Fusions of One-Variable First-Order Modal Logics

We investigate preservation results for the independent fusion of one-variable first-order modal logics. We show that, without equality, Kripke completeness and decidability of the global and local consequence relation are preserved, under both expanding and constant domain semantics. By contrast, Kripke completeness and decidability are not preserved for fusions with equality and non-rigid constants (or, equivalently, counting up to one), again for the global and local consequence and under both expanding and constant domain semantics. This result is shown by encoding Diophantine equations. Even without equality, the finite model property is only preserved in the local case. Finally, we view fusions of one-variable modal logics as fusions of propositional modal logics sharing an S5 modality and provide a general sufficient condition for transfer of Kripke completeness and decidability (but not of finite model property).

💡 Research Summary

The paper investigates how the independent fusion of one‑variable first‑order modal logics affects key meta‑logical properties such as Kripke completeness, decidability, and the finite model property (FMP). The authors consider two main settings: (i) logics without equality, and (ii) logics with equality together with non‑rigid constants (or, equivalently, counting quantifiers that enforce “exactly one” semantics). They also explore a more abstract perspective by viewing one‑variable modal logics as propositional multimodal logics that share an S5 modality.

In the equality‑free case, the authors prove a positive transfer theorem (Theorem A). If L₁ and L₂ are Kripke‑complete, decidable one‑variable modal logics (under either expanding‑domain or constant‑domain semantics), then their fusion L₁⊗L₂ remains Kripke‑complete and decidable for both local and global consequence. The proof adapts the cactus model construction, originally developed for propositional modal logics, to the one‑variable setting via a quasimodel technique. Types (maximal Boolean‑consistent sets of sub‑formulas) and quasistates (finite collections of types satisfying the existential closure condition) are used to simulate arbitrarily many domain elements with a finite structure. While the local consequence inherits the finite model property, the global consequence does not: for any non‑trivial fusion, the global FMP fails under both expanding and constant domain semantics. This asymmetry mirrors known results for propositional fusions, where global reasoning is more demanding.

The situation changes dramatically when equality and non‑rigid constants are allowed (Theorem B). By encoding Diophantine equations using the counting quantifier ∃=1x P(x) (which forces a predicate to hold for exactly one element), the authors simulate Minsky machines inside the fused logic. This yields undecidability of both local and global consequence, as well as non‑recursive axiomatizability, for any non‑trivial fusion, regardless of whether the underlying semantics are expanding or constant domains. Moreover, global consequence is undecidable for all such fusions even under constant‑domain semantics. The authors stress that this negative result does not imply that all one‑variable logics with equality are undecidable; for example, when the underlying propositional base is K or S5, the one‑variable fragment with equality remains decidable (as shown in the CONEXP TIME system).

To bridge the gap between first‑order and propositional perspectives, the paper introduces the notion of E‑homogeneous models for an equivalence relation E interpreting an S5 modality. An E‑homogeneous model, for sufficiently large infinite cardinals κ, guarantees that within each E‑equivalence class either no world satisfies a given formula or κ many worlds do. Theorem C shows that if two propositional modal logics admit such homogeneous models for a shared S5 modality, then their fusion preserves Kripke completeness and decidability for both local and global consequence. This condition holds for many well‑studied systems, including (semi)commutators with S5 and expanding product logics with S5, thereby providing a general sufficient criterion for transferring these properties to the corresponding one‑variable first‑order fusions. However, the finite model property does not transfer under this condition, aligning with the earlier negative result for global FMP.

Methodologically, the paper contributes several technical tools: (1) a refined cactus‑model construction adapted to one‑variable logics via quasimodels; (2) the use of surrogates to separate the modal operators of the component logics within the fusion; (3) a translation from one‑variable formulas to propositional formulas enriched with an S5 modality, preserving Barcan formulas; and (4) a systematic analysis of how counting quantifiers interact with the fusion process. The work also clarifies the role of non‑rigid constants: they are the source of the counting ability that makes the fusion undecidable, whereas rigid constants do not cause such a blow‑up.

Overall, the paper delivers a comprehensive picture of preservation and failure of meta‑logical properties under fusion of one‑variable first‑order modal logics. It delineates a clear boundary: without equality (or with only rigid constants) the fusion behaves nicely, preserving completeness and decidability, while the addition of equality together with non‑rigid constants destroys these properties. The abstract sufficient condition based on E‑homogeneous models offers a unifying framework for future investigations of combined modal systems, answering an open question posed by Baader, Ghilardi, and Tineelli. The results have implications for the design of decidable description logics, temporal logics, and other modal formalisms that rely on one‑variable fragments.