Finite and Infinite Dialogues

We consider dialogues over both specific and general Kripke structures where individuals convey and acquire knowledge through statements. We show that conventional ‘proofs’ of the existence of knowledge actually correspond to optimal strategies (which may not occur when real individuals talk). Sometimes these optimal, (and hence all) strategies need to be transfinite in that knowledge can only be acquired at some infinite ordinal. However, the situation changes sharply when we consider dialogues geared not to acquiring knowledge but to taking a justified risk.