Common theories

By a theory we mean a subset of propositions that is closed under consequences and conjunctions. When the set of all propositions is a Boolean algebra, theories are filters in this algebra. When there are several theorists, they can also have theories about what theories the other theorists subscribe to. The problem is then, if we can find a “complete set of theories”, in the sense that this set contains all theories about the states of nature and about the set of theories itself. We construct such a set of theories with a Boolean algebra of propositions on it. The result is a “complete belief model”, where agents’ beliefs are modeled by filters. We show that this model can be constructed in such a way that each possible theory corresponds to a hierarchy of theories. The first order theories are about the state of nature, the second order theories are about the state of nature and the first order theories, and so on.