Ventura Verdú

Logics preserving degrees of truth from varieties of residuated lattices

A wrong argument in the proof of one of the main results in the paper is corrected. The result itself remains true. The right proof incorporates the basic ideas in the originally alleged proof, but in a more restricted construction.

Algebraic logic for classical conjunction and disjunction

In this paper we study the relations between the fragment L of classical logic having just conjunction and disjunction and the variety D of distributive lattices, within the context of Algebraic Logic. We prove that these relations cannot be fully expressed either with the tools of Blok and Pigozzis theory of algebraizable logics or with the use of reduced matrices for L. However, these relations can be naturally formulated when we introduce a new notion of model of a sequent calculus. When applied to a certain natural calculus for L, the resulting models are equivalent to a class of abstract logics (in the sense of Brown and Suszko) which we call distributive. Among other results, we prove that D is exactly the class of the algebraic reducts of the reduced models of L, that there is an embedding of the theories of L into the theories of the equational consequence (in the sense of Blok and Pigozzi) relative to D, and that for any algebra A of type (2,2) there is an isomorphism between the D-congruences of A and the models of L over A. In the second part of this paper (which will be published separately) we will also apply some results to give proofs with a logical flavour for several new or well-known lattice-theoretical properties.

On the infinite-valued Łukasiewicz logic that preserves degrees of truth

AbstractŁukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.

On Gentzen Systems Associated with the Finite Linear MV-algebras

In this paper we obtain a characterization of the algebraizability of an m-dimensional Gentzen system in line with the characterization obtained for m-dimensional deductive systems and the characterization of 2-dimensional Gentzen systems. We also prove that if S(m) is the finite linear MV-algebraof m elements, then the m-dimensional Gentzen system obtained by using the sequent calculi associated with S(m) is equivalent to the m-valued Łukasiewicz logic Ł m and to the equational consequence relation associated with S(m). Taking the two-element Boolean algebra we obtain the expected result concerning the relationship between the sequent calculus LK, the Classical Prepositional Calculus and the variety of Boolean algebras.

On a Contraction-Less Intuitionistic Propositional Logic with Conjunction and Fusion

In this paper we prove the equivalence between the Gentzen system GLJ*\c, obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems GLJ*\c in the sense of [18] and [4], respectively. The equivalence between GLJ*\c and IPC*\c is a strengthening of a result obtained by H. Ono and Y. Komori [14, Corollary 2.8.1] and the equivalence between GLJ*\c and the equational system associated with the variety RL of residuated lattices is a strengthening of a result obtained by P.M. Idziak [13, Theorem 1].An axiomatization of the restriction of IPC*\c to the formulas whose main connective is the implication connective is obtained by using an interpretation of GLJ*\c in IPC*\c.

A Finite Hilbert‐Style Axiomatization of the Implication‐Less Fragment of the Intuitionistic Propositional Calculus

In this paper we obtain a finite Hilbert-style axiomatization of the implicationless fragment of the intuitionistic propositional calculus. As a consequence we obtain finite axiomatizations of all structural closure operators on the algebra of {–}-formulas containing this fragment. Mathematics Subject Classification: 03B20, 03B22, 06D15.

On two fragments with negation and without implication of the logic of residuated lattices

The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [26], intuitionistic logic without contraction [1], HBCK [36] (nowadays called by Ono), etc. In this paper we study the -fragment and the -fragment of the logical systems associated with residuated lattices, both from the perspective of Gentzen systems and from that of deductive systems. We stress that our notion of fragment considers the full consequence relation admitting hypotheses. It results that this notion of fragment is axiomatized by the rules of the sequent calculus for the connectives involved. We also prove that these deductive systems are non-protoalgebraic, while the Gentzen systems are algebraizable with equivalent algebraic semantics the varieties of pseudocomplemented (commutative integral bounded) semilatticed and latticed monoids, respectively. All the logical systems considered are decidable.

Abstract characterization of a four-valued logic

A four-valued logic defined by the four-element De Morgan lattice together with its two prime filters is treated. For certain classes of abstract logics, several characterizations in terms of De Morgan lattices and projective generation of logics by sets of homomorphisms or by a single epimorphism (a biological morphism) are described. A similar treatment is shown for the three-valued logic generated by the three-element chain together with its two prime filters, and for the incorporation of the falsum connective, which brings out the classes of all De Morgan algebras and all Kleene algebras.<<ETX>>

Some Algebraic Structures Determined by Closure Operators

Preprint enviat per a la seva publicacio en una revista cientifica: Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 31 (14-18):275-278 (1985)

The lattice of distributive closure operators over an algebra

In our previous paper “Algebraic Logic for Classical Conjunction and Disjunction” we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA and of all distributive closure operators overA. In the present paper we study the lattice structure of this last set, give a description of its finite and infinite operations, and obtain a topological representation. We also apply the mentioned isomorphism and other results to obtain proofs with a logical flavour for several new or well-known lattice-theoretical properties, like Hashimotos characterization of distributive lattices, and Priestleys topological representation of the congruence lattice of a bounded distributive lattice.