Multiplicative conjunction and an algebraic meaning of contraction and weakening

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Abstract

We show that the elimination rule for the multiplicative (or intensional) conjunction ∧ 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 Rm (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLLm 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 RMIm. 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.

Original languageEnglish
Pages (from-to)831-859
Number of pages29
JournalJournal of Symbolic Logic
Volume63
Issue number3
DOIs
StatePublished - Sep 1998

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