V. Michele Abrusci Claudia Casadio A Geometrical Representation of the Basic Laws of Categorial Grammar Abstract. We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci (On residuation, 2014) it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investiga- tion, we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, Geach laws and Switching laws. Keywords: Categorial grammar, Cyclic linear logic, Proof-net. 1. Introduction We propose a geometrical representation of the set of laws that are at the basis of Categorial Grammar and of the Lambek Calculus, developing our analysis in the framework of Cyclic Multiplicative Linear Logic, a purely non- commutative fragment of Linear Logic [1, 4, 5]. The rules we intend to inves- tigate are known as Residuation laws, Monotonicity laws, Application laws, Expansion laws,Type-raising laws, Composition laws, Geach laws, Switching laws [6–10, 13, 14, 18, 21]. 1.1. Formulation of Basic Laws in an Algebraic Style In an algebraic style, the basic laws of Categorial Grammar involve: – a binary operation on a set M , the product or the residuated operation, denoted by · ; – two binary residual operations on the same set M : \ (the left resid- ual operation of the product) and / (the right residual operation of the product); – a partial ordering on the same set M , denoted by ≤. Presented by Jacek Malinowski; Received 16 November, 2015 Studia Logica (2017) 105: 479–520 DOI: 10.1007/s11225-016-9698-4 c  Springer Science+Business Media Dordrecht 2017