Categorical Abstract Algebraic Logic: Referential π-Institutions

Wójcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wójcicki asserts that a logic has a referential semantics if and only if it is selfextensional. Referential semantics was subsequently studied in detail by Malinowski and the concept of selfextensionality has played, more recently, an important role in the field of abstract algebraic logic in connection with the operator approach to algebraizability. We introduce and review some of the basic definitions and results pertaining to the referential semantics of π-institutions, abstracting corresponding results from the realm of propositional logics

Categorical Abstract Algebraic Logic: Pseudo-Referential Matrix System Semantics

This work adapts techniques and results first developed by Malinowski and by Marek in the context of referential semantics of sentential logics to the context of logics formalized as π-institutions. More precisely, the notion of a pseudoreferential matrix system is introduced and it is shown how this construct generalizes that of a referential matrix system. It is then shown that every π–institution has a pseudo-referential matrix system semantics. This contrasts with referential matrix system semantics which is only available for self-extensional π-institutions by a previous result of the author obtained as an extension of a classical result of Wójcicki. Finally, it is shown that it is possible to replace an arbitrary pseudoreferential matrix system semantics by a discrete pseudo-referential matrix system semantics

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

Meet-irreducible elements in the poset of all logics

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona. Curs: 2023-2024. Tutor: Tommaso Moraschini[cat] En aquest treball hem estudiat els elements meet-irreductibles del poset Log de totes les lògiques. A més, hem presentat les eines tècniques requerides, com la relació d’interpretabilitat entre lògiques i la construcció del producte no-indexat d’una família de lògiques, inspirades en l’anàlisi del lattice de les varietats de l’àlgebra universal. Finalment, hem treballat en criteris de meet-irreductibilitat per a alguns sub-semilattices de Log[eng] In this work we have studied the meet-irreducible elements in the poset Log of all logics. Additionally, we have presented the main technical tools required to do so, such as the interpretability relation between logics and the non-indexed product of a family of logics, inspired in the analysis of the lattice of varieties in universal algebra. Finally, we have worked out meet-irreducibility criteria for some sub-semilattices of Log

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

Admissible rules and the Leibniz hierarchy

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and signi cance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the non-algebraizable fragments of relevance logic are considered.This work is based on research supported in part by the National Research Foundation of South Africa (UID 85407).https://www.dukeupress.edu/notre-dame-journal-of-formal-logichb2016Mathematics and Applied Mathematic

LE MOLTE VERSIONI DELLA CONSEGUENZA LOGICA

This thesis is on the concept of logical consequence (LC) and it is divided into two parts. In the first one, I show that LC: (1) was not important in eminent logicians (like Aristotle) (2) has been described in several different ways (preservation of truth from premises to conclusion, formality, necessity of thought, following a rule, to transform what is a ground for the premises into a ground for the conclusion, \u2026) and by different methods (predication, natural language, formal language, per se entities, Theory of Types, Set-theory, derivation in a formal calculus, variation of the non-logical parts of the sentences, \u2026). I explain how LC became one of the central notions of contemporary logic, why it was not important in many authors (in certain cases, until very recent years), which forms had the logics in which LC was not important, the many and important relations among LC and extra-logical (metaphysical, epistemological, pragmatic, \u2026) notions. It shows that we cannot simply take for granted that there is an intuitive concept of LC or even a natural concept of LC, since it has always been formulated and become understandable and important only in connection with non-logical notions and different scientific aims. The authors or the schools studied in this first section are: Aristotle, Descartes, Kant, Bolzano, Frege, the algebra of logic, the axiomatic study of mathematical theories, Brouwer, Gentzen, Tarski, Etchemendy and Prawitz. In the second part of my thesis, I explore the seminal Tarski\u2019s idea to consider LC as a closure operator and further developed by \u141os, Suszko, W\uf3jcicki, Czelakowski and the Barcelona Group. I define LC as a structural closure relation on the algebra of formulas, without taking into account its syntactical or semantic definition. I examine the philosophical ideas lying behind this conception and I explore different definitions (e.g., non-monotone consequence). Then I explore how we can define LC by a calculus (Hilbert-calculus and Natural Deduction Calculus) or by a semantic system (I consider predicative language too) and I explain the philosophical implications of these different points of view. In the last chapter I explore how we can define LC by matrices and the philosophical implication of this method. I study Lindenbaum matrices, Lindenbaum bundles, Lindenbaum-Tarski algebras and I investigate the relation among some properties of logical systems and Lindenbaum matrices