What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what non-trivial identifications must hold between lambda terms, thought-of as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cut-elimination and asymmetrical interpretations of cut-free proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the Kelly-Lambek-Mac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...

In the late Fifties, Jim Lambek has started a line of investigation that accounts for the composition of form and meaning in natural language in deductive terms: formal grammar is presented as a logic — a system for reasoning about the basic form/meaning units of language and the ways they can be put together into wellformed structured configurations. The reception of the categorial grammar logics in linguistic circles has always been somewhat mixed: the mathematical elegance of the original system ([Lambek 58]) is counterbalanced by clear descriptive limitations, as Lambek has been the first to emphasize on a variety of occasions. As a result of the deepened understanding of the options for ‘substructural ’ styles of reasoning, the categorial architecture has been redesigned in recent work, in ways that suggest that mathematical elegance may indeed be compatible with linguistic sophistication. A careful separation of the logical and the structural components of the categorial inference engine leads to the identification of constants of grammatical reasoning. At the level of the basic rules of use and proof for these constants one finds an explanation for the uniformities in the composition of form and meaning across languages. Cross-linguistic variation in the realization of the form-meaning correspondence is captured in terms of structural inference packages, acting as plug-ins with respect to the base logic of the grammatical