Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or “player.” One is interested not only in each person's knowledge about substantive matters, but also in his knowledge about the others' knowledge. This paper examines two parallel approaches to the subject. The first is the semantic approach, in which knowledge is represented by a space Ω of states of the world, together with partitions ℐi of Ω for each player i; the atom of ℐi containing a given state ω of the world represents i's knowledge at that state – the set of those other states that i cannot distinguish from ω. The second is the syntactic approach, in which knowledge is embodied in sentences constructed according to certain syntactic rules. This paper examines the relation between the two approaches, and shows that they are in a sense equivalent.
In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space Ω and the partitions ℐi can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach.
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