Not every ``tabular'' predicate logic is finitely axiomatizable (Q1378430)

A well-known result in propositional intermediate logic is the theorem (by McKay-De Jongh-Hosoi) saying that every tabular logic (i.e. the logic of a finite Kripke frame) is finitely axiomatizable. The paper shows that an analogue of this result for predicate logics is not true. Namely, for a Kripke frame (p.o. set) \(M\) let \({\mathbf L}M\) be the intersection of the predicate logics \({\mathbf L}(M,U)\), where \((M,U)\) is an arbitrary predicate frame with varying domains over \(M\). It is proved that for a very simple 5-element tree \(M_0\) the logic \({\mathbf L}M_0\) is not finitely axiomatizable; an infinite axiomatization of this logic is presented. The proof essentially uses Ghilardi's functor semantics of modal and intermediate predicate logics.