{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T04:46:45Z","timestamp":1787719605007,"version":"build-2784847793"},"reference-count":7,"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":14256,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,3]]},"abstract":"<jats:p>In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited\u2014there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic <jats:italic>can<\/jats:italic> be handled in a first-order way, and precisely which modal sentences <jats:italic>are<\/jats:italic> determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed <jats:italic>LMp<\/jats:italic> \u2192 <jats:italic>MLp<\/jats:italic>.<\/jats:p><jats:p>Definition 1. A modal <jats:italic>frame<\/jats:italic> \u2131 = \u3008<jats:italic>W, R<\/jats:italic>\u3009 consists of a set <jats:italic>W<\/jats:italic> on which a binary relation <jats:italic>R<\/jats:italic> is defined. A valuation <jats:italic>V<\/jats:italic> on \u2131 is a function that associates with each propositional variable <jats:italic>p<\/jats:italic> a subset <jats:italic>V(p)<\/jats:italic> of <jats:italic>W<\/jats:italic> (the set of points at which <jats:italic>p<\/jats:italic> is \u201ctrue\u201d).<\/jats:p>","DOI":"10.2307\/2272267","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:33:55Z","timestamp":1146936835000},"page":"35-40","source":"Crossref","is-referenced-by-count":21,"title":["First-order definability in modal logic"],"prefix":"10.1017","volume":"40","author":[{"given":"R. I.","family":"Goldblatt","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200054232_ref002","volume-title":"Theoria","author":"Fine"},{"key":"S0022481200054232_ref001","volume-title":"Models and ultraproducts","author":"Bell","year":"1969"},{"key":"S0022481200054232_ref006","volume-title":"Theoria","author":"Thomason"},{"key":"S0022481200054232_ref003","volume-title":"Intensional logic","author":"Lemmon","year":"1966"},{"key":"S0022481200054232_ref004","volume-title":"Non-standard analysis","author":"Robinson","year":"1966"},{"key":"S0022481200054232_ref005","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1968.tb00335.x"},{"key":"S0022481200054232_ref007","first-page":"150","volume":"37","author":"Thomason","year":"1972","journal-title":"Semantic analysis of tense logics"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200054232","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T16:05:45Z","timestamp":1559145945000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054232\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1975,3]]}},"alternative-id":["S0022481200054232"],"URL":"https:\/\/doi.org\/10.2307\/2272267","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}