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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 \(V(L)\).

MSC:
03G25 Other algebras related to logic
03B55 Intermediate logics
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