{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T13:25:14Z","timestamp":1708953914324},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11150,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,9]]},"abstract":"<jats:p>It was brought to our attention by M. Fitting that Beth's semantic tableau system using the intuitionistic propositional rules and the classical quantifier rules produces a correct but incomplete axiomatization of the logic <jats:italic><jats:bold>CD<\/jats:bold><\/jats:italic> of constant domains. The formula<\/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=\"S0022481200037762_Uequ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where <jats:italic>T<\/jats:italic> is a truth constant, being an instance of a formula which is valid in all Kripke models with constant domains but which is not provable without cut.<\/jats:p><jats:p>From the Fitting formula one can immediately obtain that the sequent<\/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=\"S0022481200037762_equ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>although provable in the system <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic> outlined in [3], does not have a cut-free proof (in the system <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>).<\/jats:p><jats:p>If the only problem with <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic> were the sequent <jats:italic>S<\/jats:italic><jats:sub>0<\/jats:sub>, then we could extend <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic> to the system <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>+ by adding the following (correct) rule:<\/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=\"S0022481200037762_Uequ2\" \/><\/jats:disp-formula><\/jats:p><jats:p>Since the new rule still satisfies the subformula property a cut elimination theorem for <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>+ would be a step in the right direction for a syntactical proof for the interpolation theorem for the logic of constant domains (cf. Gabbay [2]; see also \u00a74). Unfortunately, one can show that the sequent<\/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=\"S0022481200037762_equ2\" \/><\/jats:disp-formula><\/jats:p><jats:p>where <jats:italic>P<\/jats:italic> is a propositional parameter (or formula without <jats:italic>x<\/jats:italic> free) has a derivation in <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>+, but does not have a cut-free derivation (in <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>+). Of course, we could extend <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>+ to <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>++ by adding the following correct (and with the subformula property) rule:<\/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=\"S0022481200037762_Uequ3\" \/><\/jats:disp-formula><\/jats:p><jats:p>But then we can find a sequent <jats:italic>S<\/jats:italic><jats:sub>2<\/jats:sub> which, although provable with cut in <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>++, does not have a cut-free derivation in <jats:italic><jats:bold>GD<\/jats:bold><\/jats:italic>++.<\/jats:p>","DOI":"10.2307\/2273451","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:04:45Z","timestamp":1146953085000},"page":"595-599","source":"Crossref","is-referenced-by-count":6,"title":["A second paper \u201cOn the interpolation theorem for the logic of constant domains\u201d"],"prefix":"10.1017","volume":"48","author":[{"given":"E.G.K.","family":"L\u00f3pez-Escobar","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200037762_ref003","first-page":"87","volume":"46","author":"Lopez-Escobar","year":"1981","journal-title":"On the interpolation theorem for the logic of constant domains"},{"key":"S0022481200037762_ref002","first-page":"269","volume":"42","author":"Gabbay","year":"1977","journal-title":"Craig interpolation theorem for intuitionistic logic and extensions. III"},{"key":"S0022481200037762_ref001","first-page":"467","volume-title":"Elements of intuitionism","author":"Dummett","year":"1977"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200037762","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T22:37:15Z","timestamp":1558651035000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200037762\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,9]]},"references-count":3,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1983,9]]}},"alternative-id":["S0022481200037762"],"URL":"https:\/\/doi.org\/10.2307\/2273451","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,9]]}}}