caa a22 4500 388040084 CHVBK 20180307125018.0 cr unu---uuuuu 161130e199809 xx s 000 0 eng 10.2307/2586715 doi S0022481200014626 pii (NATIONALLICENCE)cambridge-10.2307/2586715 Avron A. School of Mathematical Sciences, Sackxer Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail: aa@math.tau.ac.il Multiplicative conjunction and an algebraic meaning of contraction and weakening [Elektronische Daten] [A. Avron] We show that the elimination rule for the multiplicative (or intensional) conjunction Λ is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its converse, the mingle axiom.) An exception is R m (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLL m and SRm be, respectively, the systems which are obtained from LLm and Rm by adding this rule as a new rule of inference. The set of theorems of SRm is a proper extension of that of Rm , but a proper subset of the set of theorems of RMI m. Hence it still has the variable-sharing property. SRm 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 relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLLm 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 , in the second - the system SRm ). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts. Copyright © Association for Symbolic Logic 1998 The Journal of Symbolic Logic Cambridge University Press 63/3(1998-09), 831-859 0022-4812 63:3<831 1998 63 JSL https://doi.org/10.2307/2586715 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586715 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Avron A. School of Mathematical Sciences, Sackxer Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail: aa@math.tau.ac.il NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/3(1998-09), 831-859 0022-4812 63:3<831 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge