"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