Varieties of logical matrices
type: Item
instance of: scholarly article
title: Varieties of logical matrices
zbMATH Open document ID: 0617.03013
DOI: 10.1007/BF01984693
published in: Algebra and Logic
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187324
review text: The paper introduces some tools in the study of (superintuitionistic) propositional logics. In the language non-standard connectives \(f_ 1,...,f_ k\) of any arity are allowed, besides the usual ones. A logic is a set of formulas containing all theorems of some fixed ''minimal'' logic \(L(f_ 1,...,f_ k)\) (whose axioms contain some superintuitionistic logic, and whose rules are: replacement, modus ponens, and congruence rules for the \(f_ i's)\) and closed under replacement and m.p. (not necessarily under congruence rules for the \(f_ i's)\). A logical matrix is defined as usual as a pseudo-Boolean algebra with operators and with a unary predicate of membership to a filter. A variety of l.m.'s is the class of all logical matrices satisfying a given set of formulas. The lattices of all logics and of all varieties of l.m.'s are isomorphic under the correspondence ''logic- characteristic (Lindenbaum) logical matrix''. Operators on classes of l.m.'s are defined which allow the description, in Birkhoff's style, of the variety generated by a class of l.m.'s. The study of Post complete logics is reduced to that of atoms of the lattice of varieties of l.m.'s, which are the varieties generated by 0- generated matrices. Thus some generalization of results in superintuitionistic logic are obtained, e.g. that any Post complete logic derives \(p\vee \neg p\), or that any classical logic is an intersection of logics which have just one Post complete extension. With regard to modal logics, the author introduces \(\omega\)-scales as a tool enriching the usual Kripke semantics, motivated by the fact that the logic \(GL'=GL+\square p\to p\) has no Kripke semantics (it is ''Kripke contradictory'' in the sense that for no usual Kripke frame (or scale) A is \(A\vDash GL')\). It is shown that GL' is complete for \(\omega\)-scales. By building logical matrices from certain \(\omega\)-scales, it is shown that there is a continuum of Kripke contradictory logics. By a complicated example it …
Mathematics Subject Classification ID: 03B99
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03B55
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03B20
zbMATH DE Number: 3999894
zbMATH Keywords: algebraic semantics
zbMATH Keywords: lattice of varieties of logical matrices
zbMATH Keywords: propositional logics
zbMATH Keywords: superintuitionistic logic
zbMATH Keywords: logical matrix
zbMATH Keywords: pseudo- Boolean algebra with operators
zbMATH Keywords: Post complete logics
zbMATH Keywords: modal logics
zbMATH Keywords: \(\omega \)-scales
zbMATH Keywords: Kripke contradictory logics
zbMATH Keywords: logic Grz of Grzegorczyk
zbMATH Keywords: pretabular extensions of S4
author: Alexander Vasil'evich Chagrov
reviewed by: Aldo Ursini
MaRDI profile type: Publication
cites work: Five critical modal systems
cites work: Q4187817
cites work: Q3875897
cites work: Q3880841
cites work: Intuitionistic modal logics incompatible with the law of the excluded middle
cites work: Boolean Algebras with Operators. Part I
cites work: Q5650663
cites work: Some Post-complete extensions of S2 and S3
cites work: Post Completeness and Free Algebras
cites work: Provability interpretations of modal logic
cites work: Q3912553
cites work: On systems of modal logic with provability interpretations
cites work: Logics containing K4. Part I
cites work: Semantic analysis of tense logics
cites work: The logical consequence relation of propositional tense logic
cites work: Some theorems about the sentential calculi of Lewis and Heyting
cites work: Q4068699
cites work: Q4081212
description: scientific article; zbMATH DE number 3999894