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. The key to the solution is provided in [1]. Let S4.J 3 be the logic of S4-frames of depth ≤ 3. Define the frames b n = b n , by letting • • • • • • • d d d d d d d d d d d d 0 0 1 0 2 0 2 1 1 1 0 1 3 The picture shows b 3. The diagram or frame formula D(b n) of b n is D(b n) = p s → ¬p t |s = t ∧ p s → ♦p t |s t ∧ p s → ¬♦p t |s t ∧ p s |s ∈ b n ∧ p n