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. FeynmanMr. Spock’s folly:

Precision is not truthWhat 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 ProofAccording 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 ProofMeta-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.”“All Cretans are liars.”

Gödel’s ProofOther 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 ProofLogical 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 ProofInconsistent: Show that 1+1=2 and 1+12.Incomplete: We cannot show that 1+1=2 . Theorem X: Theorem X cannot be proved.

Kurt Gödel

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 .

Gödel’s Proof All theory logically developed

from an axiomatic foundation is ultimately either incomplete or inconsistent.

1Cor 13:12 “For now 1Cor 13:12 “For now we see through a we see through a glass, darkly; but glass, darkly; but then face to face: then face to face: now I know in part; now I know in part; but then shall I know but then shall I know even as also I am even as also I am known.”known.”

“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?

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

9

9 9.5

10e.g. On a scale of one to 10,

how good was the dive?

Examples include close, heavy, light, big, small, smart, fast, slow, hot, cold, tall and short.

Fuzzy ProbabilityExample #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) 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)

#1

#2