## Journal of Symbolic Logic

### Splittings and the Finite Model Property

Marcus Kracht

#### Abstract

An old conjecture of modal logics states that every splitting of the major systems $\mathbf{K4, S4, G}$ and $\mathbf{Grz}$ has the finite model property. In this paper we will prove that all iterated splittings of $\mathbf{G}$ have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these splitting frames namely that they preserve the finite model property in the following sense. Whenever an extension $\Lambda$ has fmp so does the splitting $\Lambda/f$ of $\Lambda$ by $f$. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive.

#### Article information

Source
J. Symbolic Logic, Volume 58, Issue 1 (1993), 139-157.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744182

Mathematical Reviews number (MathSciNet)
MR1217182

Zentralblatt MATH identifier
0782.03006

JSTOR