by KOSTA DOSEN in Belgrade (Yugoslavia) )

Transcription

1 A BRIEF SURVEY OF FRAMES FOR THE LAMBEK CALCULUS by KOSTA DOSEN in Belgrade (Yugoslavia) ) Abstract Models for the Lambek calculus of syntactic categories surveyed here are based on frames that are in principle of the same type as Kripke frames for intuitionistic logic. These models are extracted from the literature on models for relevant logics, in particular the ternary relationed models introduced in the early seventies. The purpose of this brief survey is to locate some open completeness problems for Variants of the Lambek calculus in the context of completeness results based on various types of ternary relational models. MSC: 03B60. Key words: Lambek calculus, Kripke models, relevant logic, linear logic. 1. Introduction Models for the Lambek calculus of syntactic categories we shall survey are based on frames that are in principle of the same type as Kripke frames for intuitionistic logic. Most of what will be said in our brief survey can be extracted with a modicum of labour from the literature on models for relevant logics from the early seventies. However, some authors (including the author of this survey) who write on the Lambek calculus, or on logics with restricted structural rules (in particular linear logic), have been making recently a number of rediscoveries, as if they were ignorant of this literature. One of our aims here is so try to straighten the record by appealing to essential references. Another, and more important, aim is to place some open completeness problems for variants of the Lambek calculus in the proper perspective. Before we reach these open problems we shall fmt see what sort of completeness is available. We shall concentrate only on the main variants of the Lambek calculus, presented in the notation of propositional logic, which amount to basic implicational logics. We will stick only to this most simple context, without the other connectives of propositional logic, and we will omit details. This we do because we are afraid that too many additions and details would distract the reader s attention from the simple essentials to which we want to draw it. 2. The Lambek calculus - Let P be a propositional language with a nonempty set of propositional letters (whose exact cardinality is not important here) and the binary connectives 0, and +. The connective 0 is called fusion in relevant logic. We use A, E, C,..., A],... as schematic letters for formulae of P. Expressions of the form A I- B will be called sequents of P. The system I., which ) I would like to thank WOJCIECH BUSZKOWSFX, AND& FUHRMAN and SAVA hsnk for discussing matters trated here. I am grateful to the Alexander von Humboldt Foundation for supporting my work with a research scholarship, and to the Universities of Konstanz and Tiibingen for their hospitality.

2 180 K.DO$EN amounts to the nonassociative Lambek calculus of LAMBEK (141, has the following axiomschemata: At-A, Ao(A--,B)I-B, Bt-A+(A A I- B, B I- (B A) +A, and the following primitive rules: from A I- B and Ct- D infer A CI- B 0 D, B+ Cl- A + D, C+B I- D+- A, from At-BandBI-CinferAI-C. The system L,, which amounts to the associative Lambek calculus of LAMBEK [13], has moreover the axiom-schemata (AoB)oCI-Ao(BoC), Ao(BoC)l-(AoB)oC; and L,, i.e. the commutative Lambek calculus, which is related to linear logic, is L, plus A EI- B A. In L, the connectives + and + become synonymous, i.e. we can always, even nonuniformly, replace one by the other so that provability is preserved. 3. Ternaryframes A ternary frame is ( W, R ), wherc W is a nonempty set and R is an arbitrary ternary relation on W. We use x, y, 2,... for members of W A ualuorion on a ternary frame is every mapping u from P into the power set of W which satisfies: (VO) (v+) u(a ob)= (x: 3y, z (Ryzx&y~ u(a) & ZE u(b))], u(a+b)=(x:vy, ~((Ryxz&y~u(A))*z~u(B))), (v-) u(b+a) = (x: Vy, z((rxyz&y~u(a))*z~u(b))). It is not difficult to prove the following completeness result: Proposition 1. ThesequentAll-AA1 isprouablein L,, iffformeryvaluation uoneueryiernary frame we haue u(aj E u(aj. Proof. From left to right we proceed by induction on the length of proof of Al I- A2 in L,,. For the other direction we construct the following canonical ternary frame. The set W is P, and R (A, B, C) holds by defmition iff Cl- A 0 B is provable in L,,. The canonical valuation y on this frame is defined by u(a) = (B: B t- A is provable in L,,). Then if Al I- A2 is not provable in L,,, with the canonical valuation on the canonical ternary frame, we have Al E u(a,) and A,~u(AJ. 0 Consider now the following conditions on ternary frames, assumed for every x, y, z, u E w (ass) (com) ~~(RXYS & RSZU) F) 3t(Rxru & Ryzr), Rwu cs Ryxu.

3 Analogously to Proposition 1 we can prove the following A BRIEF SURVEY OF FRAMES FOR THE UMBER CALCULUS 181 Prop o s it i o n 2. The sequent A I + A2 is provable in L, [ respectiuely L J iff for euery valuation u on euery ternav frame that satisfies (ass) [respectiuety (ass) and (corn)] we have u(a,) E u(a,). Models based on ternary frames with conditions as above have been introduced for various relevant logics by ROUTLEY and MEYER in [19],(201, [21], and by MAKSIMOVA in [ls] (an introductory survey may be found in DUNN [9]). In particular, the systems in (191 cover some close relatives of the Lambek calculus. 4. Operational groupoid kames and closure groupoid frames As a particular type of ternary frames we have operational groupoid frames ( W,., Cl), where W is as before a nonempty set,. is an arbitrary binary operation on W, and C1 is an arbitrary unary operation on the power set of W. Every operational groupoid frame is a ternary frame where the relation Rxyz is defined by z E Cl(x ey}. The completeness of L,, with respect to arbitrary operational groupoid frames and arbitrary valuations on them follows from the proof of Proposition 1. In the canonical frame of this proof, W, i.e. P, is a groupoid with the connective 0, and for X E W the set CLY may be defined as (B: (3A E X) Et- A is provable in L }. An operational groupoid frame (W, *, Cl) will be called a closure groupoid frame iff for every X, Y S W the operation C1 satisfies: ((211) xs Y*CLYE ClY, (C12) ClCLY E CLY, ((213) XE CLY. A closed valuation on a closure groupoid frame is a valuation v that for every A of P satisfies u(a) = Clv(A). It is easy to check that for closure groupoid frames and closed valuations u we can prove: (v +) (v +) u(a+b)={x: (Vy~v(A))y.x~u(B)}, u(b+a) = [x:(vye u(a))x.y~ u(b)}. Actually, we do not use the full power of (C1 l)-(ci 3) to prove (v +) ing weaker condition assumed for every X E W and every x, y E W: x.yeclxocl[x.y}ecly and (v +). The follow- is necessary and sufficient with closed valuations. However, valuations need not be closed to get (v +) and (v +); it is necessary and sufficient to assume that for every B of P and every X, y E W they satisfy X - ~ v(b)ocl{x.y} E E u(b). Let X. Y = {x. y : x E X & y E Y) ; then, if C1 in a closure groupoid frame satisfies for every x, YE w (C14) CIX-CIYS CI(X* Y), ((215) CLY = b: (3x E X)y E Cl(X}},

4 it will be enough to stipulate that u(p) = Clu(p) for every propositional letter p in order to infer by induction on the complexity of A that u(a) = Clu(A) for every formula A of P. Let us call closure groupoid frames which in addition to (C1 l)-(c13) satisfy (C14) and (CIS) inductive, because, as we have just seen, on them we can define closed valuations inductively; this means that we stipulate u(p) just for propositional letters p, so that u(p) = Clu(p) is satisfied, and then use (v.), (~ 4) and (#+) as clauses in an inductive definition. (The conditions (C14) and (CIS) are sufficient to enable us to define closed valuations inductively, but they are not necessary. Conditions which are necessary and sufficient are obtained by substituting CLY and CIY for, respectively, u(a) and u(b) in the sets on the right-hand side of (v o), (v-+) and (v +), and then requiring for every X, YE W that the sets Z that result from the substitution satisfy Z = C1Z. However, these conditions seem much less natural than (C14) and (C1 S).) Due to (C15), for inductive closure groupoid frames we can prove: (V.) u(a 0 B) = CI(U(A). u(b)). The completeness of L,, with respect to all closure groupoid frames or all inductive closure groupoid frames (with closed valuations) is proved as before. We only have to check that our canonical frame is an inductive closure groupoid frame, and that the canonical valuation is closed. 5. Binary groupoid frames As a particular type of closure groupoid frames we have binury groupoid frames ( W, *, S), where, as before, ( W,.) is a groupoid and S is an arbitrary binary relation on W. Every binary groupoid frame is a closure groupoid frame where, assuming that S is the reflexive and transitive closure of S, the set CLY is defined as b: (3x E X ) xs*y}. For binary groupoid frames Rxyz is equivalent with x.ys*z. In inductive binary groupoid frames condition (C14) will imply that if xs*y and zs*t, then x. zs*y. I, which is sufficient to enable us to define closed valuations inductively (condition (C15) is now an immediate consequence of the definition of C1 in terms of S). the completeness of L,, with respect to all binary groupoid frames or all inductive binary groupoid frames (with closed valuations) is proved as before. In the canonical frame, by definition, A SB holds iff B I- A is provable in h. Since in this canonical frame we have that S and S coincide, we can require that S in binary groupoid frames be reflexive and transitive, or even a partial order, and still obtain completeness. In binary groupoid frames, as we have defined them, Rxyz is equivalent with x.ys*z. Of course, we could define Rxyz by x.ysz and get that L,, is complete with respect to all binary groupoid frames with arbitrary valuations satisfying (v o), (v+) and (v +) with this new ternary relation R. If moreover we require that our valuations satisfy for every A of P and every XEW (hered) X E u(a)*vy(xsy-y~ u(a)), then we can prove (v +) and (v t), and still obtain the completeness of I., with respect to all binary groupoid frames with these special valuations. The left to right direction of (hered) is equivalent with u(a) = CIu(A), where C1 is defined in terms of S as before; this is the heredity condition typical for intuitionistic Kripke models. The other direction of (hered) is trivially satisfied if S is reflexive, but we deal with an arbitrary S that need not be reflexive.

5 A BRIEF SURVEY OF FRAMES FOR M E LAMBEK CALCULUS 183 Analogous completeness proofs with respect to appropriate classes of operational or binary groupoid frames may be obtained for L, and L, (cf. [7]). Note that if in an inductive binary groupoid frame ( W, -, S) the relation S is a partial order, then (ass) is equivalent with the associativity of the operation - and (com) is equivalent with its commutativity. In the construction of the canonical frame we may first make P go through a Lindenbaum procedure to make it a partially ordered semigroup or partially ordered commutative semigroup. The essential novelty of models based on closure groupoid frames is the new conditions (v'-+) and (v't) for implication. That implication could be modelled in such a way was foreshadowed by LAMBEK in 1131, but the fmt such explicit treatment of implication may be found in papers from the early seventies devoted to many-valued logic (Scm[23] and URQU- HART [25)) and relevant logic (URQUHART (241 and FINE [lo]). More recently, models of this type have been considered by ONO and KOMORI [16], and in [7], [8]. All these papers are concerned with models related to models based on binary groupoid frames. Models related to models based on closure groupoid frames that are not necessarily binary are envisaged by SAMBIN [22], ABRUSCI [l] and ONO [17]. In universal algebra a modelling of residuation in groupoids analogous to (v'+) and (v'e) seems to have been common knowledge for a long time (see FUCHS [ll], p. 190, and BLYM and J~~owrn [S], pp. 214, 247). 6. Simple groupoid frames Finally we come to simple groupoid frames (W,.), which are binary groupoid frames ( W, a, S), where S is the identity relation. Models based on simple groupoid frames are related to a linguistic interpretation, which stems from LAMBEK [13] and LAMBEK [14]; they have been treated in detail by BUSZKOWSKI in [6]. It is clear that simple groupoid frames are inductive and that every valuation on them is closed. However, completeness of L,, with respect to these frames is not an easy matter, as it was up to now. Presumably, we could adapt a rather difficult proof that BUSZKOWSKI has provided in [6] for L, and L, with respect to simple groupoid frames that are respectively semigroups and commutative semigroups. In BUSZKOWSKI [6] leaves as an open problem, which up to now seems to have resisted solution, whether L, is complete with respect to simple groupoid frames that are fie semigroups. The related completeness problem for L, with respect to free commutative semigroups also seems to be open. However, the related completeness problem for L, has a simple negative solution. The system L, is not complete with respect to simple groupoid frames that are free groupoids. It is easy to check that for every valuation u on every free groupoid we have u(ao(a--*(boc)))ev(a oc),thougha ~(A-'(B~C))+A~CisnotprovableinL,,(as can be shown by using the Gentzen-style decision procedure for L,J. A related counterexample may be given with the unprovable sequent A 0 (A --* (B 0 C)) + B 0 (A+ (B 0 C)). A natural extension of L,, is complete with respect to simple groupoid frames that are free semilattices, provided we restrict ourselves to certain specific valuations. This extension, which we shall call Lhr is obtained by adding to the axiomatization of L, the following axiomschemata: AI- A.A, A BI- A. Because of the commutativity of 0 we also have A 0 B I- B in Lh. The system Lh corresponds to the implication-conjunction fragment of the Heyting propositional calculus, and the prop-

6 184 K. DOSEN ositional constant T is definable in it by A+ A, for an arbitrary A. Of course, in Lh the connectives + and +- are synonymous. A simple groupoid frame ( W,.) is a semilattice iff. is associative, commutative and idempotent. Let us call a valuation u on a simple groupoid frame multiplicative iff for every A of P and every x, y E W we have (mult) XE u(a)=x.y~ u(a). For simple groupoid frames that are commutative semigroups it is enough to assume (mult) for every propositional letter A in order to infer by induction on the complexity of B that (mult) holds for every B of P; i.e. multiplicative valuations can be defined inductively. With multiplicative valuations u on semilattices we have: ~(A+B)=u(B+A)={x:Vr((3y(x*y=~)& zev(r))*zeu(b))}, u(a B) = u(a) n u(b), which shows that our models are a kind of Kripke models for intuitionistic logic. Then we prove the following Proposition 3. The sequent A l- A2 is provable in Lh ifl for every multiplicative valuation u on every simple groupoid frame that is a free semilattice we have v(aj E u(a2). Proof. From left to right we proceed by induction on the length of proof of Al I- A2 in Lh. We need (mult) in this induction in order to show that u(a B) E u(a). To prove the other direction we first introduce the following terminology. Let a formula of P be called prime iff it is not of the form B 0 C, and let a prime factor of a formula A be a prime subformula of A that is not in the scope of an +, or an +. Next, let [A] be the set of all formulae B whose set of prime factors is identical with the set of prime factors of A (in other words, B is obtained from A by using the associativity, commutativity and idempotence of the connective 0 binding the prime factors of A). It is clear that [A] = [B] implies that A l- B and B l- A are provable in Lb (the converse need not hold). Next, let [A].[B] be defined as [A B], and let W= {[A]: A is in P}. It is clear that so defined is an operation, that W is closed under this operation, and that ( W,.) is a semilattice freely generated from all the sets [A] such that A is prime. This ( W, -) will be our canonical simple groupoid frame, on which we define the canonical valuation u by u(a) = {[C]: CI- A is provable in Lh}. It is clear that Cl- A is provable in Lh iff (VB E [C]) B t- A is provable in Lh, ift(3b E [C]) B l- A is provable in Lh. It remains to check that the canonical valuation is a multiplicative valuation. (For example, we use A 0 B t- A and A B l- B in order to show that from [C] E u(a B) it follows that [C]* [C] = [C],[C] E u(a) and [C] E u(b).) Then suppose that Al t- A2 is not provable in Lh; with the canonical valuation u on our canonical simple groupoid frame we have [A,] E u(a,) and [A,] e u(a2). 0 Of course, an analogous proof shows that with multiplicative valuations, Lh is complete with respect to all simple groupoid frames that are semilattices, not necessarily free. Simple groupoid frames that are semilattices are closely related to semilattice semantics of URQUHART (24); but a significant difference is that URQUHART S semilattices, taken as join

7 A BRIEF SURVEY OF FRAMES FOR THE LAMBEK CALCULUS 185 semilattices, have a zero element. Without multiplicative valuations we obtain models for Lh minus A 0 B + A, which is a system related to the relevant logic R. In [24] URQUHART does not show completeness with the connective 0, and in the absence of principles corresponding to A 0 B I- A and A 0 B I- B his simple and elegant method (which could also be adapted for the proof of our Proposition 3) seems to be blocked. Perhaps something like the method of BUSZKOWSKI [6] could be applied for system with related to R. 7. Two-dimensional ternary frames We shall now consider the following special type of ternary frames (W, R), where, for a nonempty set D, the set W is a nonempty subset of D X D, and R~ryz holds iff for some a, b, CE D we have x = (a, b),y= (b, c) and z= (u, c). Ofcourse, with our standard definition of valuation on ternary frames, these frames, which we call two-dimensional ternary frames, give rise to models for La. We get (ass) and, hence, models for L, if moreover we assume that W is a transitive relation. We do not get (com) simply by assuming that W is a symmetric relation, but if we assume that W is transitive and symmetric, and we assume more- over that for every A of P the relation u(a) is symmetric, then we get models for L,. Recently, ORLOWSKA, BUSZKOWSKI and VAN BENTHEM have considered models for the Lambek calculus where, for an nonempty set D, we have W = D X D, and u assigns to every formula of P a subrelation of W so that: (~ 0) u(aob)={(u, b):3c((u,c)~u(a)&(c, b)eu(b))}, (v +) (v + u(a+b) = ((u, b): Vc((c, a) E u (A)a(e, b) E u(b))}, u(b+a) = {(u, b): Vc((b, c) E u(a)-(u, c) E u(b))} (see ORLOWSKA [18], VAN BENTHEM [2], 3.2.9, and VAN B ~ E [3], M 4.2; for similar conditions on valuations in two-dimensional modal logic, see, for example, KUHN [12] and references therein; cf. also the definition of residuation in terms of conversion, composition and complementation in BIRKHOFF [4], XIV, 0 14, from which (v -+) and (IT +) can be inferred). These models amount to models based on two-dimensional ternary frames where W= D x D, since from our standard definition of valuation on ternary frames we can infer (v.), (v +) and (v ~). In these two-dimensional ternary frames, W is, of course, transitive, and hence we have frames for L,. However, L, cannot be complete with respect to these frames since with them for every valuation u we have (*I u(a) E u(a o(b4b)) and u((b+b)-+a) E u(a), whereas neither A I- A 0 (B+ B) nor (B+ B)+ A I- A is provable in L,. We are able to prove the inclusions of (*) because W is a reflexive relation; if we do not want (*), we should not have the reflexivity of W 8. The Lambek calculus with T Though the sequents A I- A 0 (B+ B) and ( B4 B) + A I- A are not provable in L,, they are provable in a very natural extension of L,. Let Pt be the extension of the language P with the propositional constant T and let L1, L,, and La be systems in PI obtained from the axiomatizations of L, L, and L,, respectively, by adding the axiom-schemata:

8 186 LDOSEN ToAI-A, AoTI-A, AI-ToA, AI-AoT. The sequents A I- A 0 (B+ B) and (B- B)d A I- A are provable in L,,,,and a fortiori in kt and L. (That the second sequent is provable in a system that amounts to L,, was noted in BUSZKOWSK~ [6], 4.2.) Let now L, be I., L,, or La, and let us consider how we could formulate the (0, +, +) fragment of L. Gentzen-type sequent formulations of this fragment, which may also be cutfree, can easily be constructed by proceeding as in [8]. Via these Gentzen formulations we can show that a Hilbert-type axiomatization of all formulae A of P (i.e. formulae in the connectives 0, + and +) such that T I- A is provable in L may be obtained from our axiomatization of L, by replacing every I- in the axiom-schemata and rules by + and adding the rules A+B B+A A A+B B+A A+B B With this Hilbert-type axiomatization we have a rather good control over the (0, +, +) fragment of L, since A t- B is provable in & iff T I- A+ B is. But, of course, if T I- C is provable in L,, then C is not necessarily of the form A+B (or B+A); for example, we can prove T I- (C1+ C,) 0 (C2- CJ in L,. An axiomatization of the (0, +, +) fragment of given uniquely for sequents of the form A I- B may be obtained by extending L, with the rules A I- B h(a I- B) I- h(c1 I- 01) h(cni- On) Cit- 0; (n 2 1,ls is n), where h (A, I- A2) is either Al + A2 or A2 + Al, and the bracketing on the right-hand side of the right premise is arbitrary. This can be inferred without much dificulty from our Hilberttype axiomatization, for which the rules AoB AoB A B A B AoB are admissible, and where A B cannot be provable if A or B is a propositional letter. We shall not go here into considering models based on our frames for L,,,,L,, and L, and for extensions with other connectives (indications on how to proceed may be obtained from our references). Let us only note that in models based on two-dimensional ternary frames, u(t) will be the identity relation on 0, which is included in W if W is reflexive. We leave as an open problem whether any variant of the Lambek calculus we have considered is complete with respect to a certain class of two-dimensional ternary frames. References [I] ABRUSCI, V. M., Sequent calculus for intuitionistic linear propositional logic. In: Mathematical Logic (P. P. PETKOV, ed.), Plenum Press, New York 1990, pp [21 VAN BENIXEM, J. Semantic parallels in natural language and computation. In: Logic Colloquium 87 (H.-D. EBBINGHAUS et al., eds.), North-Holland hbl. Comp., Amsterdam 1989, pp [3] VAN BEMHEM, J., Language in action. J. Philosophical Logic 20 (1991), [4] BIRRHOFF, G., Lattice Theory. 3rd edition. American Mathematical Society, Providence, R.I., 1967.

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