Rules with parameters in modal logic I

TL;DR

This paper extends the theory of admissibility and unification in transitive modal logics to include parameters, providing new characterizations, decision procedures, and explicit bases for these rules, especially in cluster-extensible logics.

Contribution

It generalizes parameter-free results to parameterized cases, offers explicit bases of admissible rules, and characterizes properties of cluster-extensible logics.

Findings

Admissibility with parameters is decidable in certain logics.

Finitary unification is established for cluster-extensible logics.

Explicit bases of admissible rules with parameters are constructed.

Abstract

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.