{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T07:49:21Z","timestamp":1773820161455,"version":"3.50.1"},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,12,18]],"date-time":"2014-12-18T00:00:00Z","timestamp":1418860800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Rev. symb. logic"],"published-print":{"date-parts":[[2015,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is a classic result (McKinsey &amp; Tarski, 1944; Rasiowa &amp; Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the \u2283 relation) to the ordinal<jats:italic>\u03c9<\/jats:italic>+ 3:<\/jats:p><jats:p><jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1755020314000446_inline1\"\/><jats:tex-math>$S4.Gr{z_1} \\supset S4.Gr{z_2} \\supset S4.Gr{z_3} \\supset \\cdots \\,S4.Grz \\supset S4.1 \\supset S4.$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula><\/jats:p><jats:p>It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable.<\/jats:p>","DOI":"10.1017\/s1755020314000446","type":"journal-article","created":{"date-parts":[[2014,12,18]],"date-time":"2014-12-18T20:57:42Z","timestamp":1418936262000},"page":"178-191","source":"Crossref","is-referenced-by-count":10,"title":["MODAL LOGICS OF METRIC SPACES"],"prefix":"10.1017","volume":"8","author":[{"given":"GURAM","family":"BEZHANISHVILI","sequence":"first","affiliation":[]},{"given":"DAVID","family":"GABELAIA","sequence":"additional","affiliation":[]},{"given":"JOEL","family":"LUCERO-BRYAN","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,12,18]]},"reference":[{"key":"S1755020314000446_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107050884"},{"key":"S1755020314000446_ref10","doi-asserted-by":"publisher","DOI":"10.1023\/B:STUD.0000009564.00287.16"},{"key":"S1755020314000446_ref6","doi-asserted-by":"publisher","DOI":"10.2307\/1969080"},{"key":"S1755020314000446_ref8","first-page":"37","article-title":"The theory of representations for Boolean algebras","volume":"40","author":"Stone","year":"1936","journal-title":"Transactions of the American Mathematical Society"},{"key":"S1755020314000446_ref3","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198537793.001.0001","volume-title":"Modal Logic","author":"Chagrov","year":"1997"},{"key":"S1755020314000446_ref4","volume-title":"General Topology","author":"Engelking","year":"1989"},{"key":"S1755020314000446_ref9","first-page":"567","article-title":"Total paracompactness and paracompact dispersed spaces","volume":"16","author":"Telg\u00e1rsky","year":"1968","journal-title":"Bulletin de l\u2019Acad\u00e9mie Polonaise des Sciences. S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S1755020314000446_ref5","first-page":"128","volume-title":"Studies in Logic and Semantics","author":"Esakia","year":"1981"},{"key":"S1755020314000446_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s11083-011-9224-2"},{"key":"S1755020314000446_ref7","volume-title":"The Mathematics of Metamathematics","author":"Rasiowa","year":"1963"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020314000446","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,5]],"date-time":"2024-06-05T23:33:08Z","timestamp":1717630388000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020314000446\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,18]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,3]]}},"alternative-id":["S1755020314000446"],"URL":"https:\/\/doi.org\/10.1017\/s1755020314000446","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,12,18]]}}}