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Algebraic semantics for superintuitionistic predicate logics. (English. Russian original) Zbl 0924.03112
Algebra Logika 38, No. 1, 68-95 (1999); translation in Algebra Logic 38, No. 1, 36-50 (1999).
Predicate superintuitionistic logic (p.s.i. logic) [see H. Ono, Rep. Math. Logic 21, 55-67 (1987; Zbl 0676.03016)] is a set of formulas which contains the predicate intuitionistic logic Int and is closed under the substitution rule and standard rules of modus ponens and generalization. The author suggests a universal algebraic semantics for p.s.i. logics and exploits the idea of algebraization of predicate logics [see W. J. Blok and D. Pigozzi, Algebraizable logics, Mem. Am. Math. Soc. 396 (1989; Zbl 0664.03042)]. With an arbitrary predicate Int-theory \(T\), the author associates the deductive system \(\text{DS}(T)\) in the propositional language \(\{\top, \bot, \vee, \wedge, \supset, \neg, \forall_{i}, \exists_{i}, s^{i}_{j}\) \((i,j<\omega)\}\), where the symbol \(\exists_{i}\) (\(\forall_{i}\)) corresponds to binding the variable \(x_{i}\) by the existential (universal) quantifier in the formulas of the original predicate language and \(s^{i}_{j}\) corresponds to substituting \(x_{j}\) for \(x_{i}\).
Further, the author introduces quasi-cylindrical algebras whose reducts to the language \(\{\vee, \wedge, \supset, \neg\}\) are pseudoboolean algebras. For every p.s.i. logic \(L\), the variety \(V(L)\) of quasi-cylindrical algebras is defined by the identities of the form \(\Phi =\top\), where \(\Phi\) is an axiom scheme of \(\text{DS}(L)\). The main result of the article states that an arbitrary p.s.i. logic \(L\) is strongly complete with respect to the variety \(V(L)\).

03G25 Other algebras related to logic
03B55 Intermediate logics
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