On Ordering of the Family of Logics with Skolem-Löwenheim Property and Countable Compactness Property

Abstract

In this paper the family of abstract logics with Skolem-Löwenheim property — DLS, and the family of logics with countable compactness property — ℂℂOM are considered. It is proved that DLS and ℂℂOM with a natural ordering between logics form a downward semilattice but not a lattice.

In [4] Krynicki and Väänänen considered an ordering of the family of all abstract logics, as well as the family of all abstract logics having Souslin-Kleene interpolation property. They proved that these orderings form a non-modular lattice such that every distributive lattice can be embedded in it as a sublattice. Moreover, each logic in these orderings has a class of noncomparable extensions and a class of extensions being a chain. In our paper we consider an ordering of the family of logics having downward Skolem-Löwenheim property and of the family of logics having countable compactness property. We show that these orderings are downward semilattices but not lattices. In the first ordering the length of chains are limited and there are maximal elements in it. In the second ordering we have another situation. Namely, every partial ordering can be embedded into it.

Prepared for publication by Michal Krynicki

The author of this paper died soon after writing his Master Thesis [8] in 1978. We think that the results and problems discussed in it are still interesting and actual. They were never published and are little known.

The paper is an English version of [8]. It was prepared for publication by Michal Krynicki. Notes added by him are printed in small letters. We would like to thank Kerko Luosto for valuable remarks and comments concerning the first version of this paper.