# Fixed-point properties for predicate modal logics

## TL;DR

This paper investigates fixed-point properties in predicate modal logics, showing certain systems lack fixed points while others, under specific conditions, do possess them, and explores related properties like Craig interpolation.

## Contribution

It demonstrates the absence of fixed-point property in several predicate modal logic extensions and establishes fixed-point theorems under particular conditions.

## Key findings

- Extensions like $ extbf{NQGL}$ lack fixed-point property.
- Logic $ extbf{QK} + oxdot^{n+1} ot$ has fixed points.
- Kripke frames of finite transitive height satisfy fixed-point property.

## 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 $\mathsf{FH}$ of Kripke frames which are transitive and finite height satisfies the fixed-point property locally. We also show the failure of the Craig interpolation property for $\mathbf{NQGL}$. Finally, we give a sufficient condition for formulas to have a fixed-point in $\mathbf{QGL}$.