Beth Definability in the Guarded Fragment

Eva Hoogland, Maarten Marx and Martin Otto

Institute for Logic, Language and Computation
University of Amsterdam
Plantage Muidergracht 24
1018 TV Amsterdam
The Netherlands,,,


The guarded fragment (GF) was introduced in abn2 as a fragment of first order logic which combines a great expressive power with nice modal behavior. It consists of relational first order formulas whose quantifiers are relativized by atoms in a certain way. While GF has been established as a particularly well-behaved fragment of first order logic in many respects, interpolation fails in restriction to GF, homa. In this paper we consider the Beth property of first order logic and show that, despite the failure of interpolation, it is retained in restriction to GF. The Beth property for GF is here established on the basis of a limited form of interpolation, which more closely resembles the interpolation property that is usually studied in modal logics. From this we obtain that, more specifically, even every $n$-variable guarded fragment with up to $n$-ary relations has the Beth property.

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