{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T08:10:10Z","timestamp":1707034210577},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6310,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,12]]},"abstract":"<jats:p>In [1], we established Gentzenizations for a good range of relevant logics with distribution, but, in the process, we added inversion rules, which involved extra structural connectives, and also added the sentential constant<jats:italic>t<\/jats:italic>. Instead of eliminating them, we used conservative extension results to relate them back to the original logics. In [4], we eliminated the inversion rules and<jats:italic>t<\/jats:italic>and established a much simpler Gentzenization for the weak sentential relevant logic<jats:italic>DW<\/jats:italic>, and also for its quantificational extension<jats:italic>DWQ<\/jats:italic>, but a restriction to normal formulae (defined below) was required to enable these results to be proved. This method was quite general and hope was expressed about extending it to other relevant logics.<\/jats:p><jats:p>In this paper, we develop an innovative method, which makes essential use of this restriction to normality, to establish two simple Gentzenizations for the normal formulae of the slightly weaker logic B, and then extend the method to other sentential contraction-less logics. To obtain the first of these Gentzenizations, for the logics<jats:italic>B<\/jats:italic>and<jats:italic>DW<\/jats:italic>, we remove the two branching rules (<jats:italic>F<\/jats:italic><jats:italic>&amp;<\/jats:italic>) and (<jats:italic>T\u2228<\/jats:italic>), together with the structural connective \u2018,\u2019, to simplify the elimination of the inversion rules and<jats:italic>t<\/jats:italic>. We then eliminate the rules (<jats:italic>T<\/jats:italic><jats:italic>&amp;<\/jats:italic>) and (<jats:italic>F\u2228<\/jats:italic>), thus reducing the Gentzen system to one containing only \u02dc and \u2192 and their four associated rules, and reduce the remaining types of structures to four simple finite types. Subsequently, we re-introduce (<jats:italic>T<\/jats:italic><jats:italic>&amp;<\/jats:italic>) and (<jats:italic>F\u2228<\/jats:italic>), and also (<jats:italic>F<\/jats:italic><jats:italic>&amp;<\/jats:italic>) and (<jats:italic>T\u2228<\/jats:italic>), to obtain the second Gentzenization, which contains \u2018,\u2019 but no structural rules.<\/jats:p>","DOI":"10.2307\/2275819","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:59:54Z","timestamp":1146956394000},"page":"1321-1346","source":"Crossref","is-referenced-by-count":0,"title":["Simple Gentzenizations for the normal formulae of contraction-less logics"],"prefix":"10.1017","volume":"61","author":[{"given":"Ross T.","family":"Brady","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200016923_ref010","volume-title":"Relevant logics and their rivals","volume":"1","author":"Routley","year":"1982"},{"key":"S0022481200016923_ref008","first-page":"45","article-title":"Simplified Gentzenizations for contraction-less logics","volume":"137","author":"Brady","year":"1992","journal-title":"Logique et Analyse"},{"key":"S0022481200016923_ref009","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1093\/oso\/9780198537779.003.0006","volume-title":"Substructural logics","author":"Kron","year":"1993"},{"key":"S0022481200016923_ref001","unstructured":"Brady R. T. , Gentzenizations of relevant logics with distribution, this Journal, forthcoming."},{"key":"S0022481200016923_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF00211185"},{"key":"S0022481200016923_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF00454743"},{"key":"S0022481200016923_ref002","unstructured":"Brady R. T. , Gentzenizations of relevant logics without distribution\u2014I, this Journal, forthcoming."},{"key":"S0022481200016923_ref003","unstructured":"Brady R. T. , Gentzenizations of relevant logics without distribution\u2014II, this Journal, forthcoming."},{"key":"S0022481200016923_ref004","volume-title":"Towards a simple Gentzenization for relevant logics","author":"Brady","year":"1993"},{"key":"S0022481200016923_ref005","first-page":"9","article-title":"Completeness proofs for the systems RM3 and BN4","volume":"25","author":"Brady","year":"1982","journal-title":"Logique et Analyse"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200016923","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:28:46Z","timestamp":1707031726000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200016923\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,12]]},"references-count":10,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1996,12]]}},"alternative-id":["S0022481200016923"],"URL":"https:\/\/doi.org\/10.2307\/2275819","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,12]]}}}