1. Introduction

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as nondistributive RL) can be found in [1]. 1This research was partially supported by a St. Francis Xavier University Council for Research Grant 91792. 41 ...
Wendy MacCaull1

KRIPKE SEMANTICS FOR LOGICS WITH BCK IMPLICATION

Abstract We present Kripke semantics for some substructural logics with weakening, known as logics with BCK implication. This work is a continuation of the work of Allwein and Dunn on Kripke semantics for Linear Logic, which in turn rested on Dunn’s Gaggle Theory and on Urquhart’s Representation Theory for nondistributive lattices. The basic idea in the representation theory is to use maximally disjoint filter-ideal pairs (maximal pairs) to separate distinct elements. A collection of subsets of the set of maximal pairs forms the representation lattice. Ternary relations are defined on the set of maximal pairs which embody properties of the operations & and ⊃ . A three way valuation of formulas gives rise to a definition of canonical Kripke model. Properties of the ternary relations on the set of maximal pairs are used in the abstract definition of Kripke semantics. Soundness and strong completeness hold.

1. Introduction Logics with BCK implication are members of a class of logics called substructural logics. Substructural logics are currently an area of active research, as they provide abstract models of information flow (see [3], and [10]). Kripke semantics for some relevance logics, involving a ternary relation, appear in [8]. A correspondence between algebraic conditions in algebraic semantics and conditions on ternary relations for a number of substructural logics appears in [5]. Kripke semantics for Linear Logic (which can be generalized to the relational semantics for the relevance logic known as nondistributive RL) can be found in [1]. 1 This research was partially supported by a St. Francis Xavier University Council for Research Grant 91792.

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We define Kripke semantics for a logic with weakening and exchange, and additive disjunction. This logic is denoted by HBCK . It is similar to the logic of [6]: axiom 4 of [6] uses & where we use ∧; consequently, our axiom 4 implies axiom 4 of [6]. We then extend our definition of Kripke semantics to provide semantics for HDBCK , a logic where ∧ distributes over ∨. We are developing Kripke semantics for other substructural logics with weakening (and additive disjunction). We plan to use this semantics to develop relational semantics with a long term goal of providing a tableaustyle deductive system such as that found in Orlowska (see [7]). Relational semantics for Lambek calculus can be found in [2]; however, this semantics can not be extended to give semantics for a logic weaker than intuitionistic which assumes weakening and has additive disjunction. A Hilbert-style axiomatization for HBCK is as follows (binding order is &, ∨, ∧, then ⊃) : Axioms for HBCK : 1. α ⊃ (β ⊃ α) 2. ⊥⊃ α 3. (α ⊃ β) ⊃ ((γ ⊃ α) ⊃ (γ ⊃ β)) 4. ((α ⊃ γ) ∧ (β ⊃ γ)) ⊃ (α ∨ β ⊃ γ) 5. α ⊃ α ∨ β 6. β ⊃ α ∨ β 7. (α ⊃ (β ⊃ γ)) ⊃ (α&β ⊃ γ) 8. α ⊃ (β ⊃ α&β) 9. α ∧ β ⊃ α 10. α ∧ β ⊃ β 11. (γ ⊃ α) ∧ (γ ⊃ β) ⊃ (γ ⊃ α ∧ β) 12. α ⊃ (β ⊃ α ∧ β) 13. (α ⊃ (β ⊃ γ)) ⊃ (β ⊃ (α ⊃ γ)) Rules of Inference: (Modus Ponens) From α and α ⊃ β we deduce β, (Adjunction) From α and β we deduce α ∧ β.

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2. Algebraic Semantics and Canonical Kripke Models Definition 1. A BCK monoid is an 7-tuple M = (M, ∧, ∨, ⊃, &, 1, 0) such that (1) (M, ∧, ∨, 0, 1) is a lattice with the top element 1 and the bottom element 0, (a partial order ≤ is induced by ∨, so a ≤ b if and only if a ∨ b = b); (2) (M, &, 1) is a commutative monoid with identity, 1; (3) a & b ≤ a; (4) a & (b ∨ c) = a & b ∨ a & c; (5) a & b ≤ c if and only if a ≤ (b ⊃ c). If ∧ distributes over ∨, we call M a DBCK monoid. The interpretation of the language of HBCK into a BCK monoid is exactly as one would expect, with each connective mapping to the operation with the same name. We shall not distinguish between a formula A of the language and its interpretation in the monoid. ⊥ is interpreted as 0, and the partial order ≤ is defined as follows: a ≤ b iff a ⊃ b = 1. We say a formula A is valid in a BCK monoid M iff for any valuation, its value is 1. Soundness is easy to verify and strong completeness follows by the standard Lindenbaum algebra completeness proof, using the term algebra over the signature determined by the operations ∨, ∧, &, ⊃. [4] is a good general reference concerning the algebraic semantics for substructural logics. We modify the procedure found in [1], for finding Kripke semantics for Linear Logic. The reader will find it helpful to consult [1] and [8]. The canonical models are found by a generalization of the construction of Kripke models from Heyting algebras. The procedure used in [1] is a generalization of Urquhart’s lattice representation theorem for non-distributive lattices (see [9]), so it is consistent with the Stone representation of distributive and Boolean lattices. Let M be a BCK monoid. A filter-ideal pair, (∇, 4), is an ordered pair, such that ∇ is a filter of the lattice M, 4 is an ideal of the lattice M and ∇ ∩ 4 = ∅. ∇ is 4-maximal if ∇ is maximal among the collection of filters disjoint from 4. Similarly, 4 is ∇-maximal if 4 is maximal among the collection of ideals disjoint from ∇. The pair (∇, 4) is a maximal filterideal pair if ∇ is 4- maximal and 4 is ∇-maximal. Henceforth, maximal filter-ideal pairs will be called maximal pairs. Let X be the collection 43

of all maximal pairs for a lattice M. For any x ∈ X, let x1 refer to the first component and let x2 refer to the second. Define two quasiorders (preorders) on X as follows: x ≤1 y iff x1 ⊆ y1 , and x ≤2 y iff x2 ⊆ y2 . The Filter-Ideal Separation Theorem (see [9]) states that for every filter-ideal pair (∇, 4), there is a maximal pair (x1 , x2 ), which separates them; i.e., ∇ ⊆ x1 and 4 ⊆ x2 . Let X be a set with two quasiorders, let P X denote the powerset of X and let S ⊆ X. Define mappings l : P X → P X and r : P X → P X as follows: lS = {x| x ≤1 y implies y ∈ / S}, rS = {x| x ≤2 y implies y ∈ / S}. These maps define a Galois connection, so S ⊆ T implies lT ⊆ lS, rT ⊆ rS, S ⊆ rlS and S ⊆ lrS. It follows that rl and lr are closure operators on P X. If M is a lattice and X is the set of maximal pairs of M, we can define the map β from M into P X by βa = {x| a ∈ x1 }. One can show: (1) rβa = {x| x ∈ X and a ∈ x2 }, and (2) lrβa = βa. Definition 2. Call a set S ⊆ X an l-stable set iff it is closed under the operator lr; i.e., lrS = S. The sets βa are l-stable. To find the canonical Kripke models, three ternary relations, R, S and Q are defined on the maximal pairs which embody properties of the operators & and ⊃: Rxyz iff ∀a, b(a ∈ x1 and b ∈ y1 imply a & b ∈ z1 ); Sxyz iff ∀a, b(a & b ∈ z2 and b ∈ y1 imply a ∈ x2 ); Qxyz iff ∀a, b(a & b ∈ z2 and a ∈ x1 imply b ∈ y2 ). Using these relations, binary operations on the set of l-stable sets are defined: C ⊃ D = {y| ∀x, z(Rxyz and x ∈ C imply z ∈ D}; C & D = l{z| ∀x, y(Sxyz and y ∈ D imply x ∈ rC}; C ∨ D = l(rC ∩ rD); C ∧ D = C ∩ D. Finally, β1 = {x| 1 ∈ x1 } = X. The set of l-stable sets of a BCK monoid M forms a lattice under ∨ and ∧, called the representation lattice; β1 is the top and β0 is the bottom and, for all a, b ∈ M, β(a ? b) = βa ? βb, for ? ∈ {⊃, ∧, ∨, &} (see [1]). Moreover, using the fact that l and r define a Galois connection we can show that for any l-stable sets C and D, C ⊆ D 44

iff C ∨ D = D. For any x ∈ X, x ∈ β1 and x ∈ / rβ1. The l-stable sets of a commutative Girard monoid (the algebraic semantics for Linear Logic) is a commutative Girard monoid (see [1], for the proof of this). A BCK monoid M has two properties that a commutative Girard monoid does not have: (1) ∀a, b ∈ M, a & b ≤ a, and (2) the identity element is the top element of the lattice. The set of l-stable sets of a BCK monoid M has these properties. The proof requires the following lemma. Lemma 1. Let M be a BCK monoid. Then: (a) For all maximal pairs x, y and z, Sxyz implies z ≤2 x. (b) For all maximal pairs y there exists a maximal pair x such that Sxyy. (c) For all maximal pairs x, y and z, and for all l-stable sets A, (y ∈ A and Rxyz imply z ∈ A). (d) For all maximal pairs x, y and z, Rxyz implies x ≤1 y. (e) Assume M is a DBCK monoid; for all maximal pairs x, y and z, z ≤2 x implies x ≤1 z. Proof. We shall prove (a), the proof of (e) appears in [9]; the remainder of the proofs are similar to the proof of (a). Given maximal pairs x and y, suppose that Sxyz and let a ∈ z2 . Then for any b ∈ y1 , a & b ≤ a. Since z2 is an ideal, a & b ∈ z2 . By definition of Sxyz, we conclude that a ∈ x2 . 2 Theorem 1. Let M be a BCK monoid and let A, B and C be l-stable sets, of the representation lattice of M. Then: (a) A & B ⊆ A; (b) A & X = A = X & A; (c) if M is a DBCK-monoid, then A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C). Proof. We shall prove (a); the proof of (b) is clear from the definitions; the proof of (c) follows in a manner analogous to the proof of (a). To prove (a), we shall demonstrate that rA ⊆ {z|∀x, y(Sxyz and y ∈ B implies x ∈ rA)}. Then, since r and l define a Galois connection and A is an lstable set, we can conclude that A & B = l{z| ∀x, y(Sxyz and y ∈ B implies x ∈ rA)} ⊆ lrA = A. Let z ∈ rA. So for all q, if z ≤2 q, then q ∈ / A. Given an x and y, suppose Sxyz and y ∈ B. By the Lemma 1(a) z ≤2 x. Now let s be such that x ≤2 s. Since z ≤2 x, we know that z ≤2 s. So by the above, we conclude that s ∈ / A. Thus, x ∈ rA, which is what we wanted to prove. 2

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Remark. The above lemma, combined with results of [1] allow us to conclude that the set of l-stable sets of a BCK monoid M satisfy all the properties of a BCK monoid. Given a BCK monoid M, and the resulting set X of maximal pairs, each (well formed) formula A of HBCK has an interpretation βA in the set of l-stable sets of P X, defined recursively in the obvious way. The formula A also has the obvious interpretation (also called A) as an element of the BCK monoid M. The fact that an element of the lattice M may be in the filter half of a maximal pair, in the ideal half of a maximal pair or in neither, motivated Allwein and Dunn [1] to define a three way valuation of formulas as follows (here we think of the interpretation of a formula as an element of the monoid): (1) x |=T A iff A ∈ x1 iff x ∈ βA; (2) x |=F A iff A ∈ x2 iff x ∈ rβA ; (3) x |=I A iff neither x |=T A nor x |=F A. Then, binary operations on the set of l-stable sets (described after Definition 2) can be used to extend the valuation of simple formulas to compound formulas. (The three-way valuation of all formulas is written out completely in Definition 6 in the next section.) We can show that if x |=T A, then for all x0 , if x ≤1 x0 , then x0 |=T A. A formula A is said to be valid iff for all maximal pairs x, x |=T A. Thus we have a ”Kripke-style valuation” associated to each BCK monoid M. The following lemma gives the relationship between the representation lattice of a BCK monoid M and the Kripke-style valuation associated to it. Lemma 2. Let A and B be HBCK formulas and M a BCK monoid. Then x |=T A ⊃ B for all maximal pairs x of M if and only if βA ⊆ βB. Proof. We shall prove the ”if” part. The proof of the other part is similar (and easier). Suppose that βA ⊆ βB and assume that Rxyz and A ∈ x1 . We must show that B ∈ z1 . We have assumed that Rxyz, so for all A, C, if A ∈ x1 and C ∈ y1 , then A & C ∈ z1 . Since A ∈ x1 we know that A & C ∈ z1 . Since M is a BCK monoid, A & C ≤ A; z1 is a filter, so we may conclude that A ∈ z1 . The fact that βA ⊆ βB allows us to conclude that B ∈ z1 . 2

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Since the set of l-stable sets of a BCK monoid M forms a BCK monoid, it is straightforward to use the lemma to show that if A is an axiom of HBCK logic, then A is valid in the Kripke-style valuation associated to M. For any BCK monoid M, the set X of maximal pairs accompanied by its two quasiorders, the three ternary relations R, S and Q and the associated Kripke-style valuation is a Kripke model of HBCK logic. We have a special name for Kripke models arising in this manner. Definition 3. (Canonical Kripke BCK model) Associated to any BCK monoid M there is a Kripke model of HBCK called a canonical Kripke BCK model and denoted Can(M).

3. Kripke Semantics – Abstract Definition Definition 4. A frame is a 3-tuple (X, ≤1 , ≤2 ), where X is a set and ≤1 and ≤2 are quasiorders (preorders) on X. Definition 5. Let (X, ≤1, ≤2 ) be a frame. Let v be a function that maps the Cartesian product of the set of all variables and X into {T, F, I}. For every propositional variable p, let P1 (= P1 (p)) = {x| v(p, x) = T } and let P2 (= P2 (p)) = {x| v(p, x) = F }. Such a v is called a valuation exactly when P1 = lP2 and P2 = rP1 . The maps l and r define a Galois connection on the subsets of X determined by a valuation v (that is, the P1 (p)0 s and the P2 (p)0 s). The definition of the binary operations on l-stable sets given in Section 2, motivated [1] to extend the valuation to compound formulas in an arbitrary frame (X, ≤1 , ≤2 ) as follows: Definition 6. An interpretation, |= , in a frame is a function from the Cartesian product of X and the set of well-formed formulas to {T, F, I}, such that: x |=T p iff v(p, x) = T ; x |=F p iff v(p, x) = F ; x |=I p iff v(p, x) = I; x |=I A iff neither x |=T A nor x |=F A; x |=T A ∨ B iff ∀y(x ≤1 y implies y |=T I A or y |=T I B); x |=F A ∨ B iff x |=F A and x |=F B; x |=T A ∧ B iff x |=T A and x |=T B; x |=F A ∧ B iff ∀y(x ≤2 y implies [y |=F I A or y |=F I B]); 47

y |=T A ⊃ B iff ∀x, z([Rxyz and y |=T A] imply z |=T B); y |=F A ⊃ B iff ∀y 0 ∃x, z (y ≤2 y 0 implies [Rxy 0 z and x |=T A and z |=F I B]); z |=T A & B iff ∀z 0 ∃x, y (z ≤ 1 z 0 implies [Sxyz 0 and y |=T B and x |=T I A]); z |=F A & B iff ∀x, y([Sxyz and y |=T B] imply x |=F A). Remark. Let us introduce for an arbitrary formula A the sets P1 (A) and P2 (A), defined as: x ∈ P1 (A) iff x |=T A and x ∈ P2 (A) iff x |=F A. Then the maps l and r define a Galois connection on the subsets determined by the P1 (A)0 s and the P2 (A)0 s. Note that x |=T I A means either x |=T A or x |=I A. The notation x |=F I A has the analogous meaning. Here, R, S and Q are ternary relations which satisfy certain conditions depending on the logic of interest (see the following definition). Definition 7. A (propositional) Kripke HBCK model is a seventuple Ξ = (X, ≤1 , ≤2 , R, S, Q, |=) such that: (X, ≤1 , ≤2 ) is a frame, |= is an interpretation in (X, ≤1 , ≤2 ) and R, S and Q are ternary relations on X satisfying the following conditions: 1. Rxyz and x ≤2 x0 imply Rx0 yz; 2. Rxyz and z ≤1 z 0 imply Rxyz 0 ; 3. Sxyz and x ≤2 x0 imply Sx0 yz; 4. Rxyz implies ∃x0 (x ≤1 x0 and Sx0 yz); 5. Sxyz implies ∃z 0 (z ≤2 z 0 and Rxyz 0 ); 6. Rxyz implies ∃y 0 (y ≤1 y 0 and Qxy 0 z); 7. Qxyz implies ∃z 0 (z ≤2 z 0 and Rxyz 0 ); 8. R2 (uv)yz iff R2 u(vy)z (R2 (uv)yz means ∃x(Ruvx and Rxyz) and 2 R u(vy)z means ∃w(Ruwz and Rvyw)); 9. Rxyz implies Ryxz; 10. Sxyz implies z ≤2 x; 11. Rxyz implies x ≤1 z; 12. ∀y∃xSxyy; 13. y ∈ A and Rxyz imply z ∈ A. Conditions 1 through 9 were among those used to define Kripke semantics for Linear Logic in [1]. Clearly for any BCK monoid M, Can(M) forms a Kripke HBCK model. If we add the condition z ≤2 x implies x ≤1 z to the above 13 conditions, we call the resulting structure a Kripke HDBCK model. 48

A straightforward induction on the length of a formula allows one to establish the following fact for abstract models: Lemma 3. Let Ξ = (X, ≤1 , ≤2 , R, S, Q, |=) be a Kripke HBCK model, let A be an arbitrary HBCK formula and suppose that x |=T A. Then ∀x0 (x ≤1 x0 implies x0 |=T A). Definition 8. A HBCK formula φ is said to be valid at x in a Kripke HBCK model Ξ = (X, ≤1 , ≤2 , R, S, Q, |=) iff x |=T φ. The formula φ is valid in Ξ iff ∀x ∈ X, x |=T φ. We say that φ is Kripke HBCK -valid iff φ is valid in every Kripke HBCK model; in this case, we write |= φ. Theorem 2. (Soundness Theorem for HBCK ) For every HBCK formula φ, ` φ implies |= φ. Proof. The demonstration of the validity of each axiom of HBCK in all Kripke models is analogous to the proof that the l-stable sets form a BCK monoid. By Theorem 1, A&B ⊆ A holds for l-stable sets. We show how this translates into a demonstration of the validity of the weakening axiom in Kripke HBCK models. To establish that |= α ⊃ (β ⊃ α), it suffices to show that |= α&β ⊃ α. That is, we must show for all Kripke HBCK models Ξ = (X, ≤1 , ≤2 , R, S, Q, |=) and for all y ∈ X, ∀x, z([Rxyz and x |=T α&β] imply z |=T α). Suppose that we have x, y and z, such that [Rxyz and x |=T α&β]. So x ∈ P1 (A&B) and by condition 11 of Definition 7, it follows that x ≤1 z. By Lemma 3, we conclude that z ∈ P1 (A&B). We shall first prove that if z ∈ P2 (A), then z ∈ P2 (A&B), and then use the Galois connection to conclude that P1 (A&B) = lP2 (A&B) ⊆ lP2 (A) = P1 (A). Let z ∈ P2 (A). We need to prove that for all x, z, if Sxyz and y ∈ P1 (B), then x ∈ P2 (A) (which is equivalent to showing x ∈ rP1 (A)). So we must show that if Sxyz and y ∈ P1 (B), then for all s, if x ≤2 s, then s ∈ / P1 (A). By condition 10 of Definition 7, Sxyz implies z ≤2 x; so, if x ≤2 s, then z ≤2 s. Since z ∈ P2 (A) (= rP1 (A)), we conclude that s∈ / P1 (A). 2 Corollary 1. If φ is a theorem of HDBCK , then φ is valid in all Kripke HDBCK models. We shall need the following lemma in order to prove strong completeness. 49

Lemma 4. Let γ1 ,γ2 ,...,γn and α be elements of a BCK monoid M. Then there is a maximal pair containing γi , i = 1, ..., n, and not containing α. Proof. Let [γ1 &γ2 &...&γn ) denote the filter generated by γ1 &γ2 &...&γn and let (α] denote the ideal generated by α. We can show that α ∈ / [γ1 &γ2 &...&γn ) and (α] ∩ [γ1 &γ2 &...&γn ) is the empty set. By the Filter-Ideal Separation Theorem, there is a maximal pair x0 =(∇, 4), such that (α] ⊆4 and [γ1 &γ2 &...&γn ) ⊆∇. 2 Theorem 3. (Strong Completeness for HBCK ) If Γ |= α, then Γ ` α. Proof. Let Γ = {γ1 , ..., γn } and suppose that α is not a consequence in HBCK of Γ. We shall show that there is a Kripke model in which the γi are all valid but α is not valid. Let PΓ be the quotient algebra of formulas under provable equivalence and let [φ] ≤ [σ] iff Γ ` φ ⊃ σ. The set of equivalence classes forms a BCK monoid. Consider the resulting Kripke HBCK model Can(PΓ ). Then for all γi ∈ Γ, and for all maximal pairs x in PΓ , x |=T γi , but for x0 , as described in Lemma 4 (applied to PΓ ), x0 |=F α. 2

References [1] G. Allwein and J. M. Dunn, Kripke models for linear logic, Journal of Symbolic Logic 58 (1993), pp. 514–545. [2] H. Andreka and S. Mikulas, Lambek Calculus and relational semantics: completeness and incompleteness, Journal of Logic, Language and Information 3 (1994), pp. 1–38. [3] J. Barwise, D. Gabbay and C. Hartonas, On the logic of information flow, preprint, 1994. [4] K. Dosen, Sequent systems and groupoid models I, Studia Logica, 4 (1988), pp. 353–385. [5] J. M. Dunn, Partial-gaggles applied to restricted structural rules, in: Substructural Logics, edited by P. Schroeder-Heister and K. Dosen, Oxford Press, 1993. [6] H. Ono and Y. Komori, Logics without the contraction rule, Journal of Symbolic Logic 50 (1985), pp. 169–201.

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[7] E. Orlowska, Interpretation of relevant logics in a logic of ternary relations, Bulletin of the Section of Logic 19 (1990), pp. 39–48. [8] R. Routley and R. K. Meyer, The semantics of entailment, [in:] Truth, Syntax and Modality, edited by H. LeBlanc, North Holland, pp. 199–243, 1973. [9] A. Urquhart, A topological representation theory for lattices, Algebra Universalis 8 (1978), pp. 45–58. [10] H. Wansing, Informational interpretation of substructural propositional logics, Journal of Logic, Language and Information 2 (1993), pp. 285–309. The author wishes to thank the referee for his or her helpful comments.

Department of Mathematics and Computing Sciences St. Francis Xavier University P.O. Box 5000 Antigonish, Nova Scotia, B2G 2W5 Canada e-mail: [email protected]

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