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A Henkin-Style Proof of Completeness for First-Order Algebraizable Logics
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SYSNO ASEP 0428704 Druh ASEP J - Článek v odborném periodiku Zařazení RIV J - Článek v odborném periodiku Poddruh J Článek ve WOS Název A Henkin-Style Proof of Completeness for First-Order Algebraizable Logics Tvůrce(i) Cintula, Petr (UIVT-O) RID, ORCID, SAI
Noguera, Carles (UTIA-B) RID, ORCIDZdroj.dok. Journal of Symbolic Logic. - : Cambridge University Press - ISSN 0022-4812
Roč. 80, č. 1 (2015), s. 341-358Poč.str. 18 s. Jazyk dok. eng - angličtina Země vyd. US - Spojené státy americké Klíč. slova abstract algebraic logics ; algebraizable logics ; first-order logics ; completeness theorem ; Henkin theories Vědní obor RIV BA - Obecná matematika CEP GA13-14654S GA ČR - Grantová agentura ČR Institucionální podpora UIVT-O - RVO:67985807 ; UTIA-B - RVO:67985556 UT WOS 000351311600018 EID SCOPUS 84937900561 DOI https://doi.org/10.1017/jsl.2014.19 Anotace 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. Pracoviště Ústav informatiky Kontakt Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Rok sběru 2015
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