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.) 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. earl -- 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