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From: sthomas@decan.gate.net (S.F. Thomas)
Subject: [BOOK ANNOUNCEMENT] Fuzziness and Probability
Keywords: fuzzy,probability,likelihood,inference,semantics,measurement,possibility,logic,deductive,inductive,Bayesian,belief,utility,decision
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Summary: New book announced -- Fuzziness and Probability 
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Date: Mon, 17 Jul 1995 02:23:32 GMT
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		      BOOK ANNOUNCEMENT

		  FUZZINESS AND PROBABILITY

The book "Fuzziness and Probability" by S. F. Thomas is now in print.
This is a revised version of a manuscript that was made available
by internet email in December, 1994.  The internet version has been
superseded and is therefore no longer made available.

1995; 320 pages; ISBN 0-8050-2356-0; soft cover; $29.95

Publisher: ACG Press, PO Box 782948, Wichita KS 67278-2948, USA
Tel:316-777-4425
Fax:316-689-6889
Email: acg@acginc.com

                           OVERVIEW

I claim in the book:

   o to have found the extended likelihood calculus that eluded
     Fisher, and generations of statisticians since

   o to have corrected the mistake in the Zadehian fundamentals of
     fuzzy set theory which put him at odds with Aristotle -- the
     laws of excluded middle and contradiction are restored,
     without undoing the essential fuzziness of natural language
     descriptors

   o to have corrected the mistake in the Bayesian inference
     theory that treated universals (uncertainty about models) and
     particulars (uncertainty about the chance of occurrence of
     individual events) symmetrically, while keeping the essential
     idea that sources of evidence -- prior belief plus
     experimental data -- may be combined to give a posterior
     expression of belief, scientific or otherwise, about the
     universals involved

   o to have found the solution to the nagging foundational
     problem of measurement theory -- how to address errors of
     measurement within the theory itself.  First you change the
     paradigm from one of assumed precision of measurement to one
     where imprecision (fuzziness) is the general case.  It is
     easy for precision to fall out of a paradigm of imprecision,
     but quite difficult to make imprecision fall out from within
     a paradigm of precision.

     There are other claims besides, but these are the main ones,
and the last one is key: we are dealing here with a paradigm
change.  At the foundations of science, where we are addressing
issues of measurement -- including the measurement of attributes
of probability, belief, utility -- and issues of scientific
inference, where again we must confront issues of probability and
its measurement, the paradigm in place is one of precision, the
idea that the total ordering axiom applies, that sample spaces may
be continuous, that points within sample spaces may be discerned
with absolute levels of precision, and that they may be measured
in principle to an infinite number of decimal places.  Once this
notion is relaxed, everything changes.  One then finds oneself at
a point where everything in science seems to converge (or from
where everything diverges) and simultaneously in the realm of
inference throry, probability theory, measurement theory, fuzzy
set theory, decision theory, even the foundations of deductive
logic, of the philosophy of truth, of semantics, and so forth.  In
much the same way that Einstein's theory of relativity changed the
analytic paradigm for the theory of motion by putting the observer
into the picture, theories of inference regarding real-world
phenomena change when the imprecision inherent in the use of
language for the conveying of measurement reports -- the language-
use phenomenon itself -- is explicitly brought into the picture.
And, as with Einstein's theory of relativity, which did not
diminish the usefulness of the Newtonian theory in most everyday
applications, the change of paradigm here explored will not change
the usefulness of the paradigm of precision in those application
areas where "adequate" precision is capable of being achieved for
the attributes of concern.  Equally, however, the change of
paradigm appears necessary in areas where the attributes of concern
(eg., subjective probability, utility) are not susceptible of
sufficiently precise measurement/description.

     While the breadth of coverage seems immodest to say the
least, that's where I was led, purely serendipitously, as I have
indicated in the Preface and Introduction.  It's all due to Zadeh,
as the idea of fuzziness started it all, and is at the center of
the whole thing.  But surprisingly, the connectedness of all these
things seems to have eluded Zadeh, who in insisting on the
distinction between probability and fuzziness seems to have cut
himself off from exploring the intimate relation between the two.
I think I have exposed the connection, which far from weakening
the power and beauty of Zadeh's master stroke, strengthens it even
further.  

                         EXCERPTS

From the Introduction:
         ------------
 "What I offer the reader, in sum, is a philosophical essay setting
  out what insights I think I have encountered in addressing the
  problems associated with the scaling of judgmental attributes for
  use in the analysis of decision problems.  In addressing these
  problems, I have found it interesting to explore foundational
  issues in the fuzzy set theory of semantics, the theory of
  statistical inference, the theory of measurement, along with the
  concepts of probability, likelihood, possibility, logic,
  combination of evidence, and other topics typically of interest to
  those concerned with exploring the foundations of science.  The essay 
  is primarily addressed to academicians and practitioners of the
  'decision sciences' -- broadly construed to include statistics,
  economics, management science, operations research, and like
  disciplines -- who are interested in exploring foundational issues
  arising in the area of nexus between probability and fuzziness.
  It should also be of interest to social scientists, in particular
  those who address issues related to the psychometric scaling of
  judgmental attributes, and to engineers and researchers in the
  areas of artificial intelligence and cybernetic systems.

  The plan of the essay is as follows: Chapter I provides some
  motivational background, giving a brief review of the controversy 
  in the foundations of statistical inference, showing the way in 
  which fuzzy sets may enter the picture, while pointing out that
  at least one difficulty encountered by the likelihood school of 
  statistical inference -- the paradoxes sometimes engendered by 
  marginalization by maximization -- may apply equally to the fuzzy
  set theory, where this method of marginalization is commonly applied.
  Chapter II puts forward a discussion of the philosophical 
  fundamentals underlying the theory -- of semantics,
  measurement, phenomena, models, and probability.  Chapter III
  develops a non-Zadehian fuzzy-set theory of semantics, in which,
  among others, the laws of excluded middle and contradiction
  are restored.  Chapter IV is an elaboration of the possibility 
  calculus in its application to inductive (statistical) inference.  
  Chapter V continues the elaboration of the possibility calculus,
  as applied to deductive inference.  Chapter VI addresses the 
  problem of decision analysis under uncertainty, and shows how the
  possibility calculus may be applied.  Chapter VII provides a 
  comparison of the possibilistic inference and decision theory 
  with the Bayesian.  Finally, the essay is concluded in Chapter 
  VIII with an attempt to put the whole development in perspective."


From Chapter I ("Motivational Background"):
     ------------------------------------

 "This essay develops three basic propositions: first, that data
  are fuzzy sets in general, not point observations; second, that
  uncertainty about probability models or model parameters is
  structurally similar to semantic uncertainty about fuzzy data or
  fuzzy descriptions; and third, that uncertainty in data and
  uncertainty in modeling both yield to a possibility calculus which
  provides a common interpretive framework within which to construct
  a theory of inference and apply it to problems of decision
  analysis under uncertainty.
 
  The essay therefore brings together two diverse streams of
  literature, one dealing with statistical inference, the other with
  fuzzy sets and approximate reasoning.  Out of the confluence of
  these two streams flows a theory of semantics and possibilistic
  inference, freed from anomalies present in both streams.  Their
  joining should therefore be to the enrichment of both."
 

From Chapter II ("Philosophical Fundamentals")
     ----------------------------------------- 

 On semantics:   

 "It should be clear that the view of semantics being developed
  here implicitly distinguishes _calibrational_ propositions from
  propositions in _actual use_ -- what I call later, for lack of a
  better term, _descriptional_ propositions.  In the foregoing
  discussion, presenting users of the language with exemplars of
  various height values  u , and asking whether they would use the
  descriptor 'tall' to describe the height value  u  is a calibrational
  exercise.  When the descriptor 'tall' is actually used, as in the
  eyewitness's description 'the perpetrator is tall', we have a 
  _descriptional_ proposition, in actual use.  If the population
  usage of the term 'tall' has been calibrated, we have the basis for
  constructing a 'possibility distribution' or 'semantic likelihood'
  function for the unknown value of the perpetrator's height.  
  The distinction between calibrational and descriptional propositions
  is thus the distinction, as with measuring instruments, between
  calibration, and use."

 On fundamental measurement:

 "The crux of the problem lies in the treatment of error. The
  problem of equivalence intransitivity stems from the intrinsic
  accuracy or sensitivity limitations of the judge or measurement
  device -- what may be called intrinsic error. Errors of
  observation (Topping, 1955) may include as well accidental and
  systematic elements. The classical formulation of the problem of
  measurement runs into difficulty because it does not allow, within
  the formulation itself, for the treatment of error.  Intrinsic
  error is not allowed because it is inconsistent with the
  fundamental ordering axiom which is usually assumed -- a practical
  necessity within a paradigm of point representation.  Systematic
  error is not allowed because the formulation deals in principle
  with primary measurement whereas systematic error is a problem
  only for secondary measurement devices defined by reference to a
  primary standard -- more a practical problem than one of
  principle. Accidental error is treated, not as part of the problem
  of measurement in principle, but as an extra-theoretical problem
  of statistics. Once again, the premise of point measurement is too
  rigid an ideal to allow variations of measurement to be considered
  within the theoretical framework; variations must be explained
  through the invocation of random or accidental occurrences, when
  the real culprit is more usually intrinsic accuracy limitations.

  Within the fuzzy set framework being proposed for the problem of
  measurement, it is possible to treat all three forms of error.
  Systematic error and intrinsic error would be revealed by means of
  the kind of calibration exercise already discussed ...  What would
  be regarded as accidental error in the classical framework is
  treated not so much as 'error', but as variations in measurement
  reports wholly consistent with the intrinsic imprecision inherent
  in those reports."            ......

 "The point of all this is quite simply the contention that the
  character of an attribute space is determined fundamentally by an
  act of abstraction that precedes subsequent description
  (measurement) of objects with respect to the attribute, or
  comparison of objects according to the degree of possession of the
  attribute. This effectively achieves a separation between the
  issues of representation and uniqueness, on the one hand, and the
  issues pertaining to the empirical idiosyncracies of a particular
  judge or measurement device, on the other hand. The representation
  and uniqueness issues are now purely abstract, and therefore
  clear-cut (although not necessarily simple) while problems of
  error and equivalence intransitivity which reside essentially in
  the perceptional domain may be dealt with by the use of fuzzy
  descriptors, which allow for imprecision -- hence error and
  occasional intransitivity -- in judgment."

 On probability:

 "More generally, I would claim that the criterion of repeatability
  as a way of dividing the respective areas of application of the
  frequency view and the axiomatic view does not have the validity
  that it would seem to have on the surface. In the first place, no
  experiment may literally be repeated, since things change (time,
  at a minimum) from one experiment to the next. If we make another
  throw of a pair of dice, or pick another card from a pack, or grow
  another crop on an experimental plot, we have not repeated an
  earlier trial; at the very least, time has intervened, and at
  worst, some determinant of the response variable may have changed,
  accounting for the changed result. Yet it is meaningful to talk of
  a repetition, for there is something that stays the same from one
  trial to the next. That something is what I would call the
  'morphology' we mentally construct around the phenomenon. In
  typical frequentist statistical experiments, such a morphology is
  quite explicit -- the population of occurrences from which
  observations are drawn is well defined, and the variables of
  interest, response variables and explanatory variables are well
  defined. In the case of the question concerning the European war,
  nothing is defined -- neither the population of occurrences of
  which the European war may be deemed to be an instance, nor the
  precise variables that we should look at to help us make a
  judgment and answer the question posed.  Nevertheless, when we
  engage in discourse concerning the phenomenon, we implicitly
  construct such a morphology. One may offer some such discussion as
  follows: in any war, it is the relatively stronger of the
  combatants that usually wins, and the greater the relative
  strength of one combatant over the other, the faster that
  combatant wins. The strength of a combatant depends upon the
  number and quality of men, weapons, and war materiel that it
  possesses, and the cohesiveness and morale of the troops, as well
  as on the strategic deployment of such forces.  And so on. It is
  not my intention here to develop a theory of war or combat, only
  to illustrate my point that in the elaboration of discourse
  concerning a phenomenon, we implicitly construct a mental
  morphology around it, defining a population of 'objects' -- 
  combatants, considered in pairs -- and a set of attributes or
  variables which enter either as response variables of interest --
  relative strength in this case -- or as explanatory variables --
  number and quality of men, etc. Once we have such a morphology, we
  have the basis for a frequentist approach to probabilistic
  modeling of the phenomenon, for it allows us to speak in general
  terms about events that would otherwise be considered simply
  unique. Thus, on the one hand, the repeatability of unarguably
  repeatable statistical experiments really derive from the
  morphology of the situation, and on the other hand, once we do
  construct a morphology for seemingly unique events, they also
  become individual instances, or repetitions, of occurrences that
  fit into a larger pattern. I conclude that morphology is the key
  concept that allows us to close the gap between the frequency view
  and the axiomatic view in the application of the probability
  concept."

 "Once we have a morphology, the next question is on what basis do
  we assign probabilities to events within the morphology: is it
  objective or subjective? This categorization is irrelevant to the
  concept of probability that I have in mind. A probability model is
  in general purely a hypothesis which we are prepared to entertain
  as having some descriptive power in summarizing the relative
  frequencies of occurrences in the population of real or realizable
  occurrences defined by the morphology with which we structure the
  phenomenon and order our observations. As a hypothesis, it hardly
  matters how a probability model is arrived at, whether by
  subjective introspection, or by an empirical goodness-of-fit
  procedure on an experimentally observed sample from the population
  of interest. Whenever formal experimental observations are not
  possible I see no harm in making use of subjective introspection
  (of one's accumulated experience), but the status of such
  'subjective probabilities' in the present concept of probability
  is of the subjective estimation of a frequency probability. The
  probability being subjectively estimated is not assumed to have a
  psychological origin, as it would if it were taken as a primitive
  concept.

  I disagree, further, with the axiomatization of subjective
  probability which proceeds from the assumption that subjective
  probability judgments can be totally ordered, and that therefore
  sharp, numerical judgments of probability can always be elicited
  from any human judge. As discussed in the previous section, I
  would distinguish the abstract attribute space associated with the
  attribute of probability, which I have no difficulty conceiving of
  as a totally ordered linear continuum, from the empirical
  idiosyncracies and sensitivity limitations of the judge making the
  probability judgments.  That is to say probability judgments may
  be fuzzy, although the underlying abstract concept of probability
  allows probabilities to be totally ordered numerical points
  precise to an infinite number of decimal places. Furthermore, even
  in the straightforward situation where we need not rely on
  subjective probability judgments, but instead experimental sample
  data are available, probability estimates may nevertheless remain
  essentially fuzzy.  As mentioned previously in Chap. I, the
  likelihood function is essentially a fuzzy set describing what the
  data say about the 'true' unknown probability model which may be
  under investigation."

  
From Chapter III ("Fuzzy Set Theory of Semantics"):
     ---------------------------------------------

 "Zadeh (1975) has taken the position that the notion of grade of
  membership is merely a subjective estimation of the extent to
  which any given element may be said to belong to any fuzzy set in
  question.  On this view it is difficult to decide whether the set
  of tall men, for example, could not simply by represented as a
  Bayesian subjective probability distribution over the space of
  height values.  It is also difficult to establish any particular
  set of combination rules.  The minimum-maximum rules of the
  Zadehian calculus have intuitive appeal, but lead to debatable
  consequences.  In particular, the self-contradiction law, the law
  of contradiction, and the law of excluded middle are violated.
  Should they be, and if not, what other rules of combination may we
  substitute that would restore these laws.  Zadeh has also taken
  the position that the concepts of probability and fuzziness are
  distinct, raising the question what kind of statistical methods
  could one logically apply to establish either a membership
  function, or any particular rules of combination which one might
  care to propose.

  As I have indicated previously, I find it difficult to accept the
  idea that membership functions may be entirely subjective.  If I
  were to assert that 6ft. is short for a man, I think one would be
  entitled to question whether I were a competent speaker of the
  English language.  Thus there is an element of convention in the
  meaning of words in a language.  As I have tried to point out in
  the previous chapter, if a convention exists, then one's
  subjective estimation cannot be the whole story -- there must be
  an external reality out there regarding language use susceptible
  of objective characterization.

  Like Watanabe (1978), I also find it difficult to accept the
  result of the Zadehian min-max calculus that the fuzzy term 'tall
  and not tall' should be anything less than the logical absurdity,
  as the law of contradiction requires.  One would lose all
  credibility as a witness in court if one were to testify that the
  burglar was 'tall, but not tall'.  The fuzziness of the term 'tall'
  is not sufficient, in my view, to persuade a jury that such a
  contradiction could have positive meaning within the English
  language convention.  Similarly, the law of excluded middle
  requires that the disjunction 'tall or not tall' should be the
  constant tautology.  Again, I am not persuaded that the fuzziness
  of the term tall is sufficient to justify the result of the min-
  max calculus under which not all elements of the universe have
  full membership, tautologically, in this disjunction.  Finally,
  the law of self-contradiction requires that any term which implies
  its own negation must be the logical absurdity.  Under the min-max
  calculus, any term, not necessarily the absurdity, whose
  membership function is everywhere less than half must imply its
  own negation.

  In what follows I depart from the Zadehian fuzzy set theory first
  at the philosophical level.  I take the membership function to
  represent a usage convention which may in principle be objectively
  determined, using statistical methods.  Proceeding from this basic
  assumption, the concepts of probability and fuzziness may indeed
  be distinguished, but the concept of fuzziness is derived from
  that of probability, in almost exactly the same fashion that the
  Fisherian concept of likelihood derives from probability.  And in
  the same way that the concepts of probability and likelihood are
  distinct, the concepts of probability and fuzziness are distinct,
  though related concepts.  Coincidentally, it turns out that
  proceeding from such an assumption makes the min-max calculus
  quite simply untenable as a set of universal rules.  It becomes
  clear that other rules are sometimes appropriate, and it forces
  one to address the issue when does which apply.  In so doing, the
  law of contradiction and the law of excluded middle are upheld as
  a matter of necessity, a happy result if one is disposed to accept
  these as having positive empirical significance as semantic law.
  The law of self-contradiction does not hold, as in the Zadehian
  calculus, if one takes the containment relation between fuzzy sets
  as representing the implication relation.  If, however, as in fact
  is necessary in the present development, the rule of implication
  is defined with reference to possibility distributions rather than
  membership functions, and possibility distributions, like
  likelihood functions are unique only up to similarity
  transformations, then the law of self-contradiction may be
  restored.

  This result, and the restoration of the laws of contradiction and
  of excluded middle, are happy byproducts, however, rather than
  starting objectives.  My basic goal is rather to harmonize the
  essential truth of the fuzzy set concept with the essential truth
  of the concept of probability, and to try to sort out the
  respective limits of application of the two concepts in the
  representation of uncertainty.

  The approach is axiomatic -- material axiomatic rather than formal
  axiomatic: primitive concepts are first introduced and explained,
  followed by empirical postulates, followed by lemmas and theorems.
  In contrast with a formal axiomatic development, where primitive
  terms remain uninterpreted, the primitive concepts and empirical
  postulates introduced in the following development are intended
  very much to have positive, empirical significance where semantics
  is concerned.  We try to be faithful to our own conceptualization
  of phenomena and models, and give a morphology for the language-
  use phenomenon followed by an extension-set model embodied in the
  postulates which we adduce.

  In pursuing this axiomatic approach, we necessarily accept the
  rules of the two-valued set theory in our mathematical
  metalanguage, while developing rules which we hope apply to the
  fuzzy, many-valued terms which populate our object language.  This
  is a matter which itself requires some reflection.  I defer this
  discussion (see later, Sect. 3.4.5)."

 On harmony between bivalent metalanguage and fuzzy object
 language:

 "It is by no means obvious that vagueness and fuzziness in natural
  language should fall ultimately under the ambit of a two-valued
  logic.  Giles (1971, p. 322) for example has written that the
  notion of the fuzzy set is prior to that of set.  In the same
  vein, Goguen (1974, p. 514) writes: 'Ideally we would like a
  foundation for fuzzy sets which is independent of ordinary set-
  theory ...'  Goguen proceeds to axiomatize fuzzy set theory in the
  language of category theory, but as a theory of semantics it
  remains non-empirical.  Within the empirical framework which is
  proposed in this development, it would appear that what is fuzzy
  with respect to  U , e.g. 'tall', can be rendered as something
  crisp in

                             U
                        [0,1]

  e.g. mu[TALL]: U -> [0,1].

  Evidently we could define fuzzy sets of type 2, as Zadeh has done,
  corresponding to fuzzy sets with fuzzy membership values, in which
  case what is fuzzy with respect to

                             U
                        [0,1]

  could be rendered as something crisp in

                                 U
                            [0,1]
                        [0,1]

  It would appear that there is no end to the process of
  'crispification': starting with the intuitively well-accepted
  canons of classical two-valued logic we may bootstrap ourselves
  through the higher reaches of fuzziness.  For philosophers, not to
  mention computer scientists, this would no doubt be considered a
  happy result: a non-classical logic at the foundations of
  mathematical reasoning or computation is a forbidding thought
  indeed."


From Chapter IV ("Possibilistic Inductive Inference"):
     ------------------------------------------------

 "Definitions 3 and 4 correspond to Zadeh's (1978) definitions of
  possibility distribution function and of possibility distribution
  respectively, but differ in an essential way. Zadeh's notion of
  the possibility distribution function is regarded as an
  interpretation of the notion of membership function of a fuzzy
  set, and the two are set equal. The present notion is that the
  possibility distribution function represents the relative
  possibilities of a set of simple hypotheses generated by the
  entire universe of discourse U. Hence Zadeh's definition gives
  rise to an absolute function whereas in the present one,
  membership functions related by a similarity transformation would
  yield the same possibility distribution function -- fall into the
  same equivalence class. In short, Zadeh's notion of the
  possibility distribution function is absolute, while the present
  one is relative. This is all the difference, but it is key: this
  is the difference which separates the approximate reasoning
  literature from the statistical inference literature. With the
  relativist notion of the possibility distribution function, the
  gap between the literature on approximate reasoning and that on
  statistical inference is bridged, with the likelihood function
  serving as the counterpart of the (relativist) possibility
  distribution function. This relation will become clearer as we
  proceed."

 On the "rationality requirements" for inductive logic (from Chap.
 IV):

 "It will be evident that no _rationality requirements_ have been
  laid down for inductive inference, in the manner for example of
  Carnap (1971). What a rational man would or should infer about the
  value of an unknown when presented with a piece of evidence
  bearing on it depends first of all on some truth or reliability
  assessment of the speaker uttering the evidence. It seems futile
  to attempt to lay down rationality requirements that would
  encompass rules for truth assessment in general: what evidence a
  person would accept or deny as being 'true' would seem to be more
  a matter of behavioral psychology than a matter of pure logic. At
  best therefore it would seem that an inductive 'logic' by which a
  perfectly rational man may be guided would have to be predicated on
  the assumption that truth assessment of evidence is determined
  outside of the logic, which then nullifies the whole purpose of
  the exercise. This being the case, we take the slightly Popperian
  position of taking hypotheses as basic and evaluating these for
  truth conditional on truth-assessed evidence (Popper, 1972). What
  emerges is the possibility distribution, which provides a way
  first of characterizing what a speaker could mean by what he says
  taken at face-value, and second of characterizing what a listener
  may choose to infer regarding what could be the case after
  assessing the evidence for truth/reliability. Thus the possibility
  distribution may characterize belief, but it is not a subjective
  belief function in the sense of the Bayesians, since at no time is
  it necessary to invoke strength or degree of belief as a primitive
  attribute for which numeric representation is sought."

 On the product-sum rule for evaluation of composite hypotheses
 (also from Chap. IV):

 "The metaphor of Nature-as-speaker is not too far-fetched when we
  consider the frequent use of such expressions as 'What do the data
  say...?', or 'What do the data mean ...?' or 'What do the data
  tell us...?', and so forth, in reference to the results of
  experiment. It seems natural to identify 'the data' with Nature,
  in a metaphor very similar to the Bayesians' use of the phrase
  'state of Nature' in referring to the true value of the unknown
  parameter $ theta $, say, characterizing a probability model.  If
  we accept the metaphor of 'Nature asserting', then it seems
  reasonable to adopt similar meta-semantic considerations for
  making inference from assertions of Nature -- the data -- as we do
  from speakers in a natural language.  Furthermore, if we are to
  combine prior assertions of a decision-maker with assertions of
  Nature, or the data, it seems some unified interpretive framework
  is necessary. This leads us to the conception of the truth or
  likelihood of the composite hypothesis { w1, w2 } being identified
  with the probability of the calibrational affirmation 
  'w1 explains the data' \/ 'w2 explains the data'.  If we
  assume, as we have, that these affirmations should be considered
  as independent, or governed by lack of recall from one affirmation
  to the next, then we have a product-sum rule for the likelihood
  evaluation of composite hypotheses ...

  It is conjectured that the product-sum rule for composite
  hypotheses should in general resolve adequately the counter-
  example used in Chap. I ... to show how the maximum rule for
  marginal likelihood may be brought into question."


From Chapter V ("Possibilistic Deductive Inference"):
     -----------------------------------------------
   
 "In this program, the viewpoint is a little different from that of
  Zadeh (1977), who saw approximate reasoning as concerned with the
  'deduction of possibly imprecise conclusions from a set of imprecise
  premises.' Traditional logic, by restricting itself to form rather
  than meaning already allows us to reach imprecise conclusions from
  imprecise premises, a fact that is exemplified by the example 
  previously considered:

     If one is rich, then one is happy  - Premise (Theory 1)

     John is rich                       - Premise (observation)

     Therefore, John is happy           - Conclusion.


  Here both premises are imprecise, as is the conclusion, even
  though the rule of deduction which has been applied is quite
  exact, relying only on the logical form  [(P -> Q) /\ P] -> Q  in
  the standard logic notation. What we now propose to do is to take
  meaning as primary, and to allow deductive inferences to be
  validly drawn whenever meaning is preserved, in a sense to be made
  clear very shortly. From this viewpoint, the appellation
  'approximate' in 'approximate reasoning' would refer not
  so much to the rules of logic or of reasoning involved, but to the
  nature of the assertions involved, which may in general be fuzzy.
  Traditional logic in dealing primarily with form, proceeds almost
  entirely on the semantics of form, hardly at all on the semantics
  of content. Here we start with content and rules based on form
  emerge as a special case."


From Chapter VI ("Possibilistic Decision Analysis"):
     ----------------------------------------------

 ".... But the point is that judgments affecting preference or
  choice fall into a wider class of subjective judgments generally,
  so that if we could construct methods which apply generally to the
  larger problem of subjective estimation or scaling, we would _ipso
  facto_ have constructed methods for scaling decision options
  (stimuli) on the attribute of level of preference or desirability.
  Our approach to the choice problem is to treat it as a scaling
  exercise of this sort."

 On the scale properties of utility:

 "Assumption 1 supposes [the attribute of level of desirability of
  a decision option] to possess ratio-scale properties. For those
  who are persuaded that utility and therefore the present notion of
  level of desirability is at best an interval-scale attribute, a
  few remarks appear necessary. A ratio-scale attribute is one whose
  universe of discourse could be considered to possess a 'natural'
  or 'absolute' zero. This does not appear to be an unreasonable
  property for the attribute of level of desirability: the natural
  zero corresponds to a desirability level of absolute indifference
   -- the situation, say, where one may 'take it or leave it' with
  equal equanimity. To one side of the absolute indifference level,
  we have positive desirability; to the other side, we have negative
  desirability or aversion.  This seems to conform to the notion 
  of weighing the 'pleasures' and 'pains', as Plato put it.  This
  is such a natural conceptualization of the notion of desirability
  (two half-lines, one positive, the other negative with
  indifference (= natural zero) in between) that the
  fact of the interval-scale characterization of utility in
  traditional utility theory is perhaps more the one which needs
  examining. What such an examination would reveal is that the use
  of interval scale properties for utility represent merely the
  weakest assumptions necessary for the purposes of that theory. As
  everyone needs to have clarified on first acquaintance with that
  theory, 'zero' utility does not necessarily mean 'no' 
  utility, emphasizing the gap between the natural language
  characterization of notions of utility and desirability, and their
  approximation within the artificial language of utility theory.
  Within that theory, a utility function is defined to be a function
  (from a real-world set of rewards to the set of real numbers) such
  that given two probability distributions, P1 and P2 (lotteries) on
  the set of rewards, P1 is preferred to P2 if and only if the
  expectation of the function with respect to P1 is greater than
  that with respect to P2 (de Groot 1970, p. 90). This is all we
  need if we wish merely to discriminate amongst options, and this
  permits arbitrary interval-scale transformations. However it
  should not exclude a subjective estimation procedure which
  exploits ratio-scale properties. As it turns out, interval-scale
  transformations of the final scaling of decision options as to
  desirability do not affect choice of the optimum decision option.
  This vindicates the utility theory as an acceptable approach in
  principle, but it does not invalidate an approach which exploits
  ratio-scale properties in the scaling procedure, as we attempt to
  do."

 On group preferences:

  ".... Arrow proved that there is no general procedure for
  obtaining a group ordering over a set of decision options based on
  individual members' preference orderings, that is consistent with
  five seemingly reasonable conditions. When utility functions (von
  Neumann-Morgenstern type) are used in place of preference
  orderings, Harsanyi (1955), and more recently Keeney (1976) have
  shown that the impossibility result of Arrow no longer holds. A
  group utility function may be constructed using axioms analogous
  to Arrow's but stated in terms of utilities rather than preference
  orderings. Moreover, the form of the group utility function is
  shown to be restricted to the narrow class consisting of linear
  combinations of individual utilities, that is of the form W = sum
  ki *  wi where  W  is the group utility, wi (i = 1 ,..., n)  are
  individuals' utilities and ki (i = 1 ,..., n)  are constants
  reflecting individuals' relative weightings in the aggregation.
  The question which this result invites is how do we determine the
  ki (i = 1 ,..., n).  The discussion by Keeney makes it clear that
  a choice of combination weights is essentially a problem of inter-
  personal comparisons of utility, moreover it is one which must
  devolve around the members of the group as the individuals jointly
  responsible for decisions taken under an assumed group utility
  function  W . Where Assumption 2 represents a different approach
  is in side-stepping the aggregation question, and making the
  assumption right at the start that individual members have the
  capability of judging decision options on the attribute of level
  of desirability for the group as a whole.  It is therefore
  supposed that individual members of the group are capable of the
  necessary inter-personal comparisons of satisfaction, though only
  in an implicit fashion. This implicitly supposes that each
  individual is capable of distinguishing narrow self-interest --
  both of himself and of other members -- from the interest of the
  group as a whole, and is capable of judging the relative merits,
  from the group standpoint, of different compromise solutions when
  individual interests must necessarily be in conflict.  Without
  asking how or why people come to such skills, I would note only
  that the harmonious functioning of any group of individuals,
  whether as small as a family or large as a nation, seems to depend
  upon their existence."


From Chap. VII ("Bayesian vs. Possibilistic Approaches"):
     --------------------------------------------------

 "The question raised by all this is just how inevitable is the
  probabilistic characterization of prior belief.  The Bayesian
  explication hinges inordinately on the betting paradigm by which
  one's degree of belief in an uncertain proposition is identified
  with the odds at which one is willing to bet on it.  Thus the
  Bayesian argument is only inevitable if the notion of belief is
  entirely captured by the operational notion of willingness to bet.
  The Bayesians admit, as part of their development, that
  willingness to bet on an event reflects not only one's 'degree of
  belief' in the possibility of the occurrence of that event, but
  also the values one attaches to the stake and potential winnings.
  Odds remaining the same, one's willingness to bet one dollar does
  not imply willingness to bet 100,000 dollars.  Also, belief
  remaining the same, one's willingness to bet any given sum changes
  quite definitely with the odds offered.  The Bayesians fix the
  odds by requiring one to be willing either to place or take the
  bet, and they take account of value by measuring stakes and
  winnings in terms of psychological utility.  The remaining
  determinant of betting behavior, 'degree of belief', is then
  fixed, and simple rules of rationality imply that this belief
  function must obey the probability axioms.  This is a reductionist
  procedure: the notion of belief is that which is left after,
  starting with observable betting behavior, we take account of
  value and odds, the two other determinants of betting behavior.

  Could we not go the other way?  What if we were to adopt a
  constructionist approach in which the notion of belief is
  explicated, built up as it were, from other more primitive
  considerations.  And what if a constructionist belief
  characterization, when joined with consideration of odds and
  value, allowed us to explicate betting behavior?  The answer, I
  think, is that the Bayesian analysis would seem less than
  compelling.  Furthermore, if such a development preserves the
  essential truth of the Bayesian development, that subjective
  belief considerations should be incorporated where appropriate,
  together with its mathematical convenience, while not requiring
  the total commitment to subjectiveness that is an unwelcome
  intrusion when we wish to infer strictly on the basis of
  experimental evidence, then we have the basis of a compromise that
  meets the concerns of both classical and Bayesian schools.  Part
  of the burden of this essay has been to develop just such an
  explication of belief, starting with the semantic underpinnings
  provided by a (modified) fuzzy set theory."


From Chap. VIII ("Summary and Conclusion"):
     -------------------------------------

 "The end-result of all this is an extended likelihood or
  possibility calculus that covers all the Bayesian ground, without
  the Bayesian postulate of prior, subjective belief probability.
  Prior subjective belief may be linguistically expressed without
  need for the Bayesian straitjacket of 'coherence'. Moreover, the
  inferential process does not require the injection of prior
  evidence or belief, but may accommodate it whenever this is
  available, and its inclusion is warranted. Thus the classicists'
  major objection to the Bayesian procedure that it needs to suffer
  the intrusion, always, of subjective prior opinion may be met,
  while the major Bayesian advantage of a direct characterization of
  uncertainty of modelling, and an associated powerful calculus is
  not sacrificed.

  The advantages of a direct characterization of uncertainty of
  modelling are reflected most in decision analysis. Bayesian
  methods of inference combine quite attractively with an expected
  utility approach to decision analysis. The essential structure of
  such an approach to decision analysis may be retained here.
  However the intrusion of 'fuzz' into probability models does not
  conform to the axiomatic development of utility theory, in which
  the basic choices are among gambles in which the probabilities are
  non-fuzzy. Hence one could either re-develop the traditional
  axiomatic utility theory to accommodate fuzz, and its associated
  calculus, or one could proceed without utilities. In the latter
  case, the outputs of a decision analysis would display the fuzz on
  appropriate measures of consequential real-world gain, loss or
  risk.  Choices could then be made primitively from among competing
  possibility distributions on measures of gain or loss. Summary
  measures (e.g. center of gravity, area, second moment about the
  mean, etc) could of course be employed as necessary.

  A benefit of the present approach, by comparison with the Bayesian
  approach to decision analysis, is the comparative ease with which
  possibilistic prior information may be elicited from decision-
  makers.  Furthermore, evidence from several sources are treated
  symmetrically within the theory, assertions of Nature or
  experimental evidence being just one source.  This means that a
  calculus for the representation of group belief, or the
  combination of evidence, and for the aggregation of group
  preferences, is easily integrated into the framework.

  The development sheds some light, I think, on the Saaty method for
  the scaling of judgmental atributes.  This essay was motivated in
  large part by a desire to explicate the success of the Saaty
  method in fuzzy set terms.  This led us into the theory of
  measurement, and to a realization that the paradigm of point-
  numeric measurement needed to be changed to accomodate the idea
  engendered by the fuzzy set theory that data are fuzzy sets in
  general.  This has the salutary consequence that the treatment of
  error of measurement is integral to the theory, rather than being
  the conceptually difficult and bothersome afterthought that it is
  in the now-classical theory exemplified by Krantz, Luce, Suppes,
  and Tversky.

  The present development also bears on foundational questions of
  fuzzy set theory. One of the issues which has been of concern in
  the theory has been the relation between, and the line of
  demarcation that should separate, the respective applications of
  fuzzy and probabilistic calculi.  The present development puts
  forward an interpretive framework that mixes the two calculi quite
  intimately. The somewhat paradoxical result is that the
  differences between the two are made very clear, as also are the
  respective limits of application of the two concepts. Probability
  remains the more basic of the two, and stands in the same relation
  to the other, as it already does to the more familiar concept of
  likelihood."

Regards,
S.F.Thomas 

