You raise a good question. As the author in question, may I respond as follows: There really is a dualistic relationship between fuzziness and probability. They are distinct concepts, but I would argue that there is a real sense in which the former *derives* from the latter. That thought is anathema to many fuzzicists who fear the implication that if so, then there is "nothing new" to fuzzy, as falling ultimately under the ambit of probability theory. I don't think so. There is plenty new semantics in the fuzzy set theory of which probabilists have been blissfully unaware, and which in fact helps to illuminate some problems in the foundations at least of statistical inference theory. The duality is precisely analogous to the duality that exists between probability and likelihood. A probability distribution over sample space gives rise to a likelihood function over parameter space. The one is a set function, the other a point function, and they pertain to different domains. In the case of natural language semantics, it is precisely because language-use is a chance phenomenon, even in a calibrational setting, that there is fuzziness in the meanings of terms. More precisely, uncertainty in the calibrational response variable, either yes or no, to a series of calibrational propositions such as "would you use the term 'tall' to describe the height value for which John stands as exemplar, in the context of heights of adult males?" gives rise to a semantic likelihood function over height space, induced by probabilistic response uncertainty over calibrational response (yes/no) space, it being understood that many different height value exemplars (Jim, Peter, Paul, etc.) are similarly presented in calibrational setting. The affirmation probability (Bernoulli parameter) varies as a point function over height space, as opposed to a set function over calibrational response space. Thus the calibrational response rates traced out with respect to the height variable is in no sense a probability distribution, since it would in general not sum to unity; nor is it a frequency distribution over the height values of the adult male population that in an obvious sense may be rendered as a probability distribution. All you have is a characteristic function that describes, for various height values, the rate at which a relevant speaker population would use the term "tall" to describe the height values in question. It is a membership function in the obvious Zadehian sense of a point function ranging from 0 to 1, though Zadeh may or may not approve of the manner in which it is obtained. It could also have been called a *semantic likelihood* function, or a *characteristic* function of the *term* tall, as distinct from the *membership* function characterizing the associated set of tall *men*. I like the term semantic likelihood because it gets to the heart of the matter in my view. In a non-calibrational setting, eg. the use of the term "tall" by a rape victim in court to describe the height of her attacker, it is the calibrational response uncertainty in terms purely of language-use, that leads to semantic uncertainty about the precise height to which she refers. The semantic likelihood function traces out the relative possibility of various height-value hypotheses consistent with her description of her attacker as "tall". In ordinary discourse and comprehension, we don't need to have it spelt out, obviously. But is in some sense there. This analogy leads to the perhaps startling conclusion that the (absolute) likelihood function familiar from statistical inference theory is in some sense also a membership function! It certainly satisfies the minimum condition that it range on the [0,1] interval. But semantically as well, it may be construed simply as the term corresponding to what the data "say" about some unknown parameter of interest. The greater the quantity of data, the more precise, or less fuzzy, is the characterization of the unknown parameter of interest. Statistical sample data relevant to inference concerning the value of model parameters are therefore analogous to fuzzy natural-language statements about things like people's height. It is also analogous to measurement, which for continuous attributes is fuzzy in general, since no measurement may be made literally to an infinite number of decimal places, and at the digit where uncertainty enters, accidental and systematic errors of measurement, exactly like those associated with the calibrational proposition with which we started for characterizing the term "tall", may conspire to render the range of uncertainty in the ultimate digit of measurement fuzzy rather than crisp. In all of this there is an essential and unavoidable duality and interplay between uncertainty of the probabilistic sort, and uncertainty of the fuzzy (also likelihood) sort. To treat these two kinds of uncertainty in this fashion is not to exalt one over the other, rather to recognize that they are inextricably linked, as are the two faces of a single coin. We give precedence to probability naturally, because it comes first in our comprehension of types of uncertainty, and as earlier mentioned there is an obvious sense in which likelihood, including the present semantic variant, *derives* from probability. But it is precisely because statistical inference theory privileges probability over likelihood that we have on one hand the convoluted methods of classical inference seeking to render uncertainty of the likelihood sort in probabilistic terms (the use of "p-values" and the like), and on the other hand the vain Bayesian attempt to treat likelihood as though it were probability, and therefore subject to a simple integration method of disjunction (evaluation of composite hypotheses or set evaluation). Treating likelihood like the fuzzy characteristic functions that they essentially are, and using the brilliant semantics introduced by Zadeh, allows us to develop a new theory of statistical inference. But this requires both hands to clap: the right hand of probability with the left hand of likelihood/fuzzy/possibility. Privileging fuzzy in denial of a probability connection is misguided in my view, in the same way that using probability to represent a simple likelihood reality leads to difficult analytical contortions as in both classical and Bayesian statistics. It is time to get past the early tutorial exaggeration that insisted that fuzzy was different from probability. That is true, but it is also true that the twain do meet. Hope this is helpful. Regards, S. F. Thomas Joe Pfeiffer wrote: > > I'm currently reading the book mentioned above; I'm wondering about > something... > > He attempts to define the membership function of a set by using > what he calls ``calibrational propositions'' -- the idea is that if > you ask 70 people if John is tall, and 70 of them say ``yes,'' then > mu(tall) = .7. While this seems to do a good job of capturing common > word usage, it's not at all clear to me that it captures the fuzzy > behavior of the ``tall'' set; it seems probabilistic rather than > fuzzy. > > So, what are other people's reactions? > -- > Joseph J. Pfeiffer, Jr., Ph.D. Phone -- (505) 646-1605 > Department of Computer Science FAX -- (505) 646-1002 > New Mexico State University http://www.cs.nmsu.edu/~pfeiffer > SWNMRSEF: http://www.nmsu.edu/~scifair ================================================================= Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =================================================================
