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- , Infinite Languages
- Jacob Hoeksema, Re: 2.663 Is Language Finite? Polite Forms

LINGUIST has once again made it possible for a significant issue to be dissected in a way which does not seem possible in any other forum I can think of! Avery Andrews and I seem to disagree on the status of the claim that NLs are sets of sentences without an upper bound on their lengths (but always of finite length). Yet, as he points out in his latest, the real issue he is concerned with is not cardinality but rather the goals of linguistic theory as originally defined by Chomsky. I, on the other hand, was concerned specifically with the question of cardinality (and more generally with the issue of how appropriate it is to treat mathematical idealizations as literal claims about the real world). Having said this, I would like to try and convince Avery and everyone else that we must make some distinctions that are not usually made in this area: (1) To say that the limits on actual sentence length are undefined is not the same thing as saying that there aren't any. Let me use a simple analogy to illustrate this point. The set of citizens of the United States is finite and reasonably well-defined. The set of black citizens of the U.S. is not well-defined at all, yet it is clearly a subset of the former, and hence also finite. (2) The fact that sentences as they become longer and more complex also become less acceptable can be captured in a number of ways, one of which is to divide up your theory into two components, one called competence, the other performance, and let the second only worry about this fact. However, this is NOT the only reasonable alternative. Another is to assume that the device (or gadgetry, as Avery calls it) that we have is one that does things in real time, and that competence is an idealization of it. Thus, the theory of competence is not about a specific mental organ separate from performance. It is about the one mental organ that there is (but it is idealized in certain important ways, which is perfectly reasonable, by the way. I have no quibble with idealizations.) (3) We are used to devices (automata) really which do not get tired, i.e., which can do the same thing over and over without any diminution of, let us say, acceptability (or grammaticality). This is something that is commonly assumed in mathematical theories of automata, but then ALL physical properties of automata are ignored there. It would be reasonable to say that we can develop a theory of automata that do get tired, which might offer a closer analogy to human behavior (i.e., be less idealized in this respect). (4) It is by no means obvious, as a linguistic fact, that people reject long or involved utterances merely because they are not comprehensible or whatever. I have often enough had the experience of an informant who would understand perfectly what was intended and yet opine that it was "too complicated" and offer a paraphrase. The famous Dutch (and German) V-Raising constructions are a perfect example. I have had Dutch informants who would understand an utterance like: dat Jan Marie Hans heeft helpen laten zwemmen but insist on a paraphrase. In general, it seems to me that there are plenty of arguments that the division we tend to make between competence facts and performance facts is arbitrary and that it forces us to miss generalizations. (5) One particular kind of problem that the usual idealization causes is that it forces to assume that the set of morphemes of a language is a finite list, whereas the set of sentences is infinite. Yet in many respects the set of morphemes is also what used to be called OPEN (as opposed to CLOSED). In a paper about to apper, Daniel Radzinski and I argue that linguistics really needs the notions of open (alias productive) vs. closed (alias unproductive) instead of infinite vs. finite, and we offer a precise mathematical model of what we mean (on this view, then, a language which reduplicates verb stems for some grammatical function can be shown to be non-context-free, for example, whereas on the traditional view, the set of stems being finite, the language would be finite, hence regular, hence CF). As I alluded to at one point, some people in CS have explored models in which the "language" is strictly speaking not a single set of string but a set of sets of strings (each of the latter sets being finite). It is difficult to distinguish this view from the traditional one on factual grounds, though there are some cases where there is a difference. This is essentially the approach Radzinski and I employ: the set of morphemes is finite at any given point in time, but it varies from one moment to the next (and there infinitely many possible such lexicons). Again, my point is that there are, mathematically, many more possibilities than linguists have tended to assume, and that some of them may be better than what we have assumed (and in other cases it does not matter much or at all what we assume). ? ?(7) Returning now to the question of infinite strings (which is I think an example of where it does not matter much what we assume), mathematicians and computer scientists have explored automata on infinite strings (in fact, a talk was given on this at the last MOL conference by a student of mine), which means that there is no basis for the statement that "Infinite sentence lengths are on the other hand not attainableMail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue

Avery Andrews claims (following here Chomsky) that linguistics is concerned with the structure of an actual device and that actual devices do not produce infinite output. Therefore, sentences of infinite length can be ignored and the idealization of choice for formal linguistics should be that sentences are finite sequences with no upper-bound imposed on their length (contra Langendoen, Postal). However, following the footsteps of Chomsky a little further, one may ask if this is not to confuse competence and performance. After all, while I may be able to decide that n I think (that I think) is grammatical for any n, including perhaps omega, I certainly won't be able to utter more than a small finite number of "that I think"'s. Indeed, in various contexts it even may make sense to talk about actually uttering infinite sequences. (For example, if time allows loops long enough to say "that I think", or if the speed of talking can be increased at ever shorter periods -- and why not, if we are allowed to ignore speech rates in our linguistic theorizing -- or, thirdly, if we allow one sentence to be uttered by more than one speaker -- e.g. by consecutive generations (assuming that physics does not dictate an end to our universe ;-)). I hope that these objections suffice to show that unless Andrews is willing to forego the compentence/performance distinction altogether, he cannot get away with the claim that the very nature of the game of linguistics prohibits the consideration of infinite sequences. The question is rather, whether it is worthwhile to ponder about infinity in linguistics. I for one consider it to be primarily of recreational value.Mail to author|Respond to list|Read more issues|LINGUIST home page|Top of issue