Laws, Facts, and Contexts: Foundations for Multimodal Reasoning

Abstract

Leibniz’s intuition that necessity corresponds to truth in all possible worlds enabled Kripke to define a rigorous model theory for several axiomatizations of modal logic. Unfortunately, Kripke’s model structures lead to a combinatorial explosion when they are extended to all the varieties of modality and intentionality that people routinely use in ordinary language. As an alternative, any semantics based on possible worlds can be replaced by a simpler and more easily generalizable approach based on Dunn’s semantics of laws and facts and a theory of contexts based on the ideas of Peirce and McCarthy. To demonstrate consistency, this article defines a family of nested graph models, which can be specialized to a wide variety of model structures, including Kripke’s models, situation semantics, temporal models, and many variations of them. An important advantage of nested graph models is the option of partitioning the reasoning tasks into separate metalevel stages, each of which can be axiomatized in classical first-order logic. At each stage, all inferences can be carried out with well-understood theorem provers for FOL or some subset of FOL. To prove that nothing more than FOL is required, Section 6 shows how any nested graph model with a finite nesting depth can be flattened to a conventional Tarski-style model. For most purposes, however, the nested models are computationally more tractable and intuitively more understandable.