{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T15:47:03Z","timestamp":1762444023934},"reference-count":0,"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>Consider modal propositional formulae, constructed using proposition-letters, connectives and the modal operators \u25a1 and \u22c4. The semantic structures are <jats:italic>frames<\/jats:italic>, i.e., pairs &lt;<jats:italic>W, R<\/jats:italic>&gt; with <jats:italic>R<\/jats:italic> \u2286 <jats:italic>W<\/jats:italic><jats:sup>2<\/jats:sup>. Let <jats:italic>F, V<\/jats:italic> be variables ranging respectively over frames and functions from the set of proposition-letters into the powerset of <jats:italic>W<\/jats:italic>. Then the relation<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200054268_eqnU01\" \/><\/jats:disp-formula><\/jats:p><jats:p>may be defined, for arbitrary formulae \u03b1, following the Kripke truth-definition. From this relation we may further define<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200054268_eqnU02\" \/><\/jats:disp-formula><\/jats:p><jats:p>Now, to every modal formula \u03b1 there corresponds some property <jats:italic>P<\/jats:italic><jats:sub>\u03b1<\/jats:sub> of <jats:italic>R<\/jats:italic>. A particular example is obtained by considering the well-known translation of modal formulae into formulae of monadic second-order logic with a single binary first-order predicate. For these particular <jats:italic>P<\/jats:italic><jats:sub>\u03b1<\/jats:sub> we have<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200054268_eqnU03\" \/><\/jats:disp-formula><\/jats:p><jats:p>for all <jats:italic>F<\/jats:italic> and <jats:italic>w<\/jats:italic> \u2208 <jats:italic>W<\/jats:italic>. These formulae <jats:italic>P<\/jats:italic><jats:sub>\u03b1<\/jats:sub> are, however, rather intractable and more convenient ones can often be found. An especially interesting case occurs when <jats:italic>P<jats:sub>\u03b1<\/jats:sub><\/jats:italic> may be taken to be some first-order formula. For example, it can be seen that<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200054268_eqnU04\" \/><\/jats:disp-formula><\/jats:p><jats:p>for all <jats:italic>F<\/jats:italic> and <jats:italic>w<\/jats:italic> \u2208 <jats:italic>W<\/jats:italic>. It is customary to talk about a related correspondence, namely when for all <jats:italic>F<\/jats:italic> we have<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200054268_eqnU05\" \/><\/jats:disp-formula><\/jats:p><jats:p>Note that this correspondence holds whenever the first one above holds.<\/jats:p>","DOI":"10.2307\/2272270","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:33:55Z","timestamp":1146951235000},"page":"55-58","source":"Crossref","is-referenced-by-count":16,"title":["A note on modal formulae and relational properties"],"prefix":"10.1017","volume":"40","author":[{"given":"J. F. A. K.","family":"van Benthem","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200054268","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T20:06:13Z","timestamp":1559160373000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054268\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,3]]},"references-count":0,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1975,3]]}},"alternative-id":["S0022481200054268"],"URL":"https:\/\/doi.org\/10.2307\/2272270","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}