Normal forms in partial modal logic

Abstract

A “partial” generalization of Fine’s definition [Fin] of normal forms in normal minimal modal logic is given. This means quick access to complete axiomatizations and decidability proofs for partial modal logic [Thi]. Introduction. From the viewpoint of formal linguistics, cognitive science and artificial intelligence, there appears to be a natural interest in the combinations of partiality and modality [BaP] [Kam] [FaH]. Modal operators cover the uncertain status of the informational content of propositional attitudes. Partiality is the formal translation of the incomplete behavior of these attitudes. The sources of uncertainty, multiple possible worlds (interpretations), do not have a two-valued character such as in classical modal logic. We take these possible worlds to be partial. They do not necessarily assign a Boolean truth value to all propositions. In this paper we will define normal forms for partial modal logic, which is induced by a “partial” generalization of Fine’s definition of normal forms in classical (total) modal logic [Fin]. It turns out that these normal forms for partial modal logic are more difficult to handle in proving that all formulae are equivalent to a disjunction of normal forms. This bad behavior can be explained by taking normal forms to be full modal descriptions of worlds. In total possible world semantics mutually coherent normal forms must be the same for this reason. In the partial semantics this is not the case, and therefore the conjunction of two different normal forms does not guarantee inconsistency. In partial propositional logic it is guaranteed that two coherent normal forms produce a new unique normal 1991 Mathematics Subject Classification: 03B45. This research is supported by the programme “Dialogue management and knowledge acquisition” (DenK) of the Tilburg–Eindhoven Organization for Inter-University Cooperation (SOBU). The paper is in final form and no version of it will be published elsewhere.