A Study of Truth Predicates in Matrix Semantics

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F18%3A00491283" target="_blank" >RIV/67985807:_____/18:00491283 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1017/S175502031800014X" target="_blank" >http://dx.doi.org/10.1017/S175502031800014X</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1017/S175502031800014X" target="_blank" >10.1017/S175502031800014X</a>

Jazyk výsledku

angličtina

Název v původním jazyce

A Study of Truth Predicates in Matrix Semantics

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

A Study of Truth Predicates in Matrix Semantics

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA17-04630S" target="_blank" >GA17-04630S: Predikátové škálované logiky a jejich aplikace v informatice</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Review of Symbolic Logic

ISSN

1755-0203

e-ISSN

—

Svazek periodika

11

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

25

Strana od-do

780-804

Kód UT WoS článku

000451012600006

EID výsledku v databázi Scopus

2-s2.0-85048080374