Kripke semantics has revealed a powerful instrument for studying their syntactic and computational properties and also for deeper understanding of constructivity. Fundamental results in Kripke semantics were proved in the last thirty years; many of them are collected in the recent book ``Modal logic'' by A. Chagrov and M. Zakharyaschev (1997). However several important problems in this field are still open, for example, little is known about completeness/incompleteness properties of intermediate logics; this is the main subject of the thesis.
We consider several variants of completeness; in their increasing strength they are: hypercanonicity, extensive canonicity, canonicity, and strong completeness. We also introduce weaker versions of these properties, namely ω-hypercanonicity, extensive ω-canonicity, ω-strong completeness. This classification is shown to be nontrivial even within the well-known intermediate logics, such as logics axiomatized by one-variable formulas, logics of finite trees and some others. We develop new techniques, which have a general interest and can be applied when Kripke semantics is concerned.
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