"Earl Cox" <earldcox1@home.com> wrote in message news:<xJDe7.13461$L9.3523724@news1.rdc1.md.home.com>...> I really hate to continue this. But yes, the theorems of fuzzy logic, > *themselves* use multi-valued logic.
Clearly you are mistaken, as I believe I can demonstrate in few words. Let A be a fuzzy term that fuzzily characterizes point clusters belonging to some domain U, say. Then its membership function mu[A]:U -> [0,1] is an object of ordinary (bivalent) mathematics, and there is nothing fuzzy about it; nor is there anything fuzzy about about the theorems which require manipulation of it. Note that something fuzzy with respect to U, namely A, has been rendered as something non-fuzzy with respect to [0,1]^U. Fuzzy objects in the object language have non-fuzzy counterparts in the meta=language, which proceeds in the usual manner of quite ordinary mathematics, where the rules of a bivalent logic apply.> I am finished with this thread.
Too bad.> earl
Regards, S. F. Thomas> > > "S. F. Thomas" <sfrthomas@yahoo.com> wrote in message > news:66b61316.0108100550.3a699afb@posting.google.com... >> "Earl Cox" <earldcox1@home.com> wrote in message > news:<%yKc7.52091$m8.16672957@news1.rdc1.md.home.com>... >>> I suppose the statements: >>> >>>> The important distinction is not >>>> between bivalent logic and multivalent logic, but between >>>> meta-language and object language. A bivalent logic in the >>>> meta-language is perfectly adequate for the purpose of modeling the >>>> fuzziness in the object language. >>> >>> must make sense to someone. But any metalanguage >>> that can convert two-valued logic into continuous valued >>> logic must be, at heart, fuzzy logic (since this is exactly >>> what fuzzy logic, via the extension Principle, does.) >> >> Clearly you have missed the point. I can put it more simply as >> follows: ALL the theorems of fuzzy set theory and of fuzzy logic, or >> whatever flavor, are stated and proved within a framework of bivalent >> logic. More broadly, to fix in the mind the distinction between >> meta-language and object-language, a fuzzy term such as "tall" will >> populate the object language, but its membership function mu[TALL] >> belongs to the meta-language, where the bivalent rules or ordinary >> mathemtics applies. There is nothing fuzzy about fuzzy set theory, >> just as there is nothing random about probability theory. >> >>> In any case, I beg to differ in very strong terms, >>> the important distinction is exactly that -- between the >>> concepts that can be modeled with bivalent and >>> those that can be modeled with multivalent logic. >>> Obscuring the problem with lots of mumbo-jumbo >>> about meta-languages and object languages >>> contributes nothing. >> >> You've missed the point, as illustrated above. Next time you look at a >> theorem of fuzzy logic, as whether the theorem *itself* uses a >> bivalent or multivalent logic. That should clear up your evident >> confusion. >> >>> earl >> >> Regards, >> S. F. Thomas >> >>> -- >>> Earl Cox >>> VP, Research/Chief Scientist >>> Panacya, Inc. >>> 134 National Business Parkway >>> Annapolis Junction, MD 20701 >>> (410) 904-8741 >>> ------------------------------------------- >>> >>> AUTHOR: >>> "The Fuzzy Systems Handbook" (1994) >>> "Fuzzy Logic for Business and Industry" (1995) >>> "Beyond Humanity: CyberEvolution and Future Minds" >>> (1996, with Greg Paul, Paleontologist/Artist) >>> "The Fuzzy Systems Handbook, 2nd Ed." (1998) >>> "Fuzzy Tools for Data Mining and Knowledge Discovery" >>> (due Early Fall, 2001) >>> >>> >>> >>> >>> "S. F. Thomas" <sfrthomas@yahoo.com> wrote in message >>> news:66b61316.0108091708.7d6b9958@posting.google.com... >>>> robert@localhost.localdomain (Robert Dodier) wrote in message >>>> news:<9kt895$rs$1@localhost.localdomain>... >>>>> In the interest of brevity, I've indulged in wanton snippage, >>>>> but I hope what's left yields something comprehensible. >>>>> >>>>> S. F. Thomas <sfrthomas@yahoo.com> wrote: >>>>> >>>>>> Robert Dodier wrote: >>>>>>> [...] OK, now this is something I haven't heard about -- how > does > the >>>>>>> extended likelihood calculus take loss, risk, and action into > account? >>>>>> >>>>>> [...] Under the likelihood calculus, the same is possible, but the >>>>>> fact that likelihood is a point function, not a set function, >>>>>> renders the general rule for change of variable different -- >>>>>> easier in fact -- from what it is under the probability calculus. >>>>>> As to issues of risk and action, the notion of expected loss >>>>>> consequent upon any given action, is rendered as a possibility >>>>>> (or likelihood) distribution, or in effect a fuzzy set. >>>>> >>>>> Suppose, then, that I have a possibility or likelihood for two >>>>> different actions. Can I say that one action is preferable to >>>>> the other? If so, how do I determine which is more preferable? >>>> >>>> Sometimes it is very clear which is preferable, sometimes less so. If >>>> you are minimizing loss then the smaller the centroid of the >>>> possibility set, the more preferred; however, the centroid is an >>>> insufficient measure of preference, for the largeness of spread also >>>> comes into the picture, with smaller spread (i.e lesser fuzziness) >>>> being in general preferable to larger spread. In a single-criterion, >>>> single decision-maker (DM) problem, this can go to the heart of the >>>> insufficiency of the Bayesian paradigm, and indeed explain why a >>>> (rational) decision-maker may choose neither to take, nor place, the >>>> Bayesian bets. Suppose for two actions, the corresponding possibility >>>> sets on expected utility have the same centroid, but one has greater >>>> fuzziness than the other, then the rational thing to do is to opt for >>>> the action with lesser fuzziness, no? Contemplating a Bayesian >>>> betting scenario, a decision-maker always has as an option the >>>> certainty of status-quo, i.e to neither take the bet nor place the >>>> bet, either of which options would presumably carry some residual >>>> possibilistic uncertainty deriving precisely from the modeling >>>> uncertainty which is sought to be illuminated by the Bayesian >>>> analysis. That is the single-criterion, single DM problem. In the >>>> more usual case, the optimal action in any given situation must be >>>> evaluated on more than one criterion. And quite often in practical >>>> decision-making, we have multiple decision-makers, and various ways >>>> of attempting to resolve differences among them. The utility calculus >>>> will take one nowhere fast in attempting to address these questions. >>>> With the likelihood/possibility calculus, it is in fact possible to >>>> address these questions, as inter-personal comparisons, both of >>>> belief and of preference, are far easier to address within such a >>>> framework. _Fuzziness and Probability_, in the part of it that >>>> elaborates an approach to decision analysis under uncertainty, >>>> attempts to do just that. >>>> >>>> >>>>>>>> [...] But probability (over sample space) gives rise to > likelihood > (over >>>>>>>> parameter space) and the calculi required to manipulate the > two > are >>>>>>>> different. >>>>>>> >>>>>>> (i) This betrays a very limited view of what a model can be: > apparently >>>>>>> there are but samples and parameters. Many interesting models > are > not so >>>>>>> simple. >>>>>> >>>>>> What do you mean? And how does it relate to what we are > discussing? >>>>> >>>>> In the world of models implicit in your statement above, there are >>>>> sampling distributions for observable variables and there are >>>>> parameters that govern those distributions. Some models are that >>>>> simple, yes. There are many models which don't fit into this neat >>>>> division of labor. Does every class of models require its own >>>>> reasoning calculus? >>>> >>>> At the very least, there is deductive reasoning, and >>>> associated logical calculus, and there is inductive reasoning, and >>>> associated logical calculus. These two suffice in my view to carry >>>> the burden of any discourse concerning any object phenomenon, or >>>> system of >>>> inter-acting phenomena, that may be of interest. However, there is a >>>> third >>>> kind of reasoning which must remain outside either of these. It is >>>> the reasoning that derives from *insight* and which leads us in a >>>> very mysterious fashion to posit the intension models and associated >>>> premises/hypotheses that may then be subject to various kinds of >>>> deductive and inductive massaging. There is also a fourth, which is >>>> what we are here engaged in, which is a kind of meta-reasoning. >>>> >>>>>> (ii) The likelihood calculus which you state above looks >>>>>>> suspiciously similar to a rule derived from laws of probability. >>>>>> >>>>>> It *is* derived from the laws of probability. You must have missed >>>>>> large parts of the thread while feigning sleep. >>>>> >>>>> Well, I have no problem with deriving fuzzy reasoning from > probability, >>>>> but I thought that was precisely what you were arguing against. >>>> >>>> Goodness, no. What I do argue however is that the semantics of >>>> likelihood do not just fall neatly out from the semantics of >>>> probability. Probability provides some of the underpinning, but not >>>> all. Otherwise Fisher would not have been led up a blind alley by >>>> asserting that the "likelihood of a or b is like the income of Peter >>>> or Paul, we don't know what it is until we know which is meant." This >>>> leads to a likelihood calculus in which set evaluation is of the form >>>> >>>> L( {a,b} ) = L(a OR b) = Max( L(a), L(b) ) >>>> >>>> which rather quickly proves to be inadequate. Had it not been >>>> inadequate, I don't think classical statistics would have gone to all >>>> the trouble it has to develop indirect methods of describing the >>>> uncertainty in model parameters consequent upon sampling. Nor would >>>> there have been a neo-Bayesian revival intended to supplant the >>>> classicists precisely by offering a method of *direct* >>>> characterization. Indeed, Bayes offers a likelihood calculus in which >>>> >>>> L(a OR b) ~ (L(a) + L(b)) >>>> >>>> where ~ is to indicate that some normalization, appropriate to the >>>> construction of likelihood as a metaphorical (belief) probability, is >>>> necessary. It is only with the fuzzy set theory that semantics >>>> suggests itself >>>> >>>> L(a OR b) = L(a explains the data OR b explains the data) >>>> >>>> where "explains the data" is a fuzzy predicate no different in >>>> principle from "is tall", and subject to calibration in conceptually >>>> the same way. This leads, albeit with some reworking of the Zadehian >>>> fuzzy set theory along the way, to >>>> >>>> L(a OR b) = L(a) + L(b) - L(a)*L(b) >>>> >>>> where indeed the laws of probability are invoked, and at that in a >>>> very simple way, but it is the fuzzy set semantics, and the device of >>>> the calibrational proposition, that provides the essential frame that >>>> Fisher overlooked. >>>> >>>> Btw, there is a school of fuzzy which is the mirror-image of what it >>>> is you seem to wish to maintain. They maintain that fuzzy is logically >>>> prior to "crisp". This misses an essential point in my view. And that >>>> is that a bivalent logic is perfectly capable of generating >>>> ever-higher levels of fuzziness, in exactly the same way that you can >>>> run fuzzy models on binary computers. The ultimate bivalence of the >>>> computer does not disable it when it comes to elaborating computable >>>> fuzzy models; in fact the reverse. The important distinction is not >>>> between bivalent logic and multivalent logic, but between >>>> meta-language and object language. A bivalent logic in the >>>> meta-language is perfectly adequate for the purpose of modeling the >>>> fuzziness in the object language. The essential notions of membership >>>> function, and of rules of combination (fuzzy union, intersection, >>>> etc.) all belong to the bivalent meta-language we must recall, and all >>>> our theorems are cast in the bivalent meta-language. And to even >>>> suggest casting it in a logically primitive multivalent meta-language >>>> would be hopelessly confusing in my view. Instead, from the vantage >>>> point of a bivalent meta-language, it is possible to see, at the level >>>> of the object language, a class of crisp terms which are very clearly >>>> a special case of fuzzy. But that is in the object language. And that >>>> observation does not render fuzzy prior to crisp; rather it is the >>>> crispness of the meta-language that permits us to bootstrap our way to >>>> the higher reaches of fuzziness in successions of object languages. I >>>> make this point because probability stands in a similar relation to >>>> fuzzy, and to likelihood. Just because we use probability to generate >>>> the membership or likelihood function, it does not follow that there >>>> is no value-added in making the leap from the one to the other. And I >>>> maintain that there is an essential duality between the two, with the >>>> distinctness, yet connectedness, that that implies. >>>> >>>>>> [...] First you claim boredom with the discussion, and say you're >>>>>> going to sleep, only to re-appear, apparently wide awake and > engaged. >>>>> >>>>> I always feel like a million bucks after a good nap. >>>> >>>> Well, in that case, Dodier, I hope the present offering again succeeds >>>> in putting you gently to sleep. >>>> >>>>> Regards, >>>>> Robert Dodier >>>> >>>> Regards, >>>> S. F. Thomas