A Theory of Knowledge and Ignorance for Many Agents

We extend the notion of \only knowing" introduced by Halpern and Moses 11] to many agents and to a number of modal logics. In this approach, \all an agent knows is " is true in a structure M if, in M, the agent knows and has a maximum set of \possibilities". To extend this approach, we need to make precise what counts as a \possibility". In the single-agent case, we can identify a possibility with a truth assignment. In the multi-agent case, things are more complicated. We consider three notions of possibility (all related). We argue that the rst is most appropriate for non-introspective logics, such as Kn, Tn, and S4n, the second is most appropriate for K45n and KD45n, and the last is most appropriate for S5n. With the appropriate notion of possibility, we show that are reasonable extensions in all cases. Our results also shed light on the single-agent case. It was always assumed that one of the key aspects of Halpern-Moses approach in the single-agent case was its use of S5, rather than K45 or KD45. Our results show that the notion is better understood in the context of K45 (or KD45). In the single-agent case, the notion remains unchanged if we use K45 instead of S5. However, in the multi-agent case, there are signiicant diierences between K45 and S5. Moreover, in some sense, the K45 variants behave better: all results proved for the single-agent case extend more naturally to the multi-agent case of K45 than to the multi-agent case of S5. This paper is essentially identical to one that will appear in Journal of Logic and Computation. The United States Government is authorized to reproduce and distribute reprints for governmental purposes. Some of the material in this paper appeared in preliminary form in a paper entitled \Reasoning about only knowing with many agents" that appeared