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REVIEWS The vastness of natural languages. By D. Terence Langendoen and Paul M. Postal. Oxford: Blackwell, 1984. Pp. ix, 189. $35.95. Reviewed by Barbara Abbott, Michigan State University* This book is an argument that NLs (natural languages—but see below) do not form recursively enumerable sets: indeed, that they are too big to be considered sets at all. One consequence is that constructive (generative) grammars are inadequate for describing NLs. In broader terms, L&P take linguistics to be 'a logico-mathematical discipline' (159, fn. 1). Chap. 1, 'Set-theoretical background', emphasizes issues of cardinality. The general term 'collection' is used to include both sets and aggregates (e.g. the collection containing everything) which are too big to be considered as members of other collections without giving rise to inconsistency. The latter are called 'megacollections'. Cantor's Theorem, which plays a role in what follows, is sketched. As applied to sets, it says that the power set P(A), the set of all subsets of A, is of a higher cardinality than A. Chap. 2, 'The received position about NLs and their grammars', is an initial attack—on the grounds that adequate arguments have not been given—against traditional assumptions that sentences are finite in length, and that the sentences of an NL form (at least) a recursively enumerable set. Chap. 3, 'Sentence size bounds', contains two kinds of arguments that NL sentences may be not only infinitely long, but also of any transfinite length. The first holds that it would be arbitrary to set any bound, finite or transfinite, on sentence size, and that 'transfinite size laws complicate the theoretical account of NLs no less than finite size laws, and are equally subject to Occam's razor' (42). L&P thus fail to see a distinction between imposing a specific numerical limit on sentence size and simply requiring sentences to be finite in length. The second kind ofargument, illustrated in the following typical passage (43), is circular: 'But ... objects having all the defining conditions of sentencehood in an NL are sentences of that NL. Therefore, any restriction, finite or transfinite, on the length of NL sentences yields a framework unable to describe infinitely many well-formed sentences in every NL.' This assumes that being finite in length is not a defining condition of sentencehood. The main point of Chap. 4. 'The analogy with Cantor's results', is the 'NL Vastness Theorem', which states that NLs are megacollections, and hence of no fixed cardinality. This theorem depends on a principle which L&P 'take as a truth about all NLs' (53), viz. that NLs are closed under coordinate compounding . That means that, for any set of sentences in a language L, the sentence formed by conjoining all the members of that set is itself a sentence of L. (The assumption that NL sentences may be infinitely long is crucial here— *I wish to thank Lionel Bender, Herb Hendry, Grover Hudson, Polly Jacobson, and Bill Rose for discussion while I was preparing this review. 154 REVIEWS155 since it is entailed by this closure principle, plus the fact that languages have infinite subsets.) To get from here to the Vastness Theorem, note that one can, using the closure principle, project from any set S of sentences a set S* with the same cardinality as P(S). Recall that every member of P(S) is a subset of the sentences of S. Let S* contain the sentences in S, which correspond oneto -one to the singleton sets ofP(S), plus one conjoined sentence corresponding to each (non-empty) non-singleton member of P(S). Now consider an infinite set Q of non-conjoined sentences of a language L. By the closure principle, all the sentences in the set Q* must also belong to L. If Q is of cardinality Xo (the cardinality of the natural numbers), then Q* will be of a higher cardinality (in fact Ni, the cardinality of the real numbers). But from Q* we can form the set (Q*)*, which will be of a yet higher cardinality. And again, by the closure principle, each sentence in (Q*)* must also belong to L. And so forth. QED. The major consequence given in Chap. 5, 'Implications', is...

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