{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T03:37:12Z","timestamp":1649216232142},"reference-count":20,"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":12611,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,9]]},"abstract":"<jats:p>The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the form<\/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=\"S0022481200048210_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where the indices <jats:italic>k<jats:sub>i<\/jats:sub>, j<jats:sub>i<\/jats:sub><\/jats:italic> range over given countable sets and the matrix conjuncts are, say, open <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048210_inline1\" \/>-formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]\u2014[15] and Vaught [20]. Other references are [1], [3]\u2014[5], [10]\u2014[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).<\/jats:p><jats:p>Theorem 0.1. <jats:italic>Let L<\/jats:italic> = <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub>(<jats:bold>R<\/jats:bold>). <jats:italic>For every <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048210_inline1\" \/>-sentence \u03d5 there is an L<jats:sub>0<\/jats:sub>-game sentence \u0398 such that<\/jats:italic> \u22a8\u2032 \u2203<jats:bold>R<\/jats:bold>\u03d5 \u2194 \u0398.<\/jats:p><jats:p>(A word about the notations: <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub>(<jats:bold>R<\/jats:bold>) denotes the language obtained from <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub> by adding to it the sequence <jats:bold>R<\/jats:bold> of logical symbols which do not belong to <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub>; \u201c\u22a8\u2032\u03b1\u201d means that \u03b1 is true in all countable models.)<\/jats:p><jats:p>0.1 can be restated as follows.<\/jats:p><jats:p>Theorem 0.1\u2032. <jats:italic>For every<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048210_inline1\" \/>-<jats:italic>sentence<\/jats:italic> \u03d5 <jats:italic>there is an L<\/jats:italic><jats:sub>0<\/jats:sub>-<jats:italic>game sentence<\/jats:italic> \u0398 <jats:italic>such that<\/jats:italic> \u22a8\u03d5 \u2192 \u0398 <jats:italic>and for any<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048210_inline1\" \/>-<jats:italic>sentence<\/jats:italic> \u03d5 <jats:italic>if<\/jats:italic> \u22a8\u03d5 \u2192 \u03d5 <jats:italic>and L<\/jats:italic>\u2032 \u22c2 <jats:italic>L<\/jats:italic> \u2286 <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>then<\/jats:italic> \u22a8 \u0398 \u2192 \u03d5.<\/jats:p><jats:p>(We sketch the proof of the equivalence between 0.1 and 0.1\u2032.<\/jats:p><jats:p>0.1 <jats:italic>implies<\/jats:italic> 0.1\u2032. This is obvious once we realize that game sentences and their negations satisfy the downward L\u00f6wenheim-Skolem theorem and hence, \u22a8\u2032\u03b1 is equivalent to \u22a8\u03b1 whenever \u03b1 is a boolean combination of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048210_inline1\" \/> and game sentences.<\/jats:p>","DOI":"10.2307\/2273122","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:50:56Z","timestamp":1146937856000},"page":"289-306","source":"Crossref","is-referenced-by-count":1,"title":["Refinements of Vaught's normal from theorem"],"prefix":"10.1017","volume":"44","author":[{"given":"Victor","family":"Harnik","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048210_ref020","first-page":"574","volume-title":"Lecture Notes in Mathematics","author":"Vaught","year":"1973"},{"key":"S0022481200048210_ref014","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1972-0286662-3"},{"key":"S0022481200048210_ref011","first-page":"622","volume-title":"Lecture Notes in Mathematics","author":"Makkai","year":"1973"},{"key":"S0022481200048210_ref001","volume-title":"Perspectives in mathematical logic","author":"Barwise","year":"1975"},{"key":"S0022481200048210_ref015","volume-title":"Elementary induction on abstract structures","author":"Moschovakis","year":"1974"},{"key":"S0022481200048210_ref003","author":"Harnik","journal-title":"Came sentences, recursive saturation and definability"},{"key":"S0022481200048210_ref008","doi-asserted-by":"publisher","DOI":"10.4064\/fm-57-3-253-272"},{"key":"S0022481200048210_ref009","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1959.9.129"},{"key":"S0022481200048210_ref016","first-page":"271","volume":"33","author":"Oberschelp","year":"1968","journal-title":"On the Craig-Lyndon interpolation theorem"},{"key":"S0022481200048210_ref017","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90009-2"},{"key":"S0022481200048210_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71103-0"},{"key":"S0022481200048210_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/fm-73-3-219-233"},{"key":"S0022481200048210_ref005","article-title":"New axiomatizations for logics with generalized quantifiers","author":"Harnik","journal-title":"Israel Journal of Mathematics"},{"key":"S0022481200048210_ref002","first-page":"250","volume":"22","author":"Craig","year":"1957","journal-title":"Linear reasoning. A new form of the Herbrand-Gentzen theorem"},{"key":"S0022481200048210_ref006","first-page":"201","volume":"28","author":"Henkin","year":"1963","journal-title":"An extension of the Craig-Lyndon interpolation theorem"},{"key":"S0022481200048210_ref004","first-page":"171","volume":"41","author":"Harnik","year":"1976","journal-title":"Applications of Vaught sentences and the covering theorem"},{"key":"S0022481200048210_ref007","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200048210_ref019","first-page":"376","volume-title":"The theory of models","author":"Svenonius","year":"1965"},{"key":"S0022481200048210_ref018","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-86718-7"},{"key":"S0022481200048210_ref013","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-70-03744-0"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048210","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T16:52:56Z","timestamp":1558889576000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048210\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,9]]},"references-count":20,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1979,9]]}},"alternative-id":["S0022481200048210"],"URL":"https:\/\/doi.org\/10.2307\/2273122","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,9]]}}}