A solution to a problem of urquhart

In [4], Urquhart gives an example of a modal logic which has the finite model property, is recursively axiomatizable but not decidable. He poses the problem of finding such a logic containing S4. We will solve this problem by constructing a logic which in addition to the required properties also defines a locally finite variety of modal algebras. (Recall that a variety is said to be locally finite if all finitely generated algebras in that variety are finite.) This also gives an example of a locally finite variety V such that Eq(V) is recursively enumerable but not recursive.