{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,1,24]],"date-time":"2023-01-24T09:32:16Z","timestamp":1674552736573},"reference-count":13,"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":5671,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,9]]},"abstract":"Abstract<\/jats:title>We show that the elimination rule for the multiplicative (or intensional) conjunction \u039b is admissible in many important multiplicative substructural logics. These include LLm<\/jats:sub><\/jats:italic> (the multiplicative fragment of Linear Logic) and RMIm<\/jats:sub><\/jats:italic> (the system obtained from LLm<\/jats:sub><\/jats:italic> by adding the contraction axiom and its converse, the mingle axiom.) An exception is R<\/jats:italic>m<\/jats:sub> (the intensional fragment of the relevance logic R<\/jats:italic>, which is LLm<\/jats:sub><\/jats:italic> together with the contraction axiom). Let SLL<\/jats:italic>m<\/jats:sub> and SRm<\/jats:sub><\/jats:italic> be, respectively, the systems which are obtained from LLm<\/jats:sub><\/jats:italic> and Rm<\/jats:sub><\/jats:italic> by adding this rule as a new rule of inference. The set of theorems of SRm<\/jats:sub><\/jats:italic> is a proper extension of that of Rm<\/jats:sub><\/jats:italic>, but a proper subset of the set of theorems of RMI<\/jats:italic>m<\/jats:sub>. Hence it still has the variable-sharing property. SRm<\/jats:sub><\/jats:italic> has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLLm<\/jats:sub><\/jats:italic> relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLLm<\/jats:sub><\/jats:italic> corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCKm<\/jats:sub><\/jats:italic>, in the second - the system SRm<\/jats:sub><\/jats:italic>). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.<\/jats:p>","DOI":"10.2307\/2586715","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:01:28Z","timestamp":1146938488000},"page":"831-859","source":"Crossref","is-referenced-by-count":3,"title":["Multiplicative conjunction and an algebraic meaning of contraction and weakening"],"prefix":"10.1017","volume":"63","author":[{"given":"A.","family":"Avron","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200014626_ref010","volume-title":"Handbook of philosophical logic","volume":"III","author":"Dunn","year":"1986"},{"key":"S0022481200014626_ref005","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093635414"},{"key":"S0022481200014626_ref013","volume-title":"Lectures on linear logic","author":"Troelstra","year":"1992"},{"key":"S0022481200014626_ref002","volume-title":"Entailment","volume":"2","author":"Anderson","year":"1992"},{"key":"S0022481200014626_ref007","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/2.1.51"},{"key":"S0022481200014626_ref001","volume-title":"Entailment","volume":"1","author":"Anderson","year":"1975"},{"key":"S0022481200014626_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(87)90045-4"},{"key":"S0022481200014626_ref012","first-page":"23","article-title":"Axiomatization of partial system of three-valued calculus of propositions","volume":"1","author":"Soboci\u0144ski","year":"1952","journal-title":"Journal of Computing Systems"},{"key":"S0022481200014626_ref009","first-page":"1","volume-title":"Substructural logics","author":"Do\u0161en","year":"1993"},{"key":"S0022481200014626_ref008","doi-asserted-by":"publisher","DOI":"10.1093\/jigpal\/5.2.181"},{"key":"S0022481200014626_ref003","first-page":"334","volume":"49","author":"Avron","year":"1984","journal-title":"Relevant entailment\u2014semantics and formal systems"},{"key":"S0022481200014626_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(88)90037-0"},{"key":"S0022481200014626_ref006","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093635673"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200014626","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,10]],"date-time":"2019-05-10T20:20:58Z","timestamp":1557519658000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200014626\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,9]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1998,9]]}},"alternative-id":["S0022481200014626"],"URL":"https:\/\/doi.org\/10.2307\/2586715","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,9]]}}}