{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T12:41:41Z","timestamp":1648989701505},"reference-count":1,"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":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"<jats:p>This note shows that the inconsistency proof of the system <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049835_inline1\" \/> of illative combinatory logic given in [1] can be simplified as well as extended to the absolute inconsistency of a more general system.<\/jats:p><jats:p>One extension of the result in [1] lies in the fact that the following weakened form of the deduction theorem for implication will lead to the inconsistency:<\/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=\"S0022481200049835_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>Also the inconsistency follows almost as easily for<\/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=\"S0022481200049835_equ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>as it does for \u22a2 H<jats:sup>2<\/jats:sup><jats:italic>X<\/jats:italic> for arbitrary <jats:italic>X<\/jats:italic>, so we will consider the more general case.<\/jats:p><jats:p>The only properties we require other than (DT), (1) and Rule Eq for equality are modus ponens,<\/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=\"S0022481200049835_equ2\" \/><\/jats:disp-formula><\/jats:p><jats:p>and<\/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=\"S0022481200049835_eqnU3\" \/><\/jats:disp-formula><\/jats:p><jats:p>Let <jats:italic>G<\/jats:italic> = [<jats:italic>x<\/jats:italic>] H<jats:sup><jats:italic>n<\/jats:italic>\u22121<\/jats:sup><jats:italic>x<\/jats:italic>\u2283: . \u2026 H<jats:sup>2<\/jats:sup><jats:italic>x<\/jats:italic>\u2283 :H<jats:italic>x<\/jats:italic>\u2283 . <jats:italic>x<\/jats:italic> \u2283 <jats:italic>A<\/jats:italic>, where <jats:italic>A<\/jats:italic> is arbitrary. Then if Y is the paradoxical combinator and <jats:italic>X<\/jats:italic> = Y<jats:italic>G<\/jats:italic>, <jats:italic>X<\/jats:italic> = <jats:italic>GX<\/jats:italic>.<\/jats:p><jats:p>Now <jats:italic>X<\/jats:italic> \u2282 <jats:italic>X<\/jats:italic>, i.e.,<\/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=\"S0022481200049835_eqnU2\" \/><\/jats:disp-formula><\/jats:p>","DOI":"10.2307\/2271943","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:46:34Z","timestamp":1146951994000},"page":"1-2","source":"Crossref","is-referenced-by-count":3,"title":["On the inconsistency of systems similar to"],"prefix":"10.1017","volume":"43","author":[{"given":"M. W.","family":"Bunder","sequence":"first","affiliation":[]},{"given":"R. K.","family":"Meyer","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049835_ref001","first-page":"467","volume":"41","author":"Bunder","year":"1976","journal-title":"The inconsistency of"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049835","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T20:19:31Z","timestamp":1558988371000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049835\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,3]]},"references-count":1,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1978,3]]}},"alternative-id":["S0022481200049835"],"URL":"https:\/\/doi.org\/10.2307\/2271943","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,3]]}}}