{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T06:00:46Z","timestamp":1780725646909,"version":"3.54.1"},"reference-count":4,"publisher":"Wiley","issue":"1-2","license":[{"start":{"date-parts":[[2016,1,19]],"date-time":"2016-01-19T00:00:00Z","timestamp":1453161600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2016,2]]},"abstract":"<jats:p>Consider the problem which set <jats:italic>V<\/jats:italic> of propositional variables suffices for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201400099-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201400099:malq201400099-math-0001\"\/> whenever <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201400099-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201400099:malq201400099-math-0002\"\/>, where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201400099-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201400099:malq201400099-math-0003\"\/>, and \u22a2<jats:sub>c<\/jats:sub> and \u22a2<jats:sub>i<\/jats:sub> denote derivability in classical and intuitionistic implicational logic, respectively. We give a direct proof that stability for the final propositional variable of the (implicational) formula <jats:italic>A<\/jats:italic> is sufficient; as a corollary one obtains Glivenko's theorem. Conversely, using Glivenko's theorem one can give an alternative proof of our result. As an alternative to stability we then consider the Peirce formula <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201400099-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201400099:malq201400099-math-0004\"\/>. It is an easy consequence of the result above that adding a single instance of the Peirce formula suffices to move from classical to intuitionistic derivability. Finally we consider the question whether one could do the same for minimal logic. Given a classical derivation of a propositional formula not involving \u22a5, which instances of the Peirce formula suffice as additional premises to ensure derivability in minimal logic? We define a set of such Peirce formulas, and show that in general an unbounded number of them is necessary.<\/jats:p>","DOI":"10.1002\/malq.201400099","type":"journal-article","created":{"date-parts":[[2016,1,19]],"date-time":"2016-01-19T19:31:34Z","timestamp":1453231894000},"page":"94-101","source":"Crossref","is-referenced-by-count":8,"title":["Embedding classical in minimal implicational logic"],"prefix":"10.1002","volume":"62","author":[{"given":"Hajime","family":"Ishihara","sequence":"first","affiliation":[{"name":"School of Information Science Japan Advanced Institute of Science and Technoloty 1\u20101 Asahidai Nomi Ishikawa 923\u20101292 Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Helmut","family":"Schwichtenberg","sequence":"additional","affiliation":[{"name":"Mathematisches Institut Ludwig\u2010Maximilians\u2010Universit\u00e4t M\u00fcnchen Theresienstra\u00dfe 39 80333 M\u00fcnchen Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2016,1,19]]},"reference":[{"key":"e_1_2_6_2_1","first-page":"183","article-title":"Sur quelques points de la logique de M. Brouwer","volume":"15","author":"Glivenko V.","year":"1929","journal-title":"Bull. Soc. Math. Belg."},{"key":"e_1_2_6_3_1","doi-asserted-by":"publisher","DOI":"10.2168\/LMCS-10(3:1)2014"},{"key":"e_1_2_6_4_1","volume-title":"Proofs and Computations, Perspectives in Logic","author":"Schwichtenberg H.","year":"2012"},{"key":"e_1_2_6_5_1","volume-title":"Cambridge Tracts in Theoretical Computer Science Vol. 43","author":"Troelstra A. S.","year":"2000"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201400099","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201400099","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201400099","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,3]],"date-time":"2023-10-03T20:24:17Z","timestamp":1696364657000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201400099"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1,19]]},"references-count":4,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2016,2]]}},"alternative-id":["10.1002\/malq.201400099"],"URL":"https:\/\/doi.org\/10.1002\/malq.201400099","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1,19]]}}}