{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,14]],"date-time":"2026-06-14T23:29:12Z","timestamp":1781479752555,"version":"3.54.1"},"reference-count":20,"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":2384,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2007,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic<jats:bold>UL<\/jats:bold>as Multiplicative additive intuitionistic linear logic<jats:bold>MAILL<\/jats:bold>extended with the prelinearity axiom<jats:italic>((A \u2192 B) \u2227 t) V ((B \u2192 A)\u2227 t)<\/jats:italic>. Axiomatic extensions of<jats:bold>UL<\/jats:bold>include known fuzzy logics such as Monoidal<jats:italic>t<\/jats:italic>-norm logic<jats:bold>MIX<\/jats:bold>and G\u00f6del logic<jats:bold>G<\/jats:bold>, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for<jats:bold>UL<\/jats:bold>and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].<\/jats:p>","DOI":"10.2178\/jsl\/1191333844","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T16:25:04Z","timestamp":1198081504000},"page":"834-864","source":"Crossref","is-referenced-by-count":130,"title":["Substructural fuzzy logics"],"prefix":"10.1017","volume":"72","author":[{"given":"George","family":"Metcalfe","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Franco","family":"Montagna","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200005107_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-006-0011-5"},{"key":"S0022481200005107_ref016","first-page":"900","volume":"48","author":"Pottinger","year":"1983","journal-title":"Uniform, cut-free formulations of T, S4 and S5 (abstract)"},{"key":"S0022481200005107_ref003","doi-asserted-by":"crossref","first-page":"315\u2013332","DOI":"10.3233\/FUN-2004-59401","article-title":"Analytic calculi for monoidal t-norm based logic","volume":"59","author":"Baaz","year":"2004","journal-title":"Fundamenta Informaticae"},{"key":"S0022481200005107_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-5300-3"},{"key":"S0022481200005107_ref001","first-page":"939\u2013951","volume":"52","author":"Avron","year":"1987","journal-title":"A constructive analysis of RM"},{"key":"S0022481200005107_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF01531058"},{"key":"S0022481200005107_ref004","first-page":"187\u2013201","volume-title":"Proceedings of CSL 2000","author":"Baaz","year":"2000"},{"key":"S0022481200005107_ref008","doi-asserted-by":"publisher","DOI":"10.1023\/A:1016548805869"},{"key":"S0022481200005107_ref005","first-page":"441\u2013453","article-title":"T-norm based logics with n-contraction","volume":"12","author":"Ciabattoni","year":"2002","journal-title":"Neural Network World"},{"key":"S0022481200005107_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-2217(98)00325-7"},{"key":"S0022481200005107_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(01)00098-7"},{"key":"S0022481200005107_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-007-0047-1"},{"key":"S0022481200005107_ref012","doi-asserted-by":"publisher","DOI":"10.1023\/A:1015122331293"},{"key":"S0022481200005107_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-004-0225-3"},{"key":"S0022481200005107_ref017","volume-title":"An introduction to substructural logics","author":"Restall","year":"1999"},{"key":"S0022481200005107_ref019","first-page":"437\u2013461","article-title":"The structure of residuated lattices","volume":"13","author":"Tsinakis","year":"2003","journal-title":"International Journal of Algebra and Computation"},{"key":"S0022481200005107_ref020","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(95)00133-6"},{"key":"S0022481200005107_ref014","doi-asserted-by":"publisher","DOI":"10.1145\/1071596.1071600"},{"key":"S0022481200005107_ref018","first-page":"851\u2013866","volume":"49","author":"Takeuti","year":"1984","journal-title":"Intuitionistic fuzzy logic and intuitionistic fuzzy set theory"},{"key":"S0022481200005107_ref015","first-page":"169\u2013201","volume":"50","author":"Ono","year":"1985","journal-title":"Logics without the contraction rule"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200005107","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,24]],"date-time":"2025-01-24T06:00:52Z","timestamp":1737698452000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200005107\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9]]},"references-count":20,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,9]]}},"alternative-id":["S0022481200005107"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1191333844","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,9]]}}}