A QUANTITATIVE-ANALYSIS OF MODAL LOGIC

We do a quantitative analysis of modal logic. For example. for each Kr ipke structure M, we study the least ordinal mu such that for each sta te of M, the beliefs up to level mu characterize the agents' beliefs ( that is, there is only one way to extend these beliefs to higher level s). As another example, we show the equivalence of three conditions, t hat on the face of it look quite different. for what it means to say t hat the agents' beliefs have a countable description, or putting it an other way, have a ''countable amount of information''. The first condi tion says that the beliefs of the agents are those at a state of a cou ntable Kripke structure. The second condition says that the beliefs of the agents can be described in an infinitary language, where conjunct ions of arbitrary countable sets Of formulas are allowed. The third co ndition says that countably many levels of belief are sufficient to ca pture all of the uncertainty of the agents (along with a technical con dition). The fact that all of these conditions are equivalent shows th e robustness of the concept of the agents' beliefs having a ''countabl e description''.