Deductive systems and finite axiomatization properties

The notions of a deductive system, equational logic and Gentzen system can be generalized into the notion of a K-deductive system. A universal Horn logic is also a K-deductive system. In Part I the relationship between the existence of equivalence K-terms in a K-deductive system and some semantical properties of these systems is studied. In particular, a K-deductive system S has a finite system of equivalence formulas with parameters if the Leibniz operator on the filter lattice of every S-matrix is monotone. An equivalent semantics theorem, characterizing K-deductive systems that are equivalent to some Birkhoff-like systems, is proved and used to characterize algebraizable K-deductive systems. The connection between the implication terms and semantical properties of one-deductive systems is investigated.;In Part II a finite basis theorem for finitely generated filter-distributive proto-quasivarieties is proved. It says that if the language has only finitely many symbols, and if a K-deductive, filter-distributive, protoalgebraic system S is determined by a finite set of finite matrices, then S has a basis consisting of finitely many axioms and rules of inference. This theorem extends Pigozzi\u27s finite basis theorem for relatively congruence-distributive quasivarieties and therefore also Baker\u27s finite basis theorem for congruence-distributive varieties. If all tautologies of a finite matrix can be derived using only finitely many axioms and rules, then the matrix is called finitely axiomatizable. In particular, a finite algebra A is called finitely axiomatizable if there is a finite set of quasi-identities of A from which every identity of A can be derived. In Part III we consider the problem of finite axiomatizability of finite matrices and finite algebras. Three-element nonfinitely axiomatizable matrices are given. This solves the problem of finding a smallest and simplest possible non-finitely axiomatizable matrix that was posed by Rautenberg, independently by Wojtylak and restated by Dziobiak. Examples, that show that the underlying algebra of a finite nonfinitely axiomatizable matrix can be finitely axiomatizable, are given. The notion of the second-order finite axiomatization is proposed and two sufficient conditions for a finite algebra to be second-order finitely axiomatizable are presented

Order algebraizable logics

AbstractThis paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic (AAL). Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: (i) it is injective, (ii) it is isotonic, (iii) it commutes with the inverse image operator of any algebraic homomorphism, and (iv) it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties (i)–(iv)

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

An algebraic study of exactness in partial contexts

DMF@?s are the natural algebraic tool for modelling reasoning with Korner@?s partial predicates. We provide two representation theorems for DMF@?s which give rise to two adjunctions, the first between DMF and the category of sets and the second between DMF and the category of distributive lattices with minimum. Then we propose a logic L"{"1"} for dealing with exactness in partial contexts, which belongs neither to the Leibniz, nor to the Frege hierarchies, and carry on its study with techniques of abstract algebraic logic. Finally a fully adequate and algebraizable Gentzen system for L"{"1"} is given

The logic of distributive bilattices

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of abstract algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattice

Leibniz hierarchy

Mestrado em MatemáticaA Lógica Algébrica Abstracta estuda o processo pelo qual uma classe de álgebras pode ser associada a uma lógica. Nesta dissertação, analisamos este processo agrupando lógicas partilhando certas propriedades em classes. O conceito central neste estudo é a congruência de Leibniz que assume o papel desempenhado pela equivalência no processo tradicional de Lindenbaum- Tarski. Apresentamos uma hierarquia entre essas classes que é designada por hierarquia de Leibniz, caracterizando as lógicas de cada classe por propriedades meta-lógicas, por exemplo propriedades do operador de Leibniz. Estudamos também a recente abordagem comportamental que usa lógicas multigénero, lógica equacional comportamental e, consequentemente, uma versão comportamental do operador de Leibniz. Neste contexto, apresentamos alguns exemplos, aos quais aplicamos esta nova teoria, capturando alguns fenómenos de algebrização que não era possível formalizar com a abordagem standard. ABSTRACT: Abstract Algebraic logic studies the process by which a class of algebras can be associated with a logic. In this dissertation, we analyse this process by grouping logics sharing certain properties into classes. The central concept in this study is the Leibniz Congruence that assumes the role developed by the equivalence in the traditional Lindenbaum-Tarski process. We show a hierarchy between these classes, designated by Leibniz hierarchy, by characterizing logics in each class by meta-logical properties, for example properties of the Leibniz operator. We also study a recent behavioral approach which uses many-sorted logics, behavioral equational logic and, consequently, a behavioral version of the Leibniz operator. In this context, we provide some examples, to which we apply this new theory, capturing some phenomena of algebraization that are not possible to formalize using the standard approach

On a substructural Gentzen system, its equivalent variety semantics and its external deductive system

Abstract It was shown in be a propositional language of type (2, 2, 2, 2, 0, 0). Let Γ, Π be finite sequences of L-formulas and ϕ, ψ, ξ be L-formulas. The sequent calculus LJ * \c is defined by the following axioms and rules