{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T03:51:31Z","timestamp":1787025091171,"version":"build-2736575974"},"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>\n                    An (\n                    <jats:italic>m<\/jats:italic>\n                    +\n                    <jats:italic>n<\/jats:italic>\n                    )-valued propositional calculus\n                    <jats:sup>2<\/jats:sup>\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200105717_inline1\"\/>\n                    may happen to be a subsystem of an\n                    <jats:italic>m<\/jats:italic>\n                    -valued propositional calculus\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200105717_inline2\"\/>\n                    , though the converse is never true. This fact may give us the impression that, as\n                    <jats:italic>m<\/jats:italic>\n                    grows, the content of\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200105717_inline2\"\/>\n                    becomes meagre. The present treatment is intended to remove this impression by constructing a complete,\n                    <jats:italic>m<\/jats:italic>\n                    -valued sub-system of any (\n                    <jats:italic>m<\/jats:italic>\n                    +\n                    <jats:italic>n<\/jats:italic>\n                    )-valued propositional calculus.\n                  <\/jats:p>\n                  <jats:p>In the following we adopt the customary, autonymous mode of speech according to which symbols belonging to the object calculi or languages are used in the syntactic language as names for themselves, and juxtaposition serves to denote juxtaposition.<\/jats:p>\n                  <jats:p>\n                    2.1 =\n                    <jats:sub>df<\/jats:sub>\n                    stands for definational identity in the syntactic language.\n                  <\/jats:p>\n                  <jats:p>2.11 \u2261 stands for definational identity in the object calculi.<\/jats:p>\n                  <jats:p>\n                    2.2 \u220a, \u2282, \u2229, {\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    , \u2026,\n                    <jats:italic>\n                      x\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    } are used in their meanings as customarily employed in the theory of sets\u2014\u220a for class membership, \u2282 for proper inclusion, \u2229 for the product operation of classes, {\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    , \u2026,\n                    <jats:italic>\n                      x\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    } for the class with\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    , \u2026,\n                    <jats:italic>\n                      x\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    as its only elements.\n                  <\/jats:p>\n                  <jats:p>\n                    2.3\n                    <jats:italic>x, y, z<\/jats:italic>\n                    are used as unspecified natural numbers including 0.\n                    <jats:italic>m, n, i, j<\/jats:italic>\n                    are used as unspecified natural numbers other than 0.\n                  <\/jats:p>\n                  <jats:p>\n                    2.401 Definition. \u03b4 =\n                    <jats:sub>df<\/jats:sub>\n                    as the function of two variables defined for any\n                    <jats:italic>x, y<\/jats:italic>\n                    such that\n                  <\/jats:p>\n                  <jats:p>\n                    <jats:disp-formula>\n                      <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200105717_eqnU1\"\/>\n                    <\/jats:disp-formula>\n                  <\/jats:p>\n                  <jats:p>\n                    2.41 Definition. For\n                    <jats:italic>m<\/jats:italic>\n                    \u2267 2, \u03b9\n                    <jats:sup>\n                      <jats:italic>m<\/jats:italic>\n                    <\/jats:sup>\n                    =\n                    <jats:sub>df<\/jats:sub>\n                    the function of two variables denned on the set {0, \u2026,\n                    <jats:italic>m<\/jats:italic>\n                    \u2212 1} such that \u03b9\n                    <jats:sup>\n                      <jats:italic>m<\/jats:italic>\n                    <\/jats:sup>\n                    (\n                    <jats:italic>x, y<\/jats:italic>\n                    ) =\n                    <jats:italic>y<\/jats:italic>\n                    \u2212 x for\n                    <jats:italic>x<\/jats:italic>\n                    \u2266\n                    <jats:italic>y<\/jats:italic>\n                    and \u03b9\n                    <jats:sup>\n                      <jats:italic>n<\/jats:italic>\n                    <\/jats:sup>\n                    (\n                    <jats:italic>x, y<\/jats:italic>\n                    ) = 0 for\n                    <jats:italic>x<\/jats:italic>\n                    &gt;\n                    <jats:italic>y<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.2307\/2267048","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T15:06:41Z","timestamp":1146928001000},"page":"177-181","source":"Crossref","is-referenced-by-count":4,"title":["<i>m<\/i>\n                    -valued sub-system of (\n                    <i>m + n<\/i>\n                    )-valued propositional calculus"],"prefix":"10.1017","volume":"14","author":[{"given":"Tzu-Hua","family":"Hoo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200105717_ref002","first-page":"30","article-title":"Untersuchungen \u00fcber den Aussagenkalk\u00fcl","volume":"23","author":"\u0141ukasiewicz","year":"1930","journal-title":"Comptes rendus des s\u00e9ances de l'Acad\u00e9mie des Sciences at des Letters de Varsovie"},{"key":"S0022481200105717_ref003","first-page":"61","volume":"10","author":"Rosser","year":"1945","journal-title":"Axiom schemes for m-valued propositional calculi"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200105717","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T06:29:27Z","timestamp":1559975367000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200105717\/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":["S0022481200105717"],"URL":"https:\/\/doi.org\/10.2307\/2267048","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1949,9]]}}}