0428704 20250112205631.320140609235959.9 84937900561 000351311600018 10.1017/jsl.2014.19 18 s. cav_un_epca*02571230022-4812801 (2015)341-358Cambridge University Press abstract algebraic logics algebraizable logics first-order logics completeness theorem Henkin theories cav_un_auth*0100737 Cintula Petr UIVT-O Oddělení teoretické informatiky Department of Theoretical Computer Science Department of Theoretical Computer Science Ústav informatiky AV ČR, v. v. i. cav_un_auth*0293476 Noguera Carles UTIA-B Matematická teorie rozhodování Department of Decision Making Theory MTR MTR Department of Decision Making Theory Ústav teorie informace a automatizace AV ČR, v. v. i. GA13-14654S GA ČR cav_un_auth*0292719 247584 EC XE cav_un_auth*0323440 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. BA 2015 4 O R hod 4o 4rh a 20231122140250.4 A 20250112205631.3 RVO:67985807 RVO:67985556 http://hdl.handle.net/11104/0234002 1 Department of Theoretical Computer Science UIVT-O 10100 LOGIC 1 Department of Decision Making Theory UTIA-B 10100 LOGIC S n.a. Article Mathematics|Logic LOGIC|MATHEMATICS 0.517 0.215 999.9 0.00408 0.724 1705 1.203 Q1 0.462 65 Q3 62.403 34.9 Q2 Q1 35.737 2015 Soubory v repozitáři: dodatecne_citace_k_0428704.pdf, 0428704.pdf, a0428704.pdf 84937900561 SCOPUS 000351311600018 WOS cav_un_epca*0257123 Journal of Symbolic Logic 0022-4812 1943-5886 Roč. 80 č. 1 2015 341 358 Cambridge University Press