Meeting a Modality? Restricted Permutation for the Lambek Calculus

This paper contributes to the theory of hybrid substructural logics, i.e. weak logics given by a Gentzen-style proof theory in which there are constraints on the application of some structural rules. In particular, we address the question how to add an operator to the Lambek Calculus in order to give it a restricted access to the rule of Permutation, an extension which is partly motivated by linguistic applications. In line with tradition, we use the operator (∇) as a label telling us how the marked formula may be used, qua structural rules. New in our approach is that we do not see ∇ as a modality. Rather, we treat a formula ∇A as the meet of A with a special type Q. In this way we can make the specific structural behaviour of marked formulas more explicit. We define a minimal proof calculus for the system and prove some nice properties of it, like cut-elimination, decidability an an embedding result. The main motivation for our approach however is that we can supply the proof system with an intuitive semantics. ∗Department of Philosophy & Research Institute for Language and Speech, Utrecht University, c/o Heidelberglaan 8, 3584 CS Utrecht. E-mail: [email protected].