About The Unification Type Of Fusions Of Modal Logics

In a modal logic L, a unifier of a formula phi is a substitution sigma such that sigma(phi) is in L. When unifiable formulas have no minimal complete sets of unifiers, they are nullary. Otherwise, they are either infinitary, or finitary, or unitary depending on the cardinality of their minimal complete sets of unifiers. The fusion L-1 circle times L-2 of modal logics L-1 and L-2 respectively based on the modal connectives square(1) and square(2) is the least modal logic based on these modal connectives and containing both L-1 and L-2. In this paper, we prove that if L-1 circle times L-2 is unitary then L-1 and L-2 are unitary and if L-1 circle times L-2 is finitary then L-1 and L-2 are either unitary, or finitary. We also prove that the fusion of arbitrary consistent extensions of S5 is nullary when these extensions are different from Triv(1).