{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,30]],"date-time":"2024-03-30T10:10:32Z","timestamp":1711793432548},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2010,9,13]],"date-time":"2010-09-13T00:00:00Z","timestamp":1284336000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2010,12]]},"abstract":"<jats:p>Here, we provide a detailed description of the mutual relation of formulas with finite propositional variables<jats:italic>p<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026,<jats:italic>p<jats:sub>m<\/jats:sub><\/jats:italic>in modal logic<jats:bold>S4<\/jats:bold>. Our description contains more information on<jats:bold>S4<\/jats:bold>than those given in Shehtman (1978) and Moss (2007); however, Shehtman (1978) also treated Grzegorczyk logic and Moss (2007) treated many other normal modal logics. Specifically, we construct normal forms, which behave like the principal conjunctive normal forms in the classical propositional logic. The results include finite and effective methods to find a normal form equivalent to a given formula<jats:italic>A<\/jats:italic>by clarifying the behavior of connectives and giving a finite method to list all exact models.<\/jats:p>","DOI":"10.1017\/s1755020310000043","type":"journal-article","created":{"date-parts":[[2010,9,13]],"date-time":"2010-09-13T08:16:21Z","timestamp":1284365781000},"page":"600-627","source":"Crossref","is-referenced-by-count":0,"title":["FORMULAS IN MODAL LOGIC<b>S4<\/b>"],"prefix":"10.1017","volume":"3","author":[{"given":"KATSUMI","family":"SASAKI","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2010,9,13]]},"reference":[{"key":"S1755020310000043_ref20","doi-asserted-by":"publisher","DOI":"10.2307\/2272850"},{"key":"S1755020310000043_ref15","volume-title":"Logics and Provability","author":"Sasaki","year":"2001"},{"key":"S1755020310000043_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/BF02945107"},{"key":"S1755020310000043_ref16","first-page":"39","article-title":"Formulas with only one variable in Lewis logic S4","volume":"5","author":"Sasaki","year":"2005","journal-title":"Academia Mathematical Sciences and Information Engineering"},{"key":"S1755020310000043_ref4","volume-title":"Sur les alg\u00e8bres de Hilbert","author":"Diego","year":"1966"},{"key":"S1755020310000043_ref2","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198537793.001.0001","volume-title":"Modal Logic","author":"Chagrov","year":"1997"},{"key":"S1755020310000043_ref5","first-page":"95","article-title":"Christmas trees. On free cyclic algebras in some varieties of closure algebras","volume":"4","author":"Esakia","year":"1975","journal-title":"Bulletin of the Section of Logic"},{"key":"S1755020310000043_ref12","doi-asserted-by":"publisher","DOI":"10.2307\/2963526"},{"key":"S1755020310000043_ref7","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093891703"},{"key":"S1755020310000043_ref17","volume-title":"A Construction of an Exact Model for S4","author":"Sasaki","year":"2010"},{"key":"S1755020310000043_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)E0084-2"},{"key":"S1755020310000043_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01201353"},{"key":"S1755020310000043_ref18","first-page":"1288","article-title":"Rieger-Nishimura ladders","volume":"241","author":"Shehtman","year":"1978","journal-title":"Doklady Akademii Nauk SSSR"},{"key":"S1755020310000043_ref14","first-page":"1","article-title":"On lattice of Brouwerian propositional logics","volume":"189","author":"Rieger","year":"1949","journal-title":"Acta Universitatis Carolinae. Mathematica et Physica"},{"key":"S1755020310000043_ref6","first-page":"46","article-title":"The criterion of Brouwerian and closure algebras to be finitely generated","volume":"6","author":"Esakia","year":"1977","journal-title":"Bulletin of the Section of Logic"},{"key":"S1755020310000043_ref10","volume-title":"Computations in Propositional Logic","author":"Hendriks","year":"1996"},{"key":"S1755020310000043_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-007-9052-4"},{"key":"S1755020310000043_ref3","volume-title":"Exact Finite Models for Minimal Propositional Calculus Over a Finite Alphabet","author":"de Bruijn","year":"1975"},{"key":"S1755020310000043_ref13","first-page":"113","article-title":"Gentzen method in modal calculi","volume":"9","author":"Ohnishi","year":"1957","journal-title":"Osaka Mathematical Journal"},{"key":"S1755020310000043_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01195140"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020310000043","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,30]],"date-time":"2024-03-30T09:41:37Z","timestamp":1711791697000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020310000043\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,9,13]]},"references-count":20,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,12]]}},"alternative-id":["S1755020310000043"],"URL":"https:\/\/doi.org\/10.1017\/s1755020310000043","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,9,13]]}}}