Formal Interative 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 and beliefs about substantive matters, but also in his knowledge and beliefs about the others' knowledge and beliefs. This paper examines two parallel approaches to the subject. The first is the semantic, in which knowledge and beliefs are represented by a space Ω of states of the world, and for each player i, partitions 𝒫i of Ω and probability distributions πi(·; ω) on Ω for each state ω of the world. The atom of 𝒫i containing a given state ω represents i's knowledge at that state – the set of those other states that i cannot distinguish from ω the probability distributions πi(·; ω) represents i's beliefs at the state ω. The second is the syntactic approach, in which beliefs are 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, the partitions 𝒫i, and the probability distributions πi(·; ω) can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach.
