Fusions of One-Variable First-Order Modal Logics
The results clarify which properties transfer when combining one-variable first-order modal logics, providing foundational insights for modal logic and automated reasoning.
The paper investigates preservation of Kripke completeness, decidability, and finite model property in independent fusions of one-variable first-order modal logics. Without equality, completeness and decidability are preserved; with equality and non-rigid constants, they are not, as shown by encoding Diophantine equations.
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 global and local consequence relations 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 preserved only 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).