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We study relationship between first order multiplicative linear logic (MLL1),which has been known to provide representations to different categorialgrammars, and the recently introduced extended tensor type calculus (ETTC). Weidentify a fragment of MLL1, which seems sufficient for many grammarrepresentations, and establish a correspondence between ETTC and this fragment.The system ETTC, thus, can be seen as an alternative syntax and intrinsicdeductive system together with a geometric representation for the latter. Wealso give a natural deduction formulation of ETTC, which might be convenient.

First order linear logic and tensor type calculus for categorial grammars