Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!mnetor!uunet!husc6!uwvax!rutgers!iuvax!pur-ee!uiucdcs!uxc.cso.uiuc.edu!osiris.cso.uiuc.edu!goldfain From: goldfain@osiris.cso.uiuc.edu Newsgroups: comp.ai Subject: Re: Langendoen and Postal (posted by: B Message-ID: <8300012@osiris.cso.uiuc.edu> Date: Thu, 12-Nov-87 03:34:00 EST Article-I.D.: osiris.8300012 Posted: Thu Nov 12 03:34:00 1987 Date-Received: Sun, 15-Nov-87 08:29:00 EST References: <8941@shemp.UCLA.EDU> Lines: 105 Nf-ID: #R:shemp.UCLA.EDU:8941:osiris.cso.uiuc.edu:8300012:000:6061 Nf-From: osiris.cso.uiuc.edu!goldfain Nov 12 02:34:00 1987 < /* Written 7:47 pm Nov 6, 1987 by spector@suvax1.UUCP in comp.ai */ < In article <8300011@osiris.cso.uiuc.edu>, goldfain@osiris.cso.uiuc.edu < comments on an article by berke@CS.UCLA.EDU: < > ... < > Such a set is countably infinite. Far from being a proper class, < > this is a very manageable set. If you move the discussion up to the < > cardinality of the set of "discourses", which would be finite sequences of < > strings in the language, you are still only up to the power set of the < > integers, which has the same cardinality as the set of Real numbers. < > Again, this is a set, and not a proper class. < > Mark Goldfain arpa: goldfain@osiris.cso.uiuc.edu < --------------------- < The set of all finite sequences of finite strings in a language (the set < of "discourses") is still just a countably infinite set (assuming that the < alphabet is finite or countably infinite, of course). The set of infinite < sequences of finite strings is uncountable, with the same cardinality as the < set of real numbers, as is the set of infinite strings. < ... < I certainly agree with the general objections raised to the idea that < natural languages are uncountably large (or, worse yet, proper classes), < although I haven't read the book in question. Maybe somebody can state < more precisely what the book claimed, but it seems at first glance to < indicate a lack of understanding of modern set theory. < ... < Mitchell Spector |"Give me a < Dept. of Computer Science & Software Eng., Seattle Univ.| ticket to < Path: ...!uw-beaver!uw-entropy!dataio!suvax1!spector | Mars!!" < or: dataio!suvax1!spector@entropy.ms.washington.edu | - Zippy the Pinhead < ------------------ OOPS---OOPS---OOPS---OOPS---OOPS---OOPS----OOPS---OOPS----OOPS---OOPS----OOPS Mitchell Spector is correct! I must have been thinking very sluggishly, and I hope none of my professors (past or present) is watching. In any case, we still have our basic objection, only it is now greatly strengthened. Indeed, it is a well-known and oft-used proposition in computing theory that the number of things that can be said in a finite-alphabet language is COUNTABLE. ENDOOPS---ENDOOPS---ENDOOPS---ENDOOPS---ENDOOPS---ENDOOPS---ENDOOPS---ENDOOPS Continuing the "attack" ... recall that the original posting said: > /* ---------- "Langendoen and Postal ---------- */ > ... > Their basic proof/conclusion holds that natural languages, as linguistics > construes them (as products of grammars), are what they call > mega-collections, Quine calls proper classes, and some people hold cannot > exist. That is, they maintain that (1) Sentences cannot be excluded from > being of any, even transfinite size, by the laws of a grammar, and (2) > Collections of these sentences are bigger than even the continuum. They are > the size of the collection of all sets: too big to be sets. ... My follow-up and Mitchell Spector's note explained why this statement is incorrect. Note that from the above, the issue is not really "real English", but the size of a language as SPECIFIED BY A GRAMMAR. I then asked: > I haven't seen the book you cite. They must make some argument as to why > they think natural languages (or linguistic theories about them) > admit infinite sentences. Even given that, we would have only the Reals > (i.e. the "Continuum") as a cardinality without some further surprising > claims. Can you summarize their argument (if it exists) ? The only response so far that hints as to their arguments is that posted by lee@uhccux.UUCP : > Concerning the length of sentences, I think Postal and Langendoen are not > very persuasive. Most of their arguments are to the effect that previously > given attempted demonstrations that sentences cannot be of infinite length > are incorrect. I think they make that point very well. But obviously this > is not enough To show that one should assume some sentences of infinite > length. > Greg Lee, lee@uhccux.uhcc.hawaii.edu Still, without more specifics, this whole argument may continue to be way out in left field. At the risk of knocking down a mere "straw man", consider: If these experiments set a size, say 100 words (or more reasonably 100 constituents), then proceeded to test subjects, finding that this was an insufficient upper bound, then tried 200, failing again, then tried 500, still finding subjects who thought the sentences were grammatical, then: 1) I am really, really surprised. I think the experiments should be replicated simply because I find the above too ludicrous to swallow. 2) We still have a LONG ways to go before we conclude that NO upper bound exists. (It is like the humorous story about the "engineer's proof" that all odd numbers > 1 are prime : "Let's see, 3 is prime, 5 is prime, 7 is prime, ... Yep! All of 'em must be!") Let them test a LARGE number like 10,000 and get back to me when the results come in ... 3) Finally, the notion of a set with no upper bound is a different mathematical beast than a set containing non-finite elements! Consider the positive integers; there is no largest integer, but that does not mean that any of them must be infinite ... in fact, none of them are. Note: The last point is rather briefly stated - if you haven't run across it in a course or somewhere before, it deserves a moment or two to sink in. These concepts generally get their first airing in Calculus courses, and many students tend to really grasp them only after about their 2nd Calculus course. ----------------------------------- From here, further discussion seems pointless, unless some of the actual data and claims from Langendoen and Postal are put forth. - Mark Goldfain