ON THE LOGIC OF DISTRIBUTIVE LATTICES

Transcription

1 Bulletin of the Section of Logic Volume 18/2 (1989), pp reedition 2006 [original edition, pp ] Josep M. Font and Ventura Verdú ON THE LOGIC OF DISTRIBUTIVE LATTICES This note is a summary of the main result contained in the paper [9], which will be published elsewhere. In that paper we study abstract logics corresponding to the classical connectives of conjunction and disjunction, in order to clarify the relations between these logics and distributive lattices, and give some lattice-theoretical applications. Given any algebra A = A,, of type (2, 2) we can define a closure operator S on A by means of a sequent calculus consisting of the {, }-fragment of the sequent calculus for classical logic. Our sequents are expressions of the form X a where a A and X A is finite (possibly empty). We say that the relation X S a holds, for any X A, a A, if and only if there is a finite X 0 X such that the sequent X 0 a is derivable in the sequent calculus which has the following axiom and rules: (Weakening) ( ) ( ) (Axiom) X, a a X a X, b a X, a, b c X, a b c X, a c X, b c X, a b c (Cut) X a X, a b X b ( ) X a X b X a b ( ) X a X a b X b X a b The notion of formal proof being defined in the usual sense. We denote by L S (A) = A, S the abstract logic determined by S on A. As a particular case we have the logic L S = I, S where I = F orm,, is the absolutely free algebra of type (2, 2), that is, the algebra of sentential formulas ; however, several of our results do not depend on the freeness of the algebra. It is easy to see that such logics do

2 80 Josep M. Font and Ventura Verdú not fit into the framework of equivalential logics of [5] nor in that of algebraizable logics of [2], anticipated in [1]. We use the tools of the theory of abstract logics to clarify the relationships between these logics and its algebraic models, and to characterize the class of distributive lattices among them. Our approach to the subject is similar to that followed in [7] for a group of modal logics, in [6] and [8] for several four-valued and three-valued logics related to De Morgan lattices and Kleene lattices, and in [11] for some two-valued logics related to the Deduction Theorem. We deal with algebras A = A,, of type (2, 2), and denote by Con(A) the lattice of all the congruence relations of A. Given an abstract logic (briefly: a logic) L = A, C or L = A, C (where C is the closure system associated with the closure operator C), we denote by θ(c) = {(a, b) A A : C(a) = C(b)} the equivalence relation naturally associated with C. The definitions of the terms and constructions used can be found mainly in [4] and related papers. Definition. We say that C satisfies the Property of Conjunction (PC) whenever C(a, b) = C(a b) a, b A; and that C satisfies the Property of Disjunction (PDI) whenever C(X, a) C(X, b) = C(X, a b) a, b A X A. We also say that a logic L satisfies any of these properties when so does C. We say that L is a distributive logic when it is finitary and satisfies PC and PDI. The most typical example of a distributive logic is the logic of all filters of a distributive lattice, and in some sense they are paradigmatic, because Theorem 3 can be read as saying that any distributive logic is (equivalent modulo a bilogical morphism to) one of these logics. The term distributive logics was already used in [10]; PC and PDI also appear, under a variety of names, in many papers. Among them we highlight Bloom s [3], which presents a unified approach to the connectives of the set {,,,, }. Theorem 1. The logic L S (A) is the least distributive logic over A. By starting from the rules of proof of a sequent calculus we have incorporated one of the key elements of the meaning of conjunction and disjunction, according to some theories of the meaning of logical constants. However, according to some more classical positions, the meaning of logical constants is determined by the truth conditions used to establish the

3 On the Logic of Distributive Lattices 81 semantical entailment. This view can also be incorporated to our treatement by taking the {, }-reduct of the two-element Boolean algebra and studying logics defined semantically from this matrix by using arbitrary families of valuations in the classical way. Let 2 be the ordinary (distributive) lattice structure on {0, 1}, and let us consider the logical matrix 2, {1}. We can obtain an equivalent abstract logic L 2 = 2, F 2 by taking the closure system F 2 = {{1}, {0, 1}} generated by {1}. Given a fixed but arbitrary algebra A we are going to deal with logics L = A, C projectively generated from L 2 by families H Hom(A, 2), that is, whose closure system has a basis of the form {h 1 ({1}) : h H}. We denote by L 2 (A) = A, = 2 the least one among them, obtained by taking the family of all homomorphisms; note that it is the logic determined semantically in the ordinary way by taking all valuations on {0, 1} and taking for and the classical truth-tables for the connectives of conjunction and disjunction. The main properties of these logics are summarized in the following Theorem 2. Let L be a logic. The following properties are equivalent: (i) L is projectively generated from L 2 by some H Hom(A, 2); (ii) C satisfies PC, PDI, and C has a basis of finitely irreducible closed sets. (iii) θ(c) Con(A), A/θ(C) is a distributive lattice, and C/θ(C) has a basis of prime filters; and (iv) There is a bilogical morphism between L and L = A, C for some distributive lattice A and some closure system C over A having a basis of prime filters. If L is a distributive logic then a fortiori it satisfies part (ii) above, but finitarity is stronger and allows us to obtain a much more precise characterization: Theorem 3. If L is any logic, then the following conditions are equivalent: (i) L is a distributive logic; (ii) θ(c) Con(A), A/θ(C) is a distributive lattice, and C/θ(C) is the set of all filters of A/θ(C); and (iii) There is a bilogical morphism between L and the logic of all filters of some distributive lattice.

4 82 Josep M. Font and Ventura Verdú This theorem turns out to be a central result in the paper. We highlight here three of its consequences in different directions of research. Firstly, it shows a relationship between each distributive logic and some distributive lattice, through the use of a logic canonically associated with any distributive lattice, namely the logic of all its filters. This relationship is global and can be expressed in categorial terms. Let L be the category having as objects all distributive logics L = A, C, where A is any algebra of type (2, 2), and having as arrows all logical morphisms (continuous mappings) between any two such logics. Let D be the category whose objects are distributive lattices and whose arrows are ordinary homomorphisms between them. Then we have: Theorem 4. There exist two covariant functors F from L to D, and G from D to L such that G is faithful, F G is the identity in D and there is a natural transformation from the identity functor of L to G F. Secondly, we see that Theorem 3 contains the proof that if L is a distributive logic then θ(c) Con(A) and A/θ(C) is a distributive lattice. Another important result in the paper is the fact that all congruences of A giving a distributive lattice in the quotient can be obtained in this way, and indeed this gives a bijection between the two sets which respects their order structure. More precisely: Theorem 5. Let A be a fixed algebra of type (2, 2). Then the set of all distributive logics over A, ordered under the relation L 1 L 2 if and only if X A, C 1 (X) C 2 (X), is a lattice isomorphic to the lattice of all congruences θ of A such that A/θ is a distributive lattice. The paper contains several lattice-theoretical applications of this result, such as a description of the operations of the lattice of distributive logics or a representation of it as a lattice of subsets. And using some of these applications we can prove the following characterization of the families of homomorphisms which projectively generate finitary operators; this is a refinement of Theorem 2. We denote by θ h the congruence determined by h Hom(A, 2), that is, we put θ h = {(x, y) A A : h(x) = h(y)}. Then: Theorem 6. Let L be a logic. Then it is a distributive logic if and only if there is a set H Hom(A, 2) such that L is projectively generated from

5 On the Logic of Distributive Lattices 83 L 2 by H and this set satisfies that for any h Hom(A, 2), if θ h inf{θ g : g H}, then h H. Thirdly, with the aid of both Theorems 2 and 3 we can prove that for any algebra A, the logic L 2 (A) is a finitary logic, and then we obtain: Theorem 7. logic over A. For any algebra A, the logic L 2 (A) is the least distributive Corollary (Semantic Completeness Theorem). any algebra A, L S (A) = L 2 (A). S = = 2, that is, for Now we are going to use a concept of model for a logic which makes smoother the characterization of distributive logics as models for L S : Definition. Let L = A, C be any logic. We say that L is a generalized matrix (g-matrix from now on) for L S if and only if for all Γ F orm, ϕ F orm, Γ S ϕ implies h(ϕ) C(h(Γ)) for all h Hom(I, A). As we see L is a g-matrix for L S if and only if for all T C, A, T is a matrix for S in the usual sense; that is, the concept of generalized matrix is a natural extension of the concept of matrix. Actually, from the purely semantical standpoint, an abstract logic L is equivalent to the bundle of matrices { A, T : T C}. However, some of the following results cannot be properly expressed in terms of logical matrices. Now we can state a very general Algebraic Completeness Theorem in the following form: Theorem 8. Let L be a family of g-matrices for L S such that L 2 L. Then the logic L S is complete with respect to the class L in the following sense: for any Γ F orm, ϕ F orm, Γ S ϕ if and only if h(ϕ) C(h(Γ)) for any L = A, C L and any h Hom(I, A). Some of the classes fulfilling the conditions in this theorem deserve more attention. First of all, we have: Theorem 9. Any distributive logic is a g-matrix for L S. Therefore we have that {L 2 } {L = A, F, where A is a distributive lattice, and F is the closure system of all its filters} {distributive logics} {g-matrices for L S }, and each of these classes gives a complete-

6 84 Josep M. Font and Ventura Verdú ness theorem for L S. From them we obtain a rather non-standard proof of a well-known property of the variety of distributive lattices, which we include in the following statement together with another fact we also use later. Theorem 10. Let α(x 1,..., x n ), β(x 1,..., x n ) F orm. Then the following conditions are equivalent. (i) The equation α(x 1,..., x n ) = β(x 1,..., x n ) holds in 2; (ii) The equation α(x 1,..., x n ) = β(x 1,..., x n ) holds in all distributive lattices; (iii) C(α(a 1,..., a n )) = C(β(a 1,..., a n )) for all g-matrices L = A, C for L S and for all a 1,..., a n A; and (iv) α(x 1,..., x n ) S β(x 1,..., x n ). Of course not all g-matrices for L S are distributive logics (take the four-element Boolean lattice M 4 with the closure system {{1}, M 4 }; they determine a finitary g-matrix for L S but it is not a distributive logic, because it does not satisfy PDI). The condition which a finitary g-matrix for L S needs to become a distributive logic is a very natural regularity condition: Theorem 11. Let L = A, C be a logic. Then L is a distributive logic if and only if L is a finitary g-matrix for L S such that θ(c) Con(A). The logics canonically associated with distributive lattices do also have a distinguished position among the class just described: Theorem 12. Let L = A, C be any logic. Then the following conditions are equivalent: (i) L is a simple distributive logic; (ii) L is a simple finitary g-matrix for L S ; and (iii) A is a distributive lattice and C is the closure system of all filters of A. References [1] W.J. Blok and D. Pigozzi, Protoalgebraic logics, Studia Logica 45 (1986), pp

7 On the Logic of Distributive Lattices 85 [2] W.J. Blok and D. Pigozzi, Algebraizable logics, Memoirs of the American Mathematical Society, (to appear). [3] S.L. Bloom, A note on Ψ-consequences, Reports on Mathematical Logic 8(1977), pp [4] D. J. Brown and R. Suszko, Abstract Logics, Dissertationes Mathematicae 102 (1973), pp [5] J. Czelakowski, Equivalential logics, I and II, Studia Logica 40 (1981), pp and [6] J.M. Font and V. Verdú, Abstract characterization of a four-valued logic, [in:] Proceedings of the 18th International Symposium on Multiple-Valued Logic (Palma de Mallorca, 1988), pp [7] J.M. Font and V. Verdú, A first approach to abstract modal logics, The Journal of Symbolic Logic (to appear). [8] J.M. Font and V. Verdú, Completeness theorems for a four-valued logic related to DeMorgan lattices, Preprint, to appear. [9] J.M. Font and V. Verdú, Abstract logics related to classical conjunction and disjunction, Preprint, to appear. [10] V. Verdú, Distributive and Boolean logics, (in Catalan) Stochastica 3 (1979), pp [11] V. Verdú, Logics projectively generated from [M] = (F 4, [{1}]) by a set of homomorphisms, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 33 (1987), pp More references, as well as references to general literature, will be found in the full version of the paper. Faculty of Mathematics University of Barcelona Gran Via Barcelona Spain