Compatibility Operators in Abstract Algebraic Logic

Abstract

This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters (which generalizes the less-known notion of Leibniz filter). We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries.

Categorical Abstract Algebraic Logic: Tarski Congruence Systems, Logical Morphisms and Logical Quotients

“…(A very good reference is [8], where the interested reader may find, apart from a description of the most important classes of this hierarchy, many more references to original works. Also, [12] provides a brief overview of the area and [1] contains a wealth of results pertaining to the so-called "operator approach" to abstract algebraic logic. )…”

“…In [4], Blok consisting of the major classes of protoalgebraic [3], equivalential [7], and algebraizable [4] logics (see also [12] for an excellent overview). In subsequent work, Font and Jansana [11] [11] and [12], [8] and the very recent [1] are other excellent expositions of the rôle that congruences with compatibility properties play in studying the interaction between logical and algebraic properties.…”

The Suszko operator relative to truth‐equational logics

“…This result shed a new light over the Leibniz hierarchy, since for the first time one of its classes was being characterized through the Suszko operator (non‐trivially, in the sense that truth‐equational logics need not be protoalgebraic), and furthermore such characterization was algebraically simpler than the respective one in terms of the Leibniz operator. In [2, § 6.2] other classes of logics within the Leibniz hierarchy were also characterized through the Suszko operator, inspired by Raftery's characterization of truth‐equational logics.…”

A note of the full generalized models of the extensions of a logic

“…The second application regards the Suszko filters of a logic. This notion has been introduced in [1,2], where it is studied in depth, as an instance of more general notions concerning compatibility operators, and for its relevance in several characterizations of some classes of the Leibniz hierarchy. Here, all we need to know about it is the following characterization [2, Theorem 5.13] in terms of the full g-models of the logic of a particular kind.…”

“…There are other examples where the conclusion of Proposition 6 does not hold and are not covered by Proposition 7. To mention just one, consider Positive Modal Logic PML as analyzed in[2, Section 4] and its strong version PML + , which turns out to be the extension of PML with the Rule of Necessitation x…”