A General Semantics for Quantified Modal Logic

In [9] we developed a semantics for quantified relevant logic that uses general frames. In this paper, we adapt that model theory to treat quantified modal logics, giving a complete semantics to the quantified extensions, both with and without the Barcan formula, of every propositional modal logic S. If S is canonical our models are based on propositional frames that validate S. We employ frames in which not every set of worlds is an admissible proposition, and an alternative interpretation of the universal quantifier using greatest lower bounds in the lattice of admissible propositions. Our models have a fixed domain of individuals, even in the absence of the Barcan formula. For systems with the Barcan formula it is possible to preserve the usual Tarskian reading of the quantifier, at the expensive of sometimes losing validity of S in the underlying propositional frames. We apply our results to a number of logics, including S4.2, S4M and KW, whose quantified extensions are incomplete ...

• Type: Conference
• Year: 2006
• Where: AIML
• Authors: Robert Goldblatt, Edwin D. Mares
• Keywords: AIML 2006, AIML 2008, Barcan Formula, Propositional Frames, Quantified Relevant Logic