Page 335 - Encyclopedia of Philosophy of Language

erences to a large and diverse literature constituting the foundations of the categorial approach. However, for current linguistic work in this rapidly evolving area one must turn to the journals and conference proceedings. Among the former, Linguistics and Phil- osophy has been a pioneer in presenting recent cat- egorial work. Among the latter, the annual proceedings of the West Coast Conference on Formal Linguistics is one important source. Much com- putational linguistic work in CG also remains uncol- lected, and here again one must turn to journals and conference proceedings, among which Computational Linguistics and the annual proceedings of the meetings of the Association for Computational Linguistics (and of its European Chapter) are important. A more com- plete bibliography can be found in Steedman (1993).
Bibliography
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Within theories of formal grammar, it has become customary to assume that linguistic expressions belong to syntactic 'categories,' whereas their interpretations inhabit semantic 'types.' This article aims to set out the basic ideas of logical syntax and semantics as they are found in categorial grammar and lambda calculus, respectively, and to focus on their convergence in theories of linguistic syntax and semantics.
The philosophical idea that the objects of thought form a hierarchy of categories is almost as old as philosophy itself. That this hierarchy may be based on function-argument relationships was realized by two eminent mathematicians/philosophers in the nine- teenth century, namely Frege and Husserl. Their influence may be traced in two subsequent streams of logical investigation. One is that of mathematical ontology, where Russell developed his theory of types which describes mathematical universes of individual objects, functions over these, functionals over func- tions, etc. Although Zermelo-style axiomatic set theory, rather than type theory, became the received
mathematical view, the type-theoretic tradition in the foundations of mathematics has persisted, inspiring important specific research programs such as lambda calculus in its wake. A second stream developed in philosophy, where Lesniewski and Ajdukiewicz developed a technical theory of categories which eventually became a paradigm of linguistics known as 'categorial grammar.'
A first convergence of both streams may already be observed in the work of Montague (1974), where mathematically inspired type theories supply the sem- antics for a categorially inspired syntax of natural languages. Since 1980, a more principled connection between categories and types has emerged from work building on Lambek's extension of classical categorial grammar.
1. Types
In 1902, Russell showed (in a famous letter to Frege) that a naive principle of set comprehension leads to a paradox for the set A. = {x\x$x}: AeA if and only if
. Russell solved the problem by simply excluding 313
Categories and Types H. L. W. Hendricks
Categories and Types