{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,13]],"date-time":"2025-09-13T15:55:42Z","timestamp":1757778942707},"reference-count":9,"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":13068,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,6]]},"abstract":"<jats:p>It has been shown that, for all rational numbers <jats:italic>r<\/jats:italic> such that 0\u2264 <jats:italic>r<\/jats:italic> \u2264 1, the \u2135<jats:sub>0<\/jats:sub>-valued \u0141ukasiewicz propositional calculus whose designated truth-values are those truth-values <jats:italic>x<\/jats:italic> such that <jats:italic>r<\/jats:italic> \u2264 <jats:italic>x<\/jats:italic> \u2264 1 may be formalised completely by means of finitely many axiom schemes and primitive rules of procedure. We shall consider now the case where <jats:italic>r<\/jats:italic> is rational, 0\u2265<jats:italic>r<\/jats:italic>\u22641 and the designated truth-values are those truth-values <jats:italic>x<\/jats:italic> such that <jats:italic>r<\/jats:italic>\u2264<jats:italic>x<\/jats:italic>\u22641.<\/jats:p><jats:p>We note that, in the subcase of the previous case where <jats:italic>r<\/jats:italic> = 1, a complete formalisation is given by the following four axiom schemes together with the rule of <jats:italic>modus ponens<\/jats:italic> (with respect to C),<\/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=\"S0022481200049549_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>the functor A being defined in the usual way. The functors B, K, L will also be considered to be defined in the usual way. Let us consider now the functor D<jats:sub>\u03b1\u03b2<\/jats:sub> such that if P, D<jats:sub>\u03b1\u03b2<\/jats:sub> take the truth-values <jats:italic>x<\/jats:italic>, d<jats:sub>\u03b1\u03b2<\/jats:sub>(<jats:italic>x<\/jats:italic>) respectively, \u03b1, \u03b2 are relatively prime integers and <jats:italic>r<\/jats:italic> = \u03b1\/\u03b2 then<\/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=\"S0022481200049549_eqnU2\" \/><\/jats:disp-formula><\/jats:p><jats:p>It follows at once from a theorem of McNaughton that the functor D<jats:sub>\u03b1\u03b2<\/jats:sub> is definable in terms of C and N in an effective way. If <jats:italic>r<\/jats:italic> = 0 we make the definition<\/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=\"S0022481200049549_eqnU3\" \/><\/jats:disp-formula><\/jats:p><jats:p>We note first that if <jats:italic>x<\/jats:italic> \u2264 \u03b1\/\u03b2 then d<jats:sub>\u03b1\u03b2<\/jats:sub>(<jats:italic>x<\/jats:italic>)\u2264(\u03b2 + 1)\u03b1\/\u03b2 \u2212 \u03b1 = \u03b1\/\u03b2. Hence<\/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=\"S0022481200049549_eqn1\" \/><\/jats:disp-formula><\/jats:p><jats:p>Let us now define the functions <jats:italic>d<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u03b1\u03b2<\/jats:sub>(<jats:italic>x<\/jats:italic>) (<jats:italic>n<\/jats:italic> = 0,1,\u2026) by<\/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=\"S0022481200049549_eqnU4\" \/><\/jats:disp-formula><\/jats:p><jats:p>Since<\/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=\"S0022481200049549_eqnU5\" \/><\/jats:disp-formula><\/jats:p><jats:p>it follows easily that<\/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=\"S0022481200049549_eqnU6\" \/><\/jats:disp-formula><\/jats:p><jats:p>and that<\/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=\"S0022481200049549_eqnU7\" \/><\/jats:disp-formula><\/jats:p><jats:p>Thus, if <jats:italic>x<\/jats:italic> is designated, <jats:italic>x<\/jats:italic> \u2212 \u03b1\/\u03b2 &gt; 0 and, if <jats:italic>n<\/jats:italic> &gt; \u2212 log(<jats:italic>x<\/jats:italic> \u2212 \u03b1\/\u03b2)\/log(\u03b2 + 1), then (\u03b2 + 1)<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>(<jats:italic>x<\/jats:italic>\u2212\u03b1\/\u03b2) &gt; 1.<\/jats:p>","DOI":"10.2307\/2272818","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:47:35Z","timestamp":1146937655000},"page":"207-210","source":"Crossref","is-referenced-by-count":3,"title":["Formalisations of further \u2135<sub>0<\/sub>-valued \u0141ukasiewicz propositional calculi"],"prefix":"10.1017","volume":"43","author":[{"given":"Alan","family":"Rose","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049549_ref004","first-page":"54","article-title":"The dependence of an axiom of \u0141ukasiewicz","volume":"87","author":"Meredith","year":"1958","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200049549_ref003","first-page":"1","volume":"16","author":"McNaughton","year":"1951","journal-title":"A theorem about infinite-valued sentential logic"},{"key":"S0022481200049549_ref007","volume-title":"Many-valued logics","author":"Rosser","year":"1952"},{"key":"S0022481200049549_ref001","first-page":"55","article-title":"Proof of an axiom of \u0141ukasiewicz","volume":"87","author":"Chang","year":"1958","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200049549_ref005","first-page":"269","volume-title":"Lecture Notes in Mathematics","volume":"70","author":"Rose","year":"1968"},{"key":"S0022481200049549_ref002","first-page":"30","article-title":"Untersuchungen \u00fcber den Aussagenkalk\u00fcl","volume":"23","author":"Lukasiewicz","year":"1930","journal-title":"Comptes Rendus des S\u00e9ances de la Soci\u00e9t\u00e9 des Sciences et des Lettres de Varsovie"},{"key":"S0022481200049549_ref008","volume-title":"Logic, semantics, metamathematics","author":"Tarski","year":"1956"},{"key":"S0022481200049549_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1958-0094299-1"},{"key":"S0022481200049549_ref009","first-page":"230","article-title":"On computable numbers, with an application to the Entscheidungsproblem","volume":"42","author":"Turing","year":"1936","journal-title":"Proceedings of the London Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049549","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:57:26Z","timestamp":1558972646000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049549\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,6]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1978,6]]}},"alternative-id":["S0022481200049549"],"URL":"https:\/\/doi.org\/10.2307\/2272818","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,6]]}}}