{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T20:34:09Z","timestamp":1693514049709},"reference-count":8,"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":14437,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1974,9]]},"abstract":"<jats:p>This paper is concerned with extending some basic results from classical model theory to modal logic.<\/jats:p><jats:p>In \u00a71, we define the majority of terms used in the paper, and explain our notation. A full catalogue would be excessive, and we cite [3] and [7] as general references.<\/jats:p><jats:p>Many papers on modal logic that have appeared are concerned with (i) introducing a new modal logic, and (ii) proving a weak completeness theorem for it. Theorem 1, in \u00a72, in many cases allows us to conclude immediately that a strong completeness theorem holds for such a logic in languages of arbitrary cardinality. In particular, this is true of S4 with the Barcan formula.<\/jats:p><jats:p>In \u00a73 we strengthen Theorem 1 for a number of modal logics to deal with the satisfaction of several sets of sentences, and so obtain a realizing types theorem. Finally, an omitting types theorem, generalizing the result for classical logic (see [5]) is proved in \u00a74.<\/jats:p><jats:p>Several consequences of Theorem 1 are already to be found in the literature. [2] gives a proof of strong completeness in languages of arbitrary cardinality of various logics without the Barcan formula, and [8] for some logics in countable languages with it. In the latter case, the result for uncountable languages is cited, without proof, in [1], and there credited to Montague. Our proof was found independently.<\/jats:p>","DOI":"10.2307\/2272892","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:31:33Z","timestamp":1146936693000},"page":"496-508","source":"Crossref","is-referenced-by-count":4,"title":["Some results in modal model theory"],"prefix":"10.1017","volume":"39","author":[{"given":"Michael","family":"Mortimer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200076659_ref008","unstructured":"Thomason R. H. , Some completeness results for modal predicate calculi, in[4]."},{"key":"S0022481200076659_ref007","volume-title":"Mathematical logic","author":"Shoenfield","year":"1967"},{"key":"S0022481200076659_ref006","unstructured":"Scott Dana , Advice on modal logic, in[4]."},{"key":"S0022481200076659_ref001","unstructured":"Cocciarella Nino , Ph.D. Dissertation, University of California, Berkeley, 1965."},{"key":"S0022481200076659_ref002","volume-title":"Conference in Mathematical Logic\u20141970","author":"Gabbay"},{"key":"S0022481200076659_ref003","volume-title":"An introduction to modal logic","author":"Hughes","year":"1968"},{"key":"S0022481200076659_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-010-3272-8"},{"key":"S0022481200076659_ref005","volume-title":"Saturated model theory","author":"Sacks","year":"1972"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200076659","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T17:03:06Z","timestamp":1559149386000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200076659\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1974,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1974,9]]}},"alternative-id":["S0022481200076659"],"URL":"https:\/\/doi.org\/10.2307\/2272892","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1974,9]]}}}