Logical Omniscience in the Many Agent Case

The problem of logical omniscience arises at two levels. One is the individual level, where an agent is assumed to have reasoning powers which are unrealistic. The other, equally important one, is where two or more agents are supposed to share a state of knowledge (perhaps common knowledge) which is read off from a physical situation, but which may not hold in practice. By reducing knowledge to strategies in games (rather than the other way around) we show how to get around this problem. Essentially, we are using the same trick which Ramsey used when he derived subjective probability from an agent’s choices rather than the other way around. Preamble: Consider the following scenario. Ann is sitting on a chair in front of which there is a vase with a dozen roses in it. Bob can see both Ann and the roses. Charlie can see Ann and Bob and the roses. We could now ask: Does Ann know p? where p = There are roses in front of her. I.e., Ka(p) ? Does Bob know that she knows? (KbKa(p) ?) Does Charlie know that Bob knows that Ann knows (KcKbKa(p)) ? Both common sense and the corresponding Kripke structure tell us that the answer to all three questions is yes. Indeed if they can see each other then p is common knowledge among them. Let us now change the meaning of p. In this new example, Ann, Bob and Charlie are all as before, but what is in front of Ann is not a vase of roses, but a blackboard with the number 1243 written on it. Let p now denote the fact that the number on the blackboard is composite. Logically the situation is not changed. Since 1243 is composite (113 times 11), this is a necessary truth, Ann knows it, Bob knows that Ann knows it, and Charlie knows that Bob knows that Ann knows it. But are we sure that this is the case? It could be that Ann finds numbers greater than 100 to be a mystery. Or perhaps she is actually a number theorist but sexist Bob thinks that she is numberchallenged. Or perhaps Bob knows her quite well, but Charlie thinks that Bob is a chauvinist who has a poor opinion of the mathematical abilities of women. So we are no longer sure that Ka(p), KbKa(p) and KcKbKa(p) are all true.