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In reading about fuzzy logic it says that fuzzy logic is different from probability. Can some one please explain how these two differ. How can this be explained to a person with no mathematical background. Please explain the difference of fuzzy logic and probability with a example that can be understood in general.

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    $\begingroup$ "It says" is not helpful. What says, and give an exact quote, please. $\endgroup$ Commented Feb 25, 2016 at 2:31
  • $\begingroup$ @ThomasAndrews Sorry I've made a mistake in the post. I want to know the difference between fuzzy logic and probability. In this article academia.edu/772082/Fuzzy_Biology it talks about fuzziness vs probability. But a different example with more explanatation ould be helpful $\endgroup$
    – sam_rox
    Commented Feb 25, 2016 at 15:39
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    $\begingroup$ @sam_rox I get a 404 Error when I try to access the link. Perhaps give us the author, title, and date? $\endgroup$
    – Galen
    Commented Oct 31, 2021 at 16:56

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Let's use a simple example of your height. In probability you would define your height as a particular crisp value such as 72 inches. You might then discuss uncertainty and say that you are 90% confident that your height is 72 $\pm$0.5 inches. In probability , you assume that each person has a crisp value of height (i.e. there is a right answer to the question "How tall are you?") and we try to determine how likely we are going to be correct in making statements about these crisp values. In fuzzy logic, we would describe height using different terms such as "tall", "very tall", "moderately tall", etc. Each of these sets includes a range of heights. Tall might be from 68 to 76 inches, "very tall" might be from 73 to 80 inches, "moderately tall" might be from 67 to 72 inches. So a given person's height might be described by more than one set. If one was, for example, 74 inches he would be in the "tall" and "very tall" set. The other thing about fuzzy sets is that set membership is not binary. In classical logic, an element is either in or our of the set. In fuzzy logic membership in a set is a continuum, so one can be 40% in the "tall" set and 70% in the "very tall" set.

So to sum up, probability assumes that there is a definite numerical height that we can try to make assertions about, but there is a true value (which may not be known to us). Fuzzy logic deals with fuzzy sets which cover ranges of values and are not mutually exclusive.

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  • $\begingroup$ How would you compute that someone is 70% "very tall" from real data without using a probability measure? $\endgroup$
    – Galen
    Commented Oct 31, 2021 at 16:53
  • $\begingroup$ Their actual height when evaluated by the "very tall" membership function would yield a value of 70%. $\endgroup$
    – Tpofofn
    Commented Nov 5, 2021 at 18:10
  • $\begingroup$ Okay, let's say that $f_{\text{very tall}}(h) = 70$. For pure math purposes it is enough to say that such a mapping exists. In what sense does 70% correspond to data or observables? Where does such a number come from other than arbitrary assignment? $\endgroup$
    – Galen
    Commented Nov 5, 2021 at 19:01
  • $\begingroup$ The closest I have found to answering these questions is fuzzy clustering but I am not sure that there isn't a probability interpretation lurking in algorithms like fuzzy c-means. $\endgroup$
    – Galen
    Commented Nov 5, 2021 at 19:01
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On a theoretical level, it goes back to Aristotle's Law of the Excluded Middle that states a statement must be either True or False. This is one of the axioms (foundations) of probability. It is seen but the equation P(f) + P(t) = 1. This is also shown in "crisp" sets in which some object is either a member of a set or it is not.

This works well for things that can be empirically measured. But falls apart when trying to deal with non-empirical things, the best example being natural language. Here there are gray areas.

For example a grading scale often used 90-80-70-60. After tedious arguments about rounding,etc, an empirical line is drawn. Grades > 90 are A. Grades < 90 are Not A. This is pretty much how things work.

However, this falls apart when linguistic terms are used. Assume the highest category is "Superior". Empirically a 90 is "Superior" but an 89.999999 is not which is not an accurate reflection of the reality that both students are the same.

FINALLY, in fuzzy logic the two students would receive several "Possibilities". There is a 1% chance that one or both are "Poor", 25% chance that they are Average, 50% chance that they are "Above Average" and a 50% chance that one or both are "Superior".

Notice that the possibilities do not add up to 100%. MAYBE (a term not allowed in probability) one or both are Superior and MAYBE one or both are Above Average. There is a very small POSSIBILITY that one or both are poor and it is POSSIBLE that they are Average

Then there's some mathematical jujitsu to "Defuzzify" the score to a single number if needed.

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Fuzzy logic and probability theory are two independent sources of indeterminacy in judging if an element $x\in A\,$ is in a set. If $\,x\,$ is precisely known and $A\,$ is clearly defined, then $\,x\in A\,$ is either true or false, following the "law of excluded middle". This is how classical logic works.

Probability theory points out one way to turn the answer into a "maybe". If the value of $\,x\,$ is not known exactly (or not known at all) but we only have a sense of what values are more likely than others, then we can describe what we know about the value of $\,x\,$ not using one particular real number, but using a distribution over the possible real numbers to tell how probable they are. An example would be throwing two fair dices. The sum $\,x\,$ of the points would be more likely to be a $7$ than to be a $2$ or $12$ simply because there are more ways to get a $7$.

Fuzzy logic offers another way to have a "maybe" answer $-$ if the set $A\,$ is fuzzy. Still using the dice example, if we consider events like "$x=7$", "$x>9$", "$x\leq 3$", "$x$ is odd", "$x$ is prime", etc., we find that the only indeterminacy is that we don't know $x$. Once $x$ is known (e.g. after you throw the two dices), whether these events happen or not are clearly defined. Now consider this event:

$$x\mbox{ is big.}$$

This time, even if the value of $\,x\,$ is given (e.g. $x=10$), whether it is considered big is not clearly defined, because the set of "big numbers" is fuzzy. $x=12$ would impress almost all of your friends in a party; $x=11$ may still impress the majority of them; $x=10$ may impress only half of them or less, and so on. So every point sum $\,x\,$ is "big" to some extent. For some $\,x\,$, the truth value is close to $1$ (e.g.$\,x=12$), and for some $\,x\,$, the truth value is close to $0$ (e.g.$\,x=2$).

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    $\begingroup$ Isn't this example of "$x$ is big" still a probability problem? You speak of the fraction of your friends which are impressed over the possible values of $x$, which would be a frequentist probability measure. $\endgroup$
    – Galen
    Commented Oct 31, 2021 at 16:43
  • $\begingroup$ @Galen in that case you're defining the probability as "The chance that any randomly selected friend would say that x is big", right? Which seems different from "What is the average of all your friends' opinion of whether x is big?" $\endgroup$
    – naught101
    Commented Mar 7 at 4:25
  • $\begingroup$ @naught101 Specifically in response to the above answer, yes, their description in the first part of the last paragraph is clearly ratios of friends conditioned on the outcome $x$, which in terms of probably might briefly be denoted as $Pr[\text{x is big} \mid x]$. The last part of that same paragraph appears to change from that frequency perspective to fuzzy description, which is different, and hence my original comment. $\endgroup$
    – Galen
    Commented Mar 7 at 5:11

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