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

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    0428704 - ÚI 2015 RIV US eng J - Journal Article
    Cintula, Petr - Noguera, Carles
    A Henkin-Style Proof of Completeness for First-Order Algebraizable Logics.
    Journal of Symbolic Logic. Roč. 80, č. 1 (2015), s. 341-358. ISSN 0022-4812. E-ISSN 1943-5886
    R&D Projects: GA ČR GA13-14654S
    EU Projects: European Commission(XE) 247584 - MATOMUVI
    Institutional support: RVO:67985807 ; RVO:67985556
    Keywords : abstract algebraic logics * algebraizable logics * first-order logics * completeness theorem * Henkin theories
    Subject RIV: BA - General Mathematics
    Impact factor: 0.510, year: 2015

    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.
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