{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T16:57:02Z","timestamp":1773248222683,"version":"3.50.1"},"reference-count":3,"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":20646,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1957,9]]},"abstract":"<jats:p>In Herbrand's Theorem [2] or Gentzen's Extended Hauptsatz [1], a certain relationship is asserted to hold between the structures of A and A\u2032, whenever A <jats:italic>implies<\/jats:italic> A\u2032 (i.e., A \u2283 A\u2032 is valid) and moreover A is a conjunction and A\u2032 an alternation of first-order formulas in prenex normal form. Unfortunately, the relationship is described in a roundabout way, by relating A and A\u2032 to a quantifier-free tautology. One purpose of this paper is to provide a description which in certain respects is more direct. Roughly speaking, ascent to A \u2283 A\u2032 from a quantifier-free level will be replaced by movement from A to A\u2032 on the quantificational level. Each movement will be closely related to the ascent it replaces.<\/jats:p><jats:p>The new description makes use of a set <jats:italic>L<\/jats:italic> of rules of inference, the <jats:italic>L-rules<\/jats:italic>. <jats:italic>L<\/jats:italic> is <jats:italic>complete<\/jats:italic> in the sense that, if A is a conjunction and A\u2032 an alternation of first-order formulas in prenex normal form, and if A \u2283 A\u2032 is valid, then A\u2032 can be obtained from A by an <jats:italic>L-deduction<\/jats:italic>, i.e., by applications of <jats:italic>L<\/jats:italic>-rules only. The distinctive feature of <jats:italic>L<\/jats:italic> is that each <jats:italic>L<\/jats:italic>-rule possesses two characteristics which, especially in combination, are desirable. First, each <jats:italic>L<\/jats:italic>-rule yields only conclusions implied by the premisses.<\/jats:p>","DOI":"10.2307\/2963593","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T15:50:23Z","timestamp":1146930623000},"page":"250-268","source":"Crossref","is-referenced-by-count":302,"title":["Linear reasoning. A new form of the Herbrand-Gentzen theorem"],"prefix":"10.1017","volume":"22","author":[{"given":"William","family":"Craig","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200069784_ref002","first-page":"128","volume-title":"Recherches sur la th\u00e9orie de la d\u00e9monstration","author":"Herbrand"},{"key":"S0022481200069784_ref003","first-page":"550","volume-title":"Introduction to metamathematics","author":"Kleene","year":"1952"},{"key":"S0022481200069784_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01201363"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200069784","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T00:41:32Z","timestamp":1559868092000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200069784\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1957,9]]},"references-count":3,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1957,9]]}},"alternative-id":["S0022481200069784"],"URL":"https:\/\/doi.org\/10.2307\/2963593","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1957,9]]}}}