{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:34:58Z","timestamp":1759638898031},"reference-count":4,"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":11699,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,3]]},"abstract":"<jats:p>In 1959 M. Dummett [3] introduced the logic LC obtained by adding the axiom <jats:italic>ACpqCqp<\/jats:italic> to the formalization of the intuitionistic prepositional calculus having modus ponens and substitution as its rules of inference. Later on R. A. Bull [1] showed, by quite a roundabout way, that the implicational theses of LC were precisely the theses of the implicational calculus obtained by adding the axiom <jats:italic>CCCpqrCCCqprr<\/jats:italic> to the system of positive implication. In 1964 Bull [2] gave another proof, this time using results of Birkhoff concerning subdirectly reducible algebras.<\/jats:p><jats:p>The aim of this short note is to emphasize that the use of Gentzen's natural deduction systems (see Prawitz [4]) allows us to give a much more direct proof.<\/jats:p>","DOI":"10.2307\/2273391","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:58:32Z","timestamp":1146952712000},"page":"184-186","source":"Crossref","is-referenced-by-count":6,"title":["Implicational logics in natural deduction systems"],"prefix":"10.1017","volume":"47","author":[{"given":"E.G.K.","family":"L\u00f3pez-Escobar","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044807_ref001","first-page":"189","volume":"27","author":"Bull","year":"1962","journal-title":"The implicational fragment of Dummett's LC"},{"key":"S0022481200044807_ref004","volume-title":"Natural deduction, A proof theoretic study","author":"Prawitz","year":"1965"},{"key":"S0022481200044807_ref002","first-page":"33","volume":"29","author":"Bull","year":"1964","journal-title":"Some results for implicational calculi"},{"key":"S0022481200044807_ref003","first-page":"96","volume":"24","author":"Dummett","year":"1959","journal-title":"A propositional calculus with denumerable matrix"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044807","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T21:42:48Z","timestamp":1558734168000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044807\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1982,3]]}},"alternative-id":["S0022481200044807"],"URL":"https:\/\/doi.org\/10.2307\/2273391","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,3]]}}}