Ideal and real belief about belief

The goal of this paper is to provide a formalization of monotonic belief and belief about belief in a multiagent environment. We distinguish between ideal beliefs, i.e. those beliefs which satisfy certain 'idealized' properties which are unlikely to be possessed by real agents, and real beliefs. Our formalization is based on a set-theoretic specification of beliefs and, then, on the definition of the appropriate constructors which intentionally present the sets identified. This allows us to provide a uniform and taxonomic characterization of the possible ways in which ideal and real beliefs can arise. We compare our notion of ideal with the notion of logical omniscience from the modal literature, and show that the first is much weaker and more granular than the second. We provide intuitions about the conceptual importance of the cases analysed by proving and discussing some equivalence results with some important modal systems modelling (non) logical omniscience.