{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T23:25:23Z","timestamp":1648855523014},"reference-count":5,"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>This note shows that the argument used in the proof of the inconsistency of Curry's system <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049586_inline1\" \/> (see [1]) can also be applied to Fitch's system <jats:italic>QD<\/jats:italic> (see [3, Chapter 6]). As one vital rule of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049586_inline1\" \/> is not present in <jats:italic>QD<\/jats:italic> this argument does not lead to an actual contradiction, but it does lead to a theorem which is not a proposition if the system is consistent.<\/jats:p><jats:p>The method used below is that of [2], which is simpler than that of [1].<\/jats:p><jats:p>The axioms and rules of <jats:italic>QD<\/jats:italic> that we require are also present in [1] and [2] provided that Fitch's <jats:italic>D<\/jats:italic> is replaced by H. These axioms and rules are the following:<\/jats:p><jats:p>DD int: \u22a6 <jats:italic>D<\/jats:italic>(<jats:italic>Da<\/jats:italic>),<\/jats:p><jats:p>m p: If \u22a6 <jats:italic>a<\/jats:italic> \u2283 <jats:italic>b<\/jats:italic><jats:sup>2<\/jats:sup> and \u22a6 <jats:italic>a<\/jats:italic> then \u22a6 <jats:italic>b<\/jats:italic>, and<\/jats:p><jats:p>res imp int: If <jats:italic>a<\/jats:italic> \u22a6 <jats:italic>b<\/jats:italic> then <jats:italic>Da<\/jats:italic> \u22a6 <jats:italic>a<\/jats:italic> \u2283 <jats:italic>b<\/jats:italic>.<\/jats:p><jats:p>Let <jats:italic>a<\/jats:italic> be arbitrary and let <jats:italic>G<\/jats:italic> be [<jats:italic>x<\/jats:italic>](<jats:italic>Dx<\/jats:italic> \u2283 (<jats:italic>x<\/jats:italic> \u2283 <jats:italic>a<\/jats:italic>)). Then let <jats:italic>X<\/jats:italic> be <jats:italic>BWBG<\/jats:italic>(<jats:italic>BWBG<\/jats:italic>). We then have the following proof:<\/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=\"S0022481200049586_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>Thus, we have \u22a6 X. Now suppose we also had \u22a6 <jats:italic>DX<\/jats:italic>. Then by the method of the innermost subproof in the above proof, we would have \u22a6 <jats:italic>a<\/jats:italic>, and since <jats:italic>a<\/jats:italic> is<\/jats:p><jats:p>arbitrary the system would be inconsistent. Hence, if <jats:italic>QD<\/jats:italic> is consistent, we do not have \u22a6 <jats:italic>DX<\/jats:italic>, and so <jats:italic>X<\/jats:italic> is a theorem which is not a proposition.<\/jats:p>","DOI":"10.2307\/2272822","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:47:35Z","timestamp":1146937655000},"page":"247-249","source":"Crossref","is-referenced-by-count":0,"title":["Some anomalies in Fitch's system <i>QD<\/i>"],"prefix":"10.1017","volume":"43","author":[{"given":"M. W.","family":"Bunder","sequence":"first","affiliation":[]},{"given":"Jonathan P.","family":"Seldin","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049586_ref002","first-page":"1","volume":"43","author":"Bunder","year":"1978","journal-title":"On the inconsistency of systems similar to"},{"key":"S0022481200049586_ref004","unstructured":"Fitch Frederic B. , An extension of C\u0394 (to appear)."},{"key":"S0022481200049586_ref005","volume-title":"Excluded middle and the paradoxes","author":"Fitch","year":"1975"},{"key":"S0022481200049586_ref001","first-page":"467","volume":"41","author":"Bunder","year":"1976","journal-title":"The inconsistency of"},{"key":"S0022481200049586_ref003","volume-title":"Elements of combinatory logic","author":"Fitch","year":"1974"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049586","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T15:57:34Z","timestamp":1558972654000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049586\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,6]]},"references-count":5,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1978,6]]}},"alternative-id":["S0022481200049586"],"URL":"https:\/\/doi.org\/10.2307\/2272822","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,6]]}}}