Fuzzy Logic

1. General

Fuzzy logic is an extension of conventional (Boolean) logic working with the two values {0,1} to a logic extending over the closed interval [0,1]. Founder of fuzzy logic is Lotfi Zadeh.
Zadeh being refered to as the "father" of fuzzy logic, Bertrand Russell often is called the "grandfather" of fuzzy logic. His famous antinomy (paradox) that he discovered in May 1901 and which he sent in a letter to Gotlob Frege gave a contradiction in Frege's set of axioma's. It invalidated seemingly Frege's work Grundgesetze der Arithmetik. Russell's antinomy giving a paradox for 2-valued logic is not a problem in fuzzy logic.
This is his antinomy:

Let R = { x | x Ï x }. Then R Î R iff R Ï R.

In words: Let R be the set of all sets that are not members of themselves. Then R is a member of itself if and only if R is not a member of itself. This ofcourse is a paradox.
The solution that fuzzy logic offers is that the truth value of this statement is 0.5.
In fuzzy logic paradoxes of selfreference are half-truths.

2. Explanation

A paradox is of the type: A = not-A.
In other words: the truthvalue of A = the truthvalue of not-A.
In short: t(A) = t(not-A).
In binary logic you only have the choice between 0 and 1.
So if t(A) = 1, then t(not-A) = 0

But: t(not-A) can also be written as 1 - t(A)
Then the paradox can be written as: t(A) = 1 - t(A)
In fuzzy logic the truthvalues can vary from 0 to 1, with every possible number in between.
And so the fuzzy solution is: t(A) = t(not-A) = 0.5

This example has been taken from Bart Kosko's book Fuzzy Thinking.

More information about fuzzy sets and fuzzy logic.

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