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A Henkin-Style Proof of Completeness for First-Order Algebraizable Logics

SYSNO ASEP	0428704
Document Type	J - Journal Article
R&D Document Type	Journal Article
Subsidiary J	Článek ve WOS
Title	A Henkin-Style Proof of Completeness for First-Order Algebraizable Logics
Author(s)	Cintula, Petr (UIVT-O)_{RID, ORCID, SAI} Noguera, Carles (UTIA-B)_{RID, ORCID}
Source Title	Journal of Symbolic Logic. - : Cambridge University Press - ISSN 0022-4812 Roč. 80, č. 1 (2015), s. 341-358
Number of pages	18 s.
Language	eng - English
Country	US - United States
Keywords	abstract algebraic logics ; algebraizable logics ; first-order logics ; completeness theorem ; Henkin theories
Subject RIV	BA - General Mathematics
R&D Projects	GA13-14654S GA ČR - Czech Science Foundation (CSF)
Institutional support	UIVT-O - RVO:67985807 ; UTIA-B - RVO:67985556
UT WOS	000351311600018
EID SCOPUS	84937900561
DOI	10.1017/jsl.2014.19
Annotation	This paper considers Henkin’s proof of completeness of classical first-order logic and extends its scope to the realm of algebraizable logics in the sense of Blok and Pigozzi. Given a propositional logic (for which we only need to assume that it has an algebraic semantics and a suitable disjunction) we axiomatize two natural first-order extensions and prove that one is complete with respect to all models over its algebras, while the other one is complete with respect to all models over relatively finitely subdirectly irreducible ones. While the first completeness result is relatively straightforward, the second requires non-trivial modifications of Henkin’s proof by making use of the disjunction connective. As a byproduct, we also obtain a form of Skolemization provided that the algebraic semantics admits regular completions. The relatively modest assumptions on the propositional side allow for a wide generalization of previous approaches by Rasiowa, Sikorski, Hájek, Horn, and others and help to illuminate the “essentially first-order” steps in the classical Henkin’s proof.
Workplace	Institute of Computer Science
Contact	Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800
Year of Publishing	2015

Number of the records: 1