An order-theoretic analysis of interpretations among propositional deductive systems

In this paper we study interpretations and equivalences of propositional deductive systems by using a quantale-theoretic approach introduced by Galatos and Tsinakis. Our aim is to provide a general order-theoretic framework which is able to describe and characterize both strong and weak forms of interpretations among propositional deductive systems also in the cases where the systems have different underlying languages

An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

Epimorphisms, definability and cardinalities

We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m+s+ℵ0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable ‘bridge theorems’, matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic ⊢ with suitable infinitary definability properties of ⊢, while not making the standard but awkward assumption that ⊢ comes furnished with a proper class of variables.The European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 689176 (project “Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics”). The first author was also supported by the Project GA17-04630S of the Czech Science Foundation (GAČR). The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa.http://link.springer.com/journal/112252020-02-07hj2019Mathematics and Applied Mathematic

Intuitionistic logic as a connexive logic

We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic (CHL), hereby introduced as an example of a strongly connexive logic with an intuitive semantics. We use the reverse algebraisation paradigm: CHL is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for CHL ; moreover, we suggest a possible computational interpretation of its connexive conditional, and we revisit Kapsner’s idea of superconnexivity

Intuitionistic logic is a connexive logic

We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic (CHL), hereby introduced as an example of a strongly connexive logic with an intuitive semantics. We use the reverse algebraisation paradigm: CHL is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for CHL; moreover, we suggest a possible computational interpretation of its connexive conditional, and we revisit Kapsner’s idea of superconnexivity

Compatibilitat en Àlgebra, en Lògica i en Informàtica

S'exposa una visió actual de l'estudi algebraic de la Lògica, especialment de les lògiques no clàssiques, prenent com a eix alguns conceptes purament algebraics com els de compatibilitat, congruència de Leibniz, i operador de Leibniz. Es mostra com aquests conceptes permeten definir una jerarquia de lògiques i classificar-les pel seu capteniment envers la seva algebrització, és a dir, per les relacions que mantenen amb els seus models algebraics, i per les propietats d'aquests models. Al final s'esmenten algunes de les línies de recerca més recents, en el context del camp emergent actualment anomenat Lògica Algebraica Abstracta.This paper introduces the current view on the algebraic studies in Logic, especially in the domain of non-classical logics. The paper is organized around some pure algebraic concepts such as compatibility, Leibniz congruence, and the Leibniz operator. It is shown how these concepts allow to define a hierarchy of logics and to classify them according to their behaviour as far as their algebraization is concerned, that is, by the kind of relation they have with their algebraic models, and by the properties of these models. The paper ends with a brief survey of some of the most recent research lines in the context of the emerging field now called Abstract Algebraic Logic