{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T23:07:30Z","timestamp":1654988850723},"reference-count":3,"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":12064,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1981,3]]},"abstract":"<jats:p>In Gabbay [1] it is stated as an open problem whether or not Craig's Theorem holds for the logic of constant domains <jats:bold>CD<\/jats:bold>, i.e. for the extension of the intuitionistic predicate calculus, <jats:bold>IPC<\/jats:bold>, obtained by adding the schema; <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046089_inline1\" \/>. Then in the later article, [2], Gabbay gives a proof of it. The proof given in [2] is via Robinson's (weak) consistency theorem and depends on relatively complicated (Kripke-) model-theoretical constructions developed in [1] (see p. 392 of [1] for a brief sketch of the method). The aim of this note is to show that the interpolation theorem for <jats:bold>CD<\/jats:bold> can also be obtained, by simple proof-theoretic methods, from \u00a780 of Kleene's Introduction to Metamathematics [3].<\/jats:p><jats:p><jats:bold>GI<\/jats:bold> is the classical formal system whose postulates are given on p. 442 of [3]. Let <jats:bold>GD<\/jats:bold> be the system obtained from <jats:bold>GI<\/jats:bold> by the following modifications: (1) the sequents of <jats:bold>GD<\/jats:bold> are to have at most two formulas in their succedents and (2) the intuitionistic restriction that <jats:italic>\u0398<\/jats:italic> be empty is required for the succedent rules (\u2192 \u00ac) and (\u2192 \u2283). It is a simple matter to show that: <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046089_inline2\" \/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046089_inline3\" \/>, <jats:italic>x<\/jats:italic> not free in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046089_inline4\" \/>. It then follows that, using Theorem 46 of [3], if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046089_inline5\" \/> then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200046089_inline6\" \/>.<\/jats:p>","DOI":"10.2307\/2273260","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:54:55Z","timestamp":1146938095000},"page":"87-88","source":"Crossref","is-referenced-by-count":5,"title":["On the interpolation theorem for the logic of constant domains"],"prefix":"10.1017","volume":"46","author":[{"given":"E. G. K.","family":"L\u00f3pez-Escobar","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200046089_ref002","first-page":"269","volume":"42","author":"Gabbay","year":"1977","journal-title":"Craig interpolation theorem for intuitionistic logic and extensions. Part III"},{"key":"S0022481200046089_ref003","volume-title":"Introduction to metamathematics","author":"Kleene","year":"1952"},{"key":"S0022481200046089_ref001","first-page":"403","volume-title":"Logic Colloquium '69","author":"Gabbay","year":"1970"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200046089","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,25]],"date-time":"2019-05-25T16:41:28Z","timestamp":1558802488000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200046089\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1981,3]]},"references-count":3,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1981,3]]}},"alternative-id":["S0022481200046089"],"URL":"https:\/\/doi.org\/10.2307\/2273260","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1981,3]]}}}