Josep Maria Font

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.

Modality and Possibility in Some Intuitionistic Modal Logics

«-> -1Λ/-1. However if we work on anintuitionistic nonmodal base logic, then some properties of the negation areweakened, the duality disappears, and it is commonly admitted that both equiv-alences cannot remain valid, because they lead to conclusions stronger thanwished (see [4]). Of course one could ignore one of the two modal operators,but we think this pointless, because the dual interpretation of one of them givesnatural birth to the other one.

Leibniz filters and the strong version of a protoalgebraic logic

Abstract. A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?+ and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewiczs finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics.

On łukasiewicz's four-valued modal logic

AbstractŁukasiewiczs four-valued modal logic is surveyed and analyzed, together with Łukasiewiczs motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, Łukasiewiczs own texts, and related literature.

An abstract algebraic logic view of some multiple-valued logics

Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by BLOK, PIGOZZI and CZELAKOWSKI, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by WOJCKICI and NOWAK) of defining logics using a scheme of preservation of degrees of truth (as opposed to the more usual one of preservation of truth) characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by JANSANA and myself) can give some interesting results. After the general theory is explained, I apply it to an infinite family of logics defined in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the finite case the logics so obtained are protoalgebraic, which implies they have a strong version defined from their Leibniz filters; again, the general theory helps in showing that it is the logic defined from the same subalgebra by the truth-preserving scheme, that is, the corresponding finite-valued logic in the most usual sense. However, for infinite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can finally show that this logic too has a strong version, and that it coincides with the ordinary infinite-valued logic of Lukasiewicz.

Note on a six-valued extension of three-valued logic

ABSTRACT In this paper we introduce a set of six logical values, arising in the application of three-valued logics to time intervals, find its algebraic structure, and use it to define a six-valued logic. We then prove, by using algebraic properties of the class of De Morgan algebras, that this semantically defined logic can be axiomatized as Belnaps “useful” four-valued logic. Other directions of research suggested by the construction of this set of six logical values are described.

Beyond Rasiowa's Algebraic Approach to Non-classical Logics

This paper reviews the impact of Rasiowas well-known book on the evolution of algebraic logic during the last thirty or forty years. It starts with some comments on the importance and influence of this book, highlighting some of the reasons for this influence, and some of its key points, mathematically speaking, concerning the general theory of algebraic logic, a theory nowadays called Abstract Algebraic Logic. Then, a consideration of the diverse ways in which these key points can be generalized allows us to survey some issues in the development of the field in the last twenty to thirty years. The last part of the paper reviews some recent lines of research that in some way transcend Rasiowas approach. I hope in this way to give the reader a general view of Rasiowas key position in the evolution of Algebraic Logic during the twentieth century.

A First Approach to Abstract Modal Logics

On etudie quatre systemes de la logique modale (S4, S5, IM4 et IM5) dans le contexte de la logique abstraite. Les concepts abstraits correspondant a de tels systemes sont definis comme generalisations des notions logiques associees a leurs modeles algebriques. On considere des situations en rapport avec les techniques de lemploi de modeles birelationnels en logique modale intuitionniste

Generalized Matrices in Abstract Algebraic Logic

The aim of this paper is to survey some work done recently or still in progress that applies generalized matrices (also called abstract logics by some) to the study of sentential logics. My main concern will be to emphasize the links between this line of research and other existing frameworks in Algebraic Logic, either well-established ones (such as the old theory of logical matrices and the younger theories of protoalgebraie logics, algebraizable logics and the associated hierarchy) or really new ones (such as the theory of algebraizability of Gentzen systems or the model theory of equality-free logic). I would like to convey the idea that the interaction between these neighbouring fields may be specially fruitful, as it seems to be one of the leading forces in the shaping of this emerging field called Abstract Algebraic Logic.