Talk:Fuzzy logic

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The Misrepresentations of Classical Logic and Mathematics Are Promulgated and Magnified With Impunity[edit]

There have been ``fuzzy`` reasoning systems available in logic for a very long time. The popularity of the kind of ``fuzzy logic`` mentioned here is in part, I think, an anti-mathematical, and anti-formal backlash. Some of the names of fuzzy logic systems studied by these people belies their anti-mathematical and anti-formal claim, however. For example, they refer to ``Gödelian`` or ``Łukasiewicz`` logic systems, named for ``classical`` logicians such as Kurt Gödel or Jan Łukasiewicz. In fact, the characterizations by fuzzy logic proponents of ``classical logic`` as inadequate to model ``vagueness`` is based upon a gross misrepresentation of classical logic as being ``binary``: ``Classical logic only permits propositions having a value of truth or falsity.`` In fact, predicate logic (the classical logic of relational and functional systems, that includes first-order and second-order and higher-order reasoning systems) already deviated from that limited view in the time of the classic Greek philosophers. In the twentieth century, the mathematical logic community codified these notions by developing, from George Boole's ``Laws of Thought`` and Frege's theories of classes and Cantor's naive set theory, the notion of a Boolean algebra, which can be infinite, and can be densely ordered, thereby extending the set of possible truth values to ``continuum many``, and then moved on to develop lattice theory, which generalizes Boolean algebra to provide truth value sets that are structured so as to accommodate modal and non-classical reasoning systems. (John von Neumann apparently was aware of this, and in some of his work, he referred to lattices as ``logics``, presumably because they formed the structured sets of truth values he needed for reasoning about quantum systems and other non-propositional, or non-zeroth-order, systems..)

Similarly, the view that probability logic does not help model vagueness is a misrepresentation of ``vagueness`` itself. Consider the following quote from this Wikipedia article on fuzzy logic: ``It is essential to realize that fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon while probability is a mathematical model of ignorance.`` In fact, ``vagueness`` really is a kind of ``ignorance``, based upon the fact that the Sorites paradox (and other paradoxes) cannot be EASILY resolved by a convincing computational argument using the arithmetic one learns in elementary school. The fact that one needs to use a more expressive language than the one immediately at hand to resolve any given Sorites-like issues that arise in a given application merely indicates that a more sophisticated understanding is needed in that application. ``More sophistacted understanding`` here corresponds to `less ignorance``. Thus, I suggest that proponents of ``fuzzy logic`` stop claiming that they are not modeling ignorance, as that is a tenuous philosophical claim. It is the case, however, that ```multi-valued`` logics (of which the Zadeh-type fuzzy logics form a class of examples) can have properties that are different from either two-valued logic or the inconsistent single-valued logic that no one really seems to ever mention. (Thus, in fact, it is reasonable to call even two-valued logic ``multi-valued`` since it has MULTIPLE TRUTH VALUES, although this is not the traditional terminology.)

It is good that the article includes a mention of the pre-Zadeh ``fuzzy logicians`` Tarski and Łukasiewicz: ``Fuzzy logic has been applied to many fields, from control theory to artificial intelligence. Fuzzy logics however had been studied since the 1920s as infinite-valued logics notably by Łukasiewicz and Tarski.[4]`` On the other hand, the authors seem unaware of certain kinds of modern results that do not support the flattering claims made about Zadeh-type fuzzy logic systems. For example, in the article, no mention is made of articles like the following one by Gehrke, Walker and Walker: ``A mathematical setting for fuzzy logics``, in the International Journal of Uncertainty and Fuzziness Knowledge-Based Systems 5 (1997), no. 3, 223–238. This paper seems to indicate that certain kinds of Zadeh-style fuzzy logics are unnecessarily complex, as is explained in the MathSciNet review (MR1454993): ``The same holds for the standard fuzzy logic, where the algebra of truth values is the interval [0,1], and the propositional connectives are interpreted as complement, max and min. The algebra of truth values in this case is isomorphic to standard three-valued propositional logic.``

In this talk page, the fact that fuzzy logic is a ``controversial discipline`` is mentioned above. However, it is not that fuzzy logic is ``the wrong way to do things`` that really is a problem. The problem is that its proponents neglect the value of being intellectually honest. It is probably true that Lotfi Zadeh believed he was discovering something new by taking the interval [0,1], with the usual ordering, as the set of truth values for a logic system. However, plenty of time has passed in which his followers should have begun to realize that claims like ``Fuzzy logic has been around since the mid 1960s; however, it was not until the 70s that a practical application was demonstrated.`` (see http://www.controleng.com/single-article/artificial-intelligence-fuzzy-logic-explained/8f3478c13384a2771ddb7e93a2b6243d.html) are unreasonable hyperbole. For in fact, ``fuzzy`` logic has ``been around`` much longer (from the ancient Greeks at least, in the study of predicate and modal reasoning), and long ago found applications in Mathematics and Law, and in Philosophy, etc.

Also mentioned in this talk page is the contention that ``fuzzy`` logic is inconsistent. However, the argument there neglects the fact that there are many different ``fuzzy logic`` systems. (I am not addressing the question of the correctness of the argument presented for the one fuzzy logic system mentioned there.)

espresso-hound Matt Insall 01:34, 29 July 2013 (UTC)

\t I would like to second many of the ideas put forth by Matt. Far from having any practical applications, this article shows little evidence of any novel mathematical concepts at all. I think we should delete this entire page. \tDrNoBrain (talk) 04:30, 25 October 2016 (UTC)

What about XOR?[edit]

In the article, in the section about 'Define with multiply', only AND and OR are listed. What about NOT and XOR?

I deducted myself that NOT x = 1 - x (same as in the section above) and x XOR y = 0.5 - 2 * (x - 0.5) * (y - 0.5). It would be nive if this was incorporated in the article, with a real reference of course.

Proof: Let x and y denote the probability between 0% and 100% that that variable is true, otherwise it is false. When x is 50%, then whatever y is, x XOR y will be the inverse of y with 50% probability, and itself the otherwise, so the result of x XOR y is always 50% when either input is 50%. This makes 0.5 the 'multiplivative unity' of the values. By subtracting 0.5 the values can be multiplied to each other, and the sign of the result can be seen to behave like a binary XOR. Since the product comes out half the magnitude it started with, and swings between -0.25 (for true) to 0.25 (for false), it must be multiplied by 2 and subtracted from 0.5. --Zom-B (talk) 17:25, 22 December 2015 (UTC)

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Example needed?[edit]

Might a worked example of fuzzy logic in action help illustrate it better? It could cover the three stated steps of a fuzzy-logic decision process:

  • Fuzzify all input values into fuzzy membership functions.
  • Execute all applicable rules in the rulebase to compute the fuzzy output functions.
  • De-fuzzify the fuzzy output functions to get "crisp" output values.

I suggest this since I found this article helpful for establishing the concepts but it felt hard to see how they are applied without seeing a concrete example. This external article I found on Google gives a concrete example. Jtaylor100 (talk) 15:16, 20 May 2018 (UTC)

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