## Abstract

We show that the elimination rule for the multiplicative (or intensional) conjunction ∧ is admissible in many important multiplicative substructural logics. These include LL_{m} (the multiplicative fragment of Linear Logic) and RMI_{m} (the system obtained from LL_{m} 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 LL_{m} together with the contraction axiom). Let SLL_{m} and SR_{m} be, respectively, the systems which are obtained from LL_{m} and R_{m} by adding this rule as a new rule of inference. The set of theorems of SR_{m} is a proper extension of that of R_{m}, but a proper subset of the set of theorems of RMI_{m}. Hence it still has the variable-sharing property. SR_{m} 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 SLL_{m} relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLL_{m} 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 BCK_{m}, in the second - the system SR_{m}). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.

Original language | English |
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Pages (from-to) | 831-859 |

Number of pages | 29 |

Journal | Journal of Symbolic Logic |

Volume | 63 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1998 |