{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T04:06:47Z","timestamp":1770610007278,"version":"3.49.0"},"reference-count":4,"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":10054,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,9]]},"abstract":"<jats:p>The aim of the paper is to prove the completeness theorem for biprobability models. This also solves Keisler's Problem 5.4 (see [4]).<\/jats:p><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline1\"\/> be a countable admissible set and <jats:italic>\u03c9<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline1\"\/>. The logic <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline2\"\/> is similar to the standard probability logic <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline3\"\/>. The only difference is that two types of probability quantifiers <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline4\"\/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline5\"\/> are allowed.<\/jats:p><jats:p>A biprobability model is a structure (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline6\"\/>, <jats:italic>\u03bc<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>\u03bc<\/jats:italic><jats:sub>2<\/jats:sub>) where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline7\"\/> is a classical structure without operations and <jats:italic>\u03bc<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>\u03bc<\/jats:italic><jats:sub>2<\/jats:sub> are two types of probability measures such that <jats:italic>\u03bc<\/jats:italic><jats:sub>1<\/jats:sub> is absolutely continuous with respect to <jats:italic>\u03bc<\/jats:italic><jats:sub>2<\/jats:sub>, i.e. <jats:italic>\u03bc<\/jats:italic><jats:sub>1<\/jats:sub> \u226a <jats:italic>\u03bc<\/jats:italic><jats:sub>2<\/jats:sub>.<\/jats:p><jats:p>The quantifiers are interpreted in the natural way, i.e.<\/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=\"S0022481200030784_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>for <jats:italic>i<\/jats:italic> = 1, 2. (The measure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline8\"\/> is the restriction of the completion of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline9\"\/> to the <jats:italic>\u03c3<\/jats:italic>-algebra generated by the measurable rectangles and the diagonal sets <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline10\"\/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline11\"\/><\/jats:p><jats:p>Axioms and rules of inference are those of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline3\"\/>, as listed in [2] with the axiom <jats:italic>B<\/jats:italic><jats:sub>4<\/jats:sub> from [4], with the remark that both <jats:italic>P<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>P<\/jats:italic><jats:sub>2<\/jats:sub> can play the role of <jats:italic>P<\/jats:italic>, together with the following axioms:<\/jats:p><jats:p><jats:italic>Axioms of continuity<\/jats:italic>.<\/jats:p><jats:p><jats:list list-type=\"simple\"><jats:list-item><jats:p>1) <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline12\"\/>.<\/jats:p><\/jats:list-item><jats:list-item><jats:p>2) <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline13\"\/>.<\/jats:p><\/jats:list-item><\/jats:list><\/jats:p><jats:p><jats:italic>Axiom of absolute continuity<\/jats:italic>:<\/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=\"S0022481200030784_eqnU2\"\/><\/jats:disp-formula><\/jats:p><jats:p>where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030784_inline14\"\/> and <jats:italic>\u03a6<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> = {<jats:italic>\u03c6<\/jats:italic> \u2208 <jats:italic>\u03a6<\/jats:italic>: <jats:italic>\u03c6<\/jats:italic> has <jats:italic>n<\/jats:italic> free variables}.<\/jats:p>","DOI":"10.2307\/2274015","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:18:32Z","timestamp":1146953912000},"page":"586-590","source":"Crossref","is-referenced-by-count":14,"title":["Completeness theorem for biprobability models"],"prefix":"10.1017","volume":"51","author":[{"given":"Miodrag D.","family":"Ra\u0161kovi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200030784_ref004","volume-title":"Model theoretic languages","author":"Keisler","year":"1985"},{"key":"S0022481200030784_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90022-0"},{"key":"S0022481200030784_ref003","volume-title":"Logic Colloquium '76","author":"Keisler","year":"1977"},{"key":"S0022481200030784_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200030784","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T04:41:40Z","timestamp":1558500100000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200030784\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1986,9]]}},"alternative-id":["S0022481200030784"],"URL":"https:\/\/doi.org\/10.2307\/2274015","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,9]]}}}