# The difference between the probable and the fuzzy

Discussion in 'Electronic Basics' started by Immortalist, Aug 21, 2008.

1. ### ImmortalistGuest

At this juncture it is important to point out the distinction between
fuzzy systems and probability. Both operate over the same numeric
range, and at first glance both have similar values: 0.0 representing
False (or non-membership), and 1.0 representing True (or membership).

However, there is a distinction to be made between the two statements:
The probabilistic approach yields the natural-language statement,
"There is an 80% chance that Jane is old," while the fuzzy terminology
corresponds to "Jane's degree of membership within the set of old
people is 0.80." The semantic difference is significant: the first
view supposes that Jane is or is not old (still caught in the Law of
the Excluded Middle); it is just that we only have an 80% chance of
knowing which set she is in. By contrast, fuzzy terminology supposes
that Jane is "more or less" old, or some other term corresponding to
the value of 0.80. There are more very technical ways to show how
comparing probibility with fuzzy degrees of truth and falsehood don't
match.

Natural language abounds with vague and imprecise concepts, such as
"Sally is tall," or "It is very hot today." Such statements are
difficult to translate into more precise language without losing some
of their semantic value: for example, the statement "Sally's height is
152 cm." does not explicitly state that she is tall, and the statement
"Sally's height is 1.2 standard deviations about the mean height for
women of her age in her culture" is fraught with difficulties: would a
woman 1.1999999 standard deviations above the mean be tall? Which
culture does Sally belong to, and how is membership in it defined?

While it might be argued that such vagueness is an obstacle to clarity
of meaning, only the most staunch traditionalists would hold that
there is no loss of richness of meaning when statements such as "Sally
is tall" are discarded from a language. Yet this is just what happens
when one tries to translate human language into classic logic.

"To what degree is something true or false?"

)) There are two types of thinking:
)) Approximate and Exact.

Exact is just a "degree of truth" equal to 100%. Approximate is a
degree of truth between 1% and 99%. Therefore there is only one type
of thinking: Fuzzy Think!

)) Approximate always trumps Exact.

Unlike traditional or classical logic, which attempts to categorize
information into binary patterns such as black / white, true / false,
yes / no, or all / nothing, Fuzzy Logic pays attention to the
"excluded middle" and tries to account for the "grays", the partially
true and partially false situations which make up 99.9% of human
reasoning in everyday life. It builds upon the assumption that
everything consists of degrees on a sliding scale-whether it be truth,
age, beauty, wealth, color, race, or anything else that is effected by
the dynamic nature of human behavior and perception. The question
Zadeh always insists upon asking is, "To what degree is something true
or false?"

Zadeh looks around him in the real world which he finds pervaded by
concepts which do not have sharply defined boundaries, where
information is often incomplete or sometimes unreliable. In fact, he
would classify most words as having fuzzy meanings-virtually every
seen in transition from membership to non-membership in gradual,
rather than abrupt, increments.

In quest for precision, scientists have generally attempted to
manipulate the real world into artificial mathematical models that
make no provision for gradation. They have tried to describe the laws
governing the incredibly complex behavior of humans, both singly and
in groups, in mathematical terms similar to those employed in the
analysis of inanimate systems, which, in Zadeh's view, has been, and
will continue to be, a misdirected effort.

Because the human mind can't handle so many isolated separate ideas at
one time, it tends to bundle similarly-related objects into categories
in such a way as to reduce the complexity of the information
processing task. It is this incredible capacity of the human mind to
manipulate these fuzzy or unsharp categories that distinguishes human
intelligence from the machine intelligence of current generation
computers.

Because Fuzzy Logic provides the tools to classify information into
broad, coarse categorizations or groupings, it has infinite
possibilities for application which have proven to be much cheaper,
simpler and more effective than other systems in handling complex
information. Fuzzy Logic has extremely broad implications for many
fields not just electrical engineering and computer technology which
have been fairly quick to incorporate its theoretical principles.
Numerous consumer goods especially household products and electronic
equipment-microwaves, cameras, and camcorders already incorporate
Fuzzy Logic into their design. So have computer control systems such
as the famous subway of Sendai, Japan, or numerous complex diagnostic
and monitoring biomedical systems which are starting to be used in
hospitals.

But other fields such as the social sciences-economy, finance,
psychology, sociology, politics, religion, ethics, law, medicine,
geography, folklore, anthropology that deal with the complexity of
human behavior-are just beginning to explore the infinite
possibilities of Fuzzy Logic.

such as Plato indicated that there was a third region (beyond "true"
and "false") where opposites "tumbled about". Hegel, Marx, Engels and
Lukasiewicz, among others, also dealt with middle regions. But it was
Zadeh who first developed the general theory and laid the foundation
for what Fuzzy Logic is today.

For a well-researched, very readable, popular description of Lotfi
Zadeh and the development of the field of Fuzzy Logic, refer to Daniel
McNeill and Paul Freiberger's award winning book, Fuzzy Logic: The
Revolutionary Computer Technology that is Changing our World, 1993.
For a technical introduction to the field, see Zadeh's Fuzzy Sets and
Applications: Selected Papers. Edited by Yager, Ovchnikov, Tong, and
Nguyen. New York: Wiley, 1987.

------------------------

Now remembering that in fuzzy logic the truth of any statement is a
matter of degree, how will these truth tables be altered? The input
values can be real numbers between 0 and 1. What function will
preserve the results of the AND truth table (for example) and also
extend to all real numbers between 0 and 1?

One answer is the min operation. That is, resolve the statement A AND
B, where A and B are limited to the range (0,1), by using the function
min(A,B). Using the same reasoning, we can replace the OR operation
with the max function, so that A OR B becomes equivalent to max(A,B).
Finally, the operation NOT A becomes equivalent to the operation .
Notice how the truth table above is completely unchanged by this
substitution.

http://www-rohan.sdsu.edu/doc/­matlab/toolbox/fuzzy/fuzzytu4.­html

------------------------------­---

You want to replace the truth table with the degree table below it?

"Tomorrow will be a very important day for you. It is your birthday.
We are going to invite your two best friends from your preschool.
Alice and Barbara are coming to your party."

Notice that in Western logic this last statement is just true or
false. There are no shades of gray. The result is the same (false) if
either Alice or Barbara do not come for some reason (lines 2 and 3),
and if both Alice and Barbara don't come (line 4).

A B A · B
T T T
T F F
F T F
F F F

A B Result

0 0 0
15 15 .08
30 30 .16
45 45 .25
60 60 .33
75 75 .41
90 90 .50
105 105 .58
120 120 .66
135 135 .75
150 150 .83
165 165 .91
180 180 1.0

Notice that when we say that Alice or Barbara "attends" the little
girl's party that this is potentially a fuzzy concept. Suppose the
mothers of Alice and Barbara tell the little girl's mother that they
will be very busy that day and are not sure they can make it, but they
are pretty sure they can stop by for at least a little while. So you
see from the little girl's point of view "attends" is not a black and
white (true or false) event. What if one girl comes for only fifteen
minutes and the other comes for the full time? Or some other
variation of times? This will make a big difference from the little
girl's perspective for how much truth is in the statement "Alice and
Barbara are coming to your party."

So, here is how a fuzzy logic truth table might look. If we assume
the party is three hours long and we break the party up into 15-minute
segments, we would have a table that looked like this.

...adding degrees-of-truth-reasoning to the very foundations of logic
has a drastic effect on all the other rules of logic. All the rules we
have learned become applicable to only extreme cases -- cases of
complete truth or complete falsehood. Even deductive validity and
invalidity become more like the distinction we made in Chapter 3
between "strong inductive" and "weak inductive" arguments. A fuzzy
valid argument can "blend into" an invalid deductive argument.

The huge issue all this raises is, "What is the status of logic?" The
fancy way philosophers ask the question is, "What is the ontological
status of logic and mathematics?" Are they just rules we make up like
that of a game (e.g.. basketball)? Are they simply part of a cultural
view of reality? Or, are they some kind of absolutes that the universe
obeys, like the law of gravity? Are they simply rules used by the
human mind? Are they rules in the mind of God?

http://www.hcc.hawaii.edu/~pin­e/Phil110/chapt12sup2.htm

# A OR B = MAX(m(A(x)), m(B(x)))
# A AND B = MIN(m(A(x)), m(B(x)))
# A' (NOT A) = 1 - mA(x)

http://www.generation5.org/con­tent/1999/fuzzyintro.asp

2. ### David L. JonesGuest

Why?, they are both disappointingly vague.

"Do, or do not, there is no try" - Yoda.
Yoda was obviously a digital guy.

I think I'll go have some chocolate.

Dave.

4. ### David L. JonesGuest

You appear to be seriously in need of a date.
Try here:
http://www.match.com

Dave.

5. ### Sir FrederickGuest

Here is an old out of print book on fuzzy,
my book "Fuzzy Logic A Practical Approach".
http://www.fuzzysys.com/books/FLLib/FUZZYPDF/FUZZYLOG.PDF
and the old demo programs from the book :
http://www.fuzzysys.com/books/FLLib/PC/PRACTAPP/PRACTAPP.zip
you must unzip it and run install.exe.

and here is the typo list for the book :
http://www.fuzzysys.com/typolist.html

The products mentioned are no longer for sale.
If you have any problems accessing these items,
let me know.

--
Frederick Martin McNeill
Poway, California, United States of America

******************************************
"I never cease being dumbfounded by the unbelievable things people believe."
- Leo Rosten
******************************************

6. ### ZerkonXGuest

More like: Wrong Think! Why, OH WHY can't this number tyranny be broken,
overthrown and put in it's place once and for all. Physics awaken! You
guys know somewhere deep down or up shallow this is fuzzy true, more or
less, to some degree, sorta, kinda. Your science has been pointing that
way now for years.

There are not two "types" of thinking. This is a self-serving position.
'Types' demands numerical amounts. 1-99 and 0/1 (100%) are the same
thing, a numerical concept into which all must be classified, by force if
need be. It's one way to think about thinking and only that.

Data/information retention is a primitive and base function. It is not
complex. It does not indicate real intelligence unless intelligence is to
be defined only as such. Real human work, thought, comes in the
processing of information and here quantity does not ensure quality.

A person who can recall and only recall a library of information would
not be able to function at any level. Another who was able to only
vaguely recall a small amount of information but who draws generalities
applicable and constructive to situations detached from the retained
information demonstrates human intelligence.

There is still a huge fuzzy growing on what information retention is AND
what information is usable or used information. Using computer "memory"
addresses, or the old 'warehouse boxes model' seems to be more metaphors
gone wild and wrong.

Each person as thinker defines their own 'type' of thinking. Logic
satisfies first the thinker. Then only if there is a need, other thinkers.

A shift moves now from thought to the communication of thought.
Communication becomes a standard type, an accepted methodology. So
thought goes to communication which is formed into acceptable
methodology. Now, here, the methodology, forgetting it's lowly place in
line, turns around to define thought in it's own image!

What's next? Computers telling us what to do?!?