In this chapter we go beyond the model of knowledge and belief introduced in the previous chapter. Here we look at how one might represent statements such as “Mary believes that it will rain tomorrow with probability > .7,” and even “Bill knows that John believes with probability .9 that Mary believes with probability > .7 that it will rain tomorrow.” We will also look at rules that determine how these knowledge and belief statements can change over time, more broadly at the connection between logic and games, and consider how to formalize the notion of intention.

Knowledge and probability

In a Kripke structure, each possible world is either possible or not possible for a given agent, and an agent knows (or believes) a sentence when the sentence is true in all of the worlds that are accessible for that agent. As a consequence, in this framework both knowledge and belief are binary notions in that agents can only believe or not believe a sentence (and similarly for knowledge). We would now like to add a quantitative component to the picture. In our quantitative setting we will keep the notion of knowledge as is, but will be able to make statements about the degree of an agent's belief in a particular proposition. This will allow us to express not only statements of the form “the agent believes with probability .3 that it will rain” but also statements of the form “agent i believes with probability .3 that agent j believes with probability .9 that it will rain.”