Grammar and Logical Types

This paper represents categorial grammar as an implicational type theory in the spirit of Girard's linear logic, and illustrates linguistic applications of a range of type-constructors over and above implication. The type theoretic perspective is concerned with a correspondence between the logic of types, and computational operations over the objects inhabiting types. In linguistic applications this correspondence is between rules of grammar which are theorems of type inference, and compositional operations in the various algebras in which linguistic objects, i.e. signs, are assumed to have dimensions: syntax, semantics, etc. Rule-to-rule description is familiar from Mon-tague Grammar, but the idea here is to classify signs with structured types satisfying universal type laws determined by the semantics of the type connectives, in contrast to classiication by categories satisfying stipulated rules. On this scheme an object language is to be speciied by a type assignment to its nite vocabulary: a formal grammar is just a lexicon, plus perhaps some improper type axioms, and a grammar formalism is just a metalanguage of types with its uniform logic and interpretation is each linguistic dimension. The aim is to develop a language of types which has suucient transparency, sensitivity, and generality to implement interesting descriptions of natural language. The paper will illustrate sentence grammar, and also use of the semantic term algebra as a functional programming language for presentation of lexical semantics.