{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:26:17Z","timestamp":1761611177653},"reference-count":8,"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":14164,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,6]]},"abstract":"<jats:p>We present two finitely axiomatized modal propositional logics, one between<jats:italic>T<\/jats:italic>and<jats:italic>S<\/jats:italic>4 and the other an extension of<jats:italic>S<\/jats:italic>4, which are incomplete with respect to the neighbourhood or Scott-Montague semantics.<\/jats:p><jats:p>Throughout this paper we are referring to logics which contain all the classical connectives and only one modal connective \u25a1 (unary), no propositional constants, all classical tautologies, and which are closed under the rules of modus ponens (MP), substitution, and the rule RE (from<jats:italic>A<\/jats:italic>\u2194<jats:italic>B<\/jats:italic>infer \u03b1<jats:italic>A<\/jats:italic>\u2194 \u25a1<jats:italic>B<\/jats:italic>). Such logics are called<jats:italic>classical<\/jats:italic>by Segerberg [6]. Classical logics which contain the formula \u25a1<jats:italic>p<\/jats:italic>\u2227 \u25a1<jats:italic>q<\/jats:italic>\u2192 \u25a1(<jats:italic>p<\/jats:italic>\u2227<jats:italic>q<\/jats:italic>) (denoted by<jats:italic>K<\/jats:italic>) and its \u201cconverse,\u201d \u25a1{<jats:italic>p<\/jats:italic>\u2227<jats:italic>q<\/jats:italic>)\u2192 \u25a1<jats:italic>p<\/jats:italic>\u2227 \u25a1<jats:italic>q<\/jats:italic>(denoted by<jats:italic>R<\/jats:italic>) are called regular;<jats:italic>regular<\/jats:italic>logics which are closed under the rule of necessitation, RN (from<jats:italic>A<\/jats:italic>infer \u25a1<jats:italic>A<\/jats:italic>), are called<jats:italic>normal<\/jats:italic>. The logics that we are particularly concerned with are all normal, although some of our results will be true for all regular or all classical logics. It is well known that<jats:italic>K<\/jats:italic>and<jats:italic>R<\/jats:italic>and closure under RN imply closure under RE and also that normal logics are also those logics closed under RN and containing \u25a1{<jats:italic>p<\/jats:italic>\u2192<jats:italic>q<\/jats:italic>) \u2192 {\u25a1<jats:italic>p<\/jats:italic>\u2192 \u25a1<jats:italic>q<\/jats:italic>).<\/jats:p>","DOI":"10.2307\/2271893","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:35:13Z","timestamp":1146951313000},"page":"141-148","source":"Crossref","is-referenced-by-count":25,"title":["The inadequacy of the neighbourhood semantics for modal logic"],"prefix":"10.1017","volume":"40","author":[{"given":"Martin","family":"Gerson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200053688_ref008","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00077.x"},{"key":"S0022481200053688_ref004","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1111\/j.1755-2567.1970.tb00431.x","article-title":"A generalization of the concept of a relational model for modal logic","volume":"36","author":"Makinson","year":"1970","journal-title":"Theoria"},{"key":"S0022481200053688_ref003","volume-title":"A guide to intensional semantics","author":"Hansson"},{"key":"S0022481200053688_ref002","volume-title":"Zeitschrift fur mathematische Logik und Grundlagen der Mathematik","author":"Gerson"},{"key":"S0022481200053688_ref001","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1974.tb00076.x"},{"key":"S0022481200053688_ref006","volume-title":"An essay in classical modal logic","author":"Segerberg","year":"1971"},{"key":"S0022481200053688_ref005","first-page":"1","volume":"13","author":"McKinsey","year":"1948","journal-title":"Some theorems about the sentential calculi of Lewis and Heyting"},{"key":"S0022481200053688_ref007","first-page":"150","volume":"37","author":"Thomason","year":"1972","journal-title":"Semantic analysis of tense logics"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200053688","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:22:39Z","timestamp":1707031359000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200053688\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1975,6]]}},"alternative-id":["S0022481200053688"],"URL":"https:\/\/doi.org\/10.2307\/2271893","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,6]]}}}