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Multi-modal and temporal logics with universal formula – reduction of admissibility to validity and unification. (English) Zbl 1149.03017
Summary: The article studies multi-modal (in particular temporal, tense) logics possessing universal formulas. We prove that the admissibility problem for inference rules (inference rules with parameters) is decidable for all decidable multi-modal (in particular, temporal) logics possessing a universal formula and decidable w.r.t. unification (unification with parameters). These theorems use characterizations of admissible rules in terms of valid rules and unifiable formulas. Results are applied to a wide range of multi-modal logics, including all linear transitive temporal logics, all temporal logics with bounded zigzag, and all multi-modal logics with explicit universal modality. As a consequence, we show that the admissibility problem for all such logics is decidable (e.g., for all logics of the series S4\(_nU, n\in N\)).

MSC:
03B44 Temporal logic
03B45 Modal logic (including the logic of norms)
03B25 Decidability of theories and sets of sentences
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