Leibniz filters and the strong version of a protoalgebraic logic

Abstract

A filter of a sentential logic S is Leibniz when it is the smallest one among all the S-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic S + defined by the class of all S-matrices whose filter is Leibniz, which is called the strong version of S, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of S + and of the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from S to S + . For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's 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.

The Infinite-Valued Łukasiewicz Logic and Probability

“…(Note that |= ≤ c in [10], is defined on ∅ in a different way; there the set of theses of the resulting logic is non-empty. )…”

On weakening the Deduction Theorem and strengthening Modus Ponens

“…Corollary 2.7 gives an equivalent "logical" formulation of the problem: QHA is a variety if and only if for every A ∈ QHA and every θ ∈ CoA, the generalized matrix A/θ, F (A/θ) is a model of the rule (DT1). An interesting problem from quite a different context is to investigate whether the logic H 1 is the "strong version" of G 1 in the sense of [11], which in this case amounts to asking whether a G 1 -filter on an algebra in V(QHA) is Leibniz if and only if it is closed under the rule (MP2). As a problem, this turns out to be equivalent to the first one: using non-trivial results from [11] it is possible to show that a positive answer here implies that QHA is a variety, and conversely.…”

Behavioral Algebraization of Logics

“…The particular issue of non-injectivity, staying within the realm of protoalgebraic and equivalential logics, has been carefully studied in [11,13,12]. Moreover, in [20] the authors restrict the models of the protoalgebraic logic at hand by considering just the matrices with a so-called Leibniz filter, therefore obtaining a weakly algebraizable logic. Although this is a very interesting approach, the resulting logic is, of course, different from the original one.…”

“…Our aim in this paper is precisely to propose and study an extension of the tools of AAL that may encompass some of these less orthodox logics while still associating to them meaningful and insightful algebraic counterparts. Therefore, contrarily to what is done in [20], we do not want, at all, to change the logic we start from. Our strategy is rather to change the algebraic perspective.…”