{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T11:19:20Z","timestamp":1772450360000,"version":"3.50.1"},"reference-count":12,"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":10419,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1985,9]]},"abstract":"<jats:p>This paper establishes another very general completeness result for the logics within the field of <jats:italic>K<\/jats:italic>4. With each finite transitive frame \u212d we may associate a formula \u2014 <jats:italic>B<\/jats:italic><jats:sub>\u212d<\/jats:sub> which validates just those frames \u2111 in which \u212d is not in a certain sense embeddable (to be exact, \u212d is not the <jats:italic>p<\/jats:italic>-morphic image of any subframe of \u2111. By a subframe logic we mean the result of adding such formulas as axioms to <jats:italic>K<\/jats:italic>4. The general result is that each subframe logic has the finite model property.<\/jats:p><jats:p>There are a continuum of subframe logics and they include many of the standard ones, such as <jats:italic>T<\/jats:italic>, <jats:italic>S<\/jats:italic>4, <jats:italic>S<\/jats:italic>4.3, <jats:italic>S<\/jats:italic>5 and <jats:italic>G<\/jats:italic>. It turns out that the subframe logics are exactly those complete for a condition that is closed under subframes (any subframe of a frame satisfying the condition also satisfies the condition). As a consequence, every logic complete for a condition closed under subframes has the finite model property.<\/jats:p><jats:p>It is ascertained which of the subframe logics are compact. It turns out that the compact logics are just those whose axioms express an elementary condition. Tests are given for determining whether a given axiom expresses an elementary condition and for determining what it is in case it does.<\/jats:p><jats:p>In one respect the present general completeness result differs from most of the others in the literature. The others have usually either been what one might call <jats:italic>logic based<\/jats:italic> or <jats:italic>formula based<\/jats:italic>. They have usually either been to the effect that all of the logics containing a given logic are complete or to the effect that all logics whose axioms come from a given syntactically characterized class of formulas are complete. The present result is, by contrast, what one might call <jats:italic>frame based<\/jats:italic>. The axioms of the logics to be proved complete are characterized most directly in terms of their connection with certain frames.<\/jats:p>","DOI":"10.2307\/2274318","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:14:36Z","timestamp":1146953676000},"page":"619-651","source":"Crossref","is-referenced-by-count":79,"title":["Logics containing <i>K<\/i>4. Part II"],"prefix":"10.1017","volume":"50","author":[{"given":"Kit","family":"Fine","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120003228X_ref003","volume-title":"An introduction to modal logic"},{"key":"S002248120003228X_ref004","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1002\/malq.19710170141","volume":"17","journal-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik"},{"key":"S002248120003228X_ref002","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1002\/malq.19660120129","volume":"12","journal-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik"},{"key":"S002248120003228X_ref001","volume-title":"The unprovability of consistency"},{"key":"S002248120003228X_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF00293428"},{"key":"S002248120003228X_ref005","first-page":"31","volume":"39","journal-title":"Logics containing K4"},{"key":"S002248120003228X_ref009","volume-title":"Investigations in modal and tense logics with applications to problems in philosophy and linguistics"},{"key":"S002248120003228X_ref010","first-page":"163","volume-title":"Algebraic logic","volume":"450"},{"key":"S002248120003228X_ref008","first-page":"15","volume-title":"Proceedings of the Third Scandinavian Logic Symposium"},{"key":"S002248120003228X_ref007","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00076.x"},{"key":"S002248120003228X_ref006","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00081.x"},{"key":"S002248120003228X_ref012","first-page":"572","volume":"16","journal-title":"Indagationes Mathematicae"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120003228X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T21:56:32Z","timestamp":1558562192000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120003228X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,9]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1985,9]]}},"alternative-id":["S002248120003228X"],"URL":"https:\/\/doi.org\/10.2307\/2274318","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,9]]}}}