{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T22:31:20Z","timestamp":1648852280905},"reference-count":2,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":23568,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1949,9]]},"abstract":"<jats:p>This note should be considered as an appendix to a paper by McKinsey. Familiarity with this paper is assumed; its terminology and notation will be used without explanation. McKinsey's main result (theorem 1) is that every class of sentences fulfilling P<jats:sub>1<\/jats:sub>\u2013P<jats:sub>5<\/jats:sub>, with a set of substitutions satisfying <jats:italic>A<\/jats:italic><jats:sub>1<\/jats:sub>\u2013<jats:italic>A<\/jats:italic><jats:sub>4<\/jats:sub>, is a system of modal logic in the sense that the \u201cassociated\u201d set, <jats:italic>T<\/jats:italic><jats:sub>2<\/jats:sub>, contains all theorems of S4. It is also proved (theorem 2) that for <jats:italic>T<\/jats:italic><jats:sub>2<\/jats:sub> to contain all theorems of S5, it is <jats:italic>sufficient<\/jats:italic> that the elements of <jats:italic>S<\/jats:italic> form a group. The basic idea of the construction is the formalization, by means of <jats:italic>D<\/jats:italic><jats:sub>5<\/jats:sub>, of our intuitive notion of possibility.<\/jats:p><jats:p>In view of the great generality of <jats:italic>S<\/jats:italic>, it may be of some interest to give a condition that is both <jats:italic>necessary and sufficient<\/jats:italic> for <jats:italic>T<\/jats:italic><jats:sub>2<\/jats:sub> to contain S5 after some restriction suggested by our intuitive notions about modalities has first been imposed upon <jats:italic>S<\/jats:italic>. The following postulate is related to the idea that if a sentence is possible, then its negation is not necessary. Informally speaking, it requires that <jats:italic>S<\/jats:italic> be sufficiently \u201ccomprehensive\u201d for this idea to become formalizable in terms of <jats:italic>S<\/jats:italic>.<\/jats:p><jats:p><jats:italic>P<\/jats:italic>. If \u2662\u03b1 is in <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub>, then there exists an element <jats:italic>s<jats:sub>m<\/jats:sub><\/jats:italic> of <jats:italic>S<\/jats:italic> such that \u2662\u223c<jats:italic>s<jats:sub>m<\/jats:sub><\/jats:italic>(\u03b1) is not in <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub>.<\/jats:p><jats:p>I shall now prove that if <jats:italic>P<\/jats:italic> is satisfied, the following condition is both necessary and sufficient for <jats:italic>T<\/jats:italic><jats:sub>2<\/jats:sub> to contain all theorems of S5.<\/jats:p><jats:p><jats:italic>C<\/jats:italic>. If \u03b1 is in <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub>, then \u2662<jats:italic>s<\/jats:italic>(\u03b1) is in <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub> for all elements <jats:italic>s<\/jats:italic> of <jats:italic>S<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2267046","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:06:41Z","timestamp":1146942401000},"page":"173-174","source":"Crossref","is-referenced-by-count":0,"title":["A syntactical characterization of S5"],"prefix":"10.1017","volume":"14","author":[{"given":"Gustav","family":"Bergmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200105699_ref002","first-page":"33","volume":"11","year":"1946","journal-title":"Modalities and quantification"},{"key":"S0022481200105699_ref001","first-page":"83","volume":"10","year":"1945","journal-title":"On the syntactical construction of systems of modal logic"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200105699","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T10:29:21Z","timestamp":1559989761000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200105699\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1949,9]]},"references-count":2,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1949,9]]}},"alternative-id":["S0022481200105699"],"URL":"https:\/\/doi.org\/10.2307\/2267046","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1949,9]]}}}