Since the propositions you specify are either true or false, then, indeed,
if you apply truth or truth to the compound propositions they will be
similar if not equivalent. But in fuzzy logic we are discussing the degree
to which something is a member of a class (that is, the semantics of the
proposition). Now if we have fuzzy propositions,
Bill is Tall
Bill is Short
Bill is Medium
then if Bill is Tall to a degree [.12], is Short to a degree [.88] and
Medium to a degree [.65] then the proposition AND or OR propositions will
yield dissimilar truth functions. Because fuzzy logic does not obey the Law
of the Excluded Middle, someone can be both Tall and Short and Medium at the
same time -- just with different degrees of truth. This property of fuzzy
set theory is important in fuzzy reasoning where fuzzy conditional and
unconditional propositions (often expressed as if-then rules) are run in
parallel and accumulate evidence. Thus,
if height is TALL then weight is Heavy
if height is Short then weight is Light
if height is medium then weight is Moderate
form an output fuzzy set (weight) based on the fusion of the consequent
fuzzy sets (Heavy, Light, Medium) to the degree that the predicate of the
proposition (or rule) is true. Fuzzy systems are highly sensitive to
compound truth statements since (a) there is an often non-linear translation
function between the antecedent and consequent of the proposition involving
three distinct fuzzy spaces, (b) the rules are run in parallel and
accumulate evidence that effectively AND's or OR's the propositions
(depending on your choice of inference mechanism) and (c) the outcome of the
fuzzy system reflects the amount of evidence in the supporting antecedents.
You eye color logical metaphor might have been cast as,
Consider the mayor of Ashtabula. Let A = "mayor's right eye is wide".
Let B = "mayor's left eye is narrow". Let B' = "mayor's left eye is wide".
What do you suppose is the truth value of A B ? What about A B' ?
You should ask yourself, what is the truth value of B B'? These are mutually
contradictory in Boolean logic, but simply represent a logical analysis of
the degree to which the eye might be considered both wide and narrow (which
is a real world state -- there is no point at which the width of an eye goes
from narrow to wide, it is a gradual transition.)
Anyway, reading introductions to fuzzy logic and then forming an opinion
about the robustness of its representational power is a poor way to engage
in serious debate.
But you will have to take this up with other participants, I will check back
in another two years to see if everyone is still debating the same issues.
-- Earl Cox VP, Research/Chief Scientist Panacya, Inc. 134 National Business Parkway Annapolis Junction, MD 20701 (410) 904-8741 -------------------------------------------
AUTHOR: "The Fuzzy Systems Handbook" (1994) "Fuzzy Logic for Business and Industry" (1995) "Beyond Humanity: CyberEvolution and Future Minds" (1996, with Greg Paul, Paleontologist/Artist) "The Fuzzy Systems Handbook, 2nd Ed." (1998) "Fuzzy Tools for Data Mining and Knowledge Discovery" (due Early Fall, 2001)
"Robert Dodier" <firstname.lastname@example.org> wrote in message news:email@example.com... > In response to the following example which I posted: > > > > Consider the mayor of Ashtabula. Let A = "mayor's right eye is blue". > > > Let B = "mayor's left eye is blue". Let B' = "mayor's left eye is brown". > > > What do you suppose is the truth value of A B ? What about A B' ? > > > > > > The difficulty is that rules of the kind applied in fuzzy logic > > > ignore relations between the elements of a compound proposition. > > "Earl Cox" <firstname.lastname@example.org> wrote: > > > [...] I also fail to see how your example about the eye color > > addresses anything about fuzzy logic. > > I see that I left too much implied. The truth values of B and B' in the > example above are more or less equal; I don't think you'll want to argue > that point. Yet then the truth values assigned to the compound > propositions "A and B" and "A and B'" would have to be similar also, > under the truth(A and B) = min(truth(A), truth(B)) or truth(A)*truth(B), > or any other definition of truth(A and B) which is solely a function > of truth(A) and truth(B). > > > And why do you suppose (erroneously) that fuzzy logic -- fuzzy set theory > > in particular -- ignores relationships between compound propositions?? > > Well, it's from reading introductions to fuzzy logic that give > definitions of the truth value of a compound proposition in terms of > truth values of its elements. > > Any such definition must ignore the relation between elements in a > compound: if truth(B')=truth(B), then in any proposition containing > A and B, I can swap in B' in place of B, and get exactly the same > truth value for the compound; whether the elements are redundant, > contradictory, or completely unrelated doesn't enter the calculation. > > Regards, > Robert Dodier > -- > "Nature exists once only." -- Ernst Mach
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