{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T04:25:08Z","timestamp":1648527908195},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10968,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Logics <jats:italic>L<\/jats:italic><jats:sub><jats:italic>F<\/jats:italic><\/jats:sub>(<jats:italic>M<\/jats:italic>) are considered, in which <jats:italic>M<\/jats:italic> (\u201cmost\u201d) is a new first-order quantifier whose interpretation depends on a given filter <jats:italic>F<\/jats:italic> of subsets of \u03c9. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that <jats:italic>F<\/jats:italic> is not of the form {(\u22c2<jats:italic>F<\/jats:italic>) \u222a <jats:italic>X<\/jats:italic>: \u2223\u03c9 \u2212 <jats:italic>X<\/jats:italic>\u2223 &lt; \u03c9} with \u2223\u03c9 \u2212 \u22c2<jats:italic>F<\/jats:italic>\u2223 = \u03c9. Moreover the set of validities of <jats:italic>L<\/jats:italic><jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> (<jats:italic>M<\/jats:italic>) and even of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200034101_inline1\" \/> depends only on a few basic properties of <jats:italic>F<\/jats:italic>. Similar characterizations are given of the class of filters <jats:italic>F<\/jats:italic> for which <jats:italic>L<\/jats:italic><jats:sup><jats:italic>F<\/jats:italic><\/jats:sup> (<jats:italic>M<\/jats:italic>) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200034101_inline2\" \/>, where the collection <jats:italic>q<\/jats:italic> is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstr\u00f6m's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter <jats:italic>F<\/jats:italic>). However, Lindstr\u00f6m's argument is applied to characterize the invariant sentences as just those of first-order logic.<\/jats:p>","DOI":"10.2307\/2274107","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:07:02Z","timestamp":1146938822000},"page":"241-256","source":"Crossref","is-referenced-by-count":0,"title":["Filter Logics on \u03c9"],"prefix":"10.1017","volume":"49","author":[{"given":"Matt","family":"Kaufmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200034101_ref008","first-page":"192","volume":"46","author":"Kaufmann","year":"1981","journal-title":"Filter logics"},{"key":"S0022481200034101_ref007","first-page":"469","article-title":"A many-sorted interpolation theorem for L(Q)","volume":"80","author":"Guichard","year":"1980","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200034101_ref010","volume-title":"Higher model theory: Logics of mathematical concepts","author":"Kaufmann","year":"1984"},{"key":"S0022481200034101_ref015","unstructured":"Sun X. [1981], Model theory in L\u03c91\u03c9(Q). Part I: Consistency property and its applications(preprint)."},{"key":"S0022481200034101_ref013","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1969.tb00356.x"},{"key":"S0022481200034101_ref012","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200034101_ref004","volume-title":"Higher model theory: Logics of mathematical concepts","author":"Ebbinghaus","year":"1984"},{"key":"S0022481200034101_ref003","unstructured":"Buechler S. [1980], Ideal models and uniform validity (preprint)."},{"key":"S0022481200034101_ref014","volume-title":"Higher model thtory: Logics of mathematical concepts","author":"Makowsky","year":"1984"},{"key":"S0022481200034101_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/S0003-4843(70)80005-5"},{"key":"S0022481200034101_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(81)90002-4"},{"key":"S0022481200034101_ref001","first-page":"304","volume":"43","author":"Bruce","year":"1978","journal-title":"Ideal models and some not so ideal problems in the model theory of L(Q)"},{"key":"S0022481200034101_ref006","volume-title":"Higher model theory: Logics of mathematical concepts","author":"Flum","year":"1984"},{"key":"S0022481200034101_ref005","volume-title":"Archiv f\u00fcr Mathematische Logik und Grundlagenforschung","author":"Ebbinghaus","year":"198?"},{"key":"S0022481200034101_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0003-4843(78)80001-1"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200034101","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T17:25:52Z","timestamp":1558632352000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200034101\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,3]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1984,3]]}},"alternative-id":["S0022481200034101"],"URL":"https:\/\/doi.org\/10.2307\/2274107","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,3]]}}}