PhD Thesis

Kripke Completeness for Intermediate Logics (pdf)

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.

More in detail:

• We prove some criteria to state canonicity, strong completeness, ω-canonicity, strong ω-completeness of intermediate logics.
• We give some results about classifications of intermediate logics with respect to these notions. In particular, we prove the significant result that all intermediate logics axiomatized by formulas in one variable (Nishimura formulas), except eight of them, are not strongly ω-complete.