Modal companions of superintuitionistic logics: syntax, semantics, and preservation theorems

M. V. Zakharyaschev

Sbornik. Mathematics, Tome 68 (1991) no. 1, pp. 277-289

Résumé

This paper studies the class $\mathscr I$ of superintuitionistic logics and the class $\mathscr M$ of normal extensions of the modal system S4, and the syntactic and semantic connections between the two classes, given by the mapping $\rho$ (which assigns to every modal logic its superintuitionistic fragment) and by the mappings $\tau$ and $\sigma$ (which assign to every superintuitionistic logic its smallest and its greatest companion, respectively). It is shown that from classes of relational models with respect to which a logic $L\in\mathscr I$ is complete, one can construct a class of models with respect to which the logics $\tau L$ and $\sigma L$ are complete. The relationship of inference (of canonical formulas) in logics $L$, $\tau L$ and $\sigma L$ is also described. As a consequence, preservation theorems are obtained for finite approximability, for Kripke completeness and for the disjunction property at the transition from $L$ to $\tau L$, and also for decidability at the transition to $\tau L$ and $\sigma L$. Bibliography: 21 titles.