{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T12:36:40Z","timestamp":1777639000667,"version":"3.51.4"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":23446,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1950,1]]},"abstract":"<jats:p>The calculus generated by the addition of the postulate<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200105079_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>to S2 will, after Miss Alban, be called \u201cS6\u201d, the calculus generated by the addition of the same postulate to S3, \u201cS7\u201d, and the calculus generated by the addition of the postulate<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200105079_eqnU2\"\/><\/jats:disp-formula><\/jats:p><jats:p>to S3, \u201cS8\u201d. No interesting interpretation of the calculi S6\u20138 is known, but they are of some indirect interest because of the connections between them and the five calculi S1\u2013S5. Certain questions concerning them will be treated in the following. In \u00a71 it will be shown that every method of decision for S7 can be turned into a method of decision for S3, and in \u00a72 that the number of complete extensions of S3 is equal to the number of complete extensions of S7 plus one. In \u00a73 it will be shown that McKinsey's method of decision for S2 and S4 can be modified so as to cover S6.<\/jats:p><jats:p>The letters \u201c<jats:italic>P<\/jats:italic>\u201d, \u201c<jats:italic>Q<\/jats:italic>\u201d, \u201c<jats:italic>R<\/jats:italic>\u201d, and \u201c<jats:italic>S<\/jats:italic>\u201d will be employed as syntactic variables denoting formulas. Logical expressions will sometimes be used as selfdenotative. \u201c<jats:italic>C<\/jats:italic>\u201d and \u201c<jats:italic>C<\/jats:italic>\u201d stand for arbitrary calculi. \u201c<jats:italic>C<\/jats:italic> + <jats:italic>P<\/jats:italic>\u201d is the calculus which is the result of adding to <jats:italic>C<\/jats:italic> as new postulates all formulas which can be derived from <jats:italic>P<\/jats:italic> by substitution. With \u201csubstitution\u201d I mean the operation performed on propositional variables.<\/jats:p>","DOI":"10.2307\/2269232","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:07:57Z","timestamp":1146942477000},"page":"230-236","source":"Crossref","is-referenced-by-count":8,"title":["Results concerning the decision problem of Lewis's calculi S3 and S6"],"prefix":"10.1017","volume":"14","author":[{"given":"S\u00f6ren","family":"Halld\u00e9n","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200105079_ref007","first-page":"2","volume":"13","author":"McKinsey","year":"1948","journal-title":"Some theorems about the sentential calculi of Lewis and Heyting"},{"key":"S0022481200105079_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF01708856"},{"key":"S0022481200105079_ref003","first-page":"117","volume":"6","author":"McKinsey","year":"1941","journal-title":"A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology"},{"key":"S0022481200105079_ref002","first-page":"15","volume-title":"Ergebnisse eines mathematischen Kolloquiums","author":"Parry","year":"1931\u20131932"},{"key":"S0022481200105079_ref009","first-page":"42","volume-title":"On the number of complete extensions \u2026","author":"McKinsey"},{"key":"S0022481200105079_ref001","first-page":"498","volume-title":"Symbolic logic","author":"Lewis"},{"key":"S0022481200105079_ref008","first-page":"42","volume":"9","author":"McKinsey","year":"1944","journal-title":"On the number of complete extensions of the Lewis systems of sentential calculus"},{"key":"S0022481200105079_ref006","first-page":"141","volume":"4","author":"Parry","year":"1939","journal-title":"Modalities in the \u201cSurvey\u201d system of strict implication"},{"key":"S0022481200105079_ref005","first-page":"43","volume":"11","author":"Carnap","year":"1946","journal-title":"Modalities and quantification"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200105079","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T10:24:16Z","timestamp":1559989456000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200105079\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1950,1]]},"references-count":9,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1950,1]]}},"alternative-id":["S0022481200105079"],"URL":"https:\/\/doi.org\/10.2307\/2269232","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1950,1]]}}}