{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T01:29:50Z","timestamp":1648690190126},"reference-count":1,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":23568,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1949,9]]},"abstract":"<jats:p>The concept of a <jats:italic>recursively definite<\/jats:italic> predicate of natural numbers was introduced by F. B. Fitch in his <jats:italic>An extension of basic logic<\/jats:italic> as follows:<\/jats:p><jats:p>Every recursive predicate is recursively definite. If R(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic>) is recursively definite so is (E<jats:italic>y<\/jats:italic>)R(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>, <jats:italic>y<\/jats:italic>) and (<jats:italic>y<\/jats:italic>)R(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>\u22121<\/jats:sub>, <jats:italic>y<\/jats:italic>). If R is recursively definite and S is the proper ancestral of R, then S is recursively definite, where the proper ancestral of a relation is defined as follows: if R is of even degree, say 2<jats:italic>m<\/jats:italic>, then the proper ancestral of R is the relation S such that for all <jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<jats:sub>m<\/jats:sub><\/jats:italic>, <jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>y<jats:sub>m<\/jats:sub><\/jats:italic>, S(<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<jats:sub>m<\/jats:sub><\/jats:italic>, <jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>y<jats:sub>m<\/jats:sub><\/jats:italic>) is true if and only if there is a finite sequence of sequences (<jats:italic>z<\/jats:italic><jats:sub>11<\/jats:sub>, \u2026, <jats:italic>z<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>1<\/jats:sub>), (<jats:italic>z<\/jats:italic><jats:sub>12<\/jats:sub>, \u2026, <jats:italic>Z<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>2<\/jats:sub>), \u2026, (<jats:italic>z<\/jats:italic><jats:sub>1<jats:italic>k<\/jats:italic><\/jats:sub>, \u2026, <jats:italic>Z<jats:sub>mk<\/jats:sub><\/jats:italic>) such that R(<jats:italic>Z<\/jats:italic><jats:sub>11<\/jats:sub>, \u2026, <jats:italic>Z<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>1<\/jats:sub>, <jats:italic>z<\/jats:italic><jats:sub>12<\/jats:sub>, \u2026, <jats:italic>z<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>2<\/jats:sub>), R(<jats:italic>z<\/jats:italic><jats:sub>12<\/jats:sub>, \u2026, <jats:italic>z<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>2<\/jats:sub>, <jats:italic>z<\/jats:italic><jats:sub>13<\/jats:sub>, \u2026, <jats:italic>z<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>3<\/jats:sub>), \u2026, R(<jats:italic>z<\/jats:italic><jats:sub>1<\/jats:sub>,<jats:sub><jats:italic>k<\/jats:italic>\u22121<\/jats:sub>, <jats:italic>z<\/jats:italic><jats:sub>1k<\/jats:sub>, \u2026, <jats:italic>Z<\/jats:italic><jats:sub>mk<\/jats:sub>) are all true, where (<jats:italic>z<\/jats:italic><jats:sub>11<\/jats:sub>, \u2026, <jats:italic>z<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>1<\/jats:sub>) is (<jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<jats:sub>m<\/jats:sub><\/jats:italic>) and (<jats:italic>z<\/jats:italic><jats:sub>1k<\/jats:sub>, \u2026, <jats:italic>z<jats:sub>mk<\/jats:sub><\/jats:italic>) is (<jats:italic>y<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>y<\/jats:italic><jats:sub>m<\/jats:sub>).<\/jats:p><jats:p>An arithmetic predicate is one which is definable in terms of the operations \u2018+\u2019 and \u2018\u00b7\u2019 of elementary arithmetic, the connectives of the classical prepositional calculus, and quantifiers.<\/jats:p>","DOI":"10.2307\/2267047","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:06:41Z","timestamp":1146942401000},"page":"175-176","source":"Crossref","is-referenced-by-count":1,"title":["Note on an idea of Fitch"],"prefix":"10.1017","volume":"14","author":[{"given":"John R.","family":"Myhill","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200105705_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01700692"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200105705","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T10:29:32Z","timestamp":1559989772000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200105705\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1949,9]]},"references-count":1,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1949,9]]}},"alternative-id":["S0022481200105705"],"URL":"https:\/\/doi.org\/10.2307\/2267047","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1949,9]]}}}