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. 11 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. “All Cretans are liars.”Paradox of the Liar is usually attributed to Epimenides (6 th 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. 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 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.” “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

e.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  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) 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