Notre Dame Journal of Formal Logic Volume XV, Number 3, July 1974 NDJFAM LOGICAL CONSEQUENCE IN MODAL LOGIC II: SOME SEMANTIC SYSTEMS FOR S4 GEORGE WEAVER and JOHN CORCORAN 1 This paper is a continuation of the investigations reported in Corcoran and Weaver [1] where two logics j£Π and -CDD, having natural deduction systems based on Lewis's S5, are shown to have the usually desired properties (strong soundness, strong completeness, compactness). As in [1], we desire to treat modal logic as a "clean" natural deduction system with a conceptually meaningful semantics. Here, our investigations are carried out for several S4 based logics. These logics, when regarded as logistic systems (cf. Corcoran [2], p. 154), are seen to be equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be linked. Some of the results of [l] are presupposed here and the more obvious definitions will not be repeated in detail. We consider the logics -Cl, -C2, and -£3. These logics share the same language (^DD) and deductive system (Δ'DD) but each has its own semantics system (Σl, Σ2, Σ3). Σl is an extension of the Kripke [3] semantics for S5 as modified in [l], Σ2 is largely due to Makinson [5], and Σ3 is due to Kripke [4]. -C DD is the usual modal sentential language with D, ~ and 3 as logical constants (see 2 below). Δ'DD (see 3 below), a modification of the natural deduction system given in [l], permits proofs from arbitrary sets of premises. For S a set of sentences and A & sentence, Sv-A means that A is provable from S, i.e., there is a proof (in Δ'DD) of A whose premises are among the members of S. ([-A means S\-A where S is empty.) If S\-A, we sometimes say that the argument (S, A) is demonstrable and when, in addition, S is empty we say that A is provable. Each semantic system includes a set of interpretations for ^ D D together with truth-valuations for each interpretation. As usual, if M is an interpretation (in Σi) and A is a sentence, M is a model of A iff A is true on M; if S is a set of sentences, M is a model of S iff Mis a model of every member of S. St=A (in Σi) indicates that A is a logical consequence of S Received May 12, 1971 LOGICAL CONSEQUENCE IN MODAL LOGIC II 371 (in Σi), i.e., that every model of S is a model of A (alternatively, we say that the argument (S, A) is valid in Σi). t= A indicates that A is logically true (in Σi), i.e., that A is true on every interpretation. Given two semantic systems for a given language, we can naturally ask whether they are "equivalent" either in the sense (a) that the sets of logical truths are identical, or in the sense (b) that the sets of valid arguments are identical. More precisely, we say that two semantic systems are strongly equivalent if they have the same valid arguments; and weakly equivalent if they have the same logical truths. Strong equivalence implies weak equivalence but not conversely: ΣI is weakly but not strongly equivalent to Σ3 (see 6.1 below). The notions of completeness (weak and strong), soundness (weak and strong), and compactness are defined as usual. Recall that the semantic system of any strongly sound and strongly complete logic has a compact semantics. In the usual hierarchy of non-modal logics (sentential, first order with identity, second order, etc.) weak completeness, strong completeness, and compactness all fail at the same point. This leads to the question of whether there is some "inherent" property of logics which links weak completeness (essentially enumerability of logical truths) to compactness. Although £\ is seen to be a counterexample to the positive conjecture on the issue (below), further investigation may reveal an intimate relation between compactness and the enumerability of the set of logical truths. In view of the fact that there are uncountably many valid arguments (S, A), compactness is an index of the complexity of a semantic system in much the same way that enumerability of logical truths is. One feels that these two indices are linked. We show (1) that -(Ί is strongly sound; (2) that -('1 is not compact, hence not strongly complete; (3) that for every interpretation in ΣI there is an interpretation in Σ2 making exactly the same sentences true; (4) that £2 is strongly sound and strongly complete, and hence that ΣI and Σ2 are not strongly equivalent; (5) that for every interpretation in Σ2, there is an interpretation in Σ3 making exactly the same set of sentences true; therefore £3 is strongly complete; (6) that given any interpretation in Σ3 and any infinite proper subset Cr of the set of sentence constants, there is an interpretation in ΣI agreeing with it on all sentences whose constants are not in C; hence (a) that «C3 is strongly sound and compact; (b) that Σ3 is strongly equivalent to Σ2 but not (strongly equivalent) to ΣI; (c) that Σ2 and Σ3 are both weakly equivalent but not strongly equivalent to ΣI; and (d) that ^1 is weakly complete. Finally, we note that both the Feys-von Wright system [5], M, and the Brouwerische system [5] (with semantics like ΣI) are not strongly complete. 2 ^ D Π is the result of closing C, a countably infinite set of sentential constants, under forming DA, ~A, and (A ^ B) for A, B already in the set. jQ indicates those sentences of ^ D D devoid of occurrences of D. 3 Δ'DD: The deductive system Δ'DD closely resembles the system ΔDD for <£ DD given in [l]. As in [l] proofs are written left-to-right rather than 372 GEORGE WEAVER and JOHN CORCORAN up-to-down, and brackets, ] [, are used as punctuation (see [l] for examples). The terms proof expression, open in a proof expression, closed in a proof expression, and assumption of a proof expression are defined as in [l] (p. 343). A proof expression is any string over <^DDu {],[}. Loosely, an occurrence of the sentence A is closed in a proof expression provided it occurs between matching brackets; an occurrence of A is open in a proof expression provided this occurrence is not closed; an occurrence of A is an assumption in a proof expression iff it is immediately preceded by a left bracket, [ a sentence is an assumption in a proof expression iff some occurrence of that sentence is an assumption. A is an open assumption in a proof expression provided that an occurrence of A is an assumption and open in the proof expression. In the following, A, B, and D are sentences of *C ΠD, and Π and σ are proof expressions. These symbols occur with or without subscripts and accents. Definition For every proof expression Π, Π is a proof (in Δ'DD) iff Π is constructed by a finite number of applications of the following rules: (i) The six rules A, R, DE, DI, ~E, DE (C/., [1], p. 373). (ii) (DI') If Π is a proof, A occurs open in Π, and all open assumptions in Π are of the form ΏB, then ΠDA is a proof. As in [1], C(Π) denotes the last line of Π and 0(11) denotes the open assumptions (or premises) of Π. For every set of sentences S and every sentence A, we say that A is provable from S in Δ'DD iff there exists a proof Π in Δ'DD such that O(Π) c S and C(Π) = A. (As noted above, we indicate this by St-A.) Δ'DD differs from the deductive system for ^ D D ([1], p. 373) only in the rule for the introduction of D, DI' permits the inference of DA in a proof Π provided that A has been inferred in Π and every premise in Π is of the form ΏB. The rule,1 DI, of [1] allows the inference of DA in Π provided that every premise in Π is a truth functional combination of sentences of the form ΏB (and, of course, that A is inferred in Π). Obviously, any proof in Δ'DD is a proof in ΔDD. The difference between DI and DI' is the difference between S5 and S4. DΓ is essentially the rule for the introduction of D given in [l], p. 380, [5], p. 74 and [3], p. 114. LetS c ^ D D . S is inconsistent (in Δ'DD) iff there is a sentence A such that ShA and Sh~A; S is consistent (in Δ'DD) otherwise, S is maximally consistent (in Δ'DD) iff S is consistent and for all A, either A e S or ~AeS. Let Δ be the subsystem of ΔDD consisting of just those proofs over the language <£; for S c jQ, Ae *£, we define S v-A in Δ, \-A in Δ, S is consistent in Δ, and S is maximally consistent in Δ as above {mutatis mutandis). The following facts about maximally consistent sets can easily be verified. Lemma 1 For all S c <£ DD, if S is maximally consistent in Δ'DD then for all A, Be£ 1. AeSiffS)-A; 2. AeS or ~AeS; LOGICAL CONSEQUENCE IN MODAL LOGIC II 373 3. (Ao B)eSiffAf?S or BeS; 4. S (Ί JQ is maximally consistent in Δ. Lemma 2 [Lindenbaum]. Every consistent set (mΔ'DD) can be extended to a maximally consistent set {in Δ'GD). If one has a ''standard77 enumeration of the sentences of ^ D G , each of the usual proofs of Lemma 2 provides a functional process for associating with each consistent S a unique maximally consistent S+. We assume such a standard enumeration and a functional process so that for each consistent 5. S+ indicates a maximally consistent extension of 5. For every S, we define D(S) = {A : ΏAe S}and Ώ(S) = {A : ~DAe S}. Let S + A = S U {A}. The following are easily verified. Lemma 3 For every proof ΐlin Δ'DD there exists a proof Π' such that Π' = [B&B2[. \.BnσA where O(Π) = {Bu . . ., Bn} = O(Π f) αwd C(Π) = A = C(Π'). Fact 1 For βΠ S, A, S + ~A 2s inconsistent iff S \-A. Lemma 4 Let S be maximally consistent in Δ'DD. Then (a) D(S) and D(S) «r<? disjoint) (b) For all A, AeΠ(S) iffΠ(S)\-A; (c) For αZZ A, z/ Ae D(S), £/z£?z D(S) + ~A is consistent. The first clause is obvious and the last clause is implied by the first two as follows. Suppose that Ae D(S) but that D(S) + ~A is inconsistent; then D(S) \-A and, by the second, Ae D(S). This contradicts the first. To see the second note that the "only if" part is trivial, then suppose D(S) v-A; i.e., that there is a proof Π of A from some finite subset, Bly . . ., Bn, of D(S). By Lemma 3 there exists a proof Πf = [JBitB2[. [BnσA. Let σ' be the result of deleting the left brackets ([) which occur immediately before the occurrences of Bi which are assumptions in Π'. Now form [D-B^Dîf. [ΠBnσ\ The latter is a proof of A from ΠBly . . ., ΏBn and by applying (DΓ) [D^![DJ52[. . . [ΠBnσ DA is a proof. Since S is maximally consistent and ΠBieS, ΠAeS. 4 Σl : The interpretations of Σl are all triples (a, P, R) where a is an ordinary truth-value assignment (a function from C into {t, f}), P is a set of such functions containing a, and R is a reflexive and transitive relation on P. Below a, b, ar, br', etc. are assignments, P, Pr, etc. are sets of assignments and R, Rr, etc. are relations. We use aPR to indicate interpretations (a, P, R). For each interpretation aPR we define the function VaPR from ^ D D into {t, f} as follows: Definition For every sentential constant A, VaPR(A) = α(A); if A = ~B, VaPR(A) = N(VaPR(B)); if A = (B D D), then VaPR(A) = C(VaPR(B), VaPR(D)) (where N is the truth function associated with negation and C that associated with material conditional); if A = ΠB, then VaPR(A) = t iff for all b e P such that aRb, VaPR(B) = t. The following is easily seen, 374 GEORGE WEAVER and JOHN CORCORAN Theorem 1 For all S, Sr, A, B: 50.0 ifS^AforallAeS'and Sr\=B, thenSϊB; 50.1 ifSQ Sr andSP=A, thenS'^A; 5.1 S+AϊA; 5.2 ifStA and S\=(Â> B), then S l= B; 5.3 if S +ÂB, then S t= (A =) B); 5.4 if S + ~A 1= J5 αndS + ~Al= ~ £ , ίfterc S t=A; 5.5 ifSPΏA, thenS^A; 5.6 i/ αΠ members of S are of the form DA and Sϊ=B, then S f= D-B. 4.1 Strong Soundness of «C1: The strong soundness of £\ is immediate from the following lemma whose proof using Theorem 1 parallels that of of Lemma 1.4 of [l] (p. 376). Lemma 1 For every proof Π in ΔDD, if A occurs open in Π, then O(Π)t= A. Theorem 1 {Strong Soundness of £\): S h A i n Δ ' D D implies S 1= A inΣl. 4.2 Non-compactness of Σl : In order to see that Σl is not compact it is sufficient to notice that the set T (below) has no models despite the fact that all of its finite subsets have models. T is the union of the following four sets: 7\ is the set of sentential constants, T2 is all sentences ~ Π £ for B in Tί9 T3 is all sentences ~ΠB for B in T2, and T4 is all sentences D(P, = P f +1) where the indices are given by some standard enumeration of the sentential constants. Thus, T is the union of all sets {P,-, ~ D P t , ~ D ~ D P , , • (Pf Ξ Pι+i)}. It follows that ^1 is not strongly complete. In a sense one might say that -(Ί is not compact because two assignments make the same sentences true iff they are identical.2 If assignments are considered representations of "possible worlds/' the above can be rephrased as follows: two worlds are different iff they can be internally distinguished.3 5 Σ2: The interpretations of Σ2 are just those triples (S, K, R) where K is a set of maximally consistent sets of sentences (in Δ'DD), SeK, and R is a relation on K such that for all T, Tr in K, TRT' iff for all A, if DAe T, Ae Tr; and for all TeK, if ~D£e T there exists Tr with TRT' and ~J2e T'. Note that i? is symmetric and transitive since DA IA and DA tD DA for every A.4 As above, SKR indicates (5, K, R); and for each SKR we define VSKR(A) as follows: if A is a sentential constant VSKR{A) = t iff A ε S; VS**(~B) = MV5/CR(5)); V5KR(A 3 5) = C(V5K*(A), V5K*CB)); VSKH ΏB) = t iff for all T with &RT, VTKR(B) =t . Since -Cl is sound the set of sentences true on aPR is maximally consistent in Δ'DD. Moreover, for each interpretation in Σl there is an interpretation in Σ2 making exactly the same set of sentences true. For aPR in Σl, let aPR be those sentences true on aPR. Let S = aPR and K = {bPR: b e P}. Define the relation R on K as follows where bPR R brPR iff bRbr. It is trivial to verify that SKR is an interpretation in Σ2 and for all A, VaPR(A) =VSKR{A). 5.1 The Strong Soundness of ^2: LOGICAL CONSEQUENCE IN MODAL LOGIC II 375 Lemma 1 For all SKR and A, VSKR(A) = t iff A e S. Proof: Let Z be the set of sentences which satisfy the lemma. Every sentential constant is in Z, as are all truth-functional combinations of sentences in Z. Let A e Z and let SKR be arbitrary. Suppose DA eS; then by definition for all T such that SRT, AeT. Therefore, by hypothesis, VTKR(A) = t, and VSKR( DA) = t. Suppose DA /S. Then ~DA e S and there exists T such that SRT and ~A e T. Therefore, by hypothesis, VSKR(DA) = f. Theorem 1 ^2 is strongly sound. 5.2 The Strong Completeness of £2: Consider the following relation W on sets of sentences. SWSr iff S' = (D(S) + ~A)+ for some Ae D(S). Let S be a maximally consistent set of sentences. The Makinson set for S, M(S), is the smallest class of sets which contains S (as a member) and which is closed under W. Given S and M(S), we can easily form the interpretation SM(S)R where TRT1 iff for all DA, if DA e 7\ A e V. Note that for each 5, M(5) is countable. We can easily establish the following: Theorem 1 Every consistent set of sentences in Δ'DD has a model in Σ2. Proof: Let S be consistent and let S+ be the maximally consistent extension of S. By Lemma 1 of 5.1 S+M(S+)R is a model of S. Since results of this section imply that Σ2 is compact it follows that Σl and Σ2 are not strongly equivalent. 6 Σ3: For a truth value assignment, «, and a natural number w, let an denote the pair (a, n). Interpretations in Σ3 are triples a,PR where P is a subset of pairs, R is a reflexive and transitive relation on P and α, e P. For each a,PR, we define V(t'rR as follows: Va'rR(A) = a(A) for A a sentential constant; Va'rR(A ^ B) = C(V"'i|>Λ(A), VaiPR(B))', V"iPR(~A) = N(VaiPR(A)h V"iPR(ΠA) = X iff for all b, such that a,'Rbh V hiPR(A) = t. Σ3 is essentially the semantics given by Kripke for S4 in [3j. Σ3 differs from Σl only in distinguishing "assignments" which make exactly the same sentences true. Let SKR be any interpretation in Σ2, where K is countable. For each Te K, T Π L is maximally consistent in Δ; therefore T n L has a model a. Let To = S and \ai] be an enumeration of the models of {T, Π L}. Also let P(K) = \a0, «1? . . .} and take R* to be the relation on P(K) such that a{R*aj iff TiRTj. a0P(K)R* is obviously an interpretation in Σ3. More interestingly, we have the following: Lemma 1 For all SKR in Σ2 D and all At V SKR(A) = v"f'p(l0κ*(A). Theorem 2 ^3 z\s strongly complete. Proof: This follows from the fact that for all S, A, if S\=A in Σ3, S t=/l in Σ2 and Σ2 is strongly complete. Thus Σ3 is strongly equivalent to Σ2 but not strongly equivalent to Σl. 6.1 The Weak Equivalence of Σl and Σ3: Let Cr be any subset of C, where all but a countable number of members of C are outside C f; *CCfDO indicates those sentences of ^ D D whose constants are in C\ 376 GEORGE WEAVER and JOHN CORCORAN Lemma 1 For all aiPR in Σ3, there exists ciiP'R' in Σ3 such that Pr is a countable subset of P, R1 = R\P\ and for all A in £ΏΠ, VaiP'R'(A) = VaiPR(A). Proof: Let a{PR be in Σ3. For each aj e P, we define the following: (i) D(α/PΛ) = {4:V^ PR(DΛ) = f}; (ii) for zll A eΠiajPR^Aiaj) = {ak: ajRak and V akPR(A) = f}; (iii) T(aj) = {A (α; ): A e D(afPR)} __ (iv) Ca\ the choice function for T(aj) where Cα7(A(α7)) = α7; (v) Δ0(fl;) = {Cα; (A(fl; )): A(α; ) e Γ(α, )} Δw(«/) = Δ^xία,-) U {Δofe): β^e Δ ^ f o )}; (vi) P ' = U Δ W M ; w (vii) R' = R\P\ Note: For all α7e P and all m that ΔOT(β7 ) c Δm+1(«7 ) and Δn(α7 ) is countable, hence P f is countable. A simple induction argument will show that for every α7 e P f and Ae <£ΏΠ,VaiPR(A) =VaiprR'(A). We need only consider the case where VaJPR(ΠA) =f. Suppose that_for all akeP\ V akPR{A) = VakP'R'(A)jιnd that VaiPR(\3A) = 1. Therefore, Ae D(α; P#), A(af)e T(Λ, ) and Cα/(A(α/ ))=α / eA(α / ). Hence a7i&z7.and V^' PΛ(Λ) = f. We need only show that «7 is in P ' . Since aj e Pr, there exists m with α ; eΔOT (a{) and β ; 6Δ0(α7). Obviously, α ; eΔm+1(α/) = Δ«(αf ) u U { A o t e ) : ake ΔOT(α, )}, hence δ"/ e P ' . Lemma 2 For «Z/ a{PR in Σ3 «n<i α/Z C as above, there exists aP'R' in Σl SMCA that for all A in . ( C ' D D V ^ W ) = VΛp/R/(A). Proof: Let âPi? be in Σ3. By Lemma 1 we may assume that P is countable. Let {Aj} be an enumeration of C Cr and let {(#w)7 } be an enumeration of P. Let / be defined on P so that for all j , f(an)j is a truthvalue assignment agreeing with (an)j except on Ay. Form aP'R' where a, P f and Rr are images under /, respectively, of α, , P, and E. Theorem 1 £3 is strongly sound. Proof: Let 5 HA. Then there exists Sr c 5, Sr finite, such that S'hA. Let α, PΛ be any model of Sf in Σ3. Since Sr +A is finite there are countably many constants not occurring in sentences of Sr + A. By Lemma 2 there is a'P'R' in Σl such that it agrees with a{PR o n S ' + A ; and, since j£l is sound, a'P'R' is a model of A and S NA in Σ3. As immediate consequences of Theorem 1 we obtain that Σ2 is strongly equivalent to Σ3; that Σl is weakly equivalent to Σ2 and hence that ^1 is weakly complete. 7 Remarks The strong completeness of the Kripke formulation of S4 depends essentially on treating ordinary truth-value assignments which are identical as being distinct. However, as Kripke points out [5] (p. 69), this restriction is inessential for weak completeness. LOGICAL CONSEQUENCE IN MODAL LOGIC II 377 The system M can be given a semantics where the interpretations are triples aPR where P is a set of assignments, ae P and R is a, relation which is reflexive. We can easily show that this semantics is not compact, by proving the analogues of Lemmas 1 and 2 of 4.2 for M. Moreover, the Brouwerische system, given the semantics whose interpretations are triples αPR, P as above, αeP and R a reflexive and symmetric relation on P, can also be shown to be non-compact. Let T be the union of the following infinite sets: {Λ , . . ., pn, . . .} {•Λ, . . .,ΠP», . . .} {~DDΛ, . . ., ~DDPn, . . .} T has no model although each of its finite subsets does. NOTES 1. The rule D P is a " r igorous" rule of inference ([2], pp. 171-175) in that it (1) is effective; (2) is sound; (3) introduces or eliminates just one logical constant (and not both); and (4) involves in its application just one logical constant (type). But while (DI) satisfies the first three conditions, it fails to satisfy the fourth. Thus (DI) gives a counterexample to the conjecture suggested ([2], p. 172) that any rule satisfying condition (3) would also satisfy condition (4). In connection with the notion of " r i g o r " introduced in [2] we may note that John Myhill has suggested a rule which satisfies all four conditions but which is clearly not rigorous in any (correct) informal sense. Myhill's Rule: Let Π and U[(A D B)σA be proofs where the indicated occurrence of (A D B) is the right-most open assumption in the latter. Then U[(A D B)σA] A is a proof. Myhill's Rule permits the following proof of " P e i r c e ' s Law" [{{A D β ) D A ) [(A D B) A] A](({A D B) D A) D A) 2. Thus Σl cannot be thought of as an abstraction gotten from a first order modal logic (without quantification into modal contexts) by treating quantified sentences as "unanalysed." The reason is that in such cases one will have (elementarily) equivalent interpretations (of the non-modal language) which are not identical. 3. It should be noted that Σl violates the tentative guidelines recently suggested by Dana Scott in his speculative article on modal logic ([8], esp. p. 149). Scott suggests that one should work in a framework rich enough to "distinguish" identical worlds. In this situation Scott's advice could be realized by postulating a large (at least uncountable) index set I for the set of assignments. A "general interpretation," let us say, would again be a triple (α( , P,R) but here P would be a subset of / and R would be a transitive, reflexive relation on P. The definition of truth here is obvious. Σl would then be strongly equivalent, not to the general semantics including all general interpretations, but rather to the special semantics which included exactly the interpretations α{PR where the indexing is one-one restricted to P. In light of construction of the obvious models for the finite subsets of T it is a triviality to produce a "general model' for T. 378 GEORGE WEAVER and JOHN CORCORAN 4. Our remark in section 1 (above) avowing a desire to deal with " conceptually meaningful" semantic systems should not be taken to imply that we regard Σ2 as such a system. Indeed, the traditional principle which bans use of semantic notions in the definition of a deductive system may profitably be amended to include a ban on use of deductive notions in defining semantic systems. Moreover (see below), unless soundness (or at least consistency) of Δ'DD had already been established we would have little reason for believing that the set of interpretations from Σ2 is not null. REFERENCES [1] Corcoran, J., and G. Weaver, (tLogical consequence in modal logic: Natural deduction in S5," Notre Dame Journal of Formal Logic, vol. X (1969), pp. 370384. [2] Corcoran, J., "Three logical theories," Philosophy of Science, vol. 36 (1969), pp. 153-177. [3] Curry, H. B., A Theory of Formal Deducibility, Notre Dame (1957). [4] Kripke, S., "A completeness theorem in modal logic," The Journal of Symbolic Logic, vol. 24 (1959), pp. 1-14. [5] Kripke, S., "Semantical analysis of modal logic I , " Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67-96. [6] Makinson, Donald, "On some completeness theorems in modal logic," Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 379-384. [7] Prawitz, Dag, Natural Deduction: A Proof-Theoretical Study, Stockholm (1965). [8] Scott, Dana, "Advice on modal logic," Philosophical Problems in Logic, ed. by K. Lambert, Dordrecht (1970). Bryn Mawr College Bryn Mawr, Pennsylvania and State University of New York at Buffalo Buffalo, New York