{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T20:45:40Z","timestamp":1783111540271,"version":"3.54.6"},"reference-count":28,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6493,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,6]]},"abstract":"<jats:p>This paper is a continuation of Zakharyaschev [25], where the following basic results on modal logics with transitive frames were obtained:<\/jats:p><jats:p>\u2022 With every finite rooted transitive frame <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline1\"\/> and every set <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline2\"\/> of antichains (which were called <jats:italic>closed domains<\/jats:italic>) in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline1\"\/> two formulas <jats:italic>\u03b1<\/jats:italic> (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline1\"\/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline2\"\/>, \u22a5) and <jats:italic>\u03b1<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline1\"\/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline2\"\/>) were associated. We called them the <jats:italic>canonical<\/jats:italic> and <jats:italic>negation free canonical formulas<\/jats:italic>, respectively, and proved the Refutability Criterion characterizing the constitution of their refutation general frames in terms of subreduction (alias partial p-morphism), the cofinality condition and the closed domain condition.<\/jats:p><jats:p>\u2022 We proved also the Completeness Theorem for the canonical formulas providing us with an algorithm which, given a modal formula <jats:italic>\u03c6<\/jats:italic>, returns canonical formulas <jats:italic>\u03b1<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline1\"\/><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline2\"\/><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>), \u22a5), for <jats:italic>i<\/jats:italic> = 1,\u2026, <jats:italic>n<\/jats:italic>, such that<\/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=\"S0022481200017369_Uequ1\"\/><\/jats:disp-formula><\/jats:p><jats:p>if <jats:italic>\u03c6<\/jats:italic> is negation free then the algorithm instead of <jats:italic>\u03b1<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline1\"\/><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline2\"\/><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, \u22a5) can use the negation free canonical formulas <jats:italic>\u03b1<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline1\"\/><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline2\"\/><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>). Thus, every normal modal logic containing <jats:bold>K4<\/jats:bold> can be axiomatized by a set of canonical formulas.<\/jats:p><jats:p>In this Part we apply the apparatus of the canonical formulas for establishing a number of results on the decidability, finite model property, elementarity and some other properties of modal logics within the field of <jats:bold>K4<\/jats:bold>.<\/jats:p><jats:p>Our attention will be focused on the class of logics which can be axiomatized by canonical formulas without closed domains, i.e., on the logics of the form<\/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=\"S0022481200017369_Uequ2\"\/><\/jats:disp-formula><\/jats:p><jats:p>Adapting the terminology of Fine [11], we call them the <jats:italic>cofinal subframe logics<\/jats:italic> and denote this class by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline3\"\/>. As was shown in Part I, almost all standard modal logics are in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017369_inline3\"\/>.<\/jats:p>","DOI":"10.2307\/2275669","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:58:29Z","timestamp":1146956309000},"page":"421-449","source":"Crossref","is-referenced-by-count":34,"title":["Canonical formulas for K4. 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V. , Syntax and semantics of superintuitionistic and modal logics, Ph.D. thesis , Moscow, 1984, Russian."},{"key":"S0022481200017369_ref023","first-page":"1415","article-title":"Modal companions of intermediate logics: syntax, semantics and preservation theorems","volume":"180","author":"Zakharyaschev","year":"1989","journal-title":"Mathematical Sbornik"},{"key":"S0022481200017369_ref009","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19710170141"},{"key":"S0022481200017369_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70723-7"},{"key":"S0022481200017369_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BF01463150"},{"key":"S0022481200017369_ref027","volume-title":"Modal logic audits neighbours","author":"Zakharyaschev","year":"1995"},{"key":"S0022481200017369_ref025","first-page":"1377","volume":"57","author":"Zakharyaschev","year":"1992","journal-title":"Canonical formulas for K4. part 1: Basic results"},{"key":"S0022481200017369_ref028","volume-title":"On the complexity of countermodels for intuitionistic calculus","author":"Zakharyaschev","year":"1980"},{"key":"S0022481200017369_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF01668576"},{"key":"S0022481200017369_ref018","unstructured":"Rodenburg Rh. , Intuitionistic correspondence theory, Ph.D. thesis , University of Amsterdam, 1986."},{"key":"S0022481200017369_ref007","first-page":"1261","volume":"56","author":"Chagrova","year":"1991","journal-title":"An undecidable problem in correspondence theory"},{"key":"S0022481200017369_ref026","first-page":"7","article-title":"Intermediate logics with disjunction free axioms are canonical","volume":"1","author":"Zakharyaschev","year":"1992","journal-title":"IGPL Newsletter"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200017369","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T21:28:43Z","timestamp":1557696523000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200017369\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,6]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1996,6]]}},"alternative-id":["S0022481200017369"],"URL":"https:\/\/doi.org\/10.2307\/2275669","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,6]]}}}