Fischer-Servi logic does not have interpolation

Abstract: We prove that the Fischer-Servi logic $\mathsf{IK}$ does not have the (Craig) interpolation property. This is obtained by showing that the corresponding class of modal Heyting algebras lacks the amalgamation property. We also generalize this result to some extensions of the Fischer-Servi logic such as $\mathsf{IT}$, $\mathsf{IK4}$, $\mathsf{IS4}$, and $\mathsf{IGL}$.

• The paper demonstrates that Fischer-Servi logic and its extensions fail to exhibit Craig interpolation through algebraic and duality frameworks.
• It employs explicit finite counterexamples and a diagram-chasing approach in FS-frames to reveal the lack of the amalgamation property.
• The study highlights that the inclusion of the diamond operator serves as a critical barrier to achieving interpolation in these intuitionistic modal logics.

Fischer-Servi Logic and the Failure of Interpolation

Introduction

The paper "Fischer-Servi logic does not have interpolation" (2604.02082) rigorously establishes the absence of the Craig interpolation property (CIP) for Fischer-Servi logic $\mathsf{IK}$. The authors leverage algebraic and duality-theoretic frameworks, primarily by demonstrating the failure of the amalgamation property in the associated class of modal Heyting algebras. The exposition generalizes this negative result to a range of extensions of $\mathsf{IK}$—notably $\mathsf{IT}$, $\mathsf{IK4}$, $\mathsf{IS4}$, and $\mathsf{IGL}$—thereby mapping out zones within intuitionistic modal logic where interpolation fails and contrasting these with fragments where interpolation is preserved.

Algebraic and Duality-Theoretic Preliminaries

The authors utilize the established correspondence between modal logics and varieties of (modal) Heyting algebras to study interpolation. Specifically, they focus on FS-algebras, which encode the algebraic semantics of super-Fischer-Servi logics—i.e., logics containing $\mathsf{IK}$ and closed under certain modal operators subject to Fischer-Servi’s axioms. The connection between CIP and amalgamation is exploited via the algebraizability of the logic: Craig interpolation for the logic holds if and only if the corresponding variety of algebras has (super)amalgamation.

The paper makes essential use of Palmigiano’s duality between modal Heyting algebras and finite relational Esakia spaces, which streamlines the verification of amalgamation/coamalgamation properties by working directly with finite Kripke frames. This duality is exploited to construct explicit counterexamples.

Failure of Amalgamation and Interpolation

The core of the paper is the construction of a finite counterexample witnessing the failure of coamalgamation in the category of Fischer-Servi frames and the dual lack of amalgamation for FS-algebras. Three carefully constructed finite frames $X$, $Y$, and $Z$, together with surjective $\mathsf{IK}$0-morphisms $\mathsf{IK}$1 and $\mathsf{IK}$2, are shown to lack a coamalgam. That is, there does not exist a Fischer-Servi frame $\mathsf{IK}$3 and $\mathsf{IK}$4-morphisms $\mathsf{IK}$5 making the respective diagrams commute.

The authors systematically verify that $\mathsf{IK}$6, $\mathsf{IK}$7, $\mathsf{IK}$8 are indeed $\mathsf{IK}$9-frames and $\mathsf{IT}$0 are valid morphisms, employing the confluence criteria (F1, F2) for frames and checking the preservation of relevant order and accessibility relations. The absence of a coamalgam is proved by a diagram-chasing argument, yielding an element that causes a necessary contradiction in the commuting conditions.

By the duality principles, failure of coamalgamation in the frame semantics immediately lifts to failure of amalgamation in the associated variety of modal Heyting algebras. Via the well-known algebra-logic correspondence, this demonstrates that $\mathsf{IT}$1 does not have CIP.

Generalizations to Extensions of $\mathsf{IT}$2

The construction is carefully adapted to cover several modal and provability extensions. For logics based on requiring reflexivity, transitivity, or both for the accessibility relation $\mathsf{IT}$3 (e.g., $\mathsf{IT}$4, $\mathsf{IT}$5, $\mathsf{IT}$6), the example is modified by adjusting $\mathsf{IT}$7 appropriately. For $\mathsf{IT}$8, which additionally constrains the absence of infinite $\mathsf{IT}$9-paths, it is shown that the constructed frames already satisfy the required properties.

This yields the formal claim that all these logics—including $\mathsf{IK4}$0, $\mathsf{IK4}$1, $\mathsf{IK4}$2, $\mathsf{IK4}$3, and $\mathsf{IK4}$4—do not possess Craig interpolation.

Relation to Algebraic and Syntactic Properties

The ramifications are not confined to Craig interpolation. The failure of amalgamation alongside the presence of the congruence extension property (CEP) means that deductive interpolation also fails for these logics. As uniform and Lyndon interpolation strengthen Craig/deductive interpolation, these failures propagate to them as well. The results thus map out sharp demarcations between fragments where strong interpolation properties are preserved and those where they fail.

In contrast, logics such as $\mathsf{IK4}$5 and $\mathsf{IK4}$6, which restrict the modal language or relations, are shown algebraically and syntactically to preserve amalgamation and thus enjoy interpolation.

Implications and Future Directions

The negative result for $\mathsf{IK4}$7 and its close relatives carves out a set of medium-strength intuitionistic modal logics that fail to satisfy interpolation, pinpointing the addition of the diamond operator $\mathsf{IK4}$8 and its associated compatibility axioms as critical obstructions. This diagnosis informs the design of modal systems aimed at balancing expressivity with the preservation of metatheoretic properties such as interpolation.

A significant open problem indicated by the authors is to characterize those extensions of Fischer-Servi logic that do admit interpolation, especially with additional axioms linking $\mathsf{IK4}$9 and $\mathsf{IS4}$0, or under alternative base fragments (e.g., dropping linearity constraints). The interaction of algebraic properties (e.g., superamalgamation, CEP) and logical metatheorems in this non-classical context presents an interesting landscape for future work.

Conclusion

The paper offers a definitive algebraic and duality-theoretic account of the failure of Craig interpolation for Fischer-Servi logic and its principal extensions. The results rely on explicit frame constructions and duality arguments to demonstrate the absence of amalgamation/coamalgamation, providing clarity on the limitations imposed by certain modal axioms. The study serves as a reference point for further research into the algebraic taxonomy and metatheoretic analysis of intuitionistic modal systems.