{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T19:17:35Z","timestamp":1693595855765},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13981,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,12]]},"abstract":"<jats:p>Let<jats:italic>L<\/jats:italic>be a first order logic and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline1\" \/>the infinitary logic (as described in [K, p. 6] over<jats:italic>L. Suslin logic<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline2\" \/>is obtained from<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline1\" \/>by adjoining new propositional operators<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline3\" \/>and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline4\" \/>. Let<jats:italic>f<\/jats:italic>range over elements of<jats:italic><jats:sup>\u03c9<\/jats:sup>\u03c9<\/jats:italic>and<jats:italic>n<\/jats:italic>range over elements of \u03c9.<jats:italic>Seq<\/jats:italic>is the set of all finite sequences of elements of \u03c9. If \u03b8:<jats:italic>Seq<\/jats:italic>\u2192<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline2\" \/>is a mapping into formulas of<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline2\" \/>then<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline5\" \/>and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline6\" \/>are<jats:italic>formulas<\/jats:italic>of<jats:italic>L<jats:sub>A<\/jats:sub><\/jats:italic>. If<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline7\" \/>is a structure in which we can interpret<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline1\" \/>and<jats:italic>h<\/jats:italic>is an<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline7\" \/>-assignment then we extend the notion of<jats:italic>satisfaction<\/jats:italic>from<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline1\" \/>to<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline2\" \/>by defining<\/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=\"S0022481200052051_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where<jats:italic>f<\/jats:italic>\u2223<jats:italic>n<\/jats:italic>is the finite sequence consisting of the first<jats:italic>n<\/jats:italic>values of<jats:italic>f<\/jats:italic>. We assume that<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline1\" \/>has<jats:italic>\u03c9<\/jats:italic>symbols for relations, functions, constants, and<jats:italic>\u03c9<jats:sub>1<\/jats:sub><\/jats:italic>variables. \u03b8 is<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline7\" \/>valid if<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline7\" \/>\u03b8 \u22a7 [<jats:italic>h<\/jats:italic>] for every<jats:italic>h<\/jats:italic>and is<jats:italic>valid<\/jats:italic>if<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline7\" \/>-valid for every<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline7\" \/>. We address ourselves to the problem of finding syntactical rules (or nearly so) which characterize validity<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200052051_inline2\" \/>.<\/jats:p>","DOI":"10.2307\/2271780","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:38:25Z","timestamp":1146951505000},"page":"567-575","source":"Crossref","is-referenced-by-count":4,"title":["The foundations of Suslin logic"],"prefix":"10.1017","volume":"40","author":[{"given":"Erik","family":"Ellentuck","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200052051_ref007","volume-title":"Topology","volume":"I","author":"Kurotowski","year":"1966"},{"key":"S0022481200052051_ref006","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.49.6.828"},{"key":"S0022481200052051_ref004","doi-asserted-by":"crossref","first-page":"99","DOI":"10.21136\/CMJ.1955.100135","article-title":"Concerning Suslin algebras and their representation","volume":"5","author":"Rieger","year":"1955","journal-title":"Czechoslovak Mathematical Journal"},{"key":"S0022481200052051_ref003","volume-title":"Languages with expressions of infinite length","author":"Karp","year":"1964"},{"key":"S0022481200052051_ref001","volume-title":"Infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200052051_ref005","unstructured":"Scott D. and Solovay R. M. , Boolean valued models for set theory, Lecture notes from the 1967 Summer Institute on Axiomatic Set Theory at U.C.L.A."},{"key":"S0022481200052051_ref002","unstructured":"Campbell P. J. , Suslin logic, Ph.D. Thesis, Cornell University, 1971."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200052051","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,26]],"date-time":"2021-07-26T20:18:24Z","timestamp":1627330704000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200052051\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,12]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1975,12]]}},"alternative-id":["S0022481200052051"],"URL":"https:\/\/doi.org\/10.2307\/2271780","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,12]]}}}