From - Tue Mar 12 10:27:12 1996 Received: from VMS.DC.LSOFT.COM (vms.dc.lsoft.com [205.186.43.2]) by locke.ccil.org (8.6.9/8.6.10) with ESMTP id QAA11064 for ; Mon, 11 Mar 1996 16:42:57 -0500 Message-Id: <199603112142.QAA11064@locke.ccil.org> Received: from PEACH.EASE.LSOFT.COM (205.186.43.4) by VMS.DC.LSOFT.COM (LSMTP for OpenVMS v1.0a) with SMTP id A570F5BC ; Mon, 11 Mar 1996 16:45:40 -0500 Date: Mon, 11 Mar 1996 13:39:05 -0800 Reply-To: "John E. Clifford" Sender: Lojban list From: "John E. Clifford" Subject: fuzzy threads To: lojban list X-Mozilla-Status: 0001 Content-Length: 11024 I am still not back, but I had a quiet weekend to do a little catching up, so I read up on fuzzitude, a set of threads where I remember my name coming up from time to time. Herewith some scattered comments. Although the general theory of fuzzy sets allows characteristic functions which never take the values 0 or 1 (as well as other very strange sets), for most sets we meet in practical situations, most things would receive one or the other of these values. That is, even fuzzy sets are rarely fuzzy to the core (never getting to 1) or fuzzy to the horizon (never getting to 0); for most, the class of objects which take values in the internal open interval is rather small. One of the reasons that bivalent logic has endured so well is simply that it works most of the time. Indeed, being fair to Zoroastrianism would make the fuzzy part of natural languages belong more nearly to it: most of the world belongs absolutely either to The True or The False with a small segment (human souls in the religious case) that fall between the two. The size of the fractionally valued part varies a lot from set to set. There are very few things, I think, that are not either definitely cows or definitely non-cows -- a few mutants, accidents, and hybrids (though these tend to get absorbed). On the other hand, as a matter of statistical theory, a fairly large percentage of a population P will be in the fractional area for "tall for P." Even in this latter case, though, most members of P fall definitely in or definitely out, get a 1 or a 0 from the characteristic function. To think otherwise is to confuse the categorical notion of "tall" with the the relative notion "taller than." The fact that one member of P is taller than another does not necessarily mean that the first is more into (gets a higher value from the characteristic of) the set than the other, they may both be in the 0 or 1 category. Of course, if one does get a higher characteristic value than the other, he must be taller (on this notion -- I recognize that there is another notion which takes relative width into account). And the fact that the two are of different height does allow that they may get different scores for some related sets: "very tall," "sorta tall," "short (polar opposite of tall)" and so on. Much of the talk about Guttmann scales, though carried out in the fuzzy threads, seems to me to be more relevant to the comparative notions (which might, I suppose, be fuzzy, too, though the internal interval gets very little workout in typical cases of these sorts) or, rather, to the continua that underlie them. "Ugly" is not identical with 0 or even with very low scores on the characteristic function of "beautiful," although presumably every ugly thing gets a 0 of the characteristic of "beautiful." But so do many intermediate things, too beautiful to be ugly but still too ugly to be beautiful. (The doctor will know the common complaint that I may be better but I sure ain't well.). It is usually this whole spectrum that gets chopped up in some way; though it may be named for one of its poles, it is not generally chopping up the charateristic of just that pole. So the "beauty" (better "ugly-beautiful") spectrum, if scaled somehow, need not just look at values for the set "beautiful" or even just for that and "ugly," but may use any convenient factors that play a role (typically in both sets and points in between). The simplest would be a categorical scale based on the two pole sets: ugly, beautiful, and neither (which could, of course, be fuzzy or not). And the "neither" area might be subdivided ("homely, plain, attractive" comes to mind) and so on. Or, of course, we might skip the descriptive labels and just go for slices of the spectrum (I skip over the implausibility of there being a well-defined spectrum in this case, but we often operate as though we thought there were and it works linguistically) numbered from left to right, as it were. Of course, the polar sets might be subdivided, too, taking "very ugly" and "sorta beautiful" as separate slices (I suspect that even 9's on the traditional scale are 1s on the characteristic of "beautiful," though maybe not on "very beautiful" -- I am sure that 1s on the traditonal scale are still thought to be comfortably inside "ugly.") These comments also apply to Goran's (? djer's?) complaint that _da je'axiny melbi_ would not imply _da melbi_, as it seems it ought. At least some of the time, _melbi_ is being used for the whole scale and _je'axiny_ (or _ja'axiny_, for that matter) for a slice or point on that scale, at other times _melbi_ is the proper polar set and _je'axiny_ is a value (range) of its characteristic function -- or maybe even a related set with a derived characteristic function. In the first use, _je'axiny melbi_ need not be _melbi_ but might be all the way down into _to'e melbi_ or in the middle or.... In the second, it would be in _melbi_ (presumably scoring n on the characteristic function indeed -- if I understand THIS intended use of subscripts, which is not a favorite of mine). On the third, its relation to _melbi_ would be determined by the derivation of its characteristic function -- "very", for example is entirely in the base function, "somewhat" tends to pick up some from the 0 range, and so on. At least in some of the fuzzy discussions these three notions (at least) have been thoroughly muddled together. Or, to be fair to the linguistic situation in English, where the same words are often used for all of them indifferently, have not been adequately sorted out. Lojban seems to have the resources to already in hand to do the sorting in a systematic way but this has not been carried out and adhered to consistently. stivn keeps insisting that natural languages use fuzzy logic a lot and that Lojban doesn't. The problem with this line is that all the evidence that natural languages use fuzzy concepts, etc., can equally well be taken as evidence that Lojban does too, since Lojban matches natural language usages very well indeed. The fact is that the basic form of fuzzy logic (and, indeed, all the Wooky logics out of slash-L-sub-inf) LOOK the same in terms of grammar and vocabulary; you can't generally tell by looking what kind of logic you are up against at the elementary level. As you get deeper into it, patterns start to emerge that will indicate what the game is. Insofar as any natural langauge has features of the sort that indicate fuzziness, Lojban can match them and, often, beat a particular language out by a large degree (since it has tricks drawn from several langauges). On the other hand, insofar as Lojban vocabulary is more carefully defined than most natural languages, some cases of clear rejection of fuzziness may emerge. For example, the fact that the predicate for "is a member of" does not have a place for a value in [0,1], nor does the predicate for "is true" (but see the discussion above about the difference between polar sets and spectra), indicates that these notions have not been thought of fuzzily. However, for most concepts, no such overt indications are appropriate, and thus their absence leaves Lojban as open to fuzzy interpretations as any other language. Of the most obvious fuzzy patterns in logic, Lojban has good facilities for the derived sets: forming compounds with expressions for "very," "slightly," "extremely," and the like, which can be (and have been for English) correlated with various derivative characteristic functions. Not so obviously fuzzy (but here often muddled in), Lojban also has a system for a five-way (I think I've counted) lopping up a spectrum defined by one of its poles, the NAhE set. Finer granularity is presumably available by compounds or the apparently now acceptable devise of ordinal subscripting (hawk ptui). (I do not know quite what to say about cases where the spectrum does not run from pole to pole but just from one class to another. Partly, there is a problem about what can lie between some pairs of classes and partly it is a matter of expressing the general patterns of dividng up the space. On the latter, I suppose that the ordinal words would be of some use, since they specify at least one terminus and might be jiggled to express the other as well. That still leaves granularity and scale-type, however, unexpressed. As noted, this does not seem to me to be a fuzzy matter per se.) I am not really convinced that the device of dividing characteristic values of a set into smaller sets is ever actually used (as opposed to the two notions above), but, if it is, then I do not see a Lojban device ready-made for it. I do think, however, that it can be handled simply by using tanru with _meliny_ (assuming that _me_ has not gotten so kakked up as no longer to be usable in its original basic way), maybe with a fuzzy n, if wanted. If we add a place to the structure of "is a member of" or just agree that the standard "extra place" flag with it is to stand for a fuzzy value, then we seem to have fuzzy set theory -- basic fuzzy logic -- in hand. Aside wanting to say what the fuzzy truth value of a sentence is -- which seems to be again just a matter of extending the place structure of a gismu, there seems to have been little discussion of fuzzy propositional or quantifier logic. And, indeed, there has not been a lot of a very specific nature anywhere. If we did come down to issue of which fuzzy connective each of our given connectives should be, I suppose we could handle that quite unnoticed. The problem of expressing some of the other connectives, if we decide we need them, might be more difficult (I would not want to get into devices that would have to affect the whole range of connectives at all the various levels that they have come to have). Happily, there is almost no evidence that languages actually use a variety of fuzzy connectives in the standard places nor a variety of non-standard ones. The actually used fuzzy set seems, if anything, less complex than the probability set, which has an array of conditionals and disjunctions. The only question with fuzzies seems to be whether the "and" and "or" are min and max or product and sum (roughly -- the last two have to be finagled, of course), so a few spare forms might be handy in case someone wanted to use both (but we need those already for probability logic, if we start getting serious about letting Wookies in). As for fuzzy arithmetic, that just needs a mark (same class as -/+) to say "fuzzy number," since, so far as I can find, no specifically fuzzy functions are used. If we need to mess more in that area, MEX is the easiest place to work, since so little of it is in a settled state (or, at least, so few of us are sure of what state it is in and firmly locked into that state). pc>|83