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Rules with parameters in modal logic. I. (English) Zbl 1408.03015
Summary: We study admissibility of inference rules and unification with parameters in transitive modal logics (extensions of K4), in particular we generalize various results on parameter-free admissibility and unification to the setting with parameters.
Specifically, we give a characterization of projective formulas generalizing Ghilardi’s characterization in the parameter-free case, leading to new proofs of Rybakov’s results that admissibility with parameters is decidable and unification is finitary for logics satisfying suitable frame extension properties (called cluster-extensible logics in this paper). We construct explicit bases of admissible rules with parameters for cluster-extensible logics, and give their semantic description. We show that in the case of finitely many parameters, these logics have independent bases of admissible rules, and determine which logics have finite bases.
As a sideline, we show that cluster-extensible logics have various nice properties: in particular, they are finitely axiomatizable, and have an exponential-size model property. We also give a rather general characterization of logics with directed (filtering) unification.
In the sequel, we will use the same machinery to investigate the computational complexity of admissibility and unification with parameters in cluster-extensible logics, and we will adapt the results to logics with unique top cluster (e.g., S4.2) and superintuitionistic logics.

##### MSC:
 03B45 Modal logic (including the logic of norms) 03B55 Intermediate logics 08B20 Free algebras 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010)
##### Keywords:
modal logic; admissible rule; equational unification
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