{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T07:57:22Z","timestamp":1775462242524,"version":"3.50.1"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12703,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,6]]},"abstract":"<jats:p>Beth's Definability Theorem, and consequently the Interpolation Lemma, fail for the version of quantified S5 that is presented in Kripke's [6]. These failures persist when the constant domain axiom-scheme \u2200<jats:italic>x<\/jats:italic>\u25a1\u03c6 \u2261 \u25a1\u2200<jats:italic>x<\/jats:italic>\u03c6 is added to S5 or, indeed, to any weaker extension of quantificational <jats:italic>K<\/jats:italic>.<\/jats:p><jats:p>\u00a71 reviews some standard material on quantificational modal logic. This is in contrast to quantified intermediate logics for, as Gabbay [6] has shown, the Interpolation Lemma holds for the logic CD with constant domains and for several of its extensions. \u00a7\u00a72\u20144 establish the negative results for the systems based upon S5. \u00a75 establishes a more general negative result and, finally, \u00a76 considers some positive results and open problems. A basic knowledge of classical and modal quantificational logic is presupposed.<\/jats:p><jats:p>Let me briefly review the relevant model theory for quantified modal logic. Further details can be found in [3] or [7].<\/jats:p><jats:p>The language is obtained from the language for classical first-order logic with identity by adding a unary operator \u25a1 for necessity. The atomic formula \u2018<jats:italic>Ex<\/jats:italic>\u2019 is used as an abbreviation for \u2018\u2203<jats:italic>y(y<\/jats:italic> = <jats:italic>x<\/jats:italic>)\u2019 and may be read as \u2018<jats:italic>x<\/jats:italic> exists\u2019.<\/jats:p>","DOI":"10.2307\/2273727","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:50:09Z","timestamp":1146937809000},"page":"201-206","source":"Crossref","is-referenced-by-count":42,"title":["Failures of the interpolation lemma in quantified modal logic"],"prefix":"10.1017","volume":"44","author":[{"given":"Kit","family":"Fine","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004891X_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF00693269"},{"key":"S002248120004891X_ref005","first-page":"111","volume-title":"Conference in Mathematical Logic, London, 1970, Lecture Notes in Mathematics","author":"Gabbay","year":"1972"},{"key":"S002248120004891X_ref006","first-page":"269","volume":"42","author":"Gabbay","year":"1977","journal-title":"Craig interpolation theorem for intuitionistic logic and extensions, Part III"},{"key":"S002248120004891X_ref003","first-page":"125","article-title":"Model theory for modal logic, Part I","volume":"7","author":"Fine","year":"1978","journal-title":"Journal of Philosophical Logic"},{"key":"S002248120004891X_ref004","article-title":"Model theory for modal logic, Part III","author":"Fine","journal-title":"Journal of Philosophical Logic"},{"key":"S002248120004891X_ref007","first-page":"83","article-title":"Semantical considerations on modal logic","volume":"16","author":"Kripke","year":"1963","journal-title":"Acta Philosophica Fennica"},{"key":"S002248120004891X_ref002","first-page":"416","volume":"39","author":"Czermak","year":"1974","journal-title":"Interpolation theorem for modal logics"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120004891X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T17:20:25Z","timestamp":1558891225000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120004891X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1979,6]]}},"alternative-id":["S002248120004891X"],"URL":"https:\/\/doi.org\/10.2307\/2273727","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,6]]}}}