{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T20:12:55Z","timestamp":1776888775095,"version":"3.51.2"},"reference-count":8,"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":12795,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,3]]},"abstract":"<jats:p>In this paper the<jats:italic>canonical<\/jats:italic>modal logics, a kind of complete modal logics introduced in K. Fine [4] and R. I. Goldblatt [5], will be characterized semantically using the concept of an<jats:italic>ultrafilter extension<\/jats:italic>, an operation on frames inspired by the algebraic theory of modal logic. Theorem 8 of R. I. Goldblatt and S. K. Thomason [6] characterizing the modally definable \u03a3\u22bf-elementary classes of frames will follow as a corollary. A second corollary is Theorem 2 of [4] which states that any complete modal logic defining a \u03a3\u22bf-elementary class of frames is canonical.<\/jats:p><jats:p>The main tool in obtaining these results is the duality between modal algebras and general frames developed in R. I. Goldblatt [5]. The relevant notions and results from this theory will be stated in \u00a72. The concept of a canonical modal logic is introduced and motivated in \u00a73, which also contains the above-mentioned theorems. In \u00a74, a kind of appendix to the preceding discussion, preservation of first-order sentences under ultrafilter extensions (and some other relevant operations on frames) is discussed.<\/jats:p><jats:p>The modal language to be considered here has an infinite supply of proposition letters (<jats:italic>p, q, r<\/jats:italic>, \u2026), a propositional constant \u22a5 (the so-called<jats:italic>falsum<\/jats:italic>, standing for a fixed contradiction), the usual Boolean operators \u00ac (not), \u2228 (or), \u2228 (and), \u2192 (if \u2026 then \u2026), and \u2194 (if and only if)\u2014with \u00ac and \u2228 regarded as primitives\u2014and the two unary modal operators \u25c7 (possibly) and \u25a1 (necessarily)\u2014 \u25c7 being regarded as primitive. Modal formulas will be denoted by lower case Greek letters, sets of formulas by Greek capitals.<\/jats:p>","DOI":"10.2307\/2273696","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:49:34Z","timestamp":1146952174000},"page":"1-8","source":"Crossref","is-referenced-by-count":27,"title":["Canonical modal logics and ultrafilter extensions"],"prefix":"10.1017","volume":"44","author":[{"given":"J. F. A. K.","family":"van Benthem","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048647_ref008","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00077.x"},{"key":"S0022481200048647_ref006","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/BFb0062855","volume-title":"Algebra and Logic, Lecture Notes in Mathematics","author":"Goldblatt","year":"1975"},{"key":"S0022481200048647_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70723-7"},{"key":"S0022481200048647_ref001","unstructured":"van Benthem J. F. A. K. , Modal correspondence theory, Dissertation, Amsterdam, 1976."},{"key":"S0022481200048647_ref002","first-page":"436","volume":"41","author":"van Benthem","year":"1976","journal-title":"Modal formulas are either elementary or not \u03a3\u22bf-elementary"},{"key":"S0022481200048647_ref005","first-page":"41","article-title":"Metamathematics of modal logic","volume":"6","author":"Goldblatt","year":"1976","journal-title":"Reports on Mathematical Lotic"},{"key":"S0022481200048647_ref003","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00076.x"},{"key":"S0022481200048647_ref007","volume-title":"Filosofiska Studier","author":"Segerberg","year":"1971"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048647","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,14]],"date-time":"2020-04-14T14:17:18Z","timestamp":1586873838000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048647\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,3]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1979,3]]}},"alternative-id":["S0022481200048647"],"URL":"https:\/\/doi.org\/10.2307\/2273696","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,3]]}}}