{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,21]],"date-time":"2025-12-21T10:23:42Z","timestamp":1766312622740},"reference-count":28,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7316,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We do a quantitative analysis of modal logic. For example, for each Kripke structure <jats:italic>M<\/jats:italic>, we study the least ordinal \u03bc such that for each state of <jats:italic>M<\/jats:italic>, the beliefs up to level \u03bc characterize the agents' beliefs (that is, there is only one way to extend these beliefs to higher levels). As another example, we show the equivalence of three conditions, that on the face of it look quite different, for what it means to say that the agents' beliefs have a countable description, or putting it another way, have a \u201ccountable amount of information\u201d. The first condition says that the beliefs of the agents are those at a state of a countable Kripke structure. The second condition says that the beliefs of the agents can be described in an infinitary language, where conjunctions of arbitrary countable sets of formulas are allowed. The third condition says that countably many levels of belief are sufficient to capture all of the uncertainty of the agents (along with a technical condition). The fact that all of these conditions are equivalent shows the robustness of the concept of the agents' beliefs having a \u201ccountable description\u201d.<\/jats:p>","DOI":"10.2307\/2275262","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:51:08Z","timestamp":1146941468000},"page":"209-252","source":"Crossref","is-referenced-by-count":15,"title":["A quantitative analysis of modal logic"],"prefix":"10.1017","volume":"59","author":[{"given":"Ronald","family":"Fagin","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200020351_ref028","volume-title":"An essay in classical modal logic","author":"Segerberg","year":"1971"},{"key":"S0022481200020351_ref027","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200020351_ref026","first-page":"109","volume-title":"Proceedings of the 17th IEEE symposium on 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