An Axiomatic Approach to the Logical Omniscience Problem

Abstract

Standard models of knowledge have the unrealistic property that agents are logically omniscient in the sense that they know all logical implications of their information. While many nonstandard logics have been proposed to avoid this problem, none has an obvious claim as the “right” logic to use. I show how to derive such a logic as part of a representation of an agent's preferences. In this sense, the agent's logic is given the same basis as a utility function or subjective probability. I provide necessary and sufficient conditions for a given logic to be part of a representation of preferences. Unfortunately, the conditions are not easily interprettable in general. To illustrate them further, I summarize some of my earlier results (Lipman [1993a]) on when the agent's logic is a version of the logic of inconsistency proposed by Rescher and Brandom [1979]. I also discuss the difficulties of representing an agent as using Levesque's logic of implicit belief (Levesque [1984]) or some form of resource-bounded computation.