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Intuitionistic and classical satisfiability in Kripke models. (English) Zbl 1002.03009

Regarding the theory of Kripke semantics for intuitionistic logic, there arises the natural question of the relation between forcing at a node and satisfaction in the classical structure associated with that node. In Notre Dame J. Formal Logic 24, 395-398 (1983; Zbl 0487.03015), the author introduced a class of formulas with the following property: whenever a formula from this class is satisfied in a classical structure associated with a node of a Kripke model, it must be forced at that node. The paper is devoted to the other direction of the question. Namely, the author defines a class of formulas which, whenever forced at a node of a Kripke model, must be satisfied in the classical structure associated with that node. Some consequences concerning the relation between classical and intuitionistic provability of sentences from these classes are given.

MSC:

03B20 Subsystems of classical logic (including intuitionistic logic)
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03C25 Model-theoretic forcing

Citations:

Zbl 0487.03015