One or two members of the list have made points like the following one by
David Larkin:
>With standard Bayesian techniques it is possible to reason
probabilistically about variables with non-binary domains. For example, in a
standard belief network one might find a node labeled "Temperature" which
can take on the values "Low", "Medium", and "High". A fuzzy-logic based
interpretation of the variable would amount to a simple relabeling. The node
could be renamed, "The object is hot", taking a truth value of "0", "0.5",
or "1". Mathematically, there would be no difference. You would say,
according to bivalent logic, "The object is hot" can only be true or false,
and we can only reason probabilistically about the chance that it will be
true or false, and therefore partial truth cannot be addressed. But standard
belief networks are not bivalent (i. e., the variables are not binary), but
in general multi-valent. Vague and perception-based degrees of truth, such
as the temperature example given above, are commonly used to describe the
states of variables.<
I have a question (that I penned some time ago) for members of this list who
share David Larkin's sentiment:
How would I use the procedure of assessing the probabilities of multivalent
variables to determine the probability (of the proposition or hypothesis)
that "roller skates" are "motor vehicles" for purposes of an ordinance
prohibiting the operation of a motor vehicle in a park? Would you propose
that one of the hypotheses whose probability I am to assess is "roller
skates have the property of being a 'motor vehicle' only to a small extent"?
(I am taking "motor vehicle" as a primitive property or attribute, rather
than as one that is decomposable into subordinate properties or attributes.)
If this formalism were to show us anything, we would need to figure out what
events or evidence or symptoms etc. we should use in expressions such as
P(E|"'roller skates' have small amount of the property 'motor vehicle'")
Most legal scholars (if they understood probability theory well, that is)
and some probability theorists would be tempted to conclude that the concept
of a conditional probability cannot readily be used to address uncertainties
of the sort found immediately above. If so, can we use a different kind of
"formalism"? Is it _conceivable_ that we could do so? If so, what would that
way of thinking be like? Is it useful to think in terms of shifting sets
(rough sets) or varying degrees of membership in a set (fuzzy sets)? If not,
why not?
***
In the world of accounting there are often alternative ways of describing
the same financial state of affairs. The descriptions one gets often seem
different; but they are often not inconsistent, and sometimes they amount to
paraphrases, different but equivalent ways of saying the same thing.
Sometimes (but not always) the differences between probability theorists and
fuzzy set theorists strike me as being like that. In response to formulation
x -- e.g., think of this uncertainty as "degree of truth" -- there is the
response: e.g. #1: we can instead think of this problem, or we can instead
describe this problem, as a situation in which we are trying to assess our
uncertainty about a continuum of possible states, we can instead speak of
the probabilities of continuous variables; e.g. #2: we can instead speak of
the probabilities of non-continuous but multiple disjoint possible states.
Speaking as a non-mathematician: sometimes such alternative ways of
describing uncertainty seem to me to be, at bottom, "the same", and if I
were to prefer one way of talking over another, I would want to have some
_reasons_ for talking about a particular kind of uncertainty one way rather
than another way. Although I think there are in fact some advantages
(sometimes) to using the grammar of conditional probability to talk about
hypotheses about legal norms -- in this regard I am in a small minority
among legal scholars --, I also cannot escape the sense that the grammar of
conditional probability just does not "get at" some of the types of
uncertainty that are found in legal reasoning (and in other contexts as
well). Think of the underlying intuition of rough sets, the notion that some
words or classifications seem to flip and shift, that they have fluctuating
perimeters: How does one get at this type of uncertainty, effectively
capture it, by talking in the language of conditional probabilities? I do
not deny that it could be "done" if one tried "hard enough" -- standard
probability theory is protean: it can be interpreted in an immense variety
of ways -- but in the end I think I would still wonder if deployment of
notions such as conditional probabilities, likelihood ratios etc. would be
the best way to think about words that seem to have "rough edges" or to
about the appropriate categorization of things that seem to fall into two or
more categories that to some extent seem disjoint to some degree but not
strictly or completely disjoint.
Since I am not a fuzzy or rough set theorist, I could not begin to prove
that my doubts (such as those mentioned immediately above) about the
relative fertility of the standard probability calculus in some contexts are
justified or that fuzzy sets or rough sets make it possible to think in a
more orderly and more productive manner about uncertainties such as those
mentioned in the preceding paragraph. This is why I (desperately) need the
help of logicians and mathematicians and probability theorists etc. But if
you were to advise me, I would want you to keep this in mind: I would not
think you have made a good defense of your preferred theory -- your
preferred theoretical apparatus -- merely by showing that your theory or
apparatus can "handle" some problem that a rival theory also claims to
handle. The question, it seems to me, is which theory does a _better_ job of
"handling" the problem at hand.
Nothing I say here, of course, answers the last question. Here I am not even
_attempting_ to answer the last question. Furthermore: I confess that I do
not know the answer to the last question. I only know that conditional
probability talk sometimes just doesn't seem to get at the kind of
uncertainty that I sometimes see in law. (The reverse is also the case: talk
about fuzzy and rough sets also sometimes just doesn't seem to fully get at
the kind of uncertainty that I think I sometimes see in law [unless, of
course, we are forced to agree with the thesis that the standard probability
calculus is just a special case in fuzzy sets].)
When considering the general question I pose in this message, please keep in
mind that I am a legal professional and that lawyers and legal professionals
are very interested in "word wars" -- in battles over the meaning of words
and in _arguments_ about the meaning or scope of legal classifications. The
way a word or classification is conventionally used is part of the kind of
argument that a lawyer often makes. But only part. Lawyers appeal to reasons
other conventional usage when they argue about the "proper" or "appropriate"
or "correct" interpretation of some legal phrase or classification. (Much
depends on the context in which a lawyer is operating. When lawyers draft
contracts, they are more likely to emphasize conventional {legal}
interpretation. When they litigate, they are likely to emphasize
conventional usage considerably less.)
Peter T
P.S. I suppose that a discussion list -- even one as distinguished as
this -- is not the appropriate venue for a discussion of the implication of
the principle of non-contradiction for the question of the (logical)
coherence of fuzzy set-based arguments. (When I have time -- but I am
running out of time! -- I will reconsider this basic(?) question in my own
autodidactic way. [There is a crasser but more trenchant way to describe my
efforts at self-education. The adjective I have in mind is a hyphenated
expression that begins with "h" and ends with "d".])
****
Peter Tillers http://tillers.net
Professor of Law
Cardozo School of Law
55 Fifth Avenue
New York, New York 10003
U.S.
(212) 790-0334 [EMAIL PROTECTED]
FAX (212) 790-0205
