An interpolant in predicate Gödel logic

Abstract

A logic satisfies the interpolation property provided that whenever a formula ∆ is a consequence of another formula Γ, then this is witnessed by a formula Θ which only refers to the language common to Γ and ∆. That is, the relational (and functional) symbols occurring in Θ occur in both Γ and ∆, Γ has Θ as a consequence, and Θ has ∆ as a consequence. Both classical and intuitionistic predicate logic have the interpolation property, but it is a long open problem which intermediate predicate logics enjoy it. In 2013 Mints, Olkhovikov, and Urquhart showed that constant domain intuitionistic logic does not have the interpolation property, while leaving open whether predicate Gödel logic does. In this short note, we show that their counterexample for constant domain intuitionistic logic does admit an interpolant in predicate Gödel logic. While this has no impact on settling the question for predicate Gödel logic, it lends some credence to a common belief that it does satisfy interpolation. Also, our method is based on an analysis of the semantic tools of Olkhovikov and it is our hope that this might eventually be useful in settling this question.

Some Model Theoretic Properties for Pavelka-Style Gödel Logic, RGL* and Gödel Logic with Δ

“…The Craig interpolation property is an open problem for first-order Gödel logic [1]. In [2], the author expresses an approximate version of the Craig interpolation property for Pavelka-style Gödel logic, RGL* (which is called rational Gödel logic).…”