Programs, grammars and arguments: a personal view of some connections between computation, language and logic

The question of what is an effective process or algorithm arose form the statement of Hilbert´s tenth problem. It was soon seen to be related to the question of which numerical functions Nⁿ→N are computable. Among the early answers to this question the author singles out three. A numerical function is computable if and only if it is recursive, it is computable on a Turing machine, or it is definable in the untyped λ-calculus. Some aspects of the relevance of these three notions of computability to linguistics and logic, in particular, categorial logic, are discussed