Transcript Fuzzy Logic - Petra Christian University

Fuzzy Logic
A. History & Justification
Looking at Fuzzy Logic
1.
Accurate modeling of inaccuracy
“When using a mathematical model, careful
attention must be given to the uncertainties
of the model.” Richard P. Feynman
Mr. Spock’s folly:
Precision is not truth
What is Xtmprszntwlfd?
The probability that a fair die will
show six is 1/6. This is a crisp
probability.
All
credible
mathematicians will agree on this
exact number.
The weatherman's forecast of a
probability of rain tomorrow being
70% is also a fuzzy probability. Using
the same meteorological data, another
weatherman will typically announce a
different probability.
Looking at Fuzzy Logic
2. Lukasieicz Logic: The logic of half truths. 1
One justification: Bipolar Paradoxes
Gödel’s Proof
According to TIME magazine's top 100 persons of
the century, the following are the most
influential scientists and thinkers of the
twentieth century:
•Leo Baekeland, plastics pioneer
•Tim Berners-Lee, Internet designer
•Rachel Carson, environmentalist
•Albert Einstein, physicist
•Philo Farnsworth, inventor of
electronic television
•Enrico Fermi, atomic physicist
•Alexander Fleming, bacteriologist
•Sigmund Freud, psychoanalyst
•Robert Goddard, rocket scientist
•Kurt Gödel, mathematician
•Edwin Hubble, astronomer
•John Maynard Keynes, economist
•The Leakey family, anthropologists
•Jean Piaget, child psychologist
•Jonas Salk, virologist
•William Shockley, solid-state physicist
•Alan Turing, computer scientist
•James Watson amp;& Francis Crick,
molecular biologists
•Ludwig Wittgenstein, philosopher
•The Wright brothers, visionary aviators
Gödel’s Proof
Meta-language paradoxes
• Meta-language: Language that
refers to itself: “This sentence
contains five words.” is true.
“This sentence contains six
words.” is false.
• Paradox of the Liar is usually
attributed to Epimenides (6th
Century BC), who was a Cretan:
“All Cretans are liars.”
Gödel’s Proof
Other Meta-Language Paradoxes
“What I am telling you now is a lie.”
“If this sentence is true, the next
sentence is false. The previous sentence
is true.”
“I minister only to those who do not
minister to themselves.”
“Nothing is impossible.” (Can God make
a rock so large he cannot move it?)
“Everything is possible.”
Russell’s Paradox (1901)
Gödel’s Proof
Logical Development from
Axioms (self-evident reality
– or assumptions of truth)
is a foundation of
mathematics.
From Axioms, we get
lemmas, theorems and
corollaries that builds to a
theory.
Gödel showed all such
theories are either
incomplete or inconsistent.
Gödel & Einstein (Princeton:
August 1950)
Gödel’s Proof
Inconsistent: Show that 1+1=2 and 1+12.
Incomplete: We cannot show that 1+1=2 .
Theorem X: Theorem X cannot be proved.
If we can proof Theorem X, then the
theory is inconsistent. (Proving
Theorem X is inconsistent with
Theorem X.)
If we cannot prove Theorem X, the
theory is incomplete. That is, there
are things we cannot prove .
Kurt Gödel
Gödel’s Proof
 All theory logically developed
from an axiomatic foundation
is ultimately either incomplete
or inconsistent.
“There is more in the world than you
have dreamt of in all of your
philosophies, Horatio.” Hamlet (Act
1Scene IV.)
Is such logic more compatible
with Asian philosophy?
1Cor 13:12 “For
now we see
through a glass,
darkly; but then
face to face: now
I know in part;
but then shall I
know even as
also I am
known.”
Looking at Fuzzy Logic
3. Probability versus Possibility (Fuzzy)
A difference:


All things probable are possible.
All things possible are not probable.
The contrapositive:


Impossible events are improbable.
Improbable events are not impossible.
Engineering for possible events is
different than engineering for probable
events.
Looking at Fuzzy Logic
4. Degree of Membership (Fuzzy Linguistic
Variables)
Fuzzy and “Crisp” Control
Examples include close, heavy, light, big, small,
smart, fast, slow, hot, cold, tall and short.
e.g. On a scale
of one to 10,
how good was
the dive?
10
9
9
9.5
Fuzzy  Probability
Example #1

Billy has ten toes. The
probability Billy has
nine toes is zero. The
fuzzy membership of
Billy in the set of people
with about nine toes,
however, is nonzero.
Example #2 (Bezdek)


#1

A bottle of liquid has a probability of ½
of being rat poison and ½ of being pure
water.
A second bottle’s contents, in the fuzzy
set of liquids containing lots of rat
poison, is ½.
The meaning of ½ for the two bottles
clearly differs significantly and would
impact your choice should you be
dying of thirst.
(cite: Bezdek)
#2