THE SEMANTICS OF MINIMA L INTUITIONISM G.N. GEORGACARAKOS. Introduction

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1 THE SEMANTICS OF MINIMA L INTUITIONISM G.N. GEORGACARAKOS Introduction There has existed some controversy in the history o f intuitionism concerning the tenth axiom o f Heyting's famous formalization o f intuitionistic logic. Fo r example, some logicians have suggested that this axiom does not have any intuitive foundation (cf. [51 and [6], p. 421). Even Heyting admitted that the axiom was perhaps not intuitively clear and, as a consequence, believed that counting it among the axioms o f intuitionistic logic required some kind o f justification (cf. 141, p. 102). Heyting's attempted justification f o r keeping it involved claiming that it added to the 'precision o f the definition of implication' (again cf. [4], p. 102). However, it has been argued more recently b y Susan Haack that its inclusion really amounts to an extension of the intuitionist's sense of 'construction' to a point where it no longer seems characteristically intuitionistic. In view of this consideration, she concludes rather emphatically that the system resulting from dropping the tenth axiom represents the set of intuitionistic logical truths better than the original Heyting axiomatization (12], p. 102). Whatever the merits of this claim, it shall not be discussed here. We mention it only because it suggests an examination of the system is perhaps worthwile, especially f ro m a semantical point o f vie w. Consequently, the purpose of this paper will be to provide a semantical interpretation of the resulting system. This system is well-known and has come to be called Johansson's 'minimal calculus' (cf. [5]). It has been studied proof-theorretically by Prawitz[9] and algebraically by Rasiowa and Sikorski [ 10]. Actually, a modeling for the system has been provided by Fitting [ I ]

2 384 G. N. GEORGACARAKOS propose differs in many important respects from his we consider it of some interest to present it here. In formulating 'possible world' semantics for intuitionistic logic, Kripke makes use of semantic tableaux for proving the completeness theorem 17l. The virtue of that approach is that it is more in keeping with the spirit of intuitionistic constructivism. Nevertheless, in proving the completeness theorem for the minimal calculus, we shall employ techniques adapted from HenkinI31, and, accordingly, shall proceed without any intuitionistic scruples. The reason for adopting this approach is that we consider it intrinsically interesting to direct attention to the two kinds of saturated sets of statement-forms in the minimal calculus. In any event, since the interpretation we shall offer involves a rather straight-forward modification of Kripke's semantics for intuitionistic logic, recovering a completeness theorem based on semantic tableaux would be easily obtainable with only a minimum of corresponding changes. It should also be mentioned that we shall only concern ourselves with statement logic : an extension of the modeling to the first-order predicate calculus could be accomplished by following after the manner of either [71 or 111 Section I : Syntax The symbolic language for Johansson's minimal calculus (JMC) is a triple < P,C,F> where P is a denumerably infinite set of statement variables, C is the set whose members are the unary connective binary connectives A, V, -> and the punctuation symbols (,), and F is the set o f statement-forms built up as usual from the statement variables in P and the connectives and punctuation symbols in C. The axiom schemata of JMC are the following: Al. A ) (A A A) A2. ( A A B ) A3. ( A A4. ((A ) B) A (13 > C)) ) (A ) C) A5. B > (A ) B) A6. (A A (A A7. A ) (A v B)

3 THE SEMANTICS OF MINIMAL INTUITIONISM A8. (A V B)-> (B VA) A9. ((A -) C) A (B-> C))-) ((A V B)-> C) A10. ((A->B) A (A-> TB))-> T A We note that the axiom-schemata for JMC are the same as for the intuitionistic logic of Heyting except of course for the conspicuous absence of T A - ) (A -) B). The only definition and rule of inference of JMC is given by: Dl. A B =d f (A- B) A (13-) A) From A-* B and A infer B. The notion o f a deduction o f a statement form A from a set o f statement-forms S is defined as in 181. Accordingly, we will write 'S H A' to indicate that there exists a deduction of A from S, and 'H A' as an abbreviation fo r 'OH A'. We now list, without proof, some metatheorems of JMC for future reference (they are easily proved in the usual sorts of ways): MT1. I f A ES, then Si-A. MT2. I f S H A, then S US' HA. MT3. HA -4 (B VA) MT4. I f S U IA1 HB, then S HA -) B. MT5. I f SH A and S HA-+ B, then S i - MT6. S U1A1- HB and S 111A1 HTB, then S i-t A. Mr7. I f S H A and S H 13, then SHA AB. MT8. I f SHA AB, then S H A and S i - Because o f the absence in JMC o f Heyting's tenth axiom, it is possible to distinguish between two kinds o f deductive consistency. Let's define a set o f S o f statement-forms o f JMC to be negation consistent just in case for some statement-form A of JMC, not both S H A and S H A. Correspondingly, a set S o f statement-forms o f JMC is said t o be absolutely consistent ju st in case f o r some statement-form A, not S A. Of course, the distinction between these two kinds of deductive consistency breaks down in Heyting's intuitionistic logic and in the classical logic, but not so in the minimal calculus.

4 386 G. N. GEORGACARAKOS Section 2 : Semantics Except for two modifications, JMC-models are essentially the same as intuitionistic models. The first modification involves distinguishing between two kinds of possible worlds. In intuitionistic models, there is only one kind of possible world, however, JMC-models have what we shall call 'negation coherent' and 'absolutely coherent' possible worlds. Negation coherent worlds are worlds which are negation consistent ; absolutely coherent worlds, however, may or may not be negation consistent. Nevertheless, we require that the latter worlds be characterized by the stipulation that there exists at least one statement-form in them which receive the valuation 'false.' Furthermore, we require that every JMC-model has either one or both of these kinds of possible worlds. The second modification has t o do with the valuation of statement-forms of the form true in a given world provided that A is false in all accessible negation coherent worlds. More formally, we define a JMC-model as a triple < W,R,V where W is a set of possible worlds, R is a reflexive and transitive relation defined on W and V is a value assignment satisfying the following conditions : SC1. Fo r any A e P and for any w e W, if both V (A,w w,rw SC2. Fo r any A EF o f the f o rm B AC and f o r any w V (A,w V (A,w SC3. Fo r a n y A cf o f the f o rm B VC and f o r any w EW, V (A,w V (A,w SC4_ Fo r any A EF o f the fo rm B > C and f o r any w V (A,w V (13,w SC5. Fo r any A e F of the form a n d for any w i f fo r every negation coherent w V (B,w,) F ; otherwise V (A,w

5 THE SEMANTICS OF MINIMAL INTUITIONISM We say that any w e W is negation coherent if and only if for no A cf is it the case that both V (A,w w e W is said to be absolutely coherent nust in case there is at least one A cf such that V (A,w JMC-logically true if and only if for every JMC-model <W,R,V > and for every w Given o u r definitions o f the two kinds o f possible worlds in JMC-models, it is obvious that every negation coherent world is also an absolutely coherent world ; but of course the converse doesn't hold. Accordingly, JMC-models allow for the possibility that there are states of affairs which are classically and intuitionistically inconsistent (in the sense of allowing A and T A to be present) but without the consequent disaster of allowing everything to be derivable. Resorting to metaphor, we might say that, unlike classical and intuitionistic models. JMC-models permit the existence of states of affairs which are negation inconsistent, but which are, nevertheless, devoid of total chaos and absolute absurdity. It is an easy matter to prove the soundness theorem for JMC and so we leave it to the reader to verify that all of the axiom-schemata of JMC are JMC-logically true and that the rule of detachment is truth preserving. As we should expect JMC-logically true since it isn't true in all JMC-models. Th is is patently evident once we consider the following JMC-model : le t W = w 2, w absolutely coherent. Furthermore, let w let V (A,w V (TA,W case that there exists some statement-form, sa y B, such that V (B,w V (A-) B,w V ( JMC-logically true. Section 3: Intuitive Interpretation Borrowing from Kripke in [7], pp. 97 ff. we shall say that, in general, in a JMC-model < W,R,V>, we interpret W as the set o f

6 388 G, N. GEORGACARAKOS 'evidential situations.' Where w w enough information to advance to w that at a particular time w thus, we might alternatively understand V (A,w has been verified at the point w means that A has not been verified at w do n o t denote intuitionistic tru th a n d fa lsity since although V (A,w latter only means that A has not, as yet, been verified at w might be later. Intuitionistically speaking, an evidential situation w V (A,w would entail that any statement whatsoever has been verified to be true at w intuitionism such an evidential situation, although negation inconsistent, would not necessarily entail that any statement whatsoever is verified to be true at w absolutely coherent evidential situation, in which case, there would be at least one statement at w true. I n intuitionistic logic, t o assert knowing not only that A has not been verified at w possibly be verified at any later time, no matter how much more information is gained ; thus in intuitionistic logic, V (7 A,w for every w negation inconsistent evidential situation, intuitionistically speaking, is disastrous since such a situation commits us to the absurd view that a given statement has been verified to be true at that situation and not verified to be true at that situation and any other situation which will eventually come at a later time. However, in JMC-models, since for any w for every negation coherent w assert there and cannot possibly be verified at any later negation coherent situation, no matter how much more information is gained. Thus, an evidential situation w (i.e., a situation which is negation inconsistent, but, nonetheless, absolutely coherent) simply entails that A has been verified to be true

7 THE SEMANTICS OF MINIMAL INTUITIONISM at w then, the absurdity mentioned above in the case o f intuitionistic models, doesn't arise in the case of JMC-models. Section 4: Completeness Given that there are two kinds o f deductively consistent sets of statement-forms in JMC, it is possible to define two corresponding kinds of saturated sets in J MC. Let's d e of JMC t o be negation-saturated just in case (i) F is negation consistent ; and (ii) if r H A, then A e l of statement-forms of JMC to be absolutely-saturated just in case (i) is absolutely consistent ; and (ii) if I' HA, then A Er. We now direct our attention to the proof of certain lemmata which will eventually be used in the proof of the completeness theorem. L I F o r every negation consistent set S of statement-forms of J MC there exists some negation-saturated set F o f statement-forms such that S F. Define the sets F (a) Fo = S (b) Enumerate the statement-forms o f JMC. Fo r each i 0, i f F otherwise 1 (c) F = Obviously there exists a set F such that S c r. What we must now show is that conditions (i) and (ii) o f a negation-saturated set hold for F. It is easily demonstrated by induction on i that if S is negation consistent in JMC then so is each F negation consistent since S is. Assume that F but that F

8 390 G. N. GEORGACARAKOS = F tent after all. Therefore, F We now show that i f I' H A, then A el'. Suppose that Furthermore, suppose for the sake of reductio that A F. In view of the way F was constructed, F 1 AI must be negation inconsistent ; i.e., for any statement-form B, both r!jim HB and F LI { Consequently (by MT6), f H TA. But now we have both r I- A and F established above. Therefore, A Er. L2. Fo r every absolutely consistent S o f statement forms o f JMC there exists some absolutely-saturated set I' of statement-forms such that S Define the sets fo, F replace 'negation consistent' by 'absolutely consistent.' Obviously there exists a set F such that S F. is absolutely consistent by way o f the same argument given in the proof of Ll Now we show that if I' HA, then A cr. Suppose that F suppose for the sake of reductio that A F. In view of the way I' was constructed, F 1.31A1 must be absolutely inconsistent ; i.e., for any statement-form B, F LI 1A1 H B. Consequently (b y MT4), B and so (by MT5) for any statement-form B, F H B. But in that case F is absolutely inconsistent contrary t o wh a t wa s established above. Therefore, A Er. The following lemma identifies the distinctive feature of absolutelysaturated sets: L3. Fo r any absolutely-saturated set F o f statement-forms o f JMC there is some statement-form A of JMC such that A F. Suppose that l is any absolutely-saturated set of statement-forms of JMC. Since it is absolutely consistent it must be the case that for some

9 THE SEMANTICS OF MINIMAL INTUITIONISM statement-form A ofjmc not I" H A. But in that case we may conclude that for some statement-form A of JMC, A F (by MT1). In most of the remaining lemmata, when we speak of a 'saturated set' we are speaking o f any and every saturated set whether it is negation-saturated or absolutely-saturated. L4. L e t A be any statement-form of JMC and IF any saturated set of statement-forms of JMC. Then H A if and only if A c By condition (ii) of either definition of a saturated set and MTI. L5. Let A be any statement-form of JMC and I" any saturated set of statement-forms o f J MC. Then 1' U {A} is consistent (in the appropriate sense) if and only if A c r. Suppose I' U { the way r was constructed A would have been added to r and so A c f Now suppose A c f. Furthermore, suppose F A I is inconsistent (in either sense). In that case, in view of the way F was constructed, it must be the case that A 0" which is contrary to the first supposition. Therefore, F U fal is consistent (in the appropriate sense). L6. Let A be any statement-form of JMC and IF any saturated set of statement-forms of JMC. Then if A F, then there exists some saturated set I" ' such that 1' çl"' and A ( r. Suppose A F. Then clearly (by L5) F (JI M is inconsistent. Now L i and L2 guarantee that there exists some saturated set F' such that But in that case F' U fal is also inconsistent. Consequently (by L5), A OF'. L7. Fo r any statement-form A of JMC and every saturated set F. if HA, then A cr.

10 392 G. N. GEORGACARAKOS Suppose H A. Also suppose for the sake of reductio A (p F. Now H A is the same as OH- A. Hence (b y MT2 ) U r HA and so, since UF = F, 1 A. But (by hypothesis) A F, thus (by L4) not I' HA and so we have a contradiction. Therefore, A c f. L8. Fo r any statement-forms A and B of JMC and for every saturated set F, if A VB Er', then either A c f or B (a) Suppose that I" is negation-saturated and that A V B e l. Furthermore, suppose fo r the sake o f reductio that neither A e l B E r; i.e., A a and B a. Then (by L5) I" U tal and U t B 1 are both negation inconsistent ; in other words: F U I AI HC and I' U1A11-7C: and U113)-HC and I' U{BH TC. Clearly (by MT4), we have in each case: HA-0C and H A ) F i - Hence (by MT7) we have: H (A ) C) A (B > C) H (A > C) A (13 ) Now since both ((A ) C) A (B * C)) ( ( A V B) > C) and ((A > TC) A (B )C)) > ((A V B) > C) are axioms of JMC, it follows (by L7) that they are both in F. Hence (by MT1): H ((A ) C) A (B C)) > ((A v B) > C) ((A > Consequently (by MT5): I' 1 T

11 THE SEMANTICS OF MINIMAL INTUITIONISM But A V B EU (by hypothesis), hence (by MTI ): FHA VB. Thus (again by MT5): HC and H Consequently, F is negation inconsistent contrary to supposition. Therefore, either A E r or B EU. (b) No w suppose I' is absolutely-saturated and A V B El for the sake of reductio, assume A el' and B Er. Then (by L5) both UfAl and I' U1131 are absolutely inconsistent; i.e., for any satement-form C, F UtAl-HC Hence (by MT4 and MT7) F F- (A->C) A (B -) C). But because ( is an axiom of JMC, we have (by L7 and MTI ): H ((A-> C) A (B-4 C)) ( ( A v B)-> C) and so (by MT5): H (A v B)-> C. but A V B El' (by hypothesis). Hence (by MT1): F HA VB Consequently (by MT5) it follows that for any statement-form C. F- C. Clearly, F is absolutely inconsistent contrary to the hypothesis of the proof. Therefore, either A Er' or B On either supposition, then, we have A e l

12 394 G. N. GEORGACARAKOS lemma is proved. L9. Fo r any statement-forms A and B of JMC and every saturated set F, if A V B a, then A a and B F. Suppose A V B F. Furthermore, suppose, for the sake of reductio, that not both A a and B ; i.e., either A E r or B c f. If A el', then r H A (by MTI). But A > (A v B) is an axiom of JMC, hence (by L7 and MT2 ) A (A V B) and so (by MT5 ) f HA v B. But (b y hypothesis and L4) not I' H A v B. Obviously we have a contradiction and so A a and B 4 yields the same conclusion. Either way, then, we have A 0 and so the lemma is proved. L b. Fo r any statement-form A ofjmc and for every saturated set I', if T A Er, then for every negation-saturated set r such that Ici', A Suppose T A Er and that r g r where I ' is negation-saturated. Obviously, T A a n d so (by MTI) H A. Now suppose for the sake o f recluctio A c r. Then (again by MT I ) A. Thus r is negation inconsistent contrary to the hypothesis. Therefore A IF". LI I. Fo r any statement form A and every saturated set F, if T A then there exists some negation-saturated set r' such that F and A e r. Suppose T A U. Then (by 16) there exists some saturated set r' such that. f _ r and T A H e n c e (by L4) not r H A. Clearly, then, (by MT6) not both r A I HB and r u { quently. r t A l is negation consistent. But in that case (by L5) A e r and r is negation-saturated. Let us now employ the set I' as the set of all negation-saturated and

13 THE SEMANTICS OF MINIMAL INTUITIONISM absolutely-saturated sets of JMC with set-theoretic inclusion defined on it. We can now easily form a JMC-model < W,R,V> on the basis of F. With each F rated associate a negation coherent world, where F, is absolutely-saturated associate a absolutely coherent world). I n other words, le t W = F. Let R be the relation such that w V be the following value assignment: for any statement-variable A and every w, E W, V (A,w,) = T if A r Clearly, < W,R,V> thus defined is a JMC-model. We are now prepared to state and prove the completeness theorem. (Completeness) Let W. R and V be defined as above. Then for every statement-form A of JMC and for every w, EW,V (A,w,) = T if A c r Case I: A is a statement-variable. Obviously the theorem holds for A by the initial value assignment to statement variables. Case 2: A is B A case F, B and F, HC (by MT8). Consequently (by L4), B el', and C e r V (B,w,) = T a n d V (C,w,) = T. Therefore ( b y SC2 ), V (A,w,) = T. (b) Let A 0",. In that case not F, H A (by L4). Hence (by M T7) either not F. B or not F, H C. If not F, B, then B 0', (by MTI). Consequently V (B,w,) = F (by inductive assumption) and so V (A,w,) = F (by SC2). If not I', C, then C f hypothesis) and so V (A,w,) = F (by SC2). Case 3: A is B v C. (a) Let A EF,. Then either B El', or C El', (by L8). But in that case, either V (B,w,) = T or V (C,w,) = T (by the hypothesis of induction). Therefore, V (A,w,) = T (by SC3). (b) Let A f induction hypothesis) both V (B,w,) = F and V (C,w,) = F. Therefore (by SC3), V (A,w,) = F. Case 4 : A is B--* C. (a) Let A c r F, g r

14 396 G. N. GEORGACARAKOS MT5) r Put differently, either B if o r C c r hypothesis o f in d u ctio n ) e i t h e r V (B,w (C,w A (tr r L6) there exists some saturated set r and C (tr induction) V (B,w SC4), V (A,w Case 5: A is TB. (a) Let A c r saturated set r sis o f induction) f o r every negation coherent world w (B,w A F set r of the induction) there exists some negation coherent world w Undoubtedly, i f we were t o define entailment along the lines mentioned in I l l, we could also, given our approach, prove the strong semantical completeness theorem without any difficulty.* Department o f Philosophy' Gustavus Adolphu St. Peter, Minnesota GEORGACARAKOS * 1 am indebted to my colleague Robin Smith for valuable discussion concerning some of the points mentioned in this paper. REFERENCES 111 Fitting WC. intuitionistic Logic Model Theory and Forcing. Studies in Logic and the Foundations of Mathematics, North-Holland: Publishing Company, Haack, Susan, Deviant Logic. Cambridge University Press, Henkin L. The Completeness of the First-Order Functional Calculus. Journal of Symbolic Logic 1949; 14:

15 THE SEMANTICS OF MINIMAL INTUITIONISM Heyting A. Intuitionism. North-Holland: Publishing Company, Johansson 1. Der Minimalkalkul, ein reduzierter intuitionistische Formalismus. Compositio Mathematica 1936; Kolmogorov A.N. On the Principle of Excluded Middle. In From Frege to Gode!. ed. J. Heijenoort, Harvard University Press, Kripk e S. Semantical Analysis o f Intuitionistic Logic I. Formal Systems and Recursive Functions, ed. J. Crossley and M. Dummett Amsterdam, 1965, [81 Montague R., Henkin. O n the Definition o f 'Formal Deduction'. Journal o f Symbolic Logic 1957; 21: Prawitz D. Natural Deduction, A Proof-Theoretical Study, Acta Universitatis Stockholmiensis. Stockholm : Studies in Philosophy No. 3, Almquis t and Wiksell, Stockholm Rasiowa H., Sikorski R. The Mathematics of Metamathematics. Warszawa 1963, 3rd ed [I 11 Thomason R.H. On the Strong Semantical Completeness of the Intuitionistic Predicate Calculus. Journal of Symbolic Logic 1968; 33: 1-7.