English and the class of context-free languages

Abstract

Let L range over all natural languages (NLs). For any L, one can consider two collections of strings of symbols, one consisting of all strings over the terminal vocabulary of L, call it W*(L), the other consisting of that always very proper subcollection of W*(L) consisting of all and only those members of W*(L) that are well-formed, that is, that correspond to sentences of L. Call the latter collection WF(L). During the early development of generative grammar, a number of attempts were made to show, for various choices of L, that WF(L) was not a context-free (CF) string collection. These attempts all had, naturally, a common logical structure. First, it was claimed that there was some mathematical property P which, if possessed by some collection of strings, C, rendered C non-CF. Second, it was claimed that WF(C) had P, so the conclusion followed. Two sorts of criticisms can be, and have • been, directed at such attempted demonstrations. One attacks the mathematical foundations and argues, for particular choices of P, that a collection manifesting P is not necessarily not CF. The other type of criticism admits that if a collection manifests a particular property P, it is thereby not CF, but contends that the WF(L)s claimed to manifest P in fact don't. A survey of the various attempts, from roughly 1960 to 1982, to prove for various L that WF(L) is not CF is provided in Pullum and Gazdar (1982). These authors conclude, justifiably we believe, that for one or the other of the reasons mentioned above, none of these attempts, including those by the present authors, stand up. Despite widespread belief to the contrary, as of 1982 there had been no demonstration that there is some NL L for which W F ( L ) is not CF. 2 However, Langendoen and Postal (1984) have obtained a result infinitely stronger than the claim that for some L, WF(L) is not CF. This work shows that for any L, WF(L) is a proper class, hence not a set, much less a recursively enumerable set. There is thus no question of WF(L) being CF. Moreover, WF(L) can then have no constructive characterization (generative grammar), although there is no reason to doubt that it can be given a nonconstructive characterization. But the demonstration of Langendoen and Postal (1984) is based on principles that determine WF(L) includes nonfinite strings corresponding to nonfinite (transfinite) sentences. It is the existence of such sentences that places complete NLs beyond generative (constructive) characterization. Nevertheless, as noted in Langendoen and Postal (1984: 103), this novel result still leaves entirely open the question of whether that subpart of WF(L) consisting of all and only the well-formed finite strings in W*(L) is CF. Let F(inite)WF(L) be that subcollection of WF(L) consisting of all and only the finite strings corresponding to the finite sentences of L. What follows shows that there are dialects of English, E1 and E2, such that: