{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T21:50:42Z","timestamp":1648936242299},"reference-count":3,"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":20646,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1957,9]]},"abstract":"<jats:p>There are axiomatizations of the sentential calculus which, in effect, assert the inferential equivalence (mutual deducibility) of classes of sentences. A well known axiomatization of this sort consists of<\/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=\"S0022481200069759_eqnU01\" \/><\/jats:disp-formula><\/jats:p><jats:p>Another one, closely connected with A1\u2013A3, is comprised of<\/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=\"S0022481200069759_eqnU02\" \/><\/jats:disp-formula><\/jats:p><jats:p>B1\u2013B3 jointly assert the inferential equivalence of a sentence of the form (<jats:italic>p<\/jats:italic> \u2283 <jats:italic>q<\/jats:italic>) \u2283 <jats:italic>r<\/jats:italic> and two sentences of the forms <jats:italic>q<\/jats:italic> \u2283 <jats:italic>r<\/jats:italic> and \u223c<jats:italic>p<\/jats:italic> \u2283 <jats:italic>r<\/jats:italic> respectively. This axiomatization requires the rule of substitution and the rule of detachment for conditionals.<\/jats:p><jats:p>The existence of such axiomatizations suggests a possibility of formulating natural deduction by means of inferential equivalence instead of the usual one-sided inference, customarily represented by \u2018\u22a6\u2019. \u2018<jats:italic>H<\/jats:italic>\u2019 will stand here for inferential equivalence. It may be looked upon as a combination of signs \u2018\u22a6\u2019 and \u2018\u22a3\u2019. \u2018<jats:italic>H<\/jats:italic>\u2019 is thought of as a metasystematic constant functor of two arguments each of which is a class of names of sentences. For purposes of this paper the first argument may be just a single name of a sentence, the second a pair (not necessarily ordered) of names of sentences. The role \u2018<jats:italic>H<\/jats:italic>\u2019 plays is determined by **1 \u2013*5 below. Besides that \u2018<jats:italic>H<\/jats:italic>\u2019 remains here uninterpreted. In particular, it is not directly postulated that it is an identity relation.<\/jats:p>","DOI":"10.2307\/2963590","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:50:23Z","timestamp":1146945023000},"page":"237-240","source":"Crossref","is-referenced-by-count":0,"title":["Inferential equivalence and natural deduction"],"prefix":"10.1017","volume":"22","author":[{"given":"H.","family":"Hi\u017c","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200069759_fn01","first-page":"153","article-title":"Ein Vollst\u00e4ndigkeitsbeweis des zweiwertigen Aussagenkalk\u00fcl","volume":"24","author":"\u0141ukasiewicz","year":"1931","journal-title":"Comptes rendus des s\u00e9ances de la Soci\u00e9t\u00e9 des Sciences et des Lettres de Varsovie"},{"key":"S0022481200069759_fn03","first-page":"160","volume":"1","author":"Church","year":"1956","journal-title":"Introduction to mathematical logic"},{"key":"S0022481200069759_fn03","volume-title":"Elementy logiki matematycznej","author":"\u0141ukasiewicz","year":"1929"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200069759","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T04:41:37Z","timestamp":1559882497000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200069759\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1957,9]]},"references-count":3,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1957,9]]}},"alternative-id":["S0022481200069759"],"URL":"https:\/\/doi.org\/10.2307\/2963590","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1957,9]]}}}