Operators and strong versions of sentential logics in Abstract Algebraic Logic

Abstract

This dissertation presents the results of our research on some recent devel-opments in Abstract Algebraic Logic (AAL), namely on the Suszko operator, the Leibniz filters, and truth-equational logics. Part I builts and develops an abstract framework which unifies under a common treatment the study of the Leibniz, Suszko, and Frege operators in AAL. Part II generalizes the theory of the strong version of protoalgebraic logics, started in, to arbitrary sentential logics.

The interplay between several Leibniz- and Suszko-related notions led us to consider a general framework based upon the notion of S-operator (inspired by that of "mapping compatible with S-filters" of Czelakowski), which encompasses the Leibniz, Suszko, and Frege operators. In particular, when applied to the Leibniz and Suszko operators, new notions of Leibniz and Suszko S-filters arise as instances of more general concepts inside the abstract framework built. The former generalizes the existing notion of Leibniz filter for protoalgebraic logics to arbitrary logics, while the latter is introduced here for the first time. Sev-eral results, both known and new, follow quite naturally inside this framework, again by instantiating it with the Leibniz and Suszko operators. Among the main new results, we prove a General Correspondence Theorem (Theorem ??), which generalizes Blok and Pigozzi's well-known Correspondence Theorem for protoalgebraic logics, as well as Czelakowski's less known Correspondence The-orem for arbitrary logics. We characterize protoalgebraic logics in terms of the Suszko operator as those logics in which the Suszko operator commutes with inverse images by surjective homomorphisms (Theorem ??). We characterize truth-equational logics in terms of their (Suszko) S-filters (Theorem ??), in terms of their full g-models (Corollary ??), and in terms of the Suszko operator, a characterization which strengthens that of Raftery, as those logics in which the Suszko operator is a structural representation from the set of S-filters to the set of AIg(S)-relative congruences, on arbitrary algebras (Theorem ??). Finally, we prove a new Isomorphism Theorem for protoalgebraic logics (Theorem ??), in the same spirit of the famous one for algebraizable logics and for weakly algebraizable logics.

Endowed with a notion of Leibniz filter applicable to any logic, we are able to generalize the theory of the strong version of a protoalgebraic logic developed by Font and Jansana to arbitrary sentential logics. Given a sentential logic 5, its strong version St is the logic induced by the class of matrices whose truth set is Leibniz filter. We study three definability criteria of Leibniz filters: equational, explicit and logical definability. Under (any of) these assumptions, we prove that the St-filters coincide with Leibniz S-filters on arbitrary algebras. Finally, we apply the general theory developed to a wealth of non-protoalgebraic log-ics covered in the literature. Namely, we consider Positive Modal Logic P,A4,C, Belnap's logic B, the subintuitionistic logics w1C, and Visser's logic VP,C, and Lukasiewicz's infinite-valued logic preserving degrees of truth. We also consider the generalization of the last example mentioned to logics preserving degrees of truth from varieties of integral commutative residuated lattices, and further generalizations to the non-integral case, as well as to the case without multi-plicative constant. We classify all the examples investigated inside the Leibniz and Frege hierarchies. While none of the logics studied is protoalgebraic, all the respective strong versions are truth-equational.

Aquesta dissertació presenta els resultats de la nostra recerca sobre alguns temes recents en Lògica Algebraica Abstracta (LAA), concretament, l'operador de Suszko, els filtres de Leibniz, i les lògiques truth-equacionals. La interacció entre vàries nocións relacionades amb els operadors de Leibniz i de Suszko ens va portar a considerar un marc general basat en la noció de S-operador, que abasta els operadors de Leibniz, de Suszko, i de Frege, unificant així aquests tres operadors paradigmàtics de la LAA sota un mateix tractament.