I really hate to continue this. But yes, the theorems of fuzzy logic, *themselves* use multi-valued logic. I am finished with this thread. earl "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