_{} **INFT 842: ALTERNATIVE SYSTEMS OF
PROBABILISTIC REASONING**

** FALL 2004**

** INSTRUCTOR: DAVE SCHUM**

**THE BASIC
SUBJECT MATTER OF THIS COURSE:**

** **In this seminar we will examine
alternative views of the process of drawing conclusions from masses of
incomplete, inconclusive, ambiguous, dissonant, and unreliable evidence. Among
the views we will discuss are the **Bayesian**,
**Baconian**, **Shafer-Dempster**,
and **Fuzzy** systems for probabilistic
reasoning. Each one of these formal systems, as well as others to be mentioned,
has interesting properties that deserve our attention. The necessity for
considering alternative views seems evident since it seems far too much to
expect that any one formal system of probability can capture all of the
richness of probabilistic reasoning. We will examine grounds for the necessity
of considering alternative views of probabilistic reasoning and, in the
process, consider applications of each of these views to non-trivial inference
problems. Of particular interest to us are issues concerning what seems to
constitute "rational" probabilistic reasoning.

A survey of alternative formal
systems of probabilistic reasoning, by itself, is not quite enough to provide
an adequate background for later applications. Many other issues arise in the
application of any of these formal systems. Some of these issues concern the
fact that most human inference problems seem to involve **mixtures** of the tasks of inductive, deductive, and what has been termed
"abductive" reasoning. One very important
issue concerns the **discovery** or
generation of the essential ingredients of probabilistic inference:
hypotheses/possibilities, evidence, and arguments providing the linkage between
evidence and hypotheses. Such generation is an exercise in imaginative,
creative, or abductive reasoning [the essential topic
of INFT 944 ]. Another important issue concerns the
coupling of probabilistic reasoning and choice. In some contexts [science, for
example] probabilistic reasoning has a life of its own. In many other contexts,
however, such reasoning is embedded in the further process of choice in which
difficult value-related issues arise. We will also examine various contemporary
views about the coupling of probabilities and and
values in the process of choice.

**TEXTUAL
AND OTHER REFERENCES ON PROBABILISTIC REASONING:**

1). There are many relevant and
interesting works on the topic of probabilistic reasoning to be found in a
variety of disciplines; our lives are simply too short to master all of them.
One trouble is that most published works on the topic of probabilistic
reasoning do not provide much discussion of the very basis for it, namely **evidence**. Many of the works we will
consider focus just upon algorithms for combining probabilistic judgments of
various sorts and say very little about the evidential foundations for such
judgments. After offering this seminar for a number of years, and at the
suggestion of many students who completed it, I decided to write a book that covers
what I regard as the basic evidential foundations of each of the formal systems
of probability we will discuss. This book is entitled: *The Evidential Foundations of Probabilistic Reasoning [*Northwestern
University Press,

2). There are, of course, many other
references to consider; these references come from many different disciplines
and frequently involve matters about which there is not widespread awareness.
At the end of this syllabus is a "seed" bibliography containing some
basic references that are keyed to the major probability systems we will
discuss in this seminar.

3). During
this seminar I will provide you with an assortment of notes and handouts on
particular topics. In most cases these notes will consist of applications of
the various probabilistic reasoning systems and additional sensitivity analyses
of the sort you will find in the text noted in 1) above.

**TOPICS AND
****READING**** ASSIGNMENTS**

Following is the sequence of topics
we will cover in this seminar together with associated reading assignments
from: *Evidential Foundations of
Probabilistic Reasoning* [EFPR]. The dates shown are only approximate; we
may wish to dwell longer on some topics than on others.

I. __SOME
PRELIMINARY MATTERS [31 August]__

To set the stage for later
discussions, I will begin by giving you a bit of history of the development of
interest in the evidential foundations of probabilistic reasoning. In the
process I will try to convince you that any theory of probabilistic reasoning
must be concerned about two major issues: (i)
structural matters in the generation of arguments from evidence to
hypotheses/possibilities, and (ii) matters concerning the assessment and
combination of ingredients associated with the force, strength, or weight of
evidence on these hypotheses/possibilities. Further, I will elaborate a bit on
my claim that human reasoning in natural settings [the "real world",
if you like] involves mixtures of the three forms of reasoning I mentioned
above. I will also give you an overview of the literature that exists in so
many disciplines that is relevant to an understanding of the complexity of
probabilistic reasoning.

II.
__EVIDENCE, PROBABILITY, AND STRUCTURAL ISSUES__ [7 - 14 September].

When most people think about
probability they think only about __numbers__. But, as I will explain,
probability theories are equally concerned about the __arguments__ that are
constructed to support the use of numbers. The construction of arguments is a
creative task concerning which most of us have very little appropriate
tutoring. This is especially true when we must construct arguments from __masses
of evidence__. We will consider some exercises showing just how difficult
this can be. One major benefit of studying structural matters is that it allows
us to discern various recurrent forms and combinations of evidence and to
observe how evidence is used during the process of probabilistic reasoning,
regardless of the view of probability one adopts. To be useful in probabilistic
reasoning, evidence must have certain credentials concerning its __relevance__,
__credibility__, and __inferential force, strength, or weight__. No
evidence comes to us with these credentials already established; they must be
established by cogent arguments. The major issue that divides the alternative
views of probabilistic reasoning we will examine involves how best to grade the
inferential force, strength, or weight of evidence and then to combine such gradings across many items of evidence.

__Reading
Assignment__: EFPR Chapters 1 - 3.

III.
__STRUCTURAL ISSUES AND COMPLEX INFERENCE [21-28 September]__

There is presently a substantial
level of research on what are called __inference networks__. In a wide
variety of applied areas, people have the difficult task of trying to make
sense out of masses of evidence that, upon examination, reveal all of the forms
and combinations of evidence we will have considered in Section II. When
arguments are constructed from a mass of evidence they begin to resemble complex
networks. Many [not all] of the important elements of complex inference can be
captured in graph theoretical terms. Naturally, there are many current attempts
to develop various forms of computer assistance in coping with the complexity
of inference based on masses of evidence. Matters we discuss here will be quite
relevant to other courses now offered in IT&E concerning inference networks
[e.g. Professor Kathy Laskey's INFT 819].

__Reading Assignment__: EFPR
Chapter 4

IV.
__ALTERNATIVE PROBABILITY SYSTEMS I: BASIC THEORETICAL ELEMENTS __

[5 October - 2 November; no class on 12
October]

Here we come to a discussion of the
different views about probabilistic reasoning. If only for historical reasons,
we begin with the conventional, Pascalian, or Bayesian
system for probabilistic reasoning. Then we will consider the Shafer-Dempster theory of belief functions, Cohen's system of Baconian probability for eliminative and variative
induction, and Zadeh's ideas on fuzzy reasoning and
fuzzy probabilities. Our basic objective here is to examine what one is
committed to when one applies any of these formal systems. A question
frequently asked is: which one of these systems do you

prefer? This is rather like asking whether
you prefer your saw or your hammer. Each system "resonates" to
particular important elements of probabilistic reasoning. Each system has
something important to tell us about probabilistic reasoning, but no system
says all there is to be said. At this point we will dwell upon what is meant by
the term "rational" probabilistic reasoning. In this section we will
also give attention to the matter of combining probabilistic and value-related
judgments in situations in which probabilistic reasoning is embedded in the
further process of choice.

__Reading Assignment: EFPR Chapters
5 - 6__

V. __ALTERNATIVE
PROBABILITY SYSTEMS II: EXAMPLES AND APPLICATIONS__

[12 - 23 November].

Having examined the theoretical
underpinnings of alternative systems of probabilistic reasoning, we will
examine what each produces in applications in different inferential contexts in
which different forms and combinations of evidence are encountered. In some
cases, our examination will consist of sensitivity analyses performed on
probabilistic expressions designed to capture various forms and combinations of
evidence. It is in this process that we begin to observe how many important and
interesting evidential and inferential subtleties lie just below the surface of
even the "simplest" of probabilistic reasoning tasks. These subtleties,
if recognized, can be exploited in our probabilistic reasoning.

__Reading Assignment: EFPR Chapters
7 - 8__

__VI. DISCOVERING THE INGREDIENTS OF PROBABILISTIC REASONING
[30 November - 7 __ __Dec.]__

As I noted, the ingredients of
probabilistic reasoning are rarely provided for us [except in classroom
exercises and examples]; they have to be discovered or generated. Stated in
other words, there is the necessity for imaginative or creative reasoning in
every probabilistic reasoning task. What we have time to do is simply to
examine some of the imaginative elements of probabilistic reasoning. The topics
of discovery and imaginative reasoning are of sufficient importance that we
devote an entire seminar to the topic [INFT 944] that is offered in the Spring semester.

__Reading Assignment: EFPR Chapters
9-10__

**PROCEDURAL
MATTERS AND METHOD OF EVALUATION**

To tell you how I believe this
seminar should proceed, I will make reference to the thoughts of Sir Francis
Bacon. Bacon argued that, in any scholarly activity, reading makes us **full**, discourse makes us **ready**, and writing makes us **accurate**. We will all do a fair amount
of each in this seminar. As far as **reading**
is concerned, you have one major text and the various handouts and other
materials I will give you. In addition, the reference list I include at the end
of this syllabus contains a wealth of information on all of the probability
systems we will discuss. You can pick and choose among these references
depending upon where your interests take you.

As far as **discourse** is concerned, I look upon this seminar as an experience
in the sharing of ideas. Indeed, my view of scholarship is that it is a
sophisticated form of sharing; we learn from each other. I have no wish to
monopolize our discussions. On each occasion we meet I will try to get things
started and will bring various matters to your attention. What happens after
this depends upon you. Often, you may believe I am saying outrageous things
with which you disagree; if so, your task is then to show me how I have been
misled. In addition, you may have questions that I have not raised; your task
is to raise them as we proceed. In short, I hope our meetings will be both
enjoyable and productive; whether or not this happens depends upon all of us.

Finally, the topic of **writing** brings us to the first of two
methods of evaluating your progress in this seminar. We will encounter
literature from a number of different disciplines; no one discipline dominates
scholarship on probabilistic reasoning. Together, the various works we will
discuss supply a breadth of view. Depth of view is provided by your focus upon
one or more of the formal systems; such focus will form the basis for a paper
you will write, __which will count 75% of your grade__. The choice of focus
is yours, provided that it concerns **probabilistic**
reasoning or some matter that directly involves such reasoning. Here are some
characteristics of your written work that I will look for. Notice that I here
mention several criteria that will also be in force as far as your dissertation
is concerned.

(1) Your work should demonstrate
that you have attempted to apply your own thoughts to the issues/problems that
you address. A simple review of literature on some topic [e.g. X says
this, Y says that] will not suffice. What **you**
think about the issues your paper addresses is what is of primary importance. I
do not insert this criterion out of perversity; it is the most important
criterion that will be applied in evaluating your doctoral research.

(2) Your written work should, of
course, demonstrate awareness of the relevant work of others. When you discuss
previous work on your chosen topic, you should give evidence of having
integrated it in a careful and sensible way and are suitably critical of ideas
you encounter. Unless you do these things, I will not be able to judge how you
drew the conclusions you did from the work of others. This also happens to be a
criterion that will be enforced in your doctoral dissertation.

(3) Perhaps many of you are now
engaged in outside work to which the essential ideas in this seminar are
relevant. In the past I have been quite free in allowing students to report on
their efforts to apply ideas from this seminar to some ongoing project of
interest in their outside work. Sadly, only a small number of these papers have
been of a level of quality one expects from doctoral students. One reason is
that students frequently spend too much time dwelling upon substantive issues
in the problem domain and too little time on the theoretical probabilistic
matters that are being applied in this domain. In some cases there was very
little evidence in a paper that the student actually mastered any probabilistic
matters that could have potential application. I certainly hope and expect that
there will be ideas in this seminar that have applicability in areas of
interest to you. If you do attempt to apply ideas from this seminar to
particular ongoing work you are doing, I will be most concerned about how well
and how completely you have mastered the ideas you are attempting to
apply.

(4) I will give **very** careful attention to the coherency with which you present your
ideas. Yes, I am concerned about your ability to communicate your ideas in
written form. I do you no service at all by ignoring grammatical and stylistic
matters.

(5) You should look upon this
written work as something that you do for yourself, not for me. It will simply
demonstrate that you have attempted to acquire some **uncommon** understanding of a probabilistic inferential issue that
captures your interest.

The remaining 25% of your grade will
be based upon some exercises and problems that I will assign as we proceed. I
expect to give you five such exercises during the semester. Many of these
exercises will involve probabilistic analyses of various forms and combinations
of evidence that appear in your text and in our class discussions.

** A
"SEED" BIBLIOGRAPHY**

** **Following is a list of basic
references to works in each of the alternative systems of probabilistic
reasoning we will discuss. In addition, I have included one or more important
references for each system that are frequently cited but, with less frequency,
actually read.

**I.
Views Based Upon The Conventional [Pascalian]
Calculus Of Probability.**

__A.
"Standard" Bayesianism__

Perhaps
the most frequently cited but infrequently read item on our list is the
original paper by Thomas Bayes that was communicated
by Richard Price to the Royal Society Of London two
years after Bayes' death.

•
**An Essay Towards
Solving A Problem In The Doctrine Of Chances.** By The Late Rev. Mr Bayes, F. R. S. Communicated by Mr. Price in a letter to
John Canton, A. M., F. R. S., **Philosophical
Transactions Of The Royal Society**, pp 370-418, 1763.

There
is now debate about whether or not what has become known as Bayes'
Rule should, in fact, be attributed to Bayes. Here is
a paper on the idea that others may have earlier tumbled to the essence of Bayes' Rule.

•
Stigler, S. M., Who Discovered Bayes's Theorem ? **The
American Statistician**__,__ November 1983, Vol. 37, No. 4, p 290 - 296.

Here
are some tutorials on the use of Bayes' rule in
probabilistic inference.

•
von Winterfeldt, D.,
Edwards, W., **Decision Analysis And
Behavioral Research**__, __

•
Schum, D., **Evidence And
Inference For The Intelligence Analyst** [Two Volumes],

There
are now many references relative to the use of Bayes'
Rule in **statistical**__ __inference.
Here are two places to start if you have interest in statistical inference.

•
Winkler, R., **Introduction To Bayesian Inference And Decision**__,__

•
O'Hagen, A., **Probability:
Methods And Measurement**,

Here
are two works that, to varying degrees, consider Bayes'
rule as __the__ normative standard for probabilistic reasoning:

•
Earman, J., **Bayes**** or Bust ?: A Critical Examination of Bayesian Confirmation
Theory**,

•
Howson, C., Urbach, P., **Scientific Reasoning: The Bayesian Approach**,
Open Court Press, 1989.

Here
are four works on the application of Bayes' rule in
complex "inferential networks"

•
**Probabilistic Reasoning In Intelligent
Systems: Networks Of Plausible Inference**__,__

•
Schum, D., **Evidence And
Inference For The Intelligence Analyst** [Two Volumes],

•
Neapolitan, R. E., **Probabilistic
Reasoning In Expert Systems**,

•
Kadane, J., Schum, D. **A Probabilistic Analysis of the Sacco and Vanzetti Evidence**.

We
should note here that there is a very large literature on various possible
interpretations of numbers that correspond to the standard or Pascalian calculus. Indeed, this is an entire topic by
itself. There are two useful summaries of alternative conceptions of a Pascalian probability, they are:

•
Fine, T., **Theories Of
Probability: An Examination Of Foundations**,

•
Weatherford, R., **Philosophical
Foundations of Probability Theory**,

__B. "Non-Standard" Bayesianism__.

It
has been claimed by several Scandinavians [and one renegade Englishman] that we
are frequently misguided in the manner in which we apply Bayes'
rule in drawing conclusions from evidence. They have put together what is
described as the ** evidentiary value
model** [EVM]. This model is basically Bayesian but it rests upon a
particular view of evidence and how it should be used. I include EVM on this
listing because it provides an interesting transition to the next major view,
the non-additive system of beliefs proposed by Glenn Shafer. Over a dozen years
ago, I challenged the EVMers to show how this system
is in any way congenial to application in cascaded or hierarchical inference; I
am still waiting for a reply. Here is the major reference to EVM research:

•
Gardenfors, P., Hansson,
B., Sahlin, N. E. [eds], **Evidentiary
Value: Philosophical, Judicial, And Psychological Aspects Of A Theory**.

There
are several references in the above work that are relevant to our next major
view of probabilistic inference.

**II.
A Theory Of Nonadditive Beliefs Based On Evidence.**

It
is easily shown how probabilities encountered in **aleatory** [games of chance] and **frequentistic**
[statistical] contexts can be trapped within the Pascalian
calculus. But what about the array of **epistemic
**contexts in which numbers are used to grade the **strength of our beliefs **about whether or not some event has
happened, is happening, or will happen ? In such
contexts we often encounter singular, unique, or nonreplicable events that can
have no frequentistic interpretation. In epistemic
contexts many commonly-encountered credal or belief
states cannot easily be trapped within the bounds of the Pascalian
system; it turns out that this has been known for centuries. Our first
reference here is a very useful and well-done treatise on the history of the
concept of probability. Discussed in this treatise are some of the difficulties
the Pascalian calculus has experienced in the
trapping of these credal states.

•
Hacking, **The**** Emergence Of Probability**,

There
are alternatives to the use of Bayes' rule in the
task of combining our beliefs based on some emerging body of evidence; this has
also been recognized for centuries. One mechanism for belief combination has
been termed "Dempster's rule"; this rule
has roots in much earlier work. In 1976 Glenn Shafer took Dempster's
rule as the cornerstone for a "new" system of probabilistic reasoning
involving what he terms a "belief function". We shall term this
species of probabilistic reasoning the "Shafer-Dempster"
system.

__A.
The Shafer-Dempster View__.

Shafer was a student of Dempster's
and has now achieved a considerable measure of fame as a result of the
following work:

•
Shafer, G., **A Mathematical Theory Of Evidence**,

Some
of us believe this work is not actually a theory of evidence but a theory of
belief based upon evidence; form your own opinion as we discuss his work and
the work of others. Here is a summary of recent thinking about the Shafer-Dempster system of belief functions:

•
Yager, R., Fedrizzi, M., Kacprzyk, J. **Advances
in the Dempster-Shafer Theory of** **Evidence.** Ney York, John Wiley & Sons,
1994.

Here
are several often cited works by Glenn Shafer that elaborate on the historical
foundations of the Shafer-Dempster view and that also
are critical of any view in which Bayes' rule is
advocated as **the** "normative"
or "prescriptive" view of probabilistic reasoning.

•
Shafer, G., Bayes's Two Arguments For The Rule Of
Conditioning, **The Annals Of Statistics**__,__
Vol. 10, No. 4, 1982.

•
Shafer, G., Conditional Probability, **International
Statistical Review**__,__ Vol. 53, 1985

•
Shafer, G., The Combination Of Evidence, **International Journal Of Intelligent
Systems**__,__ Vol. 1, 1986.

•
Shafer, G., The Construction of Probability Argument, **Boston**** ****University**** Law Review**__,__ Vol. 66, Nos,
3 & 4, May/July, 1986.

•
Shafer, G., **Readings in Uncertain Reasoning**, Morgan Kaufmann Publishers, 1989
[compares Bayes with Shafer-Dempster.
Formerly assigned as a text in this seminar].

__B.
"Potential Surprise": Another Nonadditive
System__.

More
than one person has been interested in relationships between the concepts of **probability** and **possibility**; one such person is the British economist G. L. S.
Shackle. In the following work Shackle proposed a metric, called **potential surprise**, for grading our
beliefs about the possibility of events, something he believed was not possible
within the Pascalian
calculus.

•
Shackle, G. L. S., **Decision, Order, And Time In Human Affairs**,

This
system is still being discussed, thanks to the efforts of the American
philosopher Isaac Levi. Levi claims this system to be quite general and sees in
it some parallels with the Shafer-Dempster system. In
some quarters this system of reasoning is referred to as the Levi-Shackle
system of reasoning. Here are some further, more up to date references to
potential surprise.

•
Levi., **Decision
And Revision: Philosophical Essays On Knowledge And
Value**,

•
Levi, **The**** ****Enterprise**** Of Knowledge: An Essay On Knowledge, Credal
Probability, And Chance.**

**III.
Probability And Eliminative Induction.**

In
many contexts, science for example, we subject our hypotheses to a testing
process in which only the fittest survive. In such tests evidence is used as a
basis for **eliminating** hypotheses. As
Professor L. Jonathan Cohen will tell us in his book, a particular hypothesis
seems to have increasing probability or provability as it survives our best
efforts to invalidate it. We subject hypotheses to a variety of __different__
evidential tests; the more of these tests some hypothesis survives, the more
confidence we have in it. The key word here is **variety**; the survival of any hypothesis depends upon the extent to
which it holds up under **different**
conditions. Replication of test results is important but we cannot gather
support for some hypothesis simply by performing the same test over and over
again. Drawing upon the work of Sir Francis Bacon and John Stuart Mill,
Professor Cohen has given us a system of probability that is suited to what he
terms **eliminative and** **variative**
inductive inference. In this system of "Baconian"
probabilistic reasoning, probabilities grade the extent to which some
hypothesis survives an eliminative testing process, The
"weight" of evidence, in Baconian terms, is
related to the **number** of evidential
tests we perform and to the extent to which our tests cover variables relevant
in discriminating among the hypotheses we consider. Cohen's system is the only
system that specifically grades the **completeness
or the sufficiency** of our evidence. How likely we view some hypothesis
depends upon how many relevant questions concerning our hypotheses that our
existing evidence does not answer.

The
first reference is to Cohen's work on developing means for grading the **inductive support** that evidence
provides in the eliminative testing process Cohen describes.

•
Cohen, L. J., **The**** Implications Of Induction**, London,
Methuen & Co. Ltd., 1970.

Cohen
certainly acknowledges that there is room for more than one view of
probabilistic reasoning. Cohen's "polycriterial"
account of probability is given in the following three references, the second
of which can probably be termed his major work.

•
Cohen. L. J., Probability: The One And The Many, **Proceedings Of The British Academy**, V ol. LXI, 1975.

•
Cohen. L. J., **The**** Probable And The Provable**,

•
Cohen, L. J., **An**** Introduction to the Philosophy of
Induction and Probability**,

Here
are several other of Cohen's works that are of
particular importance to anyone seeking to understand the full dimensions of
Cohen's views.

•
Cohen, L. J., Bayesianism Versus
Baconianism In The Evaluation Of Medical Diagnosis. **British Journal For
The Philosophy Of Science**, Vol. 31, 1980.

•
Cohen, L. J., Twelve Questions About Keynes's
Conception Of Weight, **British Journal
For The Philosophy Of Science,** Vol. 37, 1985.

•
Cohen, L. J., Hesse, M., **The Applications Of Inductive Logic**,

•
Cohen, L. J., **The**** Dialogue Of Reason**,

Here
are three references in which Cohen's Baconian system
is discussed and compared with other views. I have copies of these works which
you are welcome to have.

•
Schum, D., A Review Of A Case Against Blaise Pascal
And His Heirs, **University Of Michigan
Law Review**, Vol. 77, No. 3, 1979.

•
Schum, D., Probability And The Processes Of Discovery,
Proof, And Choice. **Boston**** ****University**** Law Review,** Vol. 66.,
Nos. 3 & 4, May/July 1986.

•
Schum, D., Jonathan Cohen And Thomas Bayes On The Analysis Of Chains Of Reasoning. In: **Rationality And
Reasoning: Essays In Honor Of L. Jonathan Cohen** [eds. Eells,
E., Maruszewski, T.]

**IV.
Imprecision, Fuzzy Probabilities, And Possibilities.**

The
Pascalian system of probability is rooted in a system
of two-valued logic; a statement is either true or false or a particular
element is either in some subset or it isn't. In 1965 Professor Lotfi Zadeh argued that our
inferences and decisions are often based upon information that is **imprecise or abiguous
**and for which this two-valued logic is inappropriate. He coined the term
"fuzzy sets" to describe collections of elements with indistinct,
imprecise, or "elastic" boundaries. Zadeh
has argued that a different calculus is necessary to represent reasoning based
upon fuzzy information. His early work has generated enormous enthusiasm and
there are now over 15,000 papers, books, and other materials that have been
published on fuzzy matters since 1965, the year in which Zadeh's
work on fuzzy sets saw the light of day.

•
Zadeh, L., "Fuzzy Sets", **Information And
Control**, Vol. 8, 1965.

Zadeh and his now enormous international body of followers
have published papers on the application of fuzzy sets in an array of
inferential and decisional contexts. The best single collection of Zadeh's papers is found in the following.

•
Yager, R., Ovchinnikov, S.,
Tong, R., Nguyen, H., **Fuzzy Sets And Applications: Selected Papers By L. A. Zadeh**,

Here
are several current works that may be regarded as tutorial regarding fuzzy sets
and systems.

•
Klir, G., Folger, T., **Fuzzy Sets, Uncertainty, and Information**,
Prentice Hall, 1988

•
McNeill, D., Freiberger, P., **Fuzzy Logic**, Simon & Schuster, 1993

•
Kosko, B., **Fuzzy
Thinking: The New Science of Fuzzy Logic**, Hyperion, 1993. This is an
absorbing (but frequently irritating) work by the leading spear-carrier of
fuzzy reasoning.

The
following work is critical of the idea that fuzzy logic actually extends
classical logic.

•
Haack, S. **Deviant
Logic, Fuzzy Logic: Beyond the Formalism**.

**V.
Probabilistic Inference And Its Role In Decisions**

As
noted in your syllabus, we should give attention to inferential activity in the
following two contexts. In many situations, various areas of science for
example, inferential activity is simply part of the
process of knowledge-acquisition. However, in other situations such as in law,
medicine, intelligence analysis, and so on, inferential activity is embedded in
the further process of choice. In all of these other situations, assessments of
probability have somehow to be combined with assessments of the value of
consequences that occur to us when we contemplate various choices in the face
of uncertainty. There is a substantial literature on the combination of
inferential and value-related ingredients in choice under uncertainty. Two quite
different views of this process of combination are found in the following three
references.

•
von Winterfeldt, D.,
Edwards, W., **Decision Analysis And
Behavioral Research**,

•
Lindley, D., **Making Decisions**, [2nd ed],

•
Shafer. G., Savage Revisited, **Statistical
Science**, Vol. 1, No. 4., 1986.

**LAST
BUT NOT LEAST, WHERE TO FIND YOUR INSTRUCTOR**

I
usually lurk in the vicinity of Room 111-A, Science & Technology Bldg. II;
my office phone is 703-993-1694. If I am not in my office, I am almost
certainly at home: