A BRIEF SURVEY OF FRAMES FOR THE LAMBEK CALCULUS

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184 K. DOSENositional 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 Pand 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. Withmultiplicative valuations u on semilattices we have:~(A+B)=u(B+A)={x:Vr((3y(x*y=~)& zEv(R))*zEu(B))},u(AB) = u(A) n u(B),which shows that our models are a kind of Kripke models for intuitionistic logic.Then we prove the followingProposition 3 . The sequent A l- A2 is provable in Lh ifl for every multiplicative valuation uon 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 ofP be called prime iff it is not of the form B 0 C, and let a prime factor of a formula A be aprime subformula of A that is not in the scope of an +, or an +. Next, let [A] be the set ofall formulae B whose set of prime factors is identical with the set of prime factors of A (inother words, B is obtained from A by using the associativity, commutativity and idempotenceof the connective 0 binding the prime factors of A). It is clear that [A] = [B] impliesthat A l- B and B l- A are provable in Lb (the converse need not hold). Next, let [A].[B] bedefined 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 fromall 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 isclear thatCl- 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). 0Of course, an analogous proof shows that with multiplicative valuations, Lh is completewith respect to all simple groupoid frames that are semilattices, not necessarily free. Simplegroupoid frames that are semilattices are closely related to semilattice semantics ofURQUHART (24); but a significant difference is that URQUHART’S semilattices, taken as join
A BRIEF SURVEY OF FRAMES FOR THE LAMBEK CALCULUS 185semilattices, have a zero element. Without multiplicative valuations we obtain models for Lhminus A 0 B + A, which is a system related to the relevant logic R. In [24] URQUHART doesnot show completeness with the connective 0, and in the absence of principles correspondingto A 0 B I- A and A 0 B I- B his simple and elegant method (which could also be adapted forthe proof of our Proposition 3) seems to be blocked. Perhaps something like the method ofBUSZKOWSKI [6] could be applied for system with related to R.7. Two-dimensional ternary framesWe shall now consider the following special type of ternary frames (W, R), where, for anonempty set D, the set W is a nonempty subset of D X D, and R~ryz holds iff for somea, b, CE D we have x = (a, b),y= (b, c) and z= (u, c). Ofcourse, with our standard definitionof valuation on ternary frames, these frames, which we call two-dimensional ternaryframes, give rise to models for La. We get (ass) and, hence, models for L, if moreover we assumethat W is a transitive relation. We do not get (com) simply by assuming that W is a symmetricrelation, 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 Lambekcalculus where, for an nonempty set D, we have W = D X D, and u assigns to every formulaof 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 conditionson valuations in two-dimensional modal logic, see, for example, KUHN [12] and referencestherein; cf. also the definition of residuation in terms of conversion, composition and complementationin BIRKHOFF [4], XIV, 0 14, from which (v”-+) and (IT”+) can be inferred).These models amount to models based on two-dimensional ternary frames whereW= 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 theseframes 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 toprove the inclusions of (*) because W is a reflexive relation; if we do not want (*), we shouldnot have the reflexivity of W8. The Lambek calculus with TThough the sequents A I- A 0 (B+ B) and ( B4 B) + A I- A are not provable in L,, theyare provable in a very natural extension of L,. Let Pt be the extension of the language P withthe propositional constant T and let L1, L,, and La be systems in PI obtained from the axiomatizationsof L, L, and L,, respectively, by adding the axiom-schemata:
Page 3: Analogously to Proposition 1 we can

A BRIEF SURVEY OF FRAMES FOR THE LAMBEK CALCULUS

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