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A prototypical example of categorial grammars are those based on Lambekcalculus, i.e. noncommutative intuitionistic linear logic. However, it has beennoted that purely noncommutative operations are often not sufficient formodeling even very simple natural language phenomena. Therefore a number ofalternative formalisms are considered in the literature: those using purely``commutative'' linear logic as well as combining (to some level) commutativeand non-commutative operations. The logic of tensor terms that we propose is avariant of such a combination. This logical calculus was designed specially fordefining categorial grammars. It contains Lambek calculus and multiplicativelinear logic as conservative fragments, yet the syntax is very simple, notdeparting much from that of multiplicative linear logic. The system is cut-freeand decidable, it has both intuitionistic and classical versions, besides it isequipped with a simple intuitive semantics, which is sound and complete.

Tensor term logic for categorial grammars: simple unification of commutative and noncommutative structure