Craig interpolation for Fischer–Servi logics over Gödel–Dummett base

Determine whether the Fischer–Servi intuitionistic modal logics obtained by adding the Gödel–Dummett axiom (p → q) ∨ (q → p) to the base intuitionistic calculus (i.e., extensions of the Fischer–Servi logic IK by the Gödel–Dummett axiom) have the Craig interpolation property. Since the Gödel–Dummett axiom enforces linear underlying frames, the non-linear co-amalgamation counterexample used to refute interpolation for IK does not apply, and a new analysis is required.

Background

This paper proves that the Fischer–Servi logic IK lacks the Craig interpolation property by showing that the corresponding variety of modal Heyting algebras does not have the amalgamation property. The failure is exhibited via a dual counterexample on relational Esakia spaces (FS-frames), and the result is extended to several related logics (IT, IK4, IS4, and IGL).

However, the counterexample construction relies on non-linear partial orders. The authors note that adding the Gödel–Dummett axiom (p → q) ∨ (q → p) forces linearity of the underlying frames, invalidating their specific counterexample. Consequently, whether Fischer–Servi logics over Gödel–Dummett (and similar stronger intuitionistic bases) enjoy Craig interpolation remains unresolved and is explicitly posed as an open question.

References

Another natural direction is to ask whether Fischer–Servi logics over stronger intuitionistic bases exhibit the same behaviour. For example, one could consider adding the Gödel–Dummett axiom (p → q) ∨ (q → p). We note that this axiom fails in the kind of counterexample provided here, as it forces the underlying frames to be linear. Thus, the counterexample in this paper does not apply to these logics, leaving the study of interpolation for them as an interesting open question.

— Fischer-Servi logic does not have interpolation  (2604.02082 - Almeida et al., 2 Apr 2026) in Section “Conclusions”