### Camillo Fiorentini

# PhD Thesis

Università degli Studi di Milano, January 2000
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## Kripke Completeness for Intermediate Logics
(pdf)

The thesis studies problems in model theory for nonclassical logics, more
specifically, for intermediate propositional logics. This area was opened
in the 1930s by the well-known works of Gödel and Tarski and has now
evolved into an interesting and complex mathematical field, with a strong
trend to Computer Science applications.
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*.

Camillo Fiorentini