Unification in Intermediate Logics

Abstract

This paper contains a proof theoretic treatment of some aspects of unification in intermediate logics. It is shown that many existing results can be extended to fragments that at least contain implication and conjunction. For such fragments the connection between valuations and most general unifiers is clarified, and it is shown how from the closure of a formula under the Visser rules a proof of the formula under a projective unifier can be obtained. This implies that in the logics considered, for the n-unification type to be finitary it suffices that the m-th Visser rule is admissible for a sufficiently large m. At the end of the paper it is shown how these results imply several well-known results from the literature.

A Syntactic Approach to Unification in Transitive Reflexive Modal Logics

“…The inspiration for this paper is the proof-theoretic approach to unification in intuitionistic logic as developed by Rozière in [26]. In [16] we have extended these results to intermediate logics. I thank Emil Jeřábek, George Metcalfe, and Paul Rozière for helpful remarks along the way, and an anonymous referee for many comments that helped improve the paper.…”

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.…”

Unification on Subvarieties of Pseudocomplemented Distributive Lattices

“…The unification type of various subvarieties of Heyting algebras has been determined in [6,8,9,20]. Unification in different fragments of intuitionistic logic that include the implication were investigated in [5,11,15,18]. The variety of bounded distributive lattices was proved to have nullary type (that is, there exists a unification problem that does not admit a minimal complete set of unifiers) in [7], and the type of each unification problem was calculated in [4].…”