Thanks to both of you.

Piegat had a website with a lot of useful links some years ago (unless I'm mistaking him for someone else now) but it seems to exist only in the past. I'll have a look at his book.

Andrew, unfortunately a full account of the interaction of fuzzy and statistical ideas over half a century is a project exceeding my capability, and it couldn't be squeezed into paper length (but it looks like a beautiful book project). The 2000 book I mentioned contains a 96-page chapter titled "Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps". It is reasonable to take the story at the point those books left it. Moreover, I am specially interested in the appearence of fuzzy-like objects, of course under other names, in mainstream statistical journals, which is mainly a recent phenomenon.

In the 90's there was a full-fledged attack on fuzzy sets by well-known Bayesian statisticians. It actually originated in the 80's within the artificial intelligence community, but the arrival of "fuzzy" electronic devices to the market, the hype surrounding it, and a number of overblown or wrong claims by some fuzzy proponents triggered a very strong response from respected members of the statistical community.

Since then, both communities have more or less ignored each other, to the point that Andrew Gelman recently referred to fuzzy sets as an "obscure statistical method", even if Zadeh's seminal paper is among the ten most cited ever O_o

Due to this rift, it is difficult to me to realize how the issue is viewed today. It could be hoped that fuzzy sets are more familiar now and some arguments would not find support.

Although the aim of the survey is not to debunk old arguments, I expect a quite hard time trying to publish the paper in a statistics journal. Understanding what reasons would be presented today against fuzzy sets should help me structure the paper.

Statisticians' objections twenty years ago were typically not concerned with questions such as "Are fuzzy connectives and linguistic hedges a faithful model of how language / cognition works?" or "Does fuzzy logic advance to what logicians consider is their aim?" which were raised in other communities.

Rather, the impossibility that fuzzy sets and fuzzy logic could make any sense or have any usefulness *at all* was posited. In 1996, a top statistics journal did not find it necessary to edit a line of an interview in which the interviewer, a widely known British probabilist, called working on fuzzy sets "a fate worse than death". That viciousness is really far from Susan Haack's solidly argued "We don't need fuzzy logic". But I am clearly digressing now--

1) Why that particular date? I know that objections to Zadeh's sets were raised in the 70s and that they have been heavily used in machine learning, statistics, etc., for some time while objections to fuzzy sets have, in general, been based not upon utility in analyses but more epistemic and philosophical reasons. Journals like Fuzzy Sets and Systems originate long before your 2001 date.

2) Can you clarify what you mean by "objections"? Applied mathematics, soft computing, computational intelligence, etc., have all advanced in unquestionable ways using fuzzy sets. However, objections fuzzy sets as a means of realistically modelling cognitive processes, linguistic responses, and logic in general have been attacked for reasons that have nothing to do with statistics It is simplicity itself (relatively, anyway) to show how fuzzy sets can make statistical analyses more powerful. Use of fuzzy sets as models, however, means ontological interpretations that are more controversial.

Maybe this book will be helpful to you:

Andrzej Pieagat,

Fuzzy Modeling and Control (Studies in Fuzziness and Soft Computing),

first publication date: 2001 May

Have a nice day!

Thanks to both of you.

Piegat had a website with a lot of useful links some years ago (unless I'm mistaking him for someone else now) but it seems to exist only in the past. I'll have a look at his book.

Andrew, unfortunately a full account of the interaction of fuzzy and statistical ideas over half a century is a project exceeding my capability, and it couldn't be squeezed into paper length (but it looks like a beautiful book project). The 2000 book I mentioned contains a 96-page chapter titled "Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps". It is reasonable to take the story at the point those books left it. Moreover, I am specially interested in the appearence of fuzzy-like objects, of course under other names, in mainstream statistical journals, which is mainly a recent phenomenon.

In the 90's there was a full-fledged attack on fuzzy sets by well-known Bayesian statisticians. It actually originated in the 80's within the artificial intelligence community, but the arrival of "fuzzy" electronic devices to the market, the hype surrounding it, and a number of overblown or wrong claims by some fuzzy proponents triggered a very strong response from respected members of the statistical community.

Since then, both communities have more or less ignored each other, to the point that Andrew Gelman recently referred to fuzzy sets as an "obscure statistical method", even if Zadeh's seminal paper is among the ten most cited ever O_o

Due to this rift, it is difficult to me to realize how the issue is viewed today. It could be hoped that fuzzy sets are more familiar now and some arguments would not find support.

Although the aim of the survey is not to debunk old arguments, I expect a quite hard time trying to publish the paper in a statistics journal. Understanding what reasons would be presented today against fuzzy sets should help me structure the paper.

Statisticians' objections twenty years ago were typically not concerned with questions such as "Are fuzzy connectives and linguistic hedges a faithful model of how language / cognition works?" or "Does fuzzy logic advance to what logicians consider is their aim?" which were raised in other communities.

Rather, the impossibility that fuzzy sets and fuzzy logic could make any sense or have any usefulness *at all* was posited. In 1996, a top statistics journal did not find it necessary to edit a line of an interview in which the interviewer, a widely known British probabilist, called working on fuzzy sets "a fate worse than death". That viciousness is really far from Susan Haack's solidly argued "We don't need fuzzy logic". But I am clearly digressing now--

Dear Dr. Terán:

I'd like to first point out what you are probably well aware of: I've read Haack's Philosophy of Logics and Deviant Logic, and I know I was disappointed to find that Lakoff (something of a hero/villain in my mind) criticized fuzzy set theory. However, such critiques are philosophical in nature. They may be valid criticisms to views expressed in e.g., Türksen, I. B. (2005). An ontological and epistemological perspective of fuzzy set theory. Elsevier., but they are almost entirely unrelated to the statistical use of fuzzy sets.

Also, although it is probably a matter of my non-participation of the debates you refer to in the 80s and 90s, I find it surprising that whatever acrimony, ideological differences, etc., that existed then continued to be relevant in the 21st century (or even before) at least as far as the statistical/computational community is concerned. I took a quick look to see what hardcopy material I had on fuzzy sets and Bayesian analysis published around ~2000. In case it is relevant, this meant EXCLUDING earlier work such as "Is it necessary to develop a Fuzzy Bayesian inference?" in Viertl (Ed.) Probability and Bayesian Statistics. I found several papers (including Taheri, & Behboodian's 2001 paper in Fuzzy Sets and Systems- "A Bayesian approach to fuzzy hypotheses testing"), but not much else that included (or focused) on an incorporation of Bayesian analysis/statistics and fuzzy sets. I did find Ripley's 1996 Pattern Recognition and Neural Networks, Buckley's Fuzzy Probability and Statistics published a decade later, Kandel, A. (1999). Introduction to pattern recognition: statistical, structural, neural, and fuzzy logic approaches (Series in Machine Perception and Artificial Intelligence Vol. 32). World scientific., and a few more. And again I found books and papers I hadn't looked at in I forget how long but which I exclude because they date form the 80s (e.g., Dubois & Prade's Fuzzy Sets and Systems).

Please correct me if I am wrong, but might it not be possible that you are conflating the kind of epistemological and ontological criticisms one finds in Haack and others with the application of fuzzy sets in soft computing, statistics, computational sciences, etc.? Although I don't have much more than a few dozen papers that date from around ~2000, pretty much all of my works on various applications of fuzzy sets and Bayesian networks (not to mention the broader subjects that include both as components in a more focal topic such as pattern recognition, classification algorithms, etc.) combines Bayesian statistics/analysis and fuzzy set theory. The director of my lab wrote, I think, and paper on fuzzy logic in the 70s. Perhaps I can pick his brain.

Hi Andrew, thank you for your input.

I think you are using a definition of `statistical community' which is more inclusive than mine. That is a really interesting issue, since it makes me think I am guilty of interiorizing the power structures I have been a victim of.

The Statistics department I am with successfully promoted a honoris causa doctorate for Zadeh in 1995. According to the definition of `the statistical community' I have been using above, we ourselves are not part of `the statistical community' (meaning a number of powerful people whose opinion `counts' and gets printed in good statistical journals).

All the people you cite are not a part of `the statistical community' in the restricted sense that they don't publish their work in statistics journals or conferences of any relevance to academic statisticians. Although I concur that whoever works on problems X, Y and Z *is* doing statistics, I doubt `the statistical community' considers them a part of the statistical community.

The Journal of the Americal Statistical Association, Statistical Science and Biometrika seem to be the top statistical journals to have ever published ONE fuzzy paper (the Biometrika paper contains the word `fuzzy' but it actually means `randomized' so it shouldn't even count). Those papers are from 2004, 2005 and 2007, respectively. All of them were written by non-fuzzy people who had never gone fuzzy before but already published their typical work in similar journals.

Many lesser but still good journals also seem to have never ever published a fuzzy paper, e.g. I am unaware of any in Statistica Sinica, the Scandinavian Journal of Statistics, the Canadian Journal of Statistics. The first fuzzy paper in Communications in Statistics A is scheduled to appear in 2014. The Journal of Multivariate Analysis published in 2006 two fuzzy papers by Volker Krätschmer. Extremes published one fuzzy paper in 1998.

I could tell a very similar story for probability theory journals (the only fuzzy papers I am aware of in the Annals of Probability and in Probability Theory and Related Fields are from 1985 and 1999, respectively; the Journal of Theoretical Probability published one fuzzy paper in 2002).

At the same time, many dozens of papers a year appear in computer science and mathematics journals, which is in some cases perplexing. Like in `Why do Information Sciences and Artificial Intelligence want my papers, while the journals the person next door publishes in will not even send them out for review'?

A straightforward explanation is that fuzzy papers are very bad. However, I don't think that explanation is correct, even if of course some fuzzy papers are actually very bad.

---

In summary, you can publish `statistics with fuzzy sets' papers in soft computing and artificial intelligence journals, or in subject matter journals (there is a lot of material e.g. in hydrology journals), i.e. anywhere but statistical journals. With only a few exceptions: Metrika, Statistics & Probability Letters, Computational Statistics & Data Analysis.

Hi Fausto, I really enjoy reading criticisms of fuzzy sets and logic.

So far I have had a look at "Yager...". It is clear that, if Yager's paper coincides with your account, any 7-element chain will do since only the lattice operations and involution are used. It is also clear that you can then give a geometric reformulation of the problem and make it a problem about 2-dimensional vectors.

However I think that, judging from your quotations of Yager's, your conclusions require an uncharitable reading of what he was up to doing. Also, it is faulty logic to claim that, because alternative A is proposed to overcome some shortcomings that *may* arise when using B, and you can recast A as an example of B, that those problems *will* also arise when using A. It is still conceivable that A *might* compare favourably to other forms of B when it comes to those shortcomings.

It is also clear that, in your account, what Yager does has no relationship whatsoever with fuzzy sets or logic, just with an ordered chain of linguistic labels. To explain Yager's reference to fuzzy sets in his title, it is possible, not having read Yager's paper, that he meant each linguistic label to be modelled by a fuzzy set, and then Min, Max and Neg should be read as operating on fuzzy sets, not on the set of labels anymore. The resulting quality measure would then be a fuzzy set which might not coincide with any of the labels.

---

Regarding `fuzzy' in quality control, Bob Woodall did a good critique of some approaches that made little sense.

Fuzzy p-values in latent variable problems --Thompson and Geyer

Dear Dr. Terán:

Thank you for the reply and clarification. I was indeed using a much more inclusive definition of "statistical community". Mathematics, like classics, physics, etc., is one of a handful of disciplines that have a relatively direct history going back to the early modern period and the origins of science. I recall professors who still thought of mathematics in terms of algebra, geometry, and analysis despite the fact that like the academic world in general mathematics has become interdisciplinary. So unlike e.g., number theory, when I think of statistics I tend not think of "statistics proper" (the work of those who publish in THE statistics journals, whose research is far closer to "pure mathematics" than most work that could be classified as contributing to statistics, etc.). Put simply, I tend not to distinguish between various applications of statistics or applied stats in general and statistical theory. However, this reflects my own background as one who uses mathematics, and statistics in particular, but is not a statistician in the sense you refer to (much as I love even the very theoretical aspects). I'm sure you are correct about what the statistical community considers the actual statistical community (with some notable exceptions, perhaps; as far back as 1986 ASA president John Neter was discouraging statisticians from "excluding themselves from important statistical problems in other disciplines" and encouraging a more interdisciplinary approach.

Also, a minor note: you stated that "the Biometrika paper contains the word `fuzzy' but it actually means `randomized' so it shouldn't even count". Are you referring to Thompson & Geyer's 2007) "Fuzzy p-Values in Latent Variable Problems"? Because

1) Geyer was the co-author of "Fuzzy and Randomized Confidence Intervals and P-Values" (Statistical Science, Vol. 20, No. 4 (2005), pp. 358-366)

&

2) Thompson and Geyer cite Geyer & Meeden's paper in Statistical Science when they define fuzzy p-values (p. 50).

However, to address your question about the relative paucity of statistical papers in the kind of journals you refer to, the answer is (I think) rather easy: Lectures Notes in Computer Sciences, IEEE journals, Studies in Fuzziness and Soft Computing, and countless monograph series, conference proceedings, and journals in computer science, computational intelligence, NLP, etc., are concerned with what works. If a feature extraction algorithm, fuzzy neuronal/neural model, or some other application, model, or method using fuzzy sets produces superior results (at least in certain ways, such as a far simpler method to obtain a marginally less accurate result or a slightly inferior approximation than) than we have a very concrete reason for using it. Issues relating to geometric and/or algebraic structures and possible equivalences, challenges to classical probability and by extension to "die Königin der Wissenschaften", and the philosophical issues of introducing "fuzzy" (=ill-defined) into the precision of modern mathematics that took some of the greatest minds and over 200 years just to get from the first formulations of the calculus to a sufficiently rigorous foundation do not tend to be considered, let alone considered to matter. Perhaps it is in this way analogous to the "shut up and calculate" approach to quantum mechanics that is incredibly successful in practice but poses deep concerns for many physicists who do not look with favor on a physical theory that describes physical systems as mathematical entities with no clear relation to anything outside of Hilbert space. It may be that statisticians object to fuzzy statistics for similar reasons logicians object to fuzzy logic: the concerns are more theoretic, philosophical, and general (and ultimately more wide-reaching and expansive), while the vast numbers of publications on fuzzy statistics outside of statistics proper have more immediate concerns and are judged by their utility for particular applications.

Of course, I am speculating to some extent here. I am far more familiar with statistical literature outside of the statistical community, whether in neuroscience or statistical mechanics, then I am with literature in it. I know more about Pearson, Galton, Edgeworth, Gosset, Fisher, etc,. and their work than I do about the work within the statistics community (understood more exclusively). However, I can't reconcile my experience with the ubiquitous reliance on fuzzy sets in diverse fields with your experience of its minimal existence in any other way. It is an interesting question, though.

@Jane: Thanks!

@Andrew: Yes, as I started to suspect with Jane's answer, the Biometrika paper is by both Geyer and Thompson. I seemed to recall that it was by Thompson alone. In her further papers on latent variables she used the formalism from the Geyer-Meeden paper, but her interpretation of [0,1] values is a randomized test.

I agree that the inclusive definition makes more sense. The observation that Pearson and other big names were not mathematicians regarding proving theorems as their main job (as most academic statisticians and I are) is spot on, too.

The main problems (some) Bayesians have with fuzzy sets is that they believe to have sound arguments proving that usual probability theory is necessary and sufficient for any problem involving uncertainty. That is a philosophical position but they consider it as being simply `the truth'. If you `deny' the truth when shown ample `evidence' of it, then you are not regarded as being particularly smart :)

In that sense you are right that the divide is more one of theoretical or philosophical concern. With you last answer I see much more clearly what you originally meant.

Great discussion, thank you very much.

I have been following this discussion from the start. My own findings are as follows:

(1) Two probability laws are necessary and sufficient to define a normal fuzzy law. The left reference function of a normal fuzzy number is a probability distribution function, and the right reference function is a complementary probability distribution function. Therefore trying to establish a probability law from a normal fuzzy membership function or to establish a fuzzy membership function from a probability density function is mathematically meaningless.

(2) For example, everyday there happens to be a minimum temperature and a maximum temperature at every place. The minimum temperature would follow a probability law in an interval [a, b] say, and in the same way the maximum temperature would follow another probability law in [b, c] say. The probability distribution function of the minimum temperature in [a, b], and the complementary probability distribution function in [b, c] would form the membership function of a normal fuzzy number [a, b, c].

Rainfall on any day does not have a minimum or a maximum. Therefore rainfall is never a fuzzy number.

(3) In my way, normal fuzzy numbers can be constructed. In the classical definition, the question of constructing a fuzzy number is heuristic perhaps!

Dear Ibsen,

Thank you very much for your suggestions.

I agree that more research is needed to streamline the way fuzzy or Dempster-Shafer can contribute to other areas. As you may know, despite serious efforts by Dubois and Prade in several position papers, many fuzzy researchers in decision making are content with developing fuzzy methods without ever comparing their effectiveness with non-fuzzy methods. The fallacious rationale seems to be that a fuzzy AHP method, for instance, must of course be better than using just AHP because, well, it's fuzzy and the other is not.

Ranking is a really problematic issue and many fuzzy decision making papers seem to choose their ranking methods for no good reason. However, I don't see an easy way out of that, since deciding what ranking methods make more sense in a given situation depends critically on thinking hard about the semantical part of the modelling (i.e. what the fuzzy sets are supposed to mean: constraints on the value of a variable, measures of closeness to a prototype, uncertainty distributions, representations of words from natural language, and so on). Since you can get papers published without a careful consideration of those aspects, I don't think the situation is going to improve in the near future.

---

Maybe you might be interested in the following paper?

N. Sadeghi, A. R. Fayek, W. Pedrycz (2010). Fuzzy Monte Carlo Simulation and Risk Assessment in Construction. Computer-Aided Civil and Infrastructure Engineering 25(4), 238–252.

Dear Hemanta,

To some extent I disagree with your point (1). It is a fact that you can obtain two cumulative distribution functions from the left and right slopes of a fuzzy number. It is also a fact that, this way, you can apply concepts relative to distribution functions, which is a part of probability theory. But whether such an application makes sense should not be decided using the semantics of probability theory but that of fuzzy sets.

One clear example goes in the opposite direction. Since a probability measure is a [0,1]-valued function, it can be claimed to be a fuzzy set (in fact, Bart Kosko famously did so). At the mathematical level that is true, but the question remains whether the things you can do with [0,1]-valued functions in fuzzy set theory are useful or relevant within the range of interpretations proper to probability theory.

For instance, a key concept when dealing with sets is inclusion, but if one tries to regard probability measures as fuzzy sets it turns out that one probability measure can never be 'included' in another one (if P and Q are probability measures, P

Dear Pedro,

Thank you for your answer.

In the fifth paragraph of your answer, you have mentioned that the 'pair of two cumulative distribution functions' approach does not work in fuzzy arithmetic.

It does! Please go through the attachment. You can see that this approach can actually be used to find the membership function of a fuzzy number.

Dear Pedro,

As only one attachment could be added in an answer, I am attaching herewith another. You can see that the approach can be used in fuzzy arithmetic too.

The point is: why does the method give the same result as given by the method of alpha-cuts? Indeed, in the method of alpha-cuts, a line parallel to the horizontal axis is drawn, which cuts the membership function at two points. Therefore, even if one takes the membership function as two distribution functions, the line parallel to the horizontal axis will cut the two distribution functions at the same two points. That is why this method returns the same answers.

I shall send my reply with reference to the other things mentioned in your answer later. I shall wait for your reply regarding this particular point on fuzzy arithmetic.

You have raised one point, and I am sending my answer. I hope, you would send me an unbiased comment now!

I'm not sure that your notation is correct. In paper 2, equations (3.1) and (3.2) are identical, whence X and Y would be the same fuzzy set.

What I meant is the following.

-If you are trying to reduce fuzzy numbers to pairs of distribution functions, then you would like to reduce arithmetic operations on fuzzy numbers to operations on distribution functions.

-The most natural sum-like operation between distribution functions is convolution.

-If you calculate the convolution of the distribution functions, the result is not the corresponding distribution function of the sum of the fuzzy sets.

In the papers you attach, distribution functions are not aggregated by convolution, so there is no contradiction.

While it is not absurd to aggregate distribution functions that way, I can't see a compelling reason to do it. How would you convince people?

The key step is when you say "We start with equating L(x) with L(y)" and write y as a function of x. If the distribution functions come from random variables (which is not the case in this problem, but my question is how you would convince people in a general situation), you are assuming that one of the random variables is a function of the other, i.e. that they are maximally dependent. That doesn't seem easy to sell.

Dear Pedro,

Yes, convincing people is the problem!

I know that there is a slight confusion regarding symbols. You are right.

However, it works, as you might have observed. Thanks for your comments.

My problem is not about convincing people! It is about making people realize that mathematics should not be based on beliefs, it should be based on logic. Further, axioms do not fall from the blue; this too is an important point.

Let me now ask you back one question. Is there a way to construct a fuzzy number mathematically in the Zadehian way? Once again, I am expecting an unbiased answer.

In my way, yes, fuzzy sets can be constructed mathematically. Would you like to see that?

Dear Pedro,

Regarding construction of fuzzy numbers, you have not made any comment yet! Does the existing theory of fuzzy sets say anything about construction of normal fuzzy numbers mathematically? In the two cdf approach, it can be done. I have added one attachment in this context.

For example, it is said that triangular fuzzy numbers are used because 'they are easy to deal with'. That cannot be a mathematical explanation.

On the other hand, in the two cdf approach, if there is a uniform law of randomness in [a, b], and if there is another uniform law of randomness in [b, c], then the cdf in [a, b] and the complementary cdf in [b, c] will define the membership function of a triangular fuzzy number [a, b, c].

As the uniform law of randomness is the simplest in Probability Theory, the triangular fuzzy number is the simplest fuzzy number in the Theory of Fuzzy Sets. This is the mathematical explanation behind the triangular number being the simplest!

Dear Pedro,

You have still not answered my question. What about construction of normal fuzzy numbers? Does Zadehian theory supply any method to do that?

Hemanta, why would there not be fuzzy sets applicable to rain fall on any given day the same way as with temperature? Rain fall "density" is just as fuzzy as temperature. If a single rain drop fell, it rained and thus there would be nothing fuzzy about it but we cannot be sure how many rain drops, how big, do they reach the ground or dry just before hitting the ground (typical California situation), etc. I think that rain as "density of water falling" can be fuzzy as well as temperature variations within the ranges set. Another reason why rain is suspicious becuase as clouds move, it may be raining over my head and not yours yet we walk side by side... there is something special about rain that is perhaps even harder to define than temperature. I am not very familiar with using fuzzy sets for any of my medical experiments since they all have been quite simple yes/no type experiments where even regression is meaningless, so if I am off the subject in some way, I apologize.