N P N A), abbreviated e,. 1.

N

N

N

If truth-values could be freely distributed over the 8 constituents, there would be altogether 256 combinations of them. Actually, however, some combinations are impossible in virtue of the Principle of Non-Falsification, others in virtue of the Principle of V-UCommutation, and still others in virtue of the Principle of E-VCommutation. I n virtue of the Principle of Non-Falsification, the constituent el always expresses a false proposition. (Cf. above p. 47.) This excludes 128 of the 256 combinations of truth-values. We are left whith 128 combinations. I n virtue of the Principle of Non-Falsification, further, the conjunction-sentence of constituents fi & f 2 & f 3 & f4 always expresses a false proposition. (Cf. above p. 45.) This excludes 8 of the 128 combinations of truth-values. We are left with 120 combinations. In virtue of the Principle of Non-Falsification, finally, if (el), e2, %, and e, all express false propositions, then f 4 expresses a false proposition too. For, if the Universe of Discourse is empty, then the

THE SYSTEM VE

+ EV

61

proposition that every part of the universe is empty cannot be known to be false. This excludes 7 of the 120 combinations of truth-values. We are left with 113 combinations. I n virtue of the Principle of V-U-Commutation, the proposition expressed by f l & f z entails the proposition expressed by w e 3 & m e 4 , and the proposition expressed by f l & f 3 entails the proposition expressed by N e2 & e,. This excludes 31 of the 113 combinations of truth-values. We are left with 82 combinations. I n virtue of the Principle of E-V-Commutation, the proposition expressed by (e, v)ez entails the proposition expressed by f z v f4, and the proposition expressed by (el v)e3 entails the proposition expressed by f3 v f 4 . This excludes 20 of the 82 combinations. We are thus ultimately left with 62 combinations of truth-values. They are listed in the truth-table below: N

A T T T T T T T T T T T T T T T T F F F F F F F F

= fa

fs

f4

el

ea

e3

e4

T T T T F F F F F F F F F F F F T T T T T T T

T F F F T T T F F F F F F F F F T T T T T T T T

F T F F T F F T T T T T T T F F T T T T T T T F

F F F F F F F F F F F F F F F F F F F F F F F F

F T T F F F F T T T T F F F F F T T T

F F F F T T F T T F F T T F F F T T F F T T F T

F F F F F F F T F T F T F T T F T F

T

T F F F T

T F

T F T T

52

COMBINED MODALITIES

(continued)

fl

fr

fa

f4

el

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F

T T T T T T T

T T

F F F F F F F T T T T T T

F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F

T T T T T T T T T T T F F F F F F F F F F F F F F F F F F F F

T T T

T T F F F F F F F F F F F T T

T T T T T

T T T T F F F F F F F F F

T F F F F T

T T T T T T F F F F T T

T T

T T T F F

ea

e3

e4

T

T F F T T F F T T F F T T F F F F F T T F F

F T F T F T F T F T F T F

T T F F F F T T

T T F F F T T F F T T T T F F F F F F F

T T T T F F F

F F

T T F T T F F

T T F T F T F T F T F

T T F

T F

T

T

T F F T

F T F

T F F F

T F

T T F

THE SYSTEM VE

+ EV

63

Consider next V E + E V-sentences containing two single atomic names ( A and B ) of properties. A sentence of this structure has (at most) 16 VE-constituents: I. F ( E ( A& B )& E(A & - B ) & E(-A & B )& E(-A & - B ) ) .

. . . . . . . . . .

-

F(-E(A & B) & -E(A & - B ) & - E ( - A & B) & E ( N A & B)). It has further (at most) 16 EV-constituents: 16.

N

1.

E(F(A&B)&P(A&NB)&F(NA&B)&P(-A&NB)).

. . . . . . . . . .

E(- P(A & B) &

-

- --

F(A &

B) &

-

F(- A & B) & P(-A & B)). I n virtue of the Principle of Non-Falsification, the first E V 16.

constituent expresses an always false proposition, the conjunction of all the VE-constituents expresses an always false proposition, and if all the E V-constituents express false propositions, then the last VE-constituent expresses a false proposition. I n virtue of the Principle of V-U-Commutation, the following four implication-sentences express logically true propositions: E N P(A & B). I. FE(A & B ) -+ B ) -+ E F(A & B). 2. F E ( A & 3. F E ( N . 4 & B ) -+ N E N F ( N A & B ) . 4. F E ( N . 4 & B ) --f W E F ( N A & B). The combinations of truth-values which are excluded by these four cases of entailment can immediately be written down, if we transform the four sentences above into their perfect disjunctive normal forms in terms of the V E - and EV-constituents. I n virtue of the Principle of E-V-Commutation, the following fourteen implication-sentences express logically true propositions : B V- A & B V - A & - B ) . 1. E F ( A & B ) + F - E ( A & B ) -+ P - E ( A & B v - A & B v - A & B ) . 2. E F ( A & B v N A & N B). 3. E F ( N A & B ) -+ P N E ( A & B v A & 4. E F ( N A & - B ) --f F N E ( A & B v A & B v - A & B). 5. EF(A & B V A & B ) -+ P E(-A & B v - A & B). 6. E F ( A & B v - A & B ) + 3’ - E ( A & B v - A & B). B ) -+ F E ( A & B V A & B). 7. E F ( A & B V - A &

- -N

-

N

-

N

N

N

- - N

-

-

-

N

N

54

-- - --

-

COMBINED MODALITIES

E F ( A & B V - A & B ) + F - E ( A & B v - A & B). E F ( A & B V A & B ) -+F - E ( A & B V - A & B). 10. E F ( - A & B v N A & B ) --f F N E ( A & B v A . & W B ) . B v - A & B ) + P E(- A & B). 11. E F ( A & B v A & 12. E P ( A & B v A & B v - A & B ) --f F - E ( - A & B). 13. E P ( A & B v - A & B v - A & - B ) + P - E ( A & - B ) . 14. E F ( A & BV &Bv -A & B +P - E ( A & B). The combinations of truth-values which are excluded by these fourteen cases of entailment can be immediately “read off” from the perfect disjunctive normal forms of the 14 implication-sentences in terms of the VE- and EV-constituents. Generally speaking, a V E + EV-sentence which contains n atomic predicates has (at most) 2(2n) VE- and (at most) 2(p) EVconstituents. I n virtue of the Principle of Non-Falsification, one of the E V-constituents is certainly false, not all the VE-constituents true, and one VE-constituent certainly false if all E V-constituents are false. The exclusion of truth-combinations in virtue of the Principle of V-U-Commutation is determined on the basis of 2n cases of entailment, one case for each of the 2” possible conjunctions of n properties and/or their negations. The exclusion of truth-combinations in virtue of the Principle of E-J‘-Commutation, finally, is determined on the basis of 2@)---2 cases of entailment, one case for each of the 2@)-2 possible pairs of “complementary” properties which are presence-functions of n given properties. (Two properties are called “complementary”, if their disjunction is the tautology-property ). The fact that, for the determination of the possible and the impossible combinations of truth-values, we need 2” cases of entailment in virtue of the Principle of V-U-Commutation and 2@)-2 cases of entailment in virtue of the Principle of E-VCommutation requires an explanation. It has to do with the distributability of operators. Consider, e.g., the implication-sentence PE(A & B v A & - B ) + W E F ( A & B v A & - B ) . It expresses a logically true proposition in virtue of the Principle of V - U-Commutation. If the operators are distributed we get from the above sentence the new implication-sentence 8. 9.

N

-

N

N

N

THE SYSTEM VE

+ EV

66

FE(A & B ) & FE(A & N B )-+ E - F ( A & B ) & -E - F ( A & NB). This implication-sentence, however, can be deduced from the t w o “basic” sentences FE(A & B ) -+ E N F ( A & B ) and F E ( A & N B ) -+ E N F ( A & B ) which we mentioned above on p. 53. The reader may satisfy himself that there is not a corresponding possibility of deducing the fourteen implication-sentences mentioned on pp. 53-54 from a fewer number of “basic” cases. 1 Every V E + E V-sentence can be replaced by a molecular complex of VE- and EV-constituents. The V E + EV-sentence thus expresses a truth-function of the propositions expressed by its VE- and E V-constituents. Which truth-function of its constituents the sentence expresses can be investigated and decided in a truthtable, from which have been omitted the combinations of truthvalues which are impossible in virtue of the Principles of NonFalsification, V--U-Commutation, and E-V-Commutation. If the sentence expresses the tautology of the propositions expressed by its constituents, it is said to express a V E + EV-tautology or a truth of logic in the System V E + E V . Every V E + EV-sentence has what we propose to call an absolutely perfect disjunctive normal form. This we obtain in the following way: We replace the sentence by a molecular complex of its VE- and EV-constituents. We transform the complex into its perfect disjunctive normal form. From this normal form we omit those conjunction-sentences of VE- and E V-constituents and/or their negation-sentences, which answer to impossible combinations of truth-values. What remains after we have made these omissions is the absolutely perfect disjunctive normal form of the original V E + E V-sentence. From the absolutely perfect disjunctive normal form of a V E + EV-sentence can be seen with which ones of a finite number of mutually exclusive and jointly exhaustive possibilities the sentence in question expresses agreement (and with which ones it expresses disagreement). Agreement with all possibilities is a necessary and sufficient criterion of logical truth. N

N

N

N

Cf. On the Idea of Logical Truth 11, p. 21.

56

COMBINED MODALITIES

Note. - There are various extensions of the Systems VE, E V , and V E + E V . One type of extensions are those, in which E-sentences are added to the systems. Thus we get three new Systems V E + E, and E V + E, and VE + E V + E. The expressions in these new systems have a greater number of constituents than expressions in the old systems with the same number of atomic predicates. The solution of the decision problem, however, is essentially the same for the old and the new systems. Similarly, we can extend the System M E into a System M E + E.

VII HIGHER ORDER MODALITIES I n this chapter we shall be dealing explicitly only with alethic modalities de dicto. The treatment can easily be extended, mututie mutundis, to alethic modalities de re and to epistemic modalities. There is no corresponding extension to deontic modalities, nor to existential modalities. (Cf. above p. 40.) We shall divide the treatment into two sections according to whether or not certain principles for the reduction of higher order modalities to first order modalities are accepted.

A. THE UNREDUCED

MODALITIES

1. The Systems M,, etc.

By an atomic M2-sentence we shall understand the sentence which we get when we prefix the operator M to a Ml-sentence. (Ml-sentences, it should be remembered, are molecular complexes of atomic Ml- and/or N,-sentences. Since N can be replaced by N M N, we shall throughout this chapter disregard explicit mention of N.) By a M,-sentence we shall understand a molecular complex of atomic M,-sentences. The System M , studies M,-sentences. As to the use of brackets no new rules are needed. Consider an atomic M,-sentence. It consists of the modal operator M followed by a Ml-sentence. Let us assume that the Ml-sentence has been transformed into its absolutely perfect disjunctive normal form. It is then a disjunction-sentence of conjunction-sentences of its Ml-constituents and/or their negation-sentences. I n virtue of the Principle of M-Distribution, the operator M before the Ml-sentence can be distributed. We then get a disjunction-sentence of atomic M2-sentences.Each of these new atomic

58

HIGHER ORDER MODALITIES

M,-sentences consists of the operator M followed by a conjunctionsentence of certain Ml-constituents and/or their negation-sentences. We shall call these new atomic M,-sentences the M,-constituents of the initially given atomic M,-sentence. Consider next a M,-sentence. It is a molecular complex of atomic M,-sentences. Each of the atomic M,-sentences consists of the operator M followed by a Ml-sentence. Each of the Ml-sentences is a molecular complex of atomic Ml-sentences. Each of the atomic M,-sentences consists of the operator M followed by a molecular complex of atomic (M,-)sentences. We make a complete list of all atomic HO-sentenceswhich occur in the original M,-sentence. Thereupon we transform every one of the molecular complexes of atomic M,-sentences into its perfect disjunctive normal form in terms of all atomic M,-sentences in the list. We distribute the operators M which stand in front of the normal forms. Thereupon we transform every one of the Mlsentences, thus obtained, into i t s absolutely perfect disjunctive normal form. We distribute the operators M which stand in front of t h e s e normal forms. The atomic M,-sentences, thus obtained, are the M,-constituents of the original M,-sentence. The distribution of truth-values over the propositions expressed by the M,-constituents of a given M,-sentence is - in the logic of the unrcduced higher order modalities - subject only to the restriction imposed by the Principle of Possibility. (Cf. the example given below on p. 59.) Consequently, every M,-sentence expresses a truth-function of the propositions expressed by its M,-constituents. Which truthfunction it expresses can be investigated and decided in a truthtable. This constitutes a solution of the decision problem of the System M,. A M,-sentence which expresses the tautology of the propositions expressed by its M,-constituents, will be said to express a M,tautology or a truth of logic in the System M,. Every M,-sentence also has a n absolutely perfect disjunctive normal form. This we get by first replacing the M,-sentence in question by a molecular complex of its M,-constituents and trans-

THE UNREDUCED MODALITIES. THJ3 SYSTEMS Ma, ETC.

69

forming the new molecular complex, thus obtained, into its perfect disjunctive normal form. If this perfect disjunctive normal form contains conjunction-sentences which express contradictions in virtue of the Principle of Possibility, we omit them. What remains after we have made these omissions, is the absolutely perfect disjunctive normal form of the original M2-sentence. (The rule for the occurrence of 0-termed disjunctions given above on p. 18 applies mututis mutandis.) A simple example will serve to illustrate the above general train of thought. The Ml-constituents of a Ml-sentence, which contains the sole a. These two Ml-conatomic Mo-sentence a, are Ma and M stituents can be true or false in every combination of truth-values, but one. They cannot both be false. This is proved from the Principle of Possibility. (Cf. above p. 13.) Consequently, the absolutely perfect disjunctive normal form of a Ml-sentence, which contains the sole atomic Mo-sentencea, is a disjunction-sentence of none, some, or all of the three conjunctionsentences Ma & M ,-a and Ma & ,-M - a and wMa&M-a. It will occur to the reader that the first of these three conjunction-sentences expresses the proposition that the proposition expressed by a is contingent, the second the proposition that the proposition expressed by a is necessary, and the third the proposition that the proposition expressed by a is impossible. Contingency, necessity, and impossibility form a set of mutually exclusive and jointly exhaustive alternatives as regards the modal character of a given proposition. The M,-constituents of a M2-sentence, which contains the sole atomic Mo-sentencea, are M(Ma & M ,-a ) and M(Ma & N M ,-a) and M(- M a & M - a ) . The first expresses the proposition that it is possible that the proposition expressed by a is contingent, the second the proposition that it is possible that the proposition expressed by a is necessary, and the third the proposition that it is possible that the proposition expressed by a is impossible. The crucial question from the point of view of the decision problem of the system which we are studying is, whether or not there are

-

60

HIGHER ORDER XODALITJES

restrictions to the distribution of truth-values over the propositions expressed by the three M,-constituents. It follows from the Principles of Possibility and M-Distribution that not all the three-propositions can be false. The proof is as follows: Ma & M N a v Ma & N M - a v -Ma & M aexpresses a M,-tautology and hence a true proposition. Therefore, in virtue of the Principle of Possibility, M(Ma & M N a v Ma & N M a v Ma & M N a ) also expresses a true proposition. I n virtue of the Principle of M-Distribution, this proposition can be written M(Ma&M-a) v M ( M a & N M N a ) v M(-Ma&dd-a). Since this expresses a logically true proposition, its negationsentence expresses a logically false proposition. But the false proposition in question can also be written N M(Ma & M a) & N M(Ma & M a ) & M ( N Ma & M a). The proposition that all the three propositions expressed by the M,-constituents are false is thus itself a logically false proposition., (It will occur to the reader that this result is an immediate consequence of the exhaustive nature of the alternative contingentnecessary-impossible as regards the modal character of propositions.) In the unreduced System M, the above is the only restriction on the distribution of truth-values over the propositions expressed by the three M2-constituents under consideration. Consequently, the absolutely perfect disjunctive normal form of a M,-sentence, which contains the sole atomic Mo-sentence a, is a disjunction-sentence of none, some, or all of seven conjunctionsentences of M2-constituents and/or their negation-sentences. If we had started from two atomic Mo-sentences a and b, we should have had to consider 4 (= 2 7 Ml-constituents and 15 (= 212')- 1) conjunction-sentences of them and/or their negationsentences. The number of M,-constituents would' thus have been 15 and the number of conjunction-sentences of them and/or their negation-sentences, which may occur in the absolutely perfect disjunctive normal forms of M,-sentences, would have been not less than 32767 (= 216-1). Generally speaking, if there are n atomic Mo-sentences, there

-

N

N

N

N

N

N

N

THE U N R E D U C E D MODALITIES. T H E SYSTEMS

zhn,

ETC.

61

would be 2” Ml-constituents and 2(2n)-1 M,-constituents to be considered. We can now easily ascend in the hierarchy of higher order modal systems. We shall only say a few words about the “general” System M,. Atomic M,-sentences consist of the modal operator M before a M,-,-sentence. M,-sentences are molecular complexes of atomic M,-sentences. The System M , studies M,-sentences. Every M,-sentence has a certain number of M,-constituents and an absolutely perfect disjunctive normal form in terms of its X,constituents. The M,-constituents consist of the modal operator M followed by a conjunction-sentence of certain M,-,-constituents and/or their negation-sentences. Every Ma-sentence expresses a truth-function of the propositions expressed by its M,-constituents. Which truth-function it expresses can be seen from a truth-table or from its absolutely perfect disjunctive normal form. This constitutes a solution of the decision problem for the System M,. A M,-sentence which expresses the tautology of the propositions expressed by its M,-constituents is said to express a M,-tautology or a truth of logic in the System M,. The following consequence of the Principle of M-Tautology for higher order modal systems is worth observation : If a proposition is a M,-tautology, then the proposition that the proposition in question is necessary is a M,+,-tautology. 2. The Systems ZM,, etc.

For atomic Mo-sentences we could introduce the term atomic ZM0-sentences. For Mo-sentences we could introduce the term ZMo-sentences. For atomic Ml-sentences we could introduce the term atomic ZMl-sentences. For Ml + Mo-sentences we could introduce the term ZMlsentences.

02

HIGHER ORDER MODALITIES

The System M, might also be called the System ZM,,and the System Ml + M, might also be called the System ZMl. By an atomic EM,-sentence we shall understand the operator M followed by a ZMl-sentence. By a ZM,-sentence we shall understand a molecular complex of atomic ZM2-sentencesand/or atomic ZM,-sentences and/or atomic ZMo-sentences. The System ZM, studies ZM,-sentenccs. (It follows from the “and/or” in the definitions of the expressions of the system that the System Z M , includes as fragments of itself the System M, (truth-logic of propositions), the System M,, the System M,, and the System Ml + Mo). Consider a ZM2-sentence. - It has three kinds of constituents, viz. ZM,-constituents, ZMl-constituents, and ZM,-constituents. The constituents we find in the following way: We make a complete list of all atomic ZMo-sentences which occur in the ZM,-sentence (in molecular complexes either preceded or not preceded by the operator M). We replace the ZM,-sentences everywhere in the ZM,-sentence by their perfect disjunctive normal forms in terms of all the atomic ZMo-sentences in our list. Thereupon we distribute the operators M which stand immediately in front of ZMo-sentences. We next make a complete list of all atomic ZMl-sentences which occur in the ZM,-sentence after the above transformations (in molecular complexes either preceded or not preceded by the operator M). We replace the ZMl-sentences everywhere in the ZM,-sentence by their absolutely perfect disjunctive normal forms in terms of all the atomic ZMl-sentences and atomic ZMo-sentences of our list. (Cf.above p. 24.) Thereupon we distribute the operators M which are in front of ZMl-sentences. We finally make a complete list of all atomic ZM,-sentences which occur in the ZM2-sentence after the above two sets of transformations. We have thus compiled three lists: the first contains the ZMoconstituents, the second the ZMl-constituents, and the third the ZM,-constituents of the ZM,-sentence under consideration.

63

THE UNREDUCED MODALITIES. THE: SYSTEMS z h f 2 , ETC.

If the number of ZMo-constituents is n, the number of ZMlconstituents is at most 2", and the number of ZM,-constituents at most 2(2n+n-l). Thus, if there is only one ZJf,,-constituent a, there may be two (= 2l) ZM,-constituents, viz. Ma and M a, and four (= 2(2'+1-1)) a & a) and M(Ma & M -a & ZM,-constituents, viz. M(Ma & M -a) and M ( N a & M -a &a) and M(- Ma & M -a & -a). It will occur to the reader that the first of the ZM,-constituents means that the proposition expressed by a is possibly contingently true, the second that it is possibly contingently false, the third that it is possibly necessarily true, and the fourth that it is possibly necessarily false (possibly not possibly true). It will also occur to the reader that the proposition expressed by the first ZMl-constituent is the disjunction of the propositions expressed by the first and the third ZM,-constituents. For Ma expresses the same proposition as M(Ma & M a&av Ma & N M a & a) which expresses the same proposition as M(Ma & M a & a) v M(Ma & M a & a). If a proposition is possibly true, then it is either possibly contingently true or possibly necessarily true, and vice versa. Similarly, the second ZM1-constituent expresses the disjunction of the propositions expressed by the second and fourth ZM,-constituents. If a proposition is possibly false, then it is either possibly contingently false or possibly necessarily false (impossible), and vice versa. Generally, the propositions expressed by the ZMl-constituents are disjunctions of propositions expressed by the ZM,-constituents. Hence the ZM,-constituents can be replaced by disjunctionsentences of ZM2-constituents.

-

-

N

--

- -

-

The distribution of truth-values over the propositions expressed by the ZMo- and the ZM,-constituents of a given ZM,-sentence is - in the logic of the unreduced higher order modalities - subject only to the restriction imposed by the Principle of Possibility. In order to illustrate the working of this restrictive principle, we give below a table showing the distribution of truth-values over the propositions expressed by the one ZMo-constituent and the four

64

HIGHER ORDER MODALITIES

ZM2-constituents of a ZM2-sentence with only one atomic ZMosentence a. a T T T T T T T T F F F F F F F F

M ( M a& M-a&a) T T T T T

T T F T T F F T T F F

M ( M a& M-a&-a)

-

T T F F T T F F T T T T T T

M(Ma& -M-a&a) T T T T

F F F T T

F T F T F

M(-Ma& M-a &-a) T

F T F T F T F T T T T

F F

T

T

F

F

F

T

The argument which takes us to the above table can be stated as follows: There are 4 possible combinations of truth-values in the propositions expressed by a and Ma and M-a (the ZM0-and ZMlconstituents). They are TTT and TTF and FTT and FFT. Consider first the combination TTT. I n this case the first ZM2-constituent ought to express a true proposition in virtue of the Principle of Possibility. Remembering that Ma expresses the same proposition as the disjunction-sentence of the first and the third, and M w a as the disjunction-sentence of the second and fourth ZM2-constituent, we exclude the 2 combinations FTF and F F F in the remaining ZM2-constituents. Consider next the combination TTF. I n this case the third ZMaconstituent ought to express a true proposition. I n virtue of the above fact about the ZMl-constituents, we exclude the 6 combinations TTT and TTF and TFT and FTT and FTF and FFT in the remaining ZMa-constituents.

THE U N R E D U C E D MODALITIES. T H E SYSTEMS ZM,, ETC.

65

Consider thereupon the combination FTT. I n this case the second ZM,-constituent ought t o express a true proposition. I n virtue of the above fact about the ZMl-constituents, we exclude the 2 combinations FFT and F F F in the remaining ZM2-constituents. Consider finally the combination FFT. In this case the fourth ZM2-constituent ought to express a true proposition. I n virtue of the above fact about the ZMl-constituents, we exclude the 6 combinations TTT and T T F and T F T and T F F and FTT and FFT in the remaining EM2-constituents. The above exclusions having been made, we are left with the 16 combinations listed in the truth-table. Any ZM2-sentence expresses a truth-function of the propositions expressed by its ZMo-( ,ZMl-,)and ZM,-constituents. Which truthfunction i t expresses can be investigated and decided in a truthtable. This fact constitutes a solution of the decision problem of the System ZM,. A ZM,-sentence which expresses the tautology of the propositions expressed by its ZMo-( ,ZMl-,)and ZM,-constituents, will be said to express a ZM,-tautology or a truth of logic in the System ZM,. Every EM,-sentence has what we propose to call an absolutely perfect disjunctive normal form. This we get by replacing the ZM,-sentence in question by a molecular complex of its ZMoand ZM,-constituents and then transforming the new ZM,-sentence, thus obtained, into its perfect disjunctive normal form. If this perfect disjunctive normal form contains conjunctionsentences which express contradictions in virtue of the Principle of Possibility, we omit them. What remains when we have made these omissions, is the absolutely perfect disjunctive normal form of the original ZM,-sentence.

A few words should be added about the “general” System ZM,,. ZM,-sentences are molecular complexes of atomic ZM,-, and/or atomic ZhZ,,-l-, and/or . . . and/or atomic ZMo-sentences. The System ZM,, studies ZM,,-sentences. Every ZM,,-sentence has n + 1 kinds of constituents, viz. ZMo-, ZMl-, and . . . and ZM,-constituents. Every ZMl-con-

60

HIGHER ORDER MODALITIES

stituent, however, is the disjunction of some ZM2-constituents, every ZM2-constituentof some ZM3-constituentsand . . . and every ZM,-,-constituent of some ZM,,-constituents. Hence only the ZMo- and the EM,,-constituents are of independent importance, i.e. need enter into the absolutely perfect disjunctive normal form of the ZM,,-sentences. Every ZM,,-sentence expresses a truth-function of the propositions expressed by its ZMo- (,ZMl-, . . .) and ZM,-constituents. Which truth-function it expresses can be seen from a truth-table or from its absolutely perfect disjunctive normal form. This constitutes a solution of the decision problem of the System ZM,,. A ZM,-sentence which expresses the tautology of the propositions expressed by its ZMo- and ZM,,-constituents is said to express a ZM,,-tautology or a truth of logic in the System ZM,,.

B. THE REDUCED

MODALITIES

So far we have assumed that the higher order modalities are subject to the same governing principles as the first order modalities and no others. This assumption, however, can be questioned in several ways. We shall here only consider two. 1. The Systems ZM;, etc.

The Principle of Possibility establishes a relation of entailment “upwards” from atomic M,,-sentences to atomic M,,+,-sentences or, which comes to the same, a relation of entailment “downwards” from atomic N,,+,-sentences to atomic N,-sentences. E.g., if the proposition expressed by Ma is true, then also the proposition expressed by MMa, and if the proposition expressed by N N a is true, then also the proposition expressed by Na. It has been suggested by many logicians that the same relation holds in the reverse directions too, i.e. from atomic M,,+,-sentences “downwards” to atomic M,,-sentences and from atomic N,,-sentences “upwards” to atomic N,,+,-sentences, provided, however, tha,t n is greater than 0. This suggestion would mean, e.g., that if the proposition expressed by MMa is true, then also the proposition

THE REDUCED MODALITIES. THE: SYSTEMS

ZM;,

ETC.

67

expressed by Ma, and if the proposition expressed by Nu is true, then also the proposition expressed by "a. These two suggestions about the reversal of the order of entailment both come to the same thing. (Cf. below pp. 69-71.) We can, therefore, dispense with necessity (the operator N ) and speak only of possibility (the operator M ) . The above suggestions can then be laid down as a First Principle of Reduction: If it is possible that a certain proposition is possible, then the proposition in question i s possible. By the System Z M ; we shall understand the System ZM,, to which has been added the First Principle of Reduction. Similarly, we define the System ZM;, etc. The derivation of constituents in the System ZJf; is the same as in the System EM2. Truth-tables and absolutely perfect disjunctive normal forms can be used for decision purposes in the System ZM; too. The truth-tables in the System ZM; contain a lesser number of combinations of truth-values than the truthtables in the System ZM,, and the absolutely perfect disjunctive normal forms in the System Z M ; are shorter than the absolutely perfect disjunctive normal forms in the System ZM2. This is so because of the restrictions imposed by the First Principle of Reduction. I n order to illustrate the working of this new restrictive principle, we return to our table on p. 64 for the distribution of truth-values over the propositions expressed by the one ZMo-constituent (= ZMi-constituent) (, the two ZM,-constituents (= ZMi-constituents),) and the four ZM,-constituents ( =ZMi-constituents) of a ZM,-sentence with only one atomic ZM0-sentence a. As will be remembered, the first ZMl-constituent is the disjunction of the first and third ZM,-constituents. I n virtue of the First Principle of Reduction, on the other hand, Ma expresses the same proposition as M M a which expresses the same proposition as M ( M a & M - a & a ) v M ( M a & M - a & - a ) vM(Ma&M - a & a). Thus the first ZM,-constituent is, in the case under consideration, also the disjunction of the first and the second and the third ZM2-constituent. This will preclude the second ZM,-con-

68

HIGHER ORDER MODALITIES

stituent from being true, if the first and third are both false. Of the combinations listed in the table on p. 64, in other words, the one combination FFTFT has to be omitted. By a similar argument we show that, because of the identity of the propositions expressed by M a and M M N a, the first ZM2constituent cannot be true, if the second and fourth are both false. This excludes the one combination TTFTP. We thus get a table listing 14 combinations of truth-values: N

= a

T T T T T T

T F F F F F F F

M ( M u& M-u&u)

T T T T T T F T T F T T F F

M ( M a& M-u&-u)

T T F T

T F F T T T

T T T F

M(Ma& -M-a&a)

T T T F F F T T F T T F T F

M(-Ma& M-a&-a)

T F T T F T F T T T F F F T

I n the case, when there are 2 ZMo-constituents, 4 CMl-constituents, and 32 ZM2-constituents, we should have to consider 4 applications of the First Principle of Reduction for the purpose of excluding combinations of truth-values. Generally speaking, there are as many applications of the reduction principle to be considered as there are ZMl-constitucnts. There is a “mechanical”, though technically somewhat laborious procedure for finding the impossible combinations of truth-va,lues by means of transformations of expressions (without the use of truth-tables). It can be described with sufficient generality in the following way : Let there be 2 atomic sentences (ZMo-constituents),a and b. We form the 4 (= 2 7 implication-sentences MM(a & b ) + M ( a & b ) and

THE R E D U C E D MODALITIES. T H E SYSTEMS Z M i , ETC.

-

-

69

MM(a & - b ) -+ M(a & - b ) and MM(-a & b ) -+ M(-a & b ) and MM(- a & b ) -+ M(- a & b). These 4 sentences express all that follows from the First Principle of Reduction as regards possibility in the range of the propositions expressed by the two sentences a and b, provided modalities are not of higher order than the second. Consider, e.g., the sentence MMa + Ma which also expresses a consequence of the First Principle of Reduction. If we replace a by a & b v a & b and distribute the operators, we get MM(a & b ) v MM(a & b ) --f M(a & b ) v M(a & b ) . The last expression, however, is obtainable from the first two of the above 4 implication-sentences according to rules of the truth-logic of propositions. The converse is not true; we cannot derive the two first implication-sentences from the longer expression. We hereupon form the conjunction-sentence of the 4 implicationsentences. This conjunction-sentence we transform into its absolutely perfect disjunctive normal form in the System ZM,. Of the totality of 2(2+82)conjunction-sentences of ZMo-, and ZM2constituents and/or their negation-sentences those which do not appear in the normal form answer to impossible combinations of truth-values. A ZM,-sentence which in the System ZMz expresses the tautology of the propositions expressed by its ZMo- (,ZMl-,) and ZMz-constituents is said to express a ZMi-tautology or a truth of logic in the System ZMi. If from the absolutely perfect disjunctive normal form of a ZM,-sentence in the System ZM, we omit those conjunctionsentences of the ZMo- and ZM,-constituents and/or their negation-sentences which answer to impossible combinations of truth-values in virtue of the First Principle of Reduction, we obtain the absolutely perfect disjunctive normal form of the ZM2sentence in question in the System ZM;. The First Principle of Reduction is “reflected” in the following ZMi-tautologies : i. MMa -+ Ma. Since Ma -+ MMa expresses a ZM,-tautology (and consequently a ZMi-tautology too) we have

--

-

70

HIQHJ3R ORDER MODALITIES

-

ii. MMa +Ma.

--

a for a and N for M and apply the If in i we substitute rules of double negation and contraposition, we get iii. Nu +-NNa. Since N N a --f N u expresses a ZM,-tautology, we have iv. Nu

e NNa.

The derivation of constituents in the System ZM; differs from the derivation of constituents in the System ZM; only in that we ought to take into account the effect of the First Principle of Reduction on the absolutely perfect disjunctive normal forms of ZM,-sentences. This means that, while the ZMi-constituents are the same’as the ZM0-constituents, the ZMI-constituents the same as the ZMl-constituents, and the ZMi-constituents the same as the ZM2-constituents, the ZM;-constituents are not the same as the ZM3-constituents, but fewer in number. Their number is the same as the number of possible distributions of truth-values over the ZMd- and ZMi-constituents. In the case, e.g., when a is the only EM:-constituent, there are 14 ZMi-constituents. They can be immediately “read off” from our truth-table on p. 68 as follows:

-

- -

1.

M(M(Ma & M a & a ) & M(Mu & M M(Mu&-M-a&a) & M (-Ma&M-a&

a & a) & -a) &a).

7.

M(-M(Ma&M-a&a) & -M(Ma& M-a&-a) &a). -M-aka) &-M(-Ma&M-a&-a)

& M(Ma&

. . . . . . . . . . 14.

M(-M(Ma&M-a & a )& -M(Ma & M-a& -a) & -M ( M a& WM-a &a)& M(-Ma & M-a& --a) &-a).

Having derived the constituents, the First Principle of Reduction is used for finding the impossible combinations of truth-values. This done, truth-tables or transformations into an absolutely perfect disjunctive normal form can be used for deciding whether any given ZM,-sentence expresses a ZMi-tautology, or not.

THE REDUCED MODALITIES. THE SYSTENS ZM;, ETC.

71

Is the First Principle of Reduction true or not ? An appeal to our logical “intuitions” appears not to be very helpful here. We shall not attempt to answer the question. We shall only make a few scattered remarks which are also intended as warnings against an overhasty acceptance of the reduction principle in question. The statement that, if a proposition is possibly possible, then it is possible, means the same as the statement that if a proposition is necessary, then it is necessarily necessary. It is noteworthy that the latter statement seems to possess a greater intuitive plausibility than the former. This may be so, because we tend to confuse the doubtful statement that if a proposition is necessary, then it is necessarily necessary, with the correct statement that it is necessary that, if a proposition is necessary, then it is necessary. The confusion, in other words, is between the meaning of N a - t N N a , which is a ZMi-tautology, and the meaning of N ( N u + N a ) , which is a ZM,-tautology. This confusion is an instance of what we propose to call for historical reasons the Aristotelian fallacy. The following point about epistemic modalities may be useful to consider: The equivalent t o the First Principle of Reduction in the logic of epistemic modalities states that, if it is not known that a certain proposition is known to be false, then the proposition in question is not known to be false. Or which means the same: if it is known that a proposition is true, then it is known that it is known that the proposition is true. This is a dubious assertion from the point of view of epistemology, since it obliterates the distinction between knowing and knowing that one is knowing. 2. The Systems ZM;, etc. Some logicians have suggested the following Second Principle of Reduction :

If a certain proposition is possible, then it is necessary that the proposition in question is possible. By the System ZM; we shall understand the System ZM2, to which has been added the Second Principle of Reduction. Similarly, we define the System ZM;, etc.

72

EIUHER ORDER MODALITIES

It will occur to the reader that the First Principle of Reduction is entailed by the Second Principle of Reduction. This is seen in the following way : Substituting in the above formulation of the Second Principle of Reduction the definition of “necessary” in terms of “possible” and of “possible” in terms of “necessary”, we get the following alternative formulation of the principle: If it is possible that a certain proposition is necessary, then the proposition in question is necessary. Since necessity entails truth, the “if-then” of either formulation of the principle can be strengthened into an “if and only if”. Thus, it is necessary that a certain proposition is possible, if and only if the proposition is possible, and possible that a proposition is necessary, if and only if the proposition is necessary. It follows by substitution that it is possible that it is necessary that a proposition is possible, if and only if it is necessary that the proposition is possible. Since, on the other hand, it is necessary that the proposition is possible, if and only of the proposition is possible, we conclude that it is possible that a proposition is possible, if and only if the proposition is possible. This gives us the First Principle of Reduction. The derivation of constituents in the System ZM; is the same as in the System ZM2. As the reader will remember, it was, in the System ZM2, possible to express the propositions expressed by the ZMl-constituents in terms of the ZM2-constituents. Hence the ZM,-constituents were here of no independent importance. (Cf. above p. 6 3 . ) It is characteristic of the System Z M i that it is possible to express the propositions expressed by the ZM2-constituents (= ZMi-constituents) in terms of the ZMl-constituents (= EM,”constituents). Hence the ZM2-constituents are here of no independent importance. We show this first for the one ZMo-constituent, the two .EMlconstituents, and the four ZM2-constituents of a ZM,-sentence with only one atomic ZMo-sentence a. Consider the first ZM2-constituent M ( M a & M a & a). The

-

THE REDUUED MODALITIES. THE SYSTEMS

ZMi,

73

ETC.

proposition which it expresses entails the proposition expressed by MMa & M M - a which, according to the First Principle of Reduction, is the same as the proposition expressed by N a 85 iM N U . Thus if the proposition expressed by the first .ZM,-constituent is true, the propositions expressed by the first and second ZMlconstituents are both true. We have now to show that the converse also holds. As we know, M a expresses the aame proposition as M ( M a & M - a & a ) v M ( M u & - M N u & u ) . But M ( M u & N M a & a ) expresses the same proposition as M N a which, according to the Second Principle of Reduction, expresses the same proposition as Nu or M N a. Thus if both the ZMl-constituents express true propositions, it cannot be the case that the third CM,-constituent expresses a true proposition and hence must be the case that the first ZM,-constituent expresses a true proposition. Herewith it has been shown that the first ZM2-constituent expresses the same proposition as the conjunction of the first and the second ZMl-constituent. By an exactly similar argument we show that the second ZM,constituent expresses the same proposition as tho conjunction of the first and the second ZN1-constituent. (It can be concluded that the first and the second CM2-constituents actually express the same proposition in the System N

N

- -

ZMi.).

Consider the third ZM,-constituent M ( M a & M a & a). As was already seen, it expresses the same proposition as -M-a. But M a expresses the same proposition as Ma SZ; M a. Thus the third CM,-constituent expresses the same proposition as the negation of the second ZMl-constituent or, which comes to the same thing, as the conjunction of the first ZMl-constituent with the negation of the second. By an exactly similar argument we show that the fourth ZM,constituent expresses the same proposition as the negation of the first ZM,-constituent or, which comes to the same thing, as the conjunction of the negation of the first ZMl-constituent with the second ZMl-constituent . N

N

N

-

74

HIGHER ORDER MODALITIES

The above arguments can easily be generalized so as to apply to ZM2-sentences with any number of atomic ZMo-sentences. Consider, for example, the case when we have to deal with the ZNo-sentences, a and b.-In this case there are 4 ZJfl-constituents and 32 ZM,-constituents. (Cf. above p. 63.) The first ZM2-constituent is ;ET(M(u&O) & M ( u& -6) & M ( -a & b ) &M(-a& -b) &a&b). The propositioii which it, espresses entails the proposition expressed by M M ( n 8: b ) &; MflL!(n.ck b ) & J I M ( - a & b) & M M ( - a & b) which, in virtue of the First Principle of Reduction, expresses the same proposit’,ion as J l ( a 6; 6 ) & M ( a & b ) & a ( - a & b ) & N ( - a & - b ) or the conjunction-sentence of the ZMl-constituents. We have now to show that the proposition expressed by this conjunction-sentence entails the proposition expressed by the first ZM2-constituent. -W(u & 6 ) expresses the same proposition as the disjunctionsentence of the 8 ZM2-constituents X ( M ( a & b ) 6 ; M ( a & - b ) S; M ( - a & b ) & M(-a&-6)&a&b) and M(M(a&b)&M(a&-b)&N(-a&b)&-M(-a&-b)&a&b) and . . . R nd M (M ((X $ 6 ) 6;-~V((X &; -6) & -M( -a &b)& -N( ~a& 4) &a&b). Thus, if all the ,rM,-constituents express true propositions, at least one of the 8 ZMz-constituents, just mentioned, must express a true proposition. The second one of them, however, cannot ospress a true proposition. For the proposition which it expresses M (-a&-b) which, entails the proposition expressed by M in virtue of the Second Principle of Reduction, entails the proposition expressed by -M (-a&-b) and this is incompatible with the truth of the proposition expressed by the conjunctionsentence of all the ZMl- constituents. In an exactly similar manner we show that neither the third, nor the fourth, nor . . . nor the eight of the propositions expressed by the ZM2-constituents in question can be true. Hence the first of them must be true. Herewith has been shown that the first ZM,-oonstituent expresses the same proposition as the conjunction-sentence of all the ZMlconstituents. The general nature of the “trick”, by means of which it is shown

-

-

-

-

THE REDUCED MODALITIES. TEE SYSTEMS & I : ETC. ,

75

in the System Z M i that any given ZM,-constituent of any given ZM,-sentence expresses the same proposition as a certain conjunction-sentence of ZMl-constituents and/or their negationsentences, should now be obvious to the reader. Given a EM,-sentence. Replace it by a molecular complex of its ZMo-, EMl-, and ZM,-constituents. Replace the ZM2-constituents by conjunction-sentences of ZMl-constituents and/or their negationsentences. We thus obtain a ZMl-sentence. It expresses the same proposition as the given ZM,-sentence in the System Z M i . Its absolutely perfect disjunctive normal form (cf. above p. 24) will be called the absolutely perfect disjunctive normal form of the EM,-sentence in the System ZM;. If and only if the ZMl-sentence expresses a truth of logic in the System ZMl, the ZM,-sentence expresses a truth of logic in the System ZMi. Thus the decision problem for the System Z M i can be said to be “reducible” to the decision problem for the System ZMl. The Second Principle of Reduction and its implications are “reflected” in the following ZMi-tautologies :

i. Ma -+ NMa. Since NMa --f Ma expresses a ZM,-tautology (and hence also a ZMi-tautology) we have ii. Ma tf NMa.

-

N for M and apply the If in ii we substitute a for a and rules of double negation and contraposition, (cf. abovep. 69f.), we get iii. Na tf MNa. N

N

Since a --f Ma expresses a ZMl-tautology (and hence a ZMgtautology too) and Ma ++NMa expresses a ZMi-tautology, we have iv. a --f NMa. This ZMi-tautology is sometimes referred to in the literature as the Axiom of Brouwer. If in iii we substitute Ma for a, we get in combination with ii v. MMa f~Ma.

78

HIUEER ORDER MODALITIES

From this we obtain as before (cf. above p. 69f.) vi. Nu e NNa. Combining ii and v on the one hand and iii and vi on the other hand we, finally, get vii. M M a ++NMa and

viii. MNa

tf

NNa.

The derivation of constituents in the System Z M ; differs from the derivation of constituents in the System ZM; in that we ought to take into account the effect of the Second Principle of Reduction on the absolutely perfect disjunctive normal forms of ZMz-sentences. Since, however, this effect is to “reduce” those normal forms to absolutely perfect disjunctive normal forms of ZMl-sentences, i t follows that the ZMa-constituents are the same as the ZMi-constituents, which are the same as the ZM2-constituents of ZMz-sentences. Hence the decision problem for the System ZM; “reduces” t o that of the System Z M ; which, as we have seen, “reduces” to that of the System ZNl. In a, similar manner we find that, in the “general” System ZM:, all constituents of higher order than the second of any given ZMi-sentence are the same as its constituents of the second order. Thus the decision problem for the “general” System ZM: also “reduces” to that of the System ZMl. I n assessing the intuitive plausibility of the Second Principle of Reduction we must not confuse the statement that if a proposition is possible, then it is necessarily possible, which is doubtful, with the statement that it is necessary that, if a proposition is possible, then it is possible, which is an obvious tautology. Similarly, in assessing the intuitive plausibility of the so-called Axiom of Brouwer we must not confuse the statement that if a proposition is true, then it is necessarily possible, which is doubtful, with the statement that i t is necessary that, if a proposition is true, then it is possible, which is an obvious tautology.

THE REDUCED MODALITIES. THE SYSTEMS

,?hi, ETC.

77

It may be helpful to consider epistemic modalities. The equivalent to the Second Principle of Reduction for epistemic modalities states that, if a proposition is not known to be false, then it is known that the proposition is not known to be false. This deduction of knowledge from ignorance appears plainly unacceptable, and should be considered at least a strong warning against assuming the Second Principle of Reduction to be true for the alethic modalities. Note. - The fact that, in the System ZM:, every ZM,-sentence (m > 1) expresses the same proposition as a ZMl-sentence and that, accordingly, the decision problem “reduces” to the decision problem of the System ZMl corresponds to a well-known feature of Lewis’s S 5. Cf. W. T. Parry, Zum Lewk8chen Aussagenkalkiil in Ergebnisse e k e s mathematischen Kolloquiums 4, 1933 and M. Wajsberg, Ein erweiterter Klussenkalkul in Monatshefte fiir Mathematik und Physik 40, 1933. Cf. also Appendix 11.

APPENDIX I THE MODAL SYLLOGISM We shall distinguish between existential and modal syllogisms. The two premisses and the one conclusion of an existential syllogism can be described as sentences which assert or deny the existence of either the conjunction of two properties or the conjunction of one property with the negation of another property. The sentences in question are thus either “particular and affirmative”, i.e. of the type “E(&)”,or “particular and negative”, i.e. of the type “ E ( &N ) ” , or “universal and negative”, i.e. of the type E($)”, or “universal and affirmative”, i.e. of the type E ( & N)”. (Blanks to be filled by atomic names of properties.) There are normally three atomic names of properties involved in a syllogism, known as the major, the middle, and the minor “term”. The first premiss contains the major and the middle, the second premiss contains the minor and the middle, and the conclusion contains the major and the minor. The terms may be taken in either order in the premisses, but in the conclusion the order is always the same. There are thus four different arrangements of the order of terms in a syllogism,four different “syllogistic figures”. Considering that there are four possible arrangements of the terms and four possible types of premisses and conclusion, there are in all 256 “possible” syllogisms, 64 in each figure. Of them, however, only a small minority (15) are valid syllogisms. The theory of the existential syllogism falls within the scope of the Quantified Logic of Properties or the System E . Since the decision problem of the System E can be effectively solved, there are “mechanical” methods for testing the validity of the various syllogisms. One such method of peculiar simplicity is by means of so-called Lewis Carroll diagrams. It can be described as follows: The three terms of the syllogism can be used for a sub-division of the individuals in the Universe of Discourse under consideration “ N

“ N

A&-B&C

A&-B&-C

A&B&C

-A&B&C

A&B&-C

-A&B&-C

-A&-B&C

-A&-B

&4

The (propositions expressed by the) premisses and the conclusion of a syllogism can be “pictured” in the diagram. The“picture” obeys the following three rules: 1. If the sentence (premiss or conclusion) asserts existence we insert the sign + in 8 m e sub-division of the area which pictures the existing property in question. 2. If the sentence denies existence we insert the sign - in every sub-division of the area which pictures the empty property in question. 3. The signs + and - must not occur within the same subdivision. The syllogism is valid, if every picture of both its premisses constitutes a picture of its conclusion as well. If a premiss or a conclusion in an existential syllogism is asserted or denied to express a possible proposition, we shall call the premise or conclusion in question “modalized”. A “modalized” premiss or conclusion is thus a sentence of one of the following eight types: “ME(&)” or ME( &)” or “ME(& N)” or “ N ME( & -)” or ‘‘M E( &)” or “- M E(&)” or “ M E “ N

N

-

-

80

APPENDIX I

(a N)” or “ N M N E(& N)”. (Blanks to be tilled by atomic names of properties.) If both the premisses and the conclusion in an existential syllogism are modalized, we get a pure modal syllogism. If either both the premisses but not the conclusion, or one of the premisses and the conclusion, or one of the premisses but not the conclusion, or only the conclusion in an existential syllogism are modalized, we get a mixed modal syllogism. Considering that there are two ways of modalizing a premiss or a conclusion, viz. so as to assert and so as to deny possibility, there are altogether 2048 possible pure modal syllogisms, 512 in each of the four figures. The number of possible mixed modal syllogisms is 4608, i.e. 1152 in each figure. The theory of the pure modal syllogism falls within the scope of the System M E , i.e. the system which studies combinations of alethic modalities de dicto and existential modalities. The theory of the mixed modal syllogism falls within the scope of the mixed System M E + E . (Cf. above p. 56.) The System M E + E includes the System M E and the System E and hence also the theory of the pure modal syllogism and of the existential syllogism. Since the decision problem of the Systems N E and M E + E can be effectively solved, there are “mechanical” methods of testing the validity of the various possible pure and mixed modal syllogisms. I n fact, Lewis Carroll diagrams can be used, provided that we adopt the following rules for the “picturing” of premisses and conclusions : 1. If the sentence (premiss or conclusion) asserts possible existence, weinsert + in some sub-division of the area which pictures the possibly existing property in question. 2. If the sentence denies possible existence, we insert - - - in every sub-division of the area which pictures the not possibly existing property in question. 3. If the sentence asserts existence, we insert + + in some subdivision of the area which pictures the existing property in question.

TEE MOD&

81

SYLLOGISM

4. If the sentence denies existence, we insert - - in evey subdivision of the area which pictures the empty property in question. 6. If the sentence asserts possible non-existence, we insert - in every sub-division of the area which pictures the possibly empty property in question. 6. If the sentence denies possible non-existence, we insert + + + in 8ome sub-division of the area which pictures the necessarily existing property in question. 7. If there are three + in a sub-division, there must not be any - in it. 8. If there are three - in a sub-division, there must not be any in it. 9. If there are two + in a sub-division, there must not be two in it, and vice versa. (It should be noted that if there are three + (or -) in a subdivision, then there are also two + (or -) and one + (or -) in the same sub-division. Similarly, if two + (or -) have been inserted in a sub-division, then also one + (or -) has been inserted.) The syllogism is valid, if every picture of both its premisses constitutes a picture of its conclusion as well. We shall illustrate the working of the above decision device on the following mixed modal syllogism: N E ( A & B ) & U(B+ C) + M E ( A & C). I n words: “If it is necessary that there exists an A which is B and if every B is (as a matter of f a t ) C, then it is possible that there exists an A which is C”. Replacing the operators N and U by M and E , we get N M E ( A & B ) & E ( B & C) --t M E ( A & C). The rules 1-9 above are now directly applicable. We begin with the second premiss E ( B & C). According to Rule 4 we insert -- in the sub-divisions A & B & W C and - A & B & N C in the diagram below. We then take the fht premiss M E ( A & B). According to Rule 6 we have to insert + + + in at least one of the sub-divisions A & B & C and A & B & Cin the diagram. But in A & B & N C we have already inserted - -. According to Rule 7 there must not be any - in a sub-division where there are three +. Thus we

+

N

-

N

N

N

N

-

-

82

APPENDIX I

are compelled to insert + + + in the sub-division A & B & C . We, finally, take the conclusion X E ( A & C). According to Rule 1 there must appear a + in at least one of the sub-divisions A & B &C and A & B & C. This is the case, since A & B & C already contains three +. Thus if we make a picture of the premisses, we thereby picture the conclusion also. This means that the tested syllogism is valid. N

A&-B&-C

A&-B & C

A&B&C

+++

-A&B&C

-A&-B&C

A&B&-C

-_

-A&B&-C

--

-A&-B&NC

We shall not in this essay attempt to determine the number and forms of the valid pure and mixed modal syllogisms in each of the four syllogistic figures. The theory of the modal syllogism outlined above only considers alethic modalities de dicto. It has been suggested that the notorious obscurity of Aristotle’s treatment of the modal syllogism is partly due to his failure to distinguish between modalities de dicto and de re. Actually, some of the modal syllogisms which Aristotle pretends to be valid, are invalid on the interpretation de dicto but would be valid on an interpretation de re. However, the interpretation de dicto seems to me to be, on the 1

This is the

p. I 9.

case, e.g., with the first syllogism mentioned in

An.

THE MODAL SYLLOGISM

83

whole, better in accordance with Aristotle’s probable intentions. 3 Quite apart from questions of history, the interpretation de re meets with the difficulties mentioned in a previous section of the present essay. (Cf. above p. 27f.) If for “necessary” we substitute the word “known”, for “possible” the phrase “not known not”, and for “impossible” the phrase “known not”, we get an epistemic version of the doctrine of the modal syllogism. (For example: “If it is known that there exists an A which is B, and if every B is C, then it is not known that there does not exist an A which is C.”) The epistemic version of the modal syllogism admits of an interpretation both de dicto and de re. The syllogisms de dicto fall within the scope of the System VE + E and their treatment is strictly analogous to the theory of the modal syllogism in the System M E + E . The syllogisms de re fall within the scope of the System EV E , and their treatment is, with minor modifications, analogous to the two previous cases. The syllogisms containing both epistemic modalities de dicto and de re fall within the scope of the System VE + EV + E . Their treatment presents considerable technical difficulties.

+

Cf., e.g., the reasoning by m e w of which he tries to show the validity of the syllogism mentioned in the footnote on p. 82. - The interpretation de dkto is also supported by Bochenski (La Logique de Thkophrmte, p. 98) against A. Becker. 1

APPENDIX I1

THE AXIOMATIC SYSTEMS M , M', AND M" The totality of systems for the alethic modalities, which we have developed in this essay, can also be exhibited in an,axiomaticform. This form is from many points of view convenient and facilitates a comparison with Lewis's systems Sl-S5.

signs Group Aa. The constants N, &, v, +, and tf of propositional logic. Group A& The constants M and N of modal logic. Group B. Sentence-variables a, b, c, . . . (an unlimited multitude). Rules of Formation RF-I. A sentence-variable is a formula. RF-II. A formula preceded by N, by M , or by N is a formula. RF-III. Two formulae joined by &, v, +, or + constitute a formula. As regards the use of brackets we adopt our previous conventions. (Cf. above p. 6 and p. 10.) Axioms Group A . A set of axioms for propositional logic. Group B. 1. u -+ Mu. The Axiom of Possibility. 2. M ( u v b ) -+Mu v Mb. The Axiom of Distribution. Group C . 1. M M u -+Mu. The First Axiom of Reduction. 2. M Mu + Mu. The Second Axiom of Reduction. N

N

Definitions If the axioms in Group A are so selected that not all the constants N, &, v, +,and ++occur in them, the missing constants have to be introduced by definition in the usual way.

85

THE AXIOMATIC SYSTEMS M, M‘, AND M”

The constant N we introduce by the dehition “N”

=

“NMN”.

Rules of Transformtion Group A . The rules of transformation of propositional logic. Group B. 1. If f i tf f B is provable, then Mfl += Mf, is also provable. The Rule of Extensionality. 2. If f is provable, the Nf is also provable. The Rule of Tautology. If from the above description we omit the axioms of Group C the Reduction Axioms - we obtain the System M . If to the System M we add the First Reduction Axiom, we obtain the System M‘. If t o the System M we add the Second Reduction Axiom, we obtain the System M”. The decision problem of the Systems M, M‘, and M” can be effectively solved. This is a consequence of the fact that the axioms and rules of the systems in question have exactly the same content as the “governing principles” of the previously developed M-systems. (A proof of this will not be reproduced here.) The Systems M, M’, and M” are consistent. This is easily established on the basis of the following three facts:

i. Every formula of the systems has what we have previously called an absolutely perfect disjunctive normal form.

ii. The normal forms of a formula and its negation-formula are “complementary” in the sense that the one normal form contains those and only those conjunctions which do not occur in the other normal form.

iii. All provable formulae have the same normal form. Some remarks will be made about completeness. We first consider only formulae which are MI-sentences. We thus consider that fragment only of the three axiomatic systems which answers to our previous System M I .

86

APPENDIX I1

Any formula of the kind mentioned has a (maximum) number of Ml-constituents. It also has what we have called an absolutely perfect disjunctive normal form in terms of its B,-constituents.The formula is provable in the axiomatic systems, if and only if its normal form contains all the possible conjunction-sentences of the Ml-constituents and/or their negation-sentences with the exception of the conjunction-sentence of the negation-sentence of all the Mlconstituents. That the axiomatic systems are complete with regard to Mlsentences can be defined as follows: No MI-sentence with a shorter normal form than the one just mentioned expresses a truth of logic. Or, which means the same: No conjunction-sentence of Ml-constituents and/or their negationsentences, with the exception of the conjunction-sentence of the negation-sentences of all the Ml-constituents, expresses a logically false proposition. That the axiomatic systems actually are complete with regard to Ml-sentences can be shown in the following way: Consider Ml-sentences with, say, two atomic sentences a and b. There are then four Ml-constituents, viz. M ( a & b ) and M ( a & b) and M ( N a & b ) and M ( N a & N b ) , and 15 conjunction-sentences of Ml-constituents and/or their negation-sentences to be considered. The first conjunction-sentence is M ( a & b ) & M ( a & N b ) & M ( - a & b ) & M ( N a & N b). That this expresses a logically false proposition would mean that a t least one of the four (‘possible” distributions of truth-values over two propositions is an impossibility. This we reject as “absurd”. The second conjunction-sentence is M ( a & b ) & M ( a & b) & M ( N a & b ) & M ( N a & N b ) . It can also be written N ( a v b). That it expresses a logically false proposition would thus mean that the disjunction of two propositions can never be a necessary proposition. This is (‘absurd’’ too. The thirteen remaining conjunction-sentences resemble the second conjunction-sentence in that they express that a truth-function (other than the contradiction) of the two propositions expressed by a and by b, is a necessity. Thus the assumption that some of

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N

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the conjunction-sentences expresses a logically false proposition would lead to the “absurd” consequence that a certain truthfunction (other than the contradiction) of two given propositions could never be a necessary proposition. Similar considerations apply to Ml-sentences with any number n of atomic sentences. We may thus conclude that the axiomatic systems are complete with regard to MI-sentences. The conclusion can be trivially extended to Ml + Mo-sentences. It follows from the above that for higher order modalities the question of completeness is essentially the same as the question, whether or not there are reductions of higher order modalities to lower order modalities. The question of completeness, in other words, is essentially the problem, whether we should add to the System M the Second Reduction Axiom, or only the First Reduction Axiom (or some still weaker reduction axiom), or no reduction axiom at all. The question of logical truth in the reduction axioms is obscure. An appeal to our logical “intuitions” does not seem to be helpful. Other means of illuminating it will not be discussed in this essay. We shall now say a few words about the relation of the Systems M, M’, and M“ t o the five “classical” systems XI-85 of C. I. Lewis. We shall refer to Sl-S5 in the versions given by Feys and McKinsey in their Modal Logics I . A deductive system is said to be contained in another deductive system, if every formula which can be proved in the first system can be proved in the second system too. Two systems are said to be equivalent, if they are mutually contained in one another. We shall not here discuss S1, which is contained in S2. The axioms of S2 are as follows (we use our own symbolism but Feys’s and McKinsey’s numbering) :

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N(a & b +a) N ( a & b + b & a) N ( ( a & 6 ) & c + a & ( b & c)) N(a +a & a ) N ( N ( a --f b ) & N ( b + c) --f N ( a + c)) N ( a --f M a ) N ( M ( a& b ) + M a ) All these axioms are provable formulae in our System M . I n the logic of propositions we prove a & b --f a. I n virtue of our rule B2 (Rule of Tautology) we have N ( a & b --f a). This is 30.11. 30.11 30.12 30.13 30.14 30.15 30.16 30.17

I n exactly the same way we prove 30.12 and 30.13 and 30.14. In the logic of propositions we prove M ( a & b & c) % -M(a&-b& NC) & N M ( & ~ b & N C ) & -M(-cc& b & N C ) + N M ( U & b & c) & N M ( a & b & N c). I n virtue of our axiom B2 (Axiom of Distribution) and rule B1 (Rule of Extensionality) we M ( a & b) & M ( b & c) 3 M ( a & -c). simplify this to I n virtue of the definition of N and the rule BI we get from this N ( a + b) & N ( b + c) --f N ( a --f c). I n virtue of the rule B2, finally, we have N ( N ( a+ b ) & N ( b + c) + N ( a + c)). This is 30.15. From our axiom BI (Axiom of Possibility) in combination with the rule B2 we get N ( a + Mu).This is 30.16. I n the logic of propositions we prove M ( a & b ) +M(a & b ) v M ( a & N b ) . I n virtue of our axiom B2 and rule BI we simplify this to M ( a & b ) + Ma. In virtue of our rule B2 we have N ( M ( a & b) + Ma). This is 30.17. If from the rules of inference of the System M we omit the Rule of Tautology, we get the rules of inference of 52. Herewith has been proved that the System M contains S2. N

N

N

N

N

N

N

N

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We get 53 from 52 by adding to 5 2 the new axiom N ( N ( u--t b) + N ( M a +- M b ) ) .

40.01

It is not possible to prove 40.01 in our System M . (This is seen from a transformation of 40.01 into its absolutely perfect disjunctive normal form.) The System M , in other words, does not contain 53.

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40.01 is, in fact, a reduction axiom. - N ( a -+ b ) -+ N ( M a + M b ) can easily be transformed into M ( M ( a& -b) & w M b )+M(a & w b ) . The last expression might be called a weakened form of our axiom

GI. It is difficult to concieve of any plausible grounds for admitting 40.01 but not the axiom CI. The system 53, therefore, appears to be of comparatively little independent interest. We get S4 from our System MI, if we omit the rule BI and replace the axiom B2 by 53.312

N ( a + b) -+ ( N u --f Nb).

I n 54 we can derive our rule BI. For, if f l tf f , is provable in 54, then N(fl + f,) is also provable in 54.But if N ( f , tf f , ) is provable, then, according to a double application of 53.312, Nfi tf Nf, is provable. I n virtue of the definition of N and laws of propositional logic, we get from Nfi ++ Nf, the equivalent formula Mfl + Mf,. Thus, if fl ++f z is provable in 5 4 , Mfl +Mf, is also provable. This is our rule BI. I n 54 we can further derive our axiom B2. The proof is as follows : I n propositional logic we prove b --f (a -+a & b ) . From our axiom B l we get Nb --f b. Thus we have Nb -+ (a -+ a & b). In virtue of our rule B2 we get N ( N b -+ (a --f a & b ) ) . I n virtue of 53.312 weget N N b + N ( a + a $ b ) andN(a+a&b)+(Na+N(a&b)). From our axiom CI we get Nb -+ NNb. Thus we have Nb --f (Nu --f N ( a & b ) ) which, in propositional logic, can be transformed into N u & Nb -+ N ( a & b). Further, in propositional logic we prove a & b --f a. I n virtue of our rule B2 we have N ( a & b -+ a). I n virtue of 53.312 we get from this N ( a & b) + N u . I n exactly the same way we prove N ( a & b) --f Nb. The two last formulae give, in propositional logic, N ( a & b) --f Nu & Nb. Herewith has been proved the formula Nu & Nb tf N ( a & b). I n virtue of the definition of N and our rule BI (which can be derived I am indebted for this observation to the work of Feys and McKinsey. Cf. op. cit. 53.21.

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in 5 4 ) and laws of propositional logic, we easily transform this into M ( a v b ) ++Ma v Mb or our axiom B2. 53.312 is provable in our System M'. (Cf. above p. 17f.) Thus 5 4 and our System M' are equivalent. We get S5 from 5 4 by replacing N ( M M a --f M a ) with N ( M M a --f -Ma). Once the equivalence of 5 4 and the System M' has been established, the equivalence of S5 and the System M" follows as a matter of course.

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