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A Henkin-Style Proof of Completeness for First-Order Algebraizable Logics
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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, ORCIDSource Title Journal of Symbolic Logic. - : Cambridge University Press - ISSN 0022-4812
Roč. 80, č. 1 (2015), s. 341-358Number 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 https://doi.org/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
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