Editing Fuzzy logic (section)

==Compared to other logics==

=== Probability ===
Fuzzy logic and probability address different forms of uncertainty. While both fuzzy logic and probability theory can represent degrees of certain kinds of subjective belief, [[fuzzy set theory]] uses the concept of fuzzy set membership, i.e., how much an observation is within a vaguely defined set, and probability theory uses the concept of [[subjective probability]], i.e., frequency of occurrence or likelihood of some event or condition {{clarify|date=April 2019}}. The concept of fuzzy sets was developed in the mid-twentieth century at [[University of California, Berkeley|Berkeley]]<ref>{{cite web |url=https://www2.eecs.berkeley.edu/Faculty/Homepages/zadeh.html |title=Lotfi Zadeh Berkeley |url-status=live |archive-url=https://web.archive.org/web/20170211080227/https://www2.eecs.berkeley.edu/Faculty/Homepages/zadeh.html |archive-date=2017-02-11 }}</ref> as a response to the lack of a probability theory for jointly modelling uncertainty and [[vagueness]].<ref>{{Cite journal |title=Fuzzy Sets |journal=Scholarpedia |volume=1 |issue=10 |pages=2031 |doi=10.4249/scholarpedia.2031 |year=2006 |last1=Mares |first1=Milan |bibcode=2006SchpJ...1.2031M |doi-access=free }}</ref>

[[Bart Kosko]] claims in Fuzziness vs. Probability<ref>{{cite web |last1=Kosko |first1=Bart |author-link1=Bart Kosko |title=Fuzziness vs. Probability |url=http://sipi.usc.edu/~kosko/Fuzziness_Vs_Probability.pdf |archive-url=https://web.archive.org/web/20060902032943/http://sipi.usc.edu/%7Ekosko/Fuzziness_Vs_Probability.pdf |archive-date=2006-09-02 |url-status=live |publisher=University of South California |access-date=9 November 2018 }}</ref> that probability theory is a subtheory of fuzzy logic, as questions of degrees of belief in mutually-exclusive set membership in probability theory can be represented as certain cases of non-mutually-exclusive graded membership in fuzzy theory. In that context, he also derives [[Bayes' theorem]] from the concept of fuzzy subsethood. [[Lotfi A. Zadeh]] argues that fuzzy logic is different in character from probability, and is not a replacement for it. He fuzzified probability to fuzzy probability and also generalized it to [[possibility theory]].<ref>{{cite journal | last1 = Novák | first1 = V | year = 2005 | title = Are fuzzy sets a reasonable tool for modeling vague phenomena? | journal = Fuzzy Sets and Systems | volume = 156 | issue = 3| pages = 341–348 | doi=10.1016/j.fss.2005.05.029}}</ref>

More generally, fuzzy logic is one of many different extensions to classical logic intended to deal with issues of uncertainty outside of the scope of classical logic, the inapplicability of probability theory in many domains, and the paradoxes of [[Dempster–Shafer theory]].

=== Ecorithms ===
Computational theorist [[Leslie Valiant]] uses the term ''ecorithms'' to describe how many less exact systems and techniques like fuzzy logic (and "less robust" logic) can be applied to [[learning algorithms]]. Valiant essentially redefines machine learning as evolutionary. In general use, ecorithms are algorithms that learn from their more complex environments (hence ''eco-'') to generalize, approximate and simplify solution logic. Like fuzzy logic, they are methods used to overcome continuous variables or systems too complex to completely enumerate or understand discretely or exactly.<!-- See in particular p.&nbsp;58 of the reference comparing induction/invariance, robust, mathematical and other logical limits in computing, where techniques including fuzzy logic and natural data selection (à la "computational Darwinism") can be used to short-cut computational complexity and limits in a "practical" way (such as the brake temperature example in this article). --><ref>{{cite book |last1=Valiant, Leslie |title=Probably Approximately Correct: Nature's Algorithms for Learning and Prospering in a Complex World |date=2013 |publisher=Basic Books |location=New York |isbn=978-0465032716 }}</ref> Ecorithms and fuzzy logic also have the common property of dealing with possibilities more than probabilities, although feedback and [[feed forward (control)|feed forward]], basically stochastic weights, are a feature of both when dealing with, for example, [[dynamical system]]s.

===Gödel G<sub>∞</sub> logic===
{{further|Many-valued logic#Gödel logics Gk and G∞}}

Another logical system where truth values are real numbers between 0 and 1 and where AND & OR operators are replaced with MIN and MAX is Gödel's G<sub>∞</sub> logic. This logic has many similarities with fuzzy logic but defines negation differently and has an internal implication. Negation <math>\neg_G</math> and implication <math>\xrightarrow[G]{}</math> are defined as follows:

: <math>\begin{align}
  \neg_G u &= \begin{cases}
                1, & \text{if }u = 0 \\
                0, & \text{if }u > 0
              \end{cases} \\[3pt]
  u \mathrel{\xrightarrow[G]{}} v &= \begin{cases}
                             1, & \text{if }u \leq v \\
                             v, & \text{if }u > v
                           \end{cases}
\end{align}</math>

which turns the resulting logical system into a model for [[intuitionistic logic]], making it particularly well-behaved among all possible choices of logical systems with real numbers between 0 and 1 as truth values. In this case, implication may be interpreted as "x is less true than y" and negation as "x is less true than 0" or "x is strictly false", and for any <math>x</math> and <math>y</math>, we have that <math> AND(x, x \mathrel{\xrightarrow[G]{}} y) = AND(x,y) </math>. In particular, in Gödel logic negation is no longer an involution and double negation maps any nonzero value to 1.