A REGULAR SEQUENT CALCULUS FOR QUANTUM LOGIC IN WHICH A AND v ARE DUAL N.J. CUTLAND GIBBINS Introduction In the paper[ Nishimura (1980)1 the author develops sequent calculi GO and GOM f o r orthologic and orthomodular logic, the latter usually going under the ambiguous name o f quantum logic. I n Nishimura's calculi there is a fundamental lack o f duality in the treatment of the disjunction V and the conjunction A. Moreover, his calculi are not regular in the following sense: a sequent calculus SC is regular iff for finite sets (of formulae) I' and A 1 sc s e The non-regularity of GO and GOM follows from the non-classical disjunction property established by Nishimura for these calculi (see 2 below). In this paper we develop two regular sequent calculi GOt and GOtM which are related to Nishimura's calculi in the following way. The calculi Go t and GOtM are extensions of GO and GOM in that t 0(m) I' >A imp lie s t but not conversely. However, for normal sequents > A', in which A' has at most one member GO(M) i f f Hoot(m) r - A Besides their intrinsic interest as formulations o f orthologic and quantum logic, G o t and GOt M have some relevance to the philosophical discussion of the nature of logic. We say that a sequent calculus SC tolerates a rule R when the calculus SC + R, obtained by adding the rule R to SC, has the same class o f provable sequents as SC.
222 N. J. CUTLAND and P.F. GIBBINS Neither Got nor GOIN tolerates the (full) cut rule of GO(M). There is reason to think that GOt(M) are logics if GO(M) are. Therefore toleration o f the (full) cut rule cannot be held to be a necessary condition for a sequent calculus to be a logic. We reserve discussion of this and other philosophical points to our final section. The organisation o f this paper is as follows. 1 contains some preliminaries. In 2 we discuss Nishimura's calculus GO. 3 develops Go t. We prove the regularity o f GOt and demonstrate its equivalence to GO for normal sequents. Soundness and completeness for GOt with respect to a version o f the relational semantics for orthologic devised by [ Goldblatt (1974)1 are obtained in 4. In 5 we sketch GOiN, our orthomodular extension of GOt, together with its relational semantics. We conclude in 6 with some philosophical remarks prompted by GOt(M). I Preliminaries In this paper we are concerned only with propositional sequent calculi. A (propositional) sequent calculus SC has a denumerable set of propositional variables {Po, Pl...1, a set of logical connectives (which in our case will be A I fo r GO(M) and A, V I f o r GOt(M)), and brackets ), (. The set o f wffs is the smallest set containing the propositional variables that is closed under the connectives. We use Greek letters a,13, t o denote wffs, the capitals A t o denote sets (possibly empty, possibly infinite) o f wffs. In a sequent calculus the proved objects, namely sequents, are of the form 1 (i) 'whenever all the wffs in I' are true, at least one wff in A is true', or (ii) for finite I', A, 'whenever any conjunction of all the formulae in is true, any disjunction of all the formulae in A is true'. (i) and (ii) are equivalent for the classical propositional calculus. The use o f sets o f formulae, rather than sequences, trivialises certain structural rules which appear in Gentzen's original sequent calculi, and whenever we apply a trivial rule in a proof we shall cite 1df.1 I n the usual fashion we write T, a' for F U la 1 ' T, A' for U A etc. Sequents of the form { a and 1
Definitions A REGULAR CALCULUS FOR QUANTUM LOGIC 2 2 3 (a) A proof in a sequent calculus SC is a finite tree of sequents such that (1) the topmost sequents are axiom sequents ; (2) every sequent in the tree except the topmost sequents is the lower sequent of a rule of SC whose upper sequents are in the tree. (b) The sequent proved by a proof in SC is the unique lowest sequent in the tree. We write i- For a finite set of formulae A there are many (logically equivalent) ways to define the conjunction of the formulae in A. Let [AA] denote the set o f all the possible ways o f forming conjunctions o f all the formulae in A. We will write A A to mean any of the formulae in [ A A]. For a language with V (primitive or defined), we d e similarly. When A is empty we understand both [AA] and [VA] to mean the empty set. The following notions enable us to see more clearly the distinction between our calculi and those of Nishimura. Definitions (a) A sequent calculus SC is regular iff, for all finite I' and A H > A i f F - A F - ) VA sc s c (b) A sequent calculus SC is dual iff for all finite F and A -4 A i f A * > F* sc s c where, for a formula a, we obtain a* by replacing each occurrence of A by v, and vice versa ; and for a set I', we define r* = ly* : y E ll. Finally, note that all the sequent calculi considered in this paper possess the following feature.
224 N. J. CUTLAND and P.F. GIBBINS Theorem 1.1 ( i ) I f t f"-> A' of - > A such that - > A' ; (ii) if a finite sequent 1E-3 A is provable in SC, each sequent occurring in any proof of I" A is finite. By induction on the construction of proofs in SC. [Cp. Nishimura (1980) Theorems 2.1 and 2.21. 2 The system GO (as developed in [Nishimura (1980)]) The formal language of GO lacks the logical connective V, which is introduced as an abbreviation, namely a V [3 is an abbreviation for A--3). Axioms of GO: a -> Rules of GO: f -> A ( e x t ) r i f (cut) a, f - ( A->) a A l l, 1 [ 3, f -)A (A->) a Ar3,1-'-)A 1 ( - ) A) r A, a Al3 1 - a - A ( - )
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 2 5 A -) A, a ( - ) - ) --a GO has the following non-classical features. (i) According to the normal form theorem [Nishimura (1980)1 Theorem 2.5 i f t -) A such that t proof such that every sequent occurring in it is normal. (ii) The normal form theorem has the disjunction theorem [Nishimura (1980)1 as a corollary, according to which i f t non-empty, then for some a EA, t GO may be easily seen to be non-regular in the following way. If GO were regular we would have t a V(3-> a, p as an axiom to GO results in an expansion to classical logic, as the next theorem shows. Theorem 2.1 GO + - a v 0-> a, 1 Note that we have F have Ha V a, g i v i n g 1---0 a, v i a (cut). In the extended system we can derive the following rule a, I'-. A The derivation is as follows. But GO + (-+ - ) p. 3421. -+ a, -la (cut) I' A, Hia
226 N. J. CUTLAND and P.F. G1BBINS The non-duality of GO may be easily seen as follows. Since t 0 a, 13-> a A, the duality of GO would imply that a* v r - 13* is GO-provable. But t a* v13*-> a*, 13* in general. In the next section we develop the system G o t, an alternative formulation of orthologic which is closely related to GO yet which is regular and in which A and V are treated dually. 3 The system GOt The formal language of GOt contains the full set of logical connectives A, V, and ---, Axioms of GOt : ) a Rules of Got : -> A (ext) A l, A2 a - A2 (cut-1) r i - ) a r2-> A 1 " (cut-2) -> A ( A - a A 13, I ( A - > ) a A 1 3, 1 F-)13 ( - ) A)t -> a A13 a--) A 1 3 - * a V13-4 A - >A, a ( - 4 V)1 A, a Vr3 1 A, a (-) v)1
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 2 7 -) a -na 1,, F -3. A - 4 A, --, We mark the rules specific to GOt with a 't'. In GOt the cut-rule is restricted. ( dual, as are the rules for A and V. The rule ( -, - ) t is a restriction of the corresponding rule in GO. But the rule (-> - ) of GO is a restriction of the corresponding Go t rule - ) t. Note that (i) GO + (-> --)t is equivalent to the classical propositional calculus; (ii) GO + ( V-Ot is equivalent to the classical propositional calculus. To see (i) note that we have t we 'have (-) -I), as a derived rule, as below. a, -na--> - > a ( - ) --)t a - > ( cu t ) 1E, a A (cut) To see (ii) it is sufficient to note that in GO + (v -))t we have H a V(3-> a, [3 In Theorems 3.1 and 3.10 we further consider the relation between GO and GOt and demonstrate their equivalence for normal sequents. Theorem 3.1 t By the normal form theorem for GO, t for some normal subsequent I' A ' o f f. - A. This normal subsequent ( 'Introduction to Quantum Logic'.
228 N. J. CUTLAND and P.F. GIBBINS has a normal proof. But a normal proof in GO is also a (normal) proof in GOt since in a normal proof in GO (i) any axiom appearing in the GO proof is an axiom of GOt (ii) any application of (cut) in GO is an application of (cut-2) in GOt (iii) any application of (-1 ) in GO is an application of (--1--)t (iv) any application of ( ) A) in GO is an application of ( > A)t (v) the remaining GO rules are all Gat rules. Hence A ' is provable in GOt, and r' ) A may be proved in GOt by (ext). Before we prove the converse of Theorem 3.1 for normal sequents we prove that GOt is regular. We require the following lemmas. Lemma 3.2 Fo r f i t This is trivial if T has less than two members. For largerf, the 'only if' part of the result is obtained by repeated application of the derived rule (DR-1) a, (3, F.--) A (DR-I) a A 1 3, 1 which is easily seen to be a derived rule of GOt. For the ' if ' part, use repeated application o f the following rule (DR-2) which is seen below to be a derived rule of GOt. a Af3,f *A ( D R - 2 ) a,13,1" >A Derivation o f (DR-2) a--)a a, f3-->a Lemma 3.3 Fo r finite A, [3, 13 (ext) ( e x t ) a, p ( 3 A)t ct.13 )a A p a A(3, - ( c u t - 2 ) a, (3, > A tot F >A i f t o t r > V
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 2 9 If A has fewer than 2 members this is trivial. For larger A, proceed by repeated application of the derived rule (DR-3) of GOt to obtain the 'only if part of lemma 3.3. ( D R - 3 ) A, a V13 Derivation o f (DR-3) ( -) A, _ a, p v)t f-- A, a, a V 13 ( - > V)t f - > A, a v p, a v p ( d f. ) f - * A, a v p For the ' i f part proceed by repeated application o f (DR-4), the following derived rule of Got. F -) A, a v p (DR-4) -* A, a, p Derivation o f (DR ot-> ( e a x t ) - ( e > x f t ) p (v w F--+A,a Vt3 a V (3 - (cut-1) Theorem 3.4 (The regularity of Gat) For finite F, A F-Got F -) A i f f i - - Directly from lemmas 3.2 and 3.3. Theorem 3.5 (The duality of Gat) Hoot T
230 N. J. CUTLAND and P.F. GIBBINS Since F** = F and A** = A it is sufficient to prove that i Let us call A* > r* the dual o ff-4 A. The dual of an instance of a rule is obtained by taking the dual of each of the sequents occurring in that rule. Now observe that the dual of an axiom of GOt is an axiom of GOt, and the dual of an instance of a GOt rule is also an instance of a GOt rule except for H o w e v e r, the dual o f a finite instance o f is an instance of the derived rule of GOt for finite A which has the following derivation. It is sufficient, using the derived rules (DR-3) and (DR-4) of Lemma 3.3 to consider the case A = 181. a A et, A A Hence any GOt proof of a sequent - 4 A (with finite sequents throughout) can be converted to a GOt proof of A* >r*, as required. We n o w prove the converse o f Theorem 3.1, and hence the equivalence of GO and GOt for normal sequents. We begin with the following lemmas. Lemma 3.6 The following is a derived rule in both GO and GOt : cc >A (DR-5) a * A > ( c u t - 1 ) a a A C i ( c u
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 3 1 Lemma 3.7 The following are derived rules of GO: (a) a V 13 a f' > (3 V? Y s (b) a > ( 3 ) 4 5 a V13 ) (a) Note first that lemma 3.2 for Gat holds in GO since (DR-1) and (DR-2) are both derived rules of GO. Thus we need only consider (a) in the following form: A l A r - > f i V y A F- - 1 ( - - l a A --43) --, ( - l a A --, (3) (--,a A la, AI ' (DR-1) H i A ( - > (-113 A -, y ) ( D R - 5 ) A I' ( - 0 A -1y) ) (DR-5) (DR-2) a -> y y -> ( c u t ) a -> (b) ) 76 >,b > ---,a A t3 a V 13 a V Lemma 3.8 The following hold (a) t 0 6 > f5 V a, (DR-5) H A) (b) f o r finite A (with A 0 ) t (c) ( i ) t (ii) t
232 N. J. CUTLAND and P.F. GIBBINS s ) (a) ( A - -115 A ---,a -) 0 -+ A -) 8 V a ( (DR-5) (b) By induction on the size of A. Note first that the result is trivial for A with one member. Suppose that A has at least one member; as the induction hypothesis assume that io -,A ( V A ) Leto!) = v A; then using the associative laws (c) it is sufficient to show that for any a, t as follows (ti v a) --, ô, a A la ( Theorem 3.9 --, A, ---,a -) - - ( --it) A la) (c) We leave the associative laws as an exercise for the reader. In the next theorem we must take into account the fact that V is primitive in GOt but not in GO. For any wff y we write y for the corresponding GO formula in which all occurrences of v are abbreviations, and similarly for sets of formulae. F By induction on the construction of the proof in GOt off' -) A. We may assume that all sequents occurring in the proof are finite.
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 3 3 Basis step Trivial for proofs of unit length; the axioms of GOt are axioms of GO. Induction Step Let the last step in the proof of I' ) A be an application of (i) ( e x t ) E,1 i.e. ( e x t ) By the induction hypothesis GO Y-) v i l EgF,FIcA So from lemma 3.8 (a), and the associative laws (lemma 3.8 (c)) as required t (ii) ( c u t - I) i.e. r-so, A O A2 (cut-1) By the induction hypothesis to r - - t o 6, By lemma 3.7 (a) and 3.8 (c) as required t IF ) ( v A (iii) (c u t -2 ) i.e. >A F2, a -+ A where I' = r i ur2, obvious since (cut-2) is a special case of (cut);
234 N. J. CUTLAND and P.F. GIBBINS (iv ) ( A ) ) trivial; (v ) ( > A ) t trivial; (v ) ( - - > A ) t trivial; (v i) ( V a > A 1 3 i.e. A ci v (3 ) A By the induction hypothesis c7. v A and t so, by lemma 3.7 (b) t (vii) V ) t i.e. F f l 1 By the induction hypothesis Go ( v r ) VCT and so by lemma 3.8 (a) t (viii) trivial; (ix) ( > 1)1-
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 3 5 The last line in the proof is where f = a n d A = B y the induction hypothesis G O1 > VII. By (-1, >) we have t GO we have t Using (--* --,) t i.e. t Now use lemma 3.8 (b) to obtain as required. (x) ( trivial; ( x LI, a i.e. ( r n, w h e r e A =HUI 1 By the induction hypothesis so by lemma 3.7 (a) > (V if ) Vci, and t (V i f ) V o f f ) as required. Theorem 3.10 (The equivalence of GO and GOt for normal sequents) For normal GO sequents A o r A i f H
236 N. J. CUTLAND and P.F. GIBBINS 'Only if' part from Theorem 3.1; 'if' part from Theorem 3.9. 4 The Semantics of GOt : Soundness and Completeness We now prove the soundness and (extended) completeness of GOt with respect to the relational semantics due to [Goldblatt (1974)]. The following account is similar to that of [Nishimura (1980) 3], but with a modified definition of the validity of sequents. Go t : Soundness A GOt-frame is a pair < X, i > where (1) X is a non-empty set, (2) I is an orthogonality relation on X. That is, X x X, and is irreflexive and symmetric. For x cx and Y X we say (i) x Y i f for every y EY, x y ; (ii) y * f x x Y 1 A subset Y of X is 1 A GOt-model is a triple < X, 1, D > where (1) < X, 1 > is a GOt-frame (2) D is a function assigning to each propositional variable p a closed subset D(p) of X. Given a GOt-model A, the notation II a II, where a is a wff, is defined as follows. Note that for any Y X, Y* is closed, always closed. ( (2) I I -- (3) I l a A011# = 1 1 0 / 4 ( ( SOla II if is
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 3 7 For arbitrary f (not necessarily finite) it is convenient to define: HAFjj = c l Definition Define Val follows: (1) for formulae V a l (2) for sequents Val e( F 0 otherwise. 0 if x A r ' l l a n d x M V A I l t I otherwise. Definitions We say I' -> A is GOt-realisable i f for some Gat-model /11 = < X, D> and some x ( 1 A sequent which is not GOt-realisable is said to be GOt-valid. Thus r, A is GOt-valid iff II A FL The following will be useful in discussing the semantics of GOt. It is easily seen that for any closed sets Y Hence (i) Y i g Y2 imp lie s Y 2 ( i i ) Y ( i ) a 11 H 1 1, H i m p l i e s 1 1 ( (ii) 11A ( v A i 114 U V A2 V ( Ai, A2) Ilee (iv) 11 H V ( A i j g - II V A2I vi Theorem 4.1 (Soundness of G0 t) Every GOt-provable sequent is Gat-valid.
238 N. J. CUTLAND and P.F. GIBBINS By induction on the construction of proofs in GOt. Every axiom o f GOt is GOt-valid. The rules o f GOt preserve GOt-validity. Demonstration of this is routine, but we illustrate some of the more awkward cases. The other cases are left to the reader. (cut-i) Suppose H A r c il v(a, A1)1 Hv(a, Ai)iltgll v(a,, A2)1 A (cut-2) Suppose that Then j A )) and llochtg-11 VA211ff; then by the above, and so H AI j x r2) Arillog 11,# = Ha H 11, Ar, and I HA A 1,2..ie g g II g j a n j A r 2 l l o = 11 A ( g 11 VA 1 V A H Suppose that j j a Then j VAII.#*gliallie* and H V A Ilt*Çlll311,/t* Therefore V A H a ill* n 11P 114*, and (Hallo Therefore whenever both a -4 A and (3 > A are GOt-valid, so is a V p - 4 This completes the soundness proof for GOt. GOt : Completeness The following is similar to Nishimura's treatment for GO. Definitions A sequent 1" > A is consistent iff A. The set F is consistent iff for some a, F > The set f is complete if f (a) I" is consistent and (b) = fy : 1
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 3 9 Lemma 4.2 [Lindenbaum's Lemma] Any consistent set F can be extended to some complete set F. Put r' = ty : Hoot r (i) Suppose H o f f ' Huot Y1, Y By m applications of (cut-2) H (ii) Since I' is consistent, then for some a, F ' Therefore a çt f ', so by (a) b O t r -) a and r is consistent. Hence r is complete. Lemma 4.3 (a) I f r a is consistent, then (i) y 5 F implies i s consistent, (ii) e f implies - > 6 is consistent ; (b) if I' i s consistent, then s (i) y 6 f implies a-3 i s consistent, (ii) 6 1 We do (a)(i) and leave the rest as an exercise for the reader. Since y e l -- y -3 y -) a Hence --la-3 i s consistent. ) y a ( c u t - -nu (-3 --)t (cu t- a a --3 a ( (c ut-i )
240 N. J. CUTLAND and P.F. GIBB INS Definition The GOt-canonical model, ilt is defined as follows. < X, I (1) X = IF : IF is complete} (2) f i r iff for some a, either (a) a e r and or (b) a a n d e l " ; (3) fo r propositional variables p D(p)= : p E FL Lemma 4.4 I t is a GOt-model. (a) I is an orthogonality relation. Symmetry is obvious f ro m the definition. Irreflexivity follows as below. If I' I F then for some a, a erʻ and œla F. Therefore H all 13 and I' is not consistent, as follows: a > (3 (ext) (b) D (p ) is closed, i.e. D(p) = D(p)** Since Y c Y** for all Y, it is sufficient to show that D(p)**g_ D(p). Suppose therefore that some FE4D(p). p a n d 1 LetE = {a:1 Then E ID(p). That is, E ED(p)*. But F s i n c e (i) if y f, p ) --cy is consistent by lemma 4.3 (a) (i) and therefore O E ; (ii) if - -, 6 is consistent by lemma 4.3 (a)(ii) and therefore 6 OE. Hence D(p )** D(p ), and therefore D(p) is closed. Before we prove the Main Theorem for I t lemma.
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 4 1 Lemma 4.5 (a) ( i ) F - - (ii) HGOt ( - -, a A ---,r3)-> a v (b) The following are derived rules of GOt F-» a A 13 a F - ) a AP s (a) a - a - l a A -113 ->--1 a -1 a (-). ---1) - -, A -,IT; a - ) - - -, a A - v - > --1(--ict A -13) ( A -* ) (cut-1) ( v, ) t A --, (3) We leave (ii) as an easy exercise for the reader. (b) Use ( A Theorem 4.6 (Main Theorem for ifft = < X, 1, D>1 For any F a X, y (Throughout this proof we write Hall for 1 By induction on the length of the wff y. a 11/gt) (1) y = p, some propositional variable p e r iff 1 (2) y = a A (3 a A (3E1" i f f a A13, since[' is complete, iff - (-) A)t i f a e l iff FE lla h I and fell1311, by the induction hypothesis, i f r
242 N. J. CUTLAND and P.F. GIBBINS ( There are two cases to consider: (a) suppose --lo te Then F i r " for every r such that a e i.e. : a E r} l i a l by the induction hypothesis. Hence re11a11* (b) suppose ---,a e r Then i s consistent. Let A = 16 : H - Ella 11 by the induction hypothesis. But L A, since if y E r, then a -> i s consistent by lemma 4.3 (b)(i) and therefore --lye A, and if -ny, then a y is consistent by lemma 4.3 (b)(ii), and therefore y e A. So therefore F e l l (4) y = a V [ 3 a v e l ' i f H iff H i f A - i f F Cli ( - l a A 13), from steps (2) and (3) above, iff r Theorem 4.7 (Completeness of G0t) If F A is GOt-valid, then [ - Suppose Got 1E-4 A, i.e. that 1 that 11 AI ' H VA U Let t f a : I that A is consistent. Then it is easy to show that - A -* - i t is consistent. Let l a : HGot a l ; then / is complete. We claim that (1) t E l l A F (2) / ell A - - (3) t
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 4 3 (1) and (2) are immediate, using Theorem 4.6 and the fact that F and --IA ÇE. Consider claim (3). Suppose that 'y E t and T h e n there is a finite A' ÇA with H Hence - -, ---,A', SO h - Alternatively, suppose that, y c t and y cx. Then there is a finite A' ÇA w i t h H H Thus claim (3) is verified. By (2) and (3) together, t O il A 1 e II Al" l let \II va a n d we have shown that 1 The following corollary is immediate. Theorem 4.8 (Compactness of G0 t) For arbitrary I" and A, I ' >A is GOt-realisable i f every f i subsequent of F > A is Gar-realisable. Remark I f r--4 A is a finite G o t consistent sequent, the completeness proof above can be modified to provide a finite GOt model that realises >A. This is done as in Nishimura (1980) * 41 by restriction to a suitable finite admissible set of formulae Q containing 1' and A. In our case Q is admissible means (a) Q is closed under subformulae (b) i f p then p Q (c) i f a V f 3 Q then ( - -, a A (3) Q. Thus Theorem 4.9 GOt has the finite model property. 5 G O I We obtain GOtM, the orthomodular extension of GOt, in a manner similar to Nishimura, by adding to GOt the rule (OM) -113 (OM)
244 N. J. CUTLAND and P.F. GIBBING The resulting calculus G O I for normal sequents in virtue of the following theorem. Theorem 5.1 Let R be a rule whose upper and lower sequents are all normal. Let SC normal sequents. The SC sequents. By induction on the length of proofs GOtM : Soundness and Completeness We proceed by analogy with GOM. Thus a GOtM-frame is a triple < X, I, 'V> where (1) < X, i > is a GOt-frame (2) I P is a non-empty set of I-closed subsets of X such that (a) W is closed under set-theoretic intersection and the operation * (b) for any Y, Z Y Z and 1 A GOt-model is a 4-tuple < X, I, W ( (2) < X, I, 4 D is a function assigning to each propositional variable p a closed subset in W. The notations \ T a l before. The expressions GOtM-realisable and GOtM-valid are defined similarly. Theorem 5.2 (Soundness of G O I Every G O I By induction on the construction of proofs in GOtM a s in the GOt
A REGULAR CALCULUS FOR Q UANTUM LOGIC 245 case but with the following addition. (Let i t be a GOtM-model, and write Mali for lialla0 (OM) Suppose d c il a I i Suppose also that 11 I I n II (311 = 0, so that Ha ll Then Hall = j1 3, so that 11011 g ita ll and hencell aallg11,1311 We now sketch the proof of the Completeness Theorem for GOt M. We say a sequent f, ) A is GOIM-consistent iff 4 ' ' a set I' is GOW-consistent i f for some a, - F The set F is GOt M-complete i f (a ) F is GOt M-consistent and (b) F = {y: 1 Gotm r Y 1 Lindenbaum's Lemma for GOtM can then be proved exactly as for GOt in lemma 4.2. Definition The GOtM-canonical model I t i s defined as follows. f i < X, I (1) X = : I' is G O t M - (2) r i i f f for some a, either (a) a cl" and --la e r, or (b) c f and a ; ( using (4); (4) f o r propositional variables p, D(p)= : p e r}. Theorem 5.3 i s a GOtM-model and realises every GOtM-consistent sequent. We sketch the proof. First show that < X, I exactly as in lemma 4.4 for l i t, and then establish that
246 N. J. CUTLAND and P.F. GIBBING (5.4) f e l l a H i f f c l e f exactly as in Theorem 4.6 (where we write Hall for lia Now, exactly as in the proof of completeness for GOt, we show that if 1 ( To complete the proof we have to show that i t t is a GOtM-model. Suppose that 11 a 11cHI3 Il 1a 11*n111311= (b. Then by ( obtain 1 Using (5.4) this means that 11011g H all. We then have the following theorem immediately. Theorem 5.6 (completeness of GOtM) If I" A is GOtM-valid, then i 6 Concluding Remarks As noted in I, for finite F and A, the intuitive interpretation of the sequent F ) A in classical logic may be rendered equivalently as (i) whenever all the formulae in r are true, at least one of the formulae in A is true; (ii) whenever a conjunction o f all the formulae in I ' is true, any disjunction of all the formulae in A is true. The interpretation of finite GO(M) sequents follows (I), that of finite GOt(M) sequents (ii). Since (i) and (ii) are not equivalent for both GO(M) and GOt(M) there is reason to think that the meanings o f sequents in these calculi differ from their meanings in classical logic. But there seems to be no conclusive reason for regarding either (I) or (ii) as the more fundamental intuitive interpretation. The primitive rules for the A - --Ifragment of GOt naturally differ from those o f GO. But the most interesting difference between the systems occurs in their structural rules. GO(M) has the f ill cut rule among its primitives. Owing to the normal fo rm theorem GO(M)
A REGULAR CALCULUS FOR QUANTUM LOGIC 2 4 7 requires only the restricted cut-rules of GOt(M). But clearly GO(M) tolerates full cut. GOt(M) does not tolerate full cut. GOt(M) + (cut) cla ssica l logic, as in shown in the lemma below. There is in the literature the suggestion (2) that one can expect only elastic constraints on what is to count as a logic but that toleration of full cut is one constraint required to reflect the necessary transitivity of implication. But GOt(M) illustrates the fact that one can reflect the transitivity of implication with weaker rules. I f GO(M) are to count as logics, there seems to be no good reason for denying that status to the extended calculi GOt(M). Lemma 6.1 GOt(M) + (cut) = classical logic. It is sufficient to obtain the result for finite sequents. We are required to show that ( > --) GOt + (cut). I", --la ) A a a,a > (c ut) F. a A f )A, a, --la ( c u t ) ( ) A ) (
248 N. J. CUTLAND and P.F. GIBBING for finite A F---) A, a finite number of applications of (-1-4) a - - -, A - 4 13 I finite number of applications of ( > --) a Ar3 1 REFERENCES Goldblatt R. Semantic Analysis of Orthologic. Journal of Philosophical Logic 1974; 3: 19-36. Hacking 1. What is Logic? Journal of Philosophy 1979; LXXVI: 285-319. Nishimura H. Sequential Method in Quantum Logic. Journal of Symbolic Logic 1980; 45: 339-352.