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Syntax and semantics of superintuitionistic logics. (English. Russian original) Zbl 0708.03011
Algebra Logic 28, No. 4, 262-282 (1989); translation from Algebra Logika 28, No. 4, 402-429 (1989).
The aim of this paper is to study general properties of superintuitionistic logics basing on the introduction of a special kind of canonical formulas. It is shown that any superintuitionistic logic (s.l.) can be axiomatized by means of canonical formulas. At the same time, it is proved that any set of formulas with the above mentioned property can be decreased, that is, the set of all s.l. has no axiomatical basis. By means of the introduced tools of canonical formulas, the author gives a solution of the Dummet-Lemmon hypothesis about the largest modal counterpart. It is proved that the finite model property is preserved under going to the smallest counterpart. New examples of modal counterparts of $${\mathcal I}nt$$ without finite model property are given. Properties of s.l. with axioms in restricted languages are considered.
Reviewer: V.V.Rybakov

##### MSC:
 03B55 Intermediate logics 03B45 Modal logic (including the logic of norms)
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