The author is interested in modal logics which can be interpreted as epistemic logics with “multiple agents, so that we consider not only an agent’s belief about ‘nature’, but also his beliefs about beliefs of other agents” (209). For this purpose, standard semantics for modal logic along Kripkean lines would have to be generalized so that the accessibility-relation \(R\) is replaced by a family of relations \(R_ i\), where \(wR_ i w^*\) may be read as ‘in \(w\), agent \(a_ i\) considers \(w^*\) as possible’. However, the use of a Kripke-style semantics for epistemic logic is held to be problematic because “the notion of a possible world is a primitive notion”, and in epistemic logic “it is not clear what a possible world is” (210). Therefore the author has elsewhere developed an alternative semantics along the following lines: “At the bottom, or \(O\) th level, is a truth-assignment, which can be thought of as a state of nature. The next level describes each agent’s beliefs about nature, which corresponds to a set of truth assignments. Intuitively, these are the states of nature that the agent considers possible. The \((k+1)\)st-order belief of each agent is modeled by a set of possibilities, each of which is a description of a state of nature and each agent’s \(k\) th-order belief. […] A modal structure consists of a state of nature and each agent’s hierarchy of beliefs.” (210)
Given this framework, one can imagine situations where the agent’s beliefs no longer have a countable description, i.e. where it becomes necessary to extend the hierarchy of the modal structure “to levels \(\omega\), \(\omega+1\), and beyond”. The detailed paper offers a “quantitative” analysis of modal structures (and corresponding Kripke structures \(M\)) by studying, among others, “the least ordinal \(\mu\) such that for each state \(M\), the beliefs up to level \(\mu\) characterize the agents’ beliefs (that is, there is only one way to extend these beliefs to higher levels)”.