Fixed-point properties for predicate modal logics

Abstract

It is well known that the propositional modal logic \(\mathbf{GL}\) of provability satisfies the de Jongh-Sambin fixed-point property. On the other hand, Montagna showed that the predicate modal system \(\mathbf{QGL}\), which is the natural variant of \(\mathbf{GL}\), loses the fixed-point property. In this paper, we discuss some versions of the fixed-point property for predicate modal logics. First, we prove that several extensions of \(\mathbf{QGL}\) including \(\mathbf{NQGL}\) do not have the fixed-point property. Secondly, we prove the fixed-point theorem for the logic \(\mathbf{QK} + \Box^{n+1} \bot\). As a consequence, we obtain that the class \(\mathsf{BL}\) of Kripke frames which are transitive and of bounded length satisfies the fixed-point property locally. We also show that the failure of the Craig interpolation property for \(\mathbf{NQGL}\) follows from our results. Finally, we give a sufficient condition for formulas to have a fixed-point in \(\mathbf{QGL}\).