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Modal companions of superintuitionistic logics: syntax, semantics, and preservation theorems
M. V. Zakharyaschev
Abstract:
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.
UDC:
510.6
MSC: Primary
03B20,
03B45,
03C40; Secondary
03F55 Received: 08.12.1988
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