Partially Commutative Linear Logic and Lambek Caculus with Product: Natural Deduction, Normalisation, Subformula Property

This article defines and studies a natural deduction system for partially commutative intuitionistic multiplicative linear logic, that is a combination of intuitionistic commutative linear logic with the Lambek calculus, which is noncommutative, and was first introduced as a sequent calculus by de Groote. In this logic, the hypotheses are endowed with a series-parallel partial order: the parallel composition corresponds to the commutative product, while the series composition corresponds to the noncommutative product. The relation between the two products is that a rule, called entropy, allows us to replace a series-parallel order with a sub series-parallel order — this rule (already studied by Retoré) strictly extends the entropy rule initially introduced by de Groote. A particular subsystem emerges when hypotheses are totally ordered: this is Lambek calculus with product, and when orders are empty it is is multiplicative linear logic. So far only the sequent calculus and cut-elimination have been properly studied. In this article, we define natural deduction with product elimination rules as Abramsky proposed long ago. We then give a brief illustration of its application to computational linguistics and prove normalisation, firstly for the Lambek calculus with product and then for the full partially ordered calculus. We show that normal proofs enjoy the subformula property, thus yielding another proof of decidability of these calculi. The authors wish thank Jiří Maršík (LORIA, Nancy) for his prompt and efficient rereading. Vol. 1 No. 1 2014 IFCoLog Journal of Logics and their Applications