A Study of Truth Predicates in Matrix Semantics

0491283 - ÚI 2019 RIV GB eng J - Článek v odborném periodiku
Moraschini, Tommaso
A Study of Truth Predicates in Matrix Semantics.
Review of Symbolic Logic. Roč. 11, č. 4 (2018), s. 780-804. ISSN 1755-0203. E-ISSN 1755-0211
Grant CEP: GA ČR GA17-04630S
Institucionální podpora: RVO:67985807
Klíčová slova: abstract algebraic logic * truth predicate * equational definability * truth-equational logic * protoalgebraic logic * Leibniz hierarchy * Leibniz operator * implicit definability * matrix semantics * algebraic semantics * propositional logic * protodisjunction * protoconjunction
Obor OECD: Pure mathematics
Impakt faktor: 0.731, rok: 2018

Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic L is associated with a matrix semantics Mod*L. This article is a contribution to the systematic study of the so-called truth sets of the matrices in Mod*L. In particular, we show that the fact that the truth sets of Mod*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 L. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of Mod*L are implicitly definable if and only if the Leibniz operator is injective on deductive filters of L over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of 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 Mod∗L that corresponds to the order-reflection of the Leibniz operator.
Trvalý link: http://hdl.handle.net/11104/0285293