# [FOM] Fuzziness

Wed Jul 20 19:04:15 EDT 2016

```I hope the following text will clarify the relation between fuzzy and probabilities. Of course, this is a complex well-studies topic, I will just mention the basic ideas.

Fuzzy logic was motivated by the need to describe expert knowledge – which experts often formulate by using imprecise (“fuzzy”) words from natural language – in precise computer-understandable form. When you ask a medical doctor, she will most probably not say: if a cyst is above 7 mm in diameter and its color is at 500 nm, then take 250 mg of certain medicine, but rather: if a cyst is reasonably big and its color is reddish, the start the patient first on a small dose of whatever medicine. Same way if you ask a person how they drive etc.: if a car nearby is close, and it decelerated a little but, hit the brake for a little bit.

These “close” etc. are imprecise words. First thing fuzzy methodology does is assigns to each such word a function that describes, for each possible value of the corresponding quantity (e.g., distance), a degree to which the value is, in the opinion of the expert, close. This degree can come from polling the experts or by estimating the expert’s degree of confidence as a subjective probability – in which case it can be viewed in probabilistic terms. On the other hand, it can come – like in student evaluation of faculty – simply from an expert marking his degree of belief on a scale from 0 to 10.

The big difference between fuzzy and probability starts on the next step: we need to come up with degrees that a given distance x is close AND that the given deceleration y is small – because the rule applies under these two conditions only. In the ideal world, we should ask experts of theor degree for all such combinations, but for many inputs, there are many combinations, so such request is not realistically possible. In such cases, we only have degrees of confidence a and b in statements A and B, and we need to estimate the degree of confidence in A & B. This estimate f(a,b) is called an “and’-operation (a.k.a.t-norm). The corresponding function f(a,b) should satisfy some reasonable conditions: e.g, since intuitively, A&B and B&A mean the same, this function should be commutative; similarly, it should be associative, etc.

With all these conditions in place, however, there are many such operations, so we need to select a one that best reflects expert reasoning in this particular discipline. The first time such an and-operation was empirically selected was when the first expert system MYCIN for detecting rare blood diseases was developed. The authors of MYCIN found an and- (and a similar or-) operation that best fits the reasoning of medical doctors. However, it turns out that, e.g., geophysicists think differently – which makes sense: a medical doctor is usually very cautious, and tries to get a lot of evidence before making a decision, while oil companies start drilling if there is good evidence of oil, without necessarily waiting for 99% confirmation.

This idea of using “and”- and “or”-operations – somewhat similar to copulas – is what differentiates fuzzy from the probabilistic approaches (plus the fact that the fuzzy degrees can simply come from marking on a scale and are, thus, not necessarily probabilities).

Hope this helps.

From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Charlie
Sent: Wednesday, July 20, 2016 3:08 PM
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Re: [FOM] Fuzziness

Is there anything to the charge I’ve often heard that Fuzzy Logic is really probability theory requiring arbitrary assumptions?

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