A Syntactic Approach to Unification in Transitive Reflexive Modal Logics

Abstract

This paper contains a proof-theoretic account of unification in transitive reflexive modal logics, which means that the reasoning is syntactic and uses as little semantics as possible. New proofs of theorems on unification types are presented and these results are extended to negationless fragments. In particular, a syntactic proof of Ghilardi's result that S4 has finitary unification is provided. In this approach the relation between classical valuations, projective unifiers and admissible rules is clarified.

Unification with parameters in the implication fragment of classical propositional logic

“…For a modal language based on the connectives →, ¬ and 2, it is known, for example, that S5 is unitary [12], S4 is finitary [17,20] and K is nullary [21]. This immediately leads us to ask what is the unification type of S5, S4 and K when → and 2 are the sole connectives?…”

On Rules

“…As to the constructivity of this method: [11,19,20] contain algorithms to obtain the projective approximation of a formula in several well-known logics and in [18,21] constructive proofs of the projectivity of the formulas in the approximation are given, by providing explicit proofs of the formulas under their projective unifier.…”

About the unification type of $\mathbf {K}+\square \square \bot $

“…Otherwise, it is nullary. Within the context of elementary unification, it is known that Alt 1 is nullary [8], S5 and S4.3 are unitary [10,11,12], transitive modal logics like K4 and S4 are finitary [13,15], KD45, K45 and K4.2 + are unitary [14,16], K is nullary [17] and K4D1 is unitary [18]. The unification types of the description logics EL and FL 0 are known too: both of them are nullary [2,3].…”