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From: sthomas@decan.gate.net (S. F. Thomas)
Subject: [ANNOUNCE] Fuzziness and Probability: the manuscript
Keywords:  Fuzzy sets, probability, possibility, likelihood, inference, measurement, decision, belief
Message-ID: <1994Dec20.005619.6872@decan.gate.net>
Summary:  Manuscript in postscript format available by email 
Organization: Decision Analytics, Inc.
Date: Tue, 20 Dec 1994 00:56:19 GMT
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		         ANNOUNCEMENT

		  FUZZINESS AND PROBABILITY

The manuscript _Fuzziness_and_Probability_ (1994) by S. F. Thomas,
is now available by email.  It is anticipated that the manuscript
will be published by next summer.

			*************
			   HOW-TO
			*************

TO OBTAIN an email copy of the manuscript, send email to 

	book@decan.gate.net 

with any subject or text.  This dummy account will reply to any 
incoming email by mailing out the uuencoded, compressed tar file 
containing postscript code for the manuscript.  The email response 
will consist of 20 parts, none exceeding 62Kbytes.  The total size 
of the uuencoded compressed tar file is 1.2Mbytes.  
Please allow 24 hours for the email response; I do not have direct
internet access and must rely on a commercial dial-up service.  
In case of difficulty, send email to me at sthomas@decan.gate.net.
I am trying to find an ftp repository site for the manuscript, and
will upload it to one as soon as it is accepted.

			*************
	         DESCRIPTION OF THE MANUSCRIPT
			*************

I have in recent posts to comp.ai.fuzzy published some excerpts from
the manuscript.  I am encouraged by the response received, which
included an expression of interest by a publisher, to publish the 
manuscript both in book form, and electronically.  I want to thank
all of those who responded privately, or publicly on the newsgroup,
for their expression of interest.

By way of indicating to those who might be interested what the contents
of the manuscript are, I give the following excerpts from the
preface and from each of the eight chapters of the manuscript: 
(I apologize if there is any overlap with excerpts previously posted 
to this newsgroup)

From the Preface:
         -------

   "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 addressed to students or practitioners of decision analysis,
   statistical analysis, artificial intelligence
   and to academicians in these and the other areas I have mentioned.
   
   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, and pointing out some of the foundational issues
   in the fuzzy set theory which are still unsettled.
   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, contradiction,
   and self-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 what I call an _actual-use_ proposition.  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
   actual-use 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 OR 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 begs 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."

Cheers!
S.F.Thomas
