Generalized interpolation and definability

Uniqueness, Definability and Interpolation

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points. Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If a is a nonlogical constant, Tat a set of sentences, cx* an additional constant of the same syntactical category as a and T7t* a copy of Tat with cx* in...

Definability and Interpolation in Non-Classical Logics

Generalized Rational Relations and their Logical Definability

The family of rational subsets of a direct product of free monoids Σ∗ 1×. . .×Σ ∗ n (the rational relations) is not closed under Boolean operations, except when n = 1 or when all Σi’s are empty or singletons. In this paper we introduce the family of generalized rational subsets of an arbitrary monoid as the closure of the singletons under the Boolean operations, concatenation and Kleene star (j...

Interpolation and Beth Definability over the Minimal Logic

Extensions of the Johansson minimal logic J are investigated. It is proved that the weak interpolation property WIP is decidable over J. Well-composed logics with the Graig interpolation property CIP, restricted interpolation property IPR and projective Beth property PBP are fully described. It is proved that there are only finitely many well-composed logics with CIP, IPR or PBP; for any well-c...

Interpolation, Definability and Fixed Points in Interpretability Logics

In this article we study interpolation properties for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the -modality easily follow from this result. Furthermore, these properties are extended to the system ILP. Failure of arrow interpolation for ILW is established by providing an explicit count...