A semantic approach to nonmonotonic reasoning: Inference operations and choice

Abstract

This paper presents a uniform semantic treatment of nonmonotonic inference operations that allow for inferences from infinite sets of premisses. The semantics is formulated in terms of selection functions and is a generalisation of the preferential semantics of Shoham, Kraus et al., and Makinson. A selection function picks out from a given set of possible states (worlds, situations, models) a subset consisting of those states that are, in some sense, the most preferred ones. A proposition (Formula presented.) is a nonmonotonic consequence of a set of propositions (Formula presented.) iff (Formula presented.) holds in all the most preferred (Formula presented.) -states. In the literature on revealed preference theory, there are a number of well-known theorems concerning the representability of selection functions, satisfying certain properties, in terms of underlying preference relations. Such theorems are utilised here to give corresponding representation theorems for nonmonotonic inference operations. At the end of the paper, the connection between nonmonotonic inference and belief revision, in the sense of Alchourrón, Gärdenfors, and Makinson, is explored. In this connection, infinitary belief revision operations, that allow for the revision of a theory with a possibly infinite set of propositions, are introduced and characterised axiomatically. Several semantic representation theorems are proved for operations of this kind.