Fusions of One-Variable First-Order Modal Logics

2026-03-04

arXiv.org

AI Summary

This study systematically investigates the preservation of Kripke completeness, decidability, and the finite model property under independent fusion in monadic first-order modal logic, distinguishing between languages with and without equality and examining differences between expanding and constant domain semantics. By employing modal semantic analysis, model constructions, Diophantine encoding techniques, and algebraic and model-theoretic methods for fused logics, the work provides the first precise characterization of the boundaries within which these properties are preserved under fusion. Key contributions include introducing a propositional fusion perspective based on shared S5 modalities and a general transitivity condition; proving that, in the absence of equality, both Kripke completeness and decidability are preserved under both global and local consequence, whereas they fail to be preserved when equality is present; and establishing that the finite model property is preserved only in the local setting.

Abstract

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).

Problem

• fusion
• one-variable first-order modal logic
• Kripke completeness
• decidability
• finite model property

Innovation

• one-variable first-order modal logic
• independent fusion
• Kripke completeness
• decidability
• Diophantine encoding