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*To*: John Cowan <cowan@LOCKE.CCIL.ORG>*Subject*: warm fuzzies*From*: "John E. Clifford" <pcliffje@CRL.COM>*Date*: Fri, 24 Nov 1995 09:53:53 -0800*Reply-To*: "John E. Clifford" <pcliffje@CRL.COM>*Sender*: Lojban list <LOJBAN@CUVMB.BITNET>

As I was saying about doing our homework... The discussion of fuzziness seems to have involved about three and a half different notions, which might be related but need not be -- and have mainly just been assumed to be in the discussion. (Although McCawley does not do as good a job as he might at making the relevant distinctions in the obvious section, he does so elsewhere.) First, there is fuzzy set theory, which starts from ordinary set theory by replacing the usual characteristic functions of sets, functions from objects into {0,1}, with functions into [0,1], the closed interval of the reals. Thus, rather than an object just being a member (1) of a set or not (0), it might be said to be a member to any degree within the range of the function. This is a device to try to deal with vague concepts, where things might be more or less like clear-cut (typical?) members of a set (holders of a property). (I will pass over -- mainly because the data is so inconclusive -- whether this is either a reasonable or an accurate way to deal with these concepts, indeed, whether there is a single -- or finitely partitioned -- notion of vagueness to which this approach might apply.) From this system three systems follow: one that simply states the function values: "x is F to degree 0.n", and two that make use of modifier notions, "very", "somewhat" and the like (these for adjectival F in English, nominal Fs require different vocabulary that can -- it is said - - be mapped onto this). In one of these system, "very F" refers to a sort of subset of F, one with a characteristic function that assigns non-0 values only to things that the characteristic of F does, but generally assigns lower values to object and, in particular, assigns 1s only to some of the things to which the F function assigns 1. "Somewhat F" is rather the reverse, giving non-0 values perhaps to some things to which F gives 0, but within F giving typically higher values than F does. The exact mathematical relations between these various modified F functions and the base one were once a favorite game. The second modifier version takes off from the first format, "x is very F" means "x is F to a degree adequately in the 'very' range." That is, the [0,1] interval is divided into a number of subintervals, labeled, say, "scarcely," "slightly," "somewhat," "moderately," "clearly," "very," and "absolutely" (not seriously proposed). These subintervals are themselves typically taken as fuzzy sets so that one can with almost equal justification call something either of adjoining classes. Mathematicians tend to feel most comfortable with the straight functional form, linguists developed the first modifier form most fully, engineers have worked mainly with the third, realized by analog devices or analogization (fuzzily, of course) of binary data. Then there is fuzzy logic. This is basically a logic which assigns truth values from [0,1] rather than {0,1} (or {T,F}), with some restrictions on how to compute values to compound wffs (in particular, restrictions that distinguish this system from probability valued logics, which start with the same sort of assignments -- and go through parallel developments. McCawley is rather good on the problems of building such computations in a way that gives a reasonable logic.) Two relations with fuzzy set theory suggest themselves immediately: we could take the truth value of each atomic sentence, Fa, as just the value of the characteristic function of F at a or we could take the truth value assignments as the values of the characteristic function of the fuzzy predicate on sentences, "is true." These are two independent ideas and fuzzy logic can use either, both or neither (the most complex form I know of off-hand is the one that has an independent fuzzy truth value assignment, so that the claim that a is F to degree 0.n has some value between 0 and 1 inclusive -- hopefully with some consistency requirement that successive suggested degrees of a as an F get progressively higher truth values up to a maximum and then fade smoothly away). The "very true," etc. truth values seem to come from the fuzzy partition of the range of the characteristic function version of the second approach, probably combined with the first. But I did see at least one comment that suggested the related sets notion of various levels of truth. Finally into this mix is added scalarity. The old philosophical histories of science talked about the development of theoretical concepts from the qualitative to the comparative to the scalar to the quantitative: this is hot, this is hotter than that, this is of the third degree of hotness, this is 250 degrees centigrade -- the classic example (one of the few that actually shows all stages in extensive treatment). The move from comparative to scalar is actually rather complex. It involves first a number of comparisons, to get a variety of phenomena lined out in a single dimension. Then this ordering, assuming that it is about a subjective factor like hotness, must be checked for intersubjective agreement: do other order the phenomena the same way? Then some of the phenomena in the ordering are taken as exemplars of certain regions of the ordering -- again with an intersubjective check. Finally (though not essentially in itself but only as a step toward quantitative concepts) the regions or their exemplars are numbered in order. The step to quantitative concepts requires that there be an objective correlate of this subjective ordering that provides (typically) continuous values. That objective version then gets assigned real numbers, typically with important exemplars in the old scalar form getting important numbers (I forget why Fahrenheit screwed up so badly). Again, there are some natural and some apparent relations between scalar properties and fuzziness. It is reasonable, for example, to take the degrees on the scale as fuzzy sets, with the exemplars getting 1's and other things getting less as they moved away from that central value (this is before the scale gets reinterpreted in the quantitative form, as the old 1-10 chili scale got reworked as a partition of the range of Scovill units or Beaufort in terms of wind velocity). What will not work, however, is to take the scale as an integer-ation of the fuzzy values or characteristic function. Typically, something has to get pretty close to 1 on the characteristic function of a property to even get into the running on the scale at level 1 and most of the items on the scale are uniformly 1 on the characteristic function. Nothing that is not hot tout simple is going to be called hot degree 2 or greater, say. Even "very hot" or "absolutely hot" is going to have its 1 cut in before, say, molten steel come into the picture. That is, the two patterns do not fit well together and, indeed, belong to different conceptual types. I think that there is at least occasional use for each of these various systems in any advanced language. I also think that Lojban is pretty well equipped to make those uses when called upon and to do so in distinctive ways. It is also in pretty good shape for doing all the corresponding things for probabilities and, should the need arise, for other such systems as may be presented. I welcome various expositions about how Lojban might do these things with the tools at hand and tend to be a bit put out with those who say we cannot do some of these things or that we should do more of one, especially before the proclaimer gives it a good try. pc>|83

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