On the face of it, one would think that there is a clearcut distinction between fuzziness and randomness. Why? Membership functions are defined over the interval [0,1] and not in a probability space. By contrast, random variables are defined in a probability space. A random variable is a function defined on a sample space. Functions that define random variables are limited to those for which a probability distribution exists and derivable from a probability measure that turns the sample space into a probability space.

Even so, fairly recent research has introduced fuzzy random variables. With a fuzzy random variable, a random event can only be observed in an uncertain manner. A fuzzy random variable is the fuzzy result of an uncertainty mapping. Hence, it is possible to define the exact relationship between randomness and fuzziness. For details, see the attached paper.

@Hemanta, it is the first queston that I do answer in 2014! As I did some research in the field regarding randomness and fuzziness, maybe the answer is in probability versus possibility theory. I do think this paper is relevant to Your thread! Regards and good luck!

http://www.engineering.nottingham.ac.uk/icccbe/proceedings/pdf/pf141.pdf

On the face of it, one would think that there is a clearcut distinction between fuzziness and randomness. Why? Membership functions are defined over the interval [0,1] and not in a probability space. By contrast, random variables are defined in a probability space. A random variable is a function defined on a sample space. Functions that define random variables are limited to those for which a probability distribution exists and derivable from a probability measure that turns the sample space into a probability space.

Even so, fairly recent research has introduced fuzzy random variables. With a fuzzy random variable, a random event can only be observed in an uncertain manner. A fuzzy random variable is the fuzzy result of an uncertainty mapping. Hence, it is possible to define the exact relationship between randomness and fuzziness. For details, see the attached paper.

@James, good paper! Famous Dresden School! Thanks! Happy New Year!

I agree with James. Fuzzy logic deals with statements, while probability spaces deal with events. Indeed, a statement can define an event. However, the corresponding structures are not the same. For instance, if S1 and S2 are two statements defining two events, their conjunction follows the laws,

P(S1 ⋀ S2) = S1 · S2

µ(S1 ⋀ S2) = min (µ(S1), µ(S2))

where P denotes de propability and µ the truth-value. Accordingly, both structures are different.

In addition, fuzzy logic need not be valued in [0,1], see

http://www.sciencedirect.com/science/article/pii/0165011492901949

Dear Juan-Esteban,

Many thanks for pointing out that fuzzy logic need not be valued in [0,1].

@Juan-Esteban: probability theory does not tell us that the probability of a conjunction is the product of the probabilities of the two. The probability of the intersection of two events is largest when the two overlap maximally, it is smallest when they overlap minimally. So one can easily write down a lower and an upper bound. The upper bound is min(P(S1),P(S2)). The lower bound is max(0, 1-P(S1)-P(S2)).

You are studying a population. Its members, according to their age (A), can be young (Y) or old (O) and they are asked to choose an ideology (I) between Egalitarianism (E) and Merit-based Inequality (M).

You have data about the joint distribution of A and I.

With the appropriate cautions, you can derive probs by frequencies. You could give P(Y&E) on the basis of the (relative) frequency of young and egalitarians in the joint distribution you have.

Y,O and E,M are, of course, non-clearcut notions, so that you could think of approaching the analysis using fuzzy sets (let m be the degree) and looking for m(Y&E).

Fuzzy methods suggest that it depends on m(Y) and m(E).

With reference to this example, fuzzy logic would use the marginal distributions and not the joint distribution.

This shows that:

(1) - Prob and Fuzzy are different.

(2) - Prob is more demanding.

My opinion is that the core of the difference stays in the fact that no matter what approach you accept for probability (subjective, frequency-base, classic) you have a method to produce probs. Nothing similar is given for m(.).

Lorenzo,

Good post and good points!

I add some remarks: in Prob you have computation rules, which are, substantially, approach independent and are compelling, in fuzzy logic these rules are weaker, but the higher degree of freedom has a cost (for an economist this is trivial: no Land of Cockaigne can exist). Such a cost is arbitrariness in setting rules.

A related and relevant question concerns the rationality paradigm staying behind Prob, whiel no such a paradigm stays behind fuzzy logic.

Prob is great as a normative paradigm, while poor as a descriptive one.

Fuzzy works better in several descriptive settings, but is rather lacking on the other side.

See:

I.R. Goodman, Random Sets and Fuzzy Sets: A Special Connection, ”Fusion ’98 International Conference”, 1998, p. 93-100 (http://www.isif.org/fusion/proceedings/fusion98CD/93.pdf)

Wang Pei-zhuang, Fuzziness vs. randomness, Falling shadow theory, 1990 (http://www.polytech.univ-savoie.fr/fileadmin/polytech_autres_sites/sites/listic/busefal/Papers/48.zip/48_15.pdf)

ARC_2012_Shapiro_B_01_Implementing Fuzzy Random Variables, 2012

Probability

1 - You have a box (Box A) that contains 5 small and 5 large apples. You randomly pick an apple from the box. There is uncertainty about the size of the apple that will come out. The size of the apple can be described by a random variable that takes two different values with 50% probability each. There are two distinct possible outcomes and there IS no uncertainty after the apple is taken out and observed.

Fuzziness

2 - You have a box (Box B) that contains 10 apples with different sizes ranging from very small to very large. You have one of these apples in hand. There is uncertainty in describing the size of the apple but this time the uncertainty is not about the lack of knowledge of the outcome. It is about the lack of the boundary between small and large. You use a fuzzy membership function to define the membership value of the apple in the set ``large`` or ``small``. Everything is known except how to label the objects (apples), how to describe them and draw the boundaries of the sets.

Therefore, I believe fuzziness and randomness are two totally distinct concept. This does not prevent us from observing or defining variables that have both fuzziness and randomness at the same time. For example, randomly take an apple from Box B. There is probabilistic randomness associated with the randomness of the outcome and fuzziness associated describing the apple as small and-or large. Another example is the paper that James attached above.

Here is an example of how to define probabilistic fuzzy systems.

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1008853

Amir,

Good post!

Nebi Caka, has already mentioned I. Goodman paper, along with other interesting. I would like to mention that I. Goodman proved a representation theorem for fuzzy sets. Fuzzy sets as equivalence classes of random sets, see e.g. Goodman, I.R., Fuzzy sets as equivalence classes of random sets, in Fuzzy

Sets and Possibility Theory , Yager, R. et al., Eds., Pergamon Press, New York, 1982.

Dear Juan-Esteban: The reference that you cite uses topos-theoretic models. These models they are "not exactly" fuzzy, since the internal logic of a topos is intuitionistic, and thus satisfy the idenpotent law. Fuzzy logic does not satisfy this law and thus in order to treat fuzzy logic categorically we should use Monoidal closed categories. In addition in Many-valued logic we use MV-algebras, and [0,1] is the MV-algebra which generates the whole variety of MV-algebras. Many-valued logic contains in a sense fuzzy logic, since semisimlpe MV-algebras are essentially Bold fuzzy sets.

For this thread, there may be interest in probability on tribes. Tribes were introduced by D. Butnariu and E.P. Klement (2002) in a study of games with fuzzy coalitions. See the attached pdf file.

Fuzziness and randomness work for very distinct type of uncertainty. For example you have ten apples each of them has different size. You have an apple randomly. Now the probability of the apple you draw is the big one is what?  Clearly you have no idea about the word big as the big has no clear-cut or sharp boundary. So first you face an uncertainty and the uncertainty is the lacking of some information and the information is about a clear-cut boundary of size of something.  After solving this problem, you now know which one is big and how many of them are big. You can now determine the probability that your drown apple is big.

What was wrong with the answer of respected @James? Why downvotes!

Randomens is based on σ-additive measures, whereas fuzziness is based on non-additive measures. See, e.g.

Zhenyuan Wang, GeorgeJ.Klir, GENERALIZED MEASURE THEORY, Springer, 2009

Hi Hemanta, this is an excellent question. In the new world with big data and high compute power, probability theory will be able to good predictions of historic scenarios. However without the fuzziness aspect there will be a gap as we unfold into new situations. Situations that have not been seen before. Choice models do come to help to some extent but again are limited to the choice set. I will be interested in your summary about this discussion and what you gathered out of the variety of responses.

Thank you, Tapas.

Randomness is well defined but fuzziness is not.

Good question and thank yoyu for answer of M.James too

so: Randomness occurs with some degree of probability. Getting struck by lighting would have a very low degree of probability. Either it happens or it doesn’t.Fuzziness is an measure about a concept that occurs with some degree, defined to be 0 (not at all)

good luck

Probability is a summary of what you know about something you can't predict.

A rational model (read de Finetti, please) is available from decades, but generally unknown.

Probability asks you to decide about a number between 0 and 1, you would accept a bet (non-knowing about you're the batter or the bank).

I'm old and I'll quickly disappear, but In my in my long life academic, I never understood the concrete meaning of fuzziness.

There is a detroying paper about fuzzines.

You should be aware, but I think you're not aware.

In case of need, please contact me.

Best

Perhaps the difference between randomness and fuzziness can be better understood from the application perspective. As such, fuzziness is more suitable when the decision maker is facing limited information concerning various parameters. Hence, it can be applied in situations that involve human judgement, which is often imprecise and of a qualitative nature (e.g., supplier portfolio selection) rather than of purely quantitative nature (e.g., stock portfolio selection, where the concept of probability is more representative).

The aim is respectable, but I don't share the strategy: why shoul I accept, without any explanation, "triangular" fuzzy numbers? Why should I use my residual lifetime (I'm 74) to such questions.

I was one of the first enthusiatically people interested in Zadeh formidable idea.

I must confess I have found a serious interpretation of \mu(X) in a specific Sociological problem (On request I can document).

I must confess to I do not understand why thhe idea to make authomatic \mu(X)

should be interesting.

I do not understand why \mu is plateally undefined.

As working in the maths applications in Economics, Management, Finance, I am still waiting for a striking application of fuzzy sets that has had in impact on the profession.

My personal opinion is that "fuzzy Craft" has started from being a potentially powerful Craft, but that, as present is nothing but a closed, selfe-references community.

I propose to "fuzzy craft" to establish their scientific standing w.r.t. Maths.

I hope to be able to attach an appropriate file.

This is a very interesting endeavor and I also agree that fuzzy logic has a great application potential, especially in the area of machine learning and AI. Coming back to the original question about the relation between probability and fuzziness, Zadeh gives an overview of this relation in his paper:

Lotfi A. Zadeh, The Information Principle, Information Sciences, Volume 294, 2015, Pages 540-549

If my understanding is correct, the concepts of possibility and probability are interlinked through the restriction posed by information (and, hence entropy) . According to Zadeh “possibilistic information and probabilistic information are underivable (orthogonal), meaning that probabilistic information cannot be derived from possibilistic information, and vice versa.” For the mathematical notation I refer to the above paper.

See this link is useful

https://www.quora.com/What-is-the-difference-between-randomness-and-fuzziness

Best wishes