{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T12:36:35Z","timestamp":1777638995046,"version":"3.51.4"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":20100,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1959,3]]},"abstract":"<jats:p>The present paper attempts to state and prove a completeness theorem for the system S5 of [1], supplemented by first-order quantifiers and the sign of equality. We assume that we possess a denumerably infinite list of individual variables <jats:italic>a, b, c, \u2026, x, y, z, \u2026, x<jats:sub>m<\/jats:sub>, y<jats:sub>m<\/jats:sub>, z<jats:sub>m<\/jats:sub><\/jats:italic>, \u2026 as well as a denumerably infinite list of <jats:italic>n<\/jats:italic>-adic predicate variables <jats:italic>P<jats:sup>n<\/jats:sup><\/jats:italic>, <jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic>, <jats:italic>R<jats:sup>n<\/jats:sup><\/jats:italic>, \u2026, <jats:italic>P<jats:sub>m<\/jats:sub><jats:sup>n<\/jats:sup><\/jats:italic>, <jats:italic>Q<jats:sub>m<\/jats:sub><jats:sup>n<\/jats:sup><\/jats:italic>, <jats:italic>R<jats:sub>m<\/jats:sub><jats:sup>n<\/jats:sup><\/jats:italic>,\u2026; if <jats:italic>n<\/jats:italic>=0, an <jats:italic>n<\/jats:italic>-adic predicate variable is often called a \u201cpropositional variable.\u201d A formula <jats:italic>P<jats:sup>n<\/jats:sup><\/jats:italic>(<jats:italic>x<jats:sub>1<\/jats:sub><\/jats:italic>, \u2026,<jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic>) is an <jats:italic>n<\/jats:italic>-adic prime formula; often the superscript will be omitted if such an omission does not sacrifice clarity.<\/jats:p>","DOI":"10.2307\/2964568","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T20:00:12Z","timestamp":1146945612000},"page":"1-14","source":"Crossref","is-referenced-by-count":514,"title":["A completeness theorem in modal logic"],"prefix":"10.1017","volume":"24","author":[{"given":"Saul A.","family":"Kripke","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200125058_ref006","first-page":"60","volume":"21","author":"Prior","year":"1956","journal-title":"Modality and quantification in S5"},{"key":"S0022481200125058_ref001","volume-title":"Symbolic logic","author":"Lewis","year":"1932"},{"key":"S0022481200125058_ref004","first-page":"309","volume-title":"Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Ajd Letterkunde","author":"Beth","year":"1955"},{"key":"S0022481200125058_ref008","volume-title":"A theory of formal deducibility","author":"Curry","year":"1950"},{"key":"S0022481200125058_ref005","first-page":"65","article-title":"Three grades of modal involvement","volume":"XIV","author":"Quine","journal-title":"Proceedings of the XIth International Congress of Philosophy"},{"key":"S0022481200125058_ref002","volume-title":"Logic for mathematicians","author":"Rosser","year":"1953"},{"key":"S0022481200125058_ref003","volume-title":"Introduction to semantics","author":"Carnap","year":"1942"},{"key":"S0022481200125058_ref007","volume-title":"Introduction to metamathematics","author":"Kleene","year":"1952"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200125058","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,5]],"date-time":"2019-06-05T20:13:36Z","timestamp":1559765616000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200125058\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1959,3]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1959,3]]}},"alternative-id":["S0022481200125058"],"URL":"https:\/\/doi.org\/10.2307\/2964568","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1959,3]]}}}