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A STUDY OF TRUTH PREDICATES IN MATRIX SEMANTICS

Published online by Cambridge University Press:  08 June 2018

TOMMASO MORASCHINI
TOMMASO MORASCHINI*
Affiliation:
Institute of Computer Science, Czech Academy of Sciences
*
*DEPARTMENT OF THEORATICAL COMPUTER SCIENCE INSTITUTE OF COMPUTER SCIENCE, CZECH ACADEMY OF SCIENCES POD VODÁRENSKOU VĚŽÍ 2 182 07 PRAGUE 8, CZECH REPUBLIC E-mail: [email protected]
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Abstract

Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic ${\cal L}$ is associated with a matrix semantics $Mo{d^{\rm{*}}}{\cal L}$. This article is a contribution to the systematic study of the so-called truth sets of the matrices in $Mo{d^{\rm{*}}}{\cal L}$. In particular, we show that the fact that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of ${\cal L}$. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ are implicitly definable if and only if the Leibniz operator is injective on deductive filters of ${\cal L}$ over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of ${\cal L}$ to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in $Mo{d^{\rm{*}}}{\cal L}$ that corresponds to the order-reflection of the Leibniz operator.

Keywords

03G2703G1003B22abstract algebraic logictruth predicateequational definabilitytruth-equational logicprotoalgebraic logicLeibniz hierarchyLeibniz operatorimplicit definabilitymatrix semanticsalgebraic semanticspropositional logicprotodisjunctionprotoconjunction
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 4 , December 2018 , pp. 780 - 804
Copyright
Copyright © Association for Symbolic Logic 2018 

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