Modal logics of answers

This paper aims to apply modal logics arising from partitionistic structures to the logic of questions and answers. Taking it for granted that a partition of possible worlds serves as a representation of a context, the idea of multi-contextual situation motivates the proposal of partitionistic structures. Logical properties of answers are represented by modal operators on such structures. 1 Motivating examples 1.1 Twenty questions Twenty questions is a two-player game to find what the answerer has in mind by posing at most twenty yes-no questions. In this example, suppose that the range of possible answers is a set of numbers from 1 to 1,000,000. Since a million is a bit less than 2, a wise questioner must be able to find the correct answer. How can she reach it? Questioners may or may not be wise. For example, a questioner may ask “Can it fly?” to lose a precious opportunity; the question could yield no meaningful answer. Another bad way is to ask “Is the number less than 500,000?” in the first turn and get the answer yes, and then to ask “Is it more than 750,000?” Obviously, the answer is no, and he will lose a question without obtaining any information. Another poor way is to repeat a question. Suppose the question “Is the number less than 500,000?” in the first turn gets the answer yes. Repeating the same question “Is the number less than 500,000?” would obtain the answer yes, but it does not help the questioner to squeeze the range of possibilities any more. Now, it is natural for logicians to ask the following questions. What logical properties do such poor ways of asking questions bear? Can they be characterized? Let us return to focus on a wise questioner. A systematic strategy is to cut the range in the half in each round. For example, asking “Is the number less than 500,001?” in the first turn submits a binary partition of the original range: