{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T02:30:34Z","timestamp":1648780234019},"reference-count":11,"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":9050,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1989,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Because the main difference between combinatory weak equality and <jats:italic>\u03bb\u03b2<\/jats:italic>-equality is that the rule<\/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=\"S0022481200027213_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>is valid for the latter but not the former, it is easy to assume that another way of defining combinatory <jats:italic>\u03b2<\/jats:italic>-equality is to add rule (<jats:italic>\u03be<\/jats:italic>) to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in (<jats:italic>\u03be<\/jats:italic>) very carefully. If one tries to use one of the more common abstraction algorithms, the result will be an equality, =<jats:italic><jats:sub>\u03be<\/jats:sub><\/jats:italic>, that is either equivalent to <jats:italic>\u03b2\u03b7<\/jats:italic>-equality (and so strictly stronger than <jats:italic>\u03b2<\/jats:italic>-equality) or else strictly weaker than <jats:italic>\u03b2<\/jats:italic>-equality. This paper will study the relations =<jats:italic><jats:sub>\u03be<\/jats:sub><\/jats:italic> for several commonly used abstraction-algorithms, distinguish between them, and axiomatize them.<\/jats:p>","DOI":"10.2307\/2274872","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:30:21Z","timestamp":1146954621000},"page":"590-607","source":"Crossref","is-referenced-by-count":0,"title":["On adding (<i>\u03be<\/i>) to weak equality in combinatory logic"],"prefix":"10.1017","volume":"54","author":[{"given":"Martin W.","family":"Bunder","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J. Roger","family":"Hindley","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan P.","family":"Seldin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200027213_ref011","first-page":"902","volume":"48","author":"Seldin","year":"1983","journal-title":"Remarks on \u03b2-strong equality"},{"key":"S0022481200027213_ref010","doi-asserted-by":"publisher","DOI":"10.2307\/1968669"},{"key":"S0022481200027213_ref009","first-page":"375","volume-title":"To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism","author":"Lambek","year":"1980"},{"key":"S0022481200027213_ref007","volume-title":"Combinatory logic","volume":"II","author":"Curry","year":"1972"},{"key":"S0022481200027213_ref005","first-page":"7","article-title":"A simplification of the theory of combinators","volume":"7","author":"Curry","year":"1949","journal-title":"Synthese"},{"key":"S0022481200027213_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/2370716"},{"key":"S0022481200027213_ref001","volume-title":"The lambda calculus: its syntax and semantics","author":"Barendregt","year":"1984"},{"key":"S0022481200027213_ref004","first-page":"54","volume":"6","author":"Curry","year":"1941","journal-title":"Consistency and completeness of the theory of combinators"},{"key":"S0022481200027213_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/1968167"},{"key":"S0022481200027213_ref006","volume-title":"Combinatory logic","volume":"I","author":"Curry","year":"1958"},{"key":"S0022481200027213_ref008","volume-title":"Introduction to combinators and \u03bb-calculus","author":"Hindley","year":"1986"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200027213","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,19]],"date-time":"2019-05-19T20:26:38Z","timestamp":1558297598000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200027213\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1989,6]]}},"alternative-id":["S0022481200027213"],"URL":"https:\/\/doi.org\/10.2307\/2274872","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,6]]}}}