Is there any problem which can be solved only by fuzzy theory but not by statistics?   When statistics is solving all the problems, why does one study the fuzzy theory? Is there any problem which can be solved only by fuzzy theory but not by statistics?  Please kindly answer with examples.
 A: Instead of hearing it from me, why don't you read about it from the man[1] himself. In this paper, as the title suggests, the author points out some of the shortcomings of the probability theory and suggests the Fuzzy Theory can complement probability theory in overcoming those. In fact, he goes further and publishes [2]. You can go through these and see if his arguments are convincing enough for you. If they are not, then there is nothing I can say that is going to convince you anyways :-).
[1] Discussion: Probability Theory and Fuzzy Logic Are Complementary Rather Than Competitive By Lotfi Zadeh
[2] Toward a perception-based theory of probabilistic reasoning
with imprecise probabilities By Lotfi Zadeh
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Personal Opinion (written on request of the OP, others may ignore this content):
A couple of years back, I was working on an Extraction Management System [3]. This system was to work along with a plethora of Information Extraction operators and extracts several entity relationships from the web. Now, for example, let us say, an operator extracts the unary relationship "Flat(Earth)". There are two things we need here : a] Correctness of extraction [What is the probability that the extraction has an error?] b] Correctness of the extracted relationship [What is the probability that the statement made in the extraction is incorrect?]. To clarify, let us say the webpage from which the relationship was extracted actually said, " ... made the trip to earth. Flat model was used ...". Clearly, the webpage is not saying, "earth is flat", so the extraction made an error (type a above). On the other hand, if the webpage actually said, "earth is flat", then the statement being made is incorrect (type b above). You can easily see cases where there is an correct extraction with an incorrect statement and incorrect extraction with an incorrect statement (an incorrect extraction with a correct statement is somewhat unlikely I presume). Downstream, we wanted to combine these two probabilities in a meaningful way and we did come up with a bayesian model to do it (along the lines of [4]), however, what I would like to point out here is that we did look at other possibilities at the time (like the imprecise probability stuff in [2]). Now, I would like to have given you numbers and so on and conclude by saying that systemA did better than systemB. But, I can't because I never got to doing those experiments :-). However, the takeaway here is that problems like these do exist in the real world applications and many different approaches to handling such cases in a generalized way is always nice. So, I welcome fuzzy models as much equally as Bayesian models.
[3] Purple SOX Extraction Management System
[4] Truth Discovery with Multiple Conflicting Information
Providers on the Web
A: There is a theory which seems to combine statistics and fuzzy logic. See the book Statistical methods for non-precise data by Reinhard Viertl (see also this pdf). Unfortunately I can't say more about this. I took a quick look at this book but it does not give some examples of applications (except if I missed them). I would be interested if somebody could tell more about Viertl's works.
A: I think fuzzy logic and statistics are very different things.  Fuzzy logic has to do with set membership where an element may belong to a set with a probability less than 1 associated with it.  Probability uses measure theory and ordinary set theory to develop its theory which is the basis for statistical inference.
A: For years I had no idea what people were talking about when they dropped the term "fuzzy logic". No one taught me about it; it was completely skipped in all my mathematics and artificial intelligence courses. I believe this is because it was part of a wave of attempts to recategorize pieces of probability theory, most of which no one today has heard of because academics realized they were kind of silly and stopped passing them on.
For some of that context, I recommend this vibrantly, hilariously scathing article from the 80s in which the author calls attention to a proliferation of (then) new theories for reasoning with uncertainty, calls them all balderdash, and shows why you can just use ordinary probability theory instead. (He actually calls out Zadeh specifically.)
At the end he states Probability is the proper generalization of Logic, not "fuzzy logic", and I tend to agree. So to answer your question, Probability (Bayes Theorem, marginals, conditionals, etc.) isn't exactly the same thing as Statistics (populations, distributions, etc.), so, yes, there are questions Probability theory answers that Statistics doesn't, like "70% of the people on the bus are men. 50% of men have bags. 80% of women have bags. If I choose a person at random, what is the probability they have a bag?"
