We show that some common varieties of modal K4-algebras have finitary unification type, thus providing effective best solutions for equations in free algebras. Applications to admissible inference rules are immediate.

A projective unifier for a modal formula A, over a modal logic L, is a unifier σ for A such that the equivalence of σ with the identity map is the consequence of A. Each projective unifier is a most general unifier for A. Let L be a normal modal logic containing S4. We show that every unifiable formula has a projective unifier in L iff L contains S4.3. The syntactic proof is effective. As a corollary, we conclude that all (...) normal modal logics L containing S4.3 are almost structurally complete, i.e. all admissible rules having unifiable premises are derivable in L. Moreover, L is structurally complete iff L contains McKinsey axiom M. (shrink)

We present a basis for the admissible rules of intuitionistic propositional logic. Thereby a conjecture by de Jongh and Visser is proved. We also present a proof system for the admissible rules, and give semantic criteria for admissibility.

This paper offers a brief analysis of the unification problem in modal transitive logics related to the logic S4 : S4 itself, K4, Grz and Gödel-Löb provability logic GL . As a result, new, but not the first, algorithms for the construction of ‘best’ unifiers in these logics are being proposed. The proposed algorithms are based on our earlier approach to solve in an algorithmic way the admissibility problem of inference rules for S4 and Grz . The first algorithms for (...) the construction of ‘best’ unifiers in the above mentioned logics have been given by S. Ghilardi in [ 16 ]. Both the algorithms in [ 16 ] and ours are not much computationally efficient. They have, however, an obvious significant theoretical value a portion of which seems to be the fact that they stem from two different methodological approaches. (shrink)