{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T18:37:10Z","timestamp":1648924630528},"reference-count":5,"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":15625,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1971,6]]},"abstract":"<jats:p>A second order formula is called \u03a0<jats:sub>1<\/jats:sub> if, in its prenex normal form, all second order quantifiers are universal. A sequent <jats:italic>F<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026 <jats:italic>F<jats:sub>m<\/jats:sub><\/jats:italic> \u2192 <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> \u2026, <jats:italic>G<jats:sub>n<\/jats:sub><\/jats:italic> is called \u03a0<jats:sub>1<\/jats:sub> if a 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=\"S0022481200083249_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>is \u03a0<jats:sub>1<\/jats:sub><\/jats:p><jats:p>If we consider only \u03a0<jats:sub>1<\/jats:sub> sequents, then we can easily generalize the completeness theorem for the cut-free first order predicate calculus to a cut-free \u03a0<jats:sub>1<\/jats:sub> predicate calculus.<\/jats:p><jats:p>In this paper, we shall prove two interpolation theorems on the \u03a0<jats:sub>1<\/jats:sub> sequent, and show that Chang's theorem in [2] is a corollary of our theorem. This further supports our belief that any form of the interpolation theorem is a corollary of a cut-elimination theorem. We shall also show how to generalize our results for an infinitary language. Our method is proof-theoretic and an extension of a method introduced in Maehara [5]. The latter has been used frequently to prove the several forms of the interpolation theorem.<\/jats:p>","DOI":"10.2307\/2270261","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:07:49Z","timestamp":1146949669000},"page":"262-270","source":"Crossref","is-referenced-by-count":0,"title":["Two interpolation theorems for a  predicate calculus"],"prefix":"10.1017","volume":"36","author":[{"given":"Shoji","family":"Maehara","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gaisi","family":"Takeuti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200083249_ref004","doi-asserted-by":"publisher","DOI":"10.4064\/fm-57-3-253-272"},{"key":"S0022481200083249_ref003","unstructured":"Kueker D. W. , Generalized interpolation and definability (to appear in Annals of Mathematical Logic)."},{"key":"S0022481200083249_ref005","first-page":"235","volume-title":"S\u016dgaku","author":"Maehara","year":"1961"},{"key":"S0022481200083249_ref001","first-page":"934","article-title":"A generalization of the Craig interpolation theorem","volume":"15","author":"Chang","year":"1968","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200083249_ref002","first-page":"17","volume-title":"Proceedings of a conference in model theory","author":"Chang","year":"1969"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200083249","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T20:05:32Z","timestamp":1559333132000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200083249\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1971,6]]},"references-count":5,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1971,6]]}},"alternative-id":["S0022481200083249"],"URL":"https:\/\/doi.org\/10.2307\/2270261","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1971,6]]}}}