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I keep coming across the term "fuzzy". I browsed around a bit and read a few articles.. It seems to me that there's absolutely nothing deep or foundational about fuzzy set theory or fuzzy logic. I know they are practically useful in robotics, ..etc. but since there's a lot of buzz around them, I am led to think that there might be something deep that I just don't see.

I realize that my question is perhaps vague or "not constructive", I would understand if it were closed for that reason.

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Your question is potentially controversial, but I upvoted it because I too am curious. My guess (and it is just a guess) is that it is as you say: fuzzy logic is very useful in certain parts of applied mathematics, but from the perspective of pure mathematics there is not much going on. E.g. in measure theory / probability we identify a subset $Y \subset X$ with its characteristic function $\mathbb{1}_Y$...but of course we also consider more general functions $f: X \rightarrow [0,1]$. It is not clear to me why regarding $f$ as some sort of generalized set gains anything non-semantic. –  Pete L. Clark Feb 6 '12 at 12:55
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Of course the sentiment expressed in the previous comment is both uncharitable and rather naive. Given that a MathSciNet search of "fuzzy" gets $27195$ hits, I am eagerly waiting to be informed/corrected... –  Pete L. Clark Feb 6 '12 at 13:05
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This is really a vague memory, but I recall reading somewhere that fuzzy logic was successfully reduced, in some way, to probability - specifically, that any problem in fuzzy logic can be reduced to a probability question. At this point, it became less interesting. That said, the "sub-object classifier" in topos theory can be seen as "like" a fuzzy logic, but in a way that may be broader than the definition in fuzzy logic. –  Thomas Andrews Feb 6 '12 at 13:40
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@Thomas: In fact, the opposite is true: see this question. The internal logic of a topos is always intuitionistic, which turns out to be more or less incompatible with fuzzy logic. –  Zhen Lin Feb 6 '12 at 14:22
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Just because fuzzy logic is a different meaning doesn't mean that fuzzy logic problems cannot be translated into pure probability problems. That doesn't mean that fuzzy logic isn't effective for tasks, only that it wouldn't be interesting in a foundation way. –  Thomas Andrews Feb 6 '12 at 14:28
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When I think of "Fuzzy logic" the first thing that comes to mind is engineering, where is it used in control systems, and information processing. For example, "fuzzy logic" is used in control systems for washing machines. I think it was a hot topic in 1980s and 1990s, and one article from that time is "Designing with Fuzzy Logic" by Kevin Self from the IEEE Spectrum magazine in 1990.

My opinion as a logician is that "fuzzy logic" per se has not developed significant interest in mathematical logic for the study of foundations of mathematics. I have not seen any examples of "deep" applications of fuzzy logic to classical foundational issues. One reason why it will be difficult to develop deep foundational implications of fuzzy logic is that much work in foundations is motivated by a few standard structures (ignoring any philosophical objections): the natural numbers, the real numbers, the cumulative hierarchy of sets. These are not ordinarily conceived as "fuzzy" structures.

As a personal aside, I would not use the term "fuzzy logic" to refer to multivalued logic in general, and I think it is somewhat ahistorical to refer to work of Łukasiewicz as "fuzzy logic". However, others do use "fuzzy logic" more broadly, and there is a book by Hájek entitled "Metamathematics of Fuzzy Logic", so my opinion is far from universal.

None of this means there is a lack of value in fuzzy logic, of course. It is obviously of great interest for information processing, control theory, and engineering, where it may indeed have deep aspects.

The introduction to the Stanford Encyclopedia of Philosophy article on fuzzy logic also remarks on some of these issues. In that terminology, I am apparently speaking about "fuzzy logic in the broad sense".

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The fuzzy logic texts I've looked at generally use "fuzzy logic", when used in the narrow sense, for any infinite-valued logic. If the logic is finite, they just use "multi-valued logic". –  Doug Spoonwood Feb 7 '12 at 15:57
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@Doug Spoonwood: Thanks, more input is helpful. I certainly see others working on multivalued/infinite valued logic. For example there is very interesting recent work on a sort of continuous-valued "logic of metric structures" by Ben-Yaacov, Berenstein, Henson, and Usvyatsov matematicas.unal.edu.co/~aberenst/mtfms.pdf - but they never use the word "fuzzy" in 100 pages of text, and other presentations I have seen about that stuff never connect it with "fuzzy logic". –  Carl Mummert Feb 7 '12 at 16:14
    
If any infinite-valued logic is a form of "fuzzy logic", then all work on boolean valued models of set theory would be fuzzy logic. There is certainly a lot of deep work in that area. –  Michael Greinecker Feb 7 '12 at 16:25
    
I would say it is quite common to understand "fuzzy logic" (in the mathematical sense I was talking in my answer) as "multi-valued logic where the values form a linear (i.e. total) order". –  boumol Feb 7 '12 at 18:07
    
@MichaelGreinecker I believe most fuzzy theorists would hesitate from making that sort of move. The values of a Boolean domain take their truth values form a complete Boolean Algebra (BA). Any theorem which holds for "2", the two-element BA also holds for any BA and conversely. So, in some sense, "2" and any other BA come as equivalent. Fuzzy logics don't come as equivalent to "2". Perhaps better, any infinite-valued logic which does not have the same structure as "2", seems to qualify as a fuzzy logic (though not all authors will use the term "fuzzy logic"). –  Doug Spoonwood Feb 7 '12 at 19:03
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I know quite a lot of mathematicians who claim that there is nothing deep about "formal logic" (in the sense of mathematical logic): they believe it is just a waste of time to spend so much time in formalizing a problem. What these people do not realize is that the interesting part of logic is not the process of formalization, what it is really interesting is the metamathematical reasoning we can do once a problem is formalized. Particular examples of useful results concerning metamathematics are: compactness theorem, Löwenheim-Skolem theorems, completeness theorem, incompleteness theorems, etc.

My interpretation about your question is that you share the same prejudices I have explained above, but towards fuzzy logic insted of logic. So, let me try to point out that what is interesting about fuzzy logic for me, at least, is not the process of formalizing using fuzzy sets (I agree that this is not worth spending time), it is the metamathematics that can be developed around this alternative semantics. To emphasize this distinction some people has coined the term "mathematical fuzzy logic" (you can find much more details in the link http://www.mathfuzzlog.org/index.php/Mathematical_Fuzzy_Logic and the references there in).

Let me just summarize some statements about fuzzy logic in this metamathematical sense:

  1. First-order logic using an infinite-valued Lukasiewicz semantics (this is a natural choice of connectives) is not recursively axiomatizable, indeed it is $\Pi_2^0$-complete. The first part was proved by Bruno Scarpellini, and the second by Mathias Ragaz. Other choices of interpreting connectives give different situations (e.g., using what is called "product logic" you get a non-arithmetical set of validities, etc.).

  2. The comprehension axiom is consistent in case you restrict your logical rules from classical logic to Lukasiewicz fuzzy logic. Thus, an alternative approach to restricting comprehension axiom (this is method used in ZFC) consists on restricting the logical rules. To explore a bit more on this topic look at http://mathoverflow.net/questions/86151/boolean-valued-models-vs-the-infinite-valued-logic-of-lukasiewicz-and-set-theory

  3. Finally, let me point out that while it is known that Peano arithmetic cannot be consistenly expanded with a truth predicate, this is not so clear in case you allow the truth predicate to be fuzzy. Unfortunately, the situation is the same than in classical logic, but the proof is more tricky than the one given for classical logics. You can find the details of this in the paper http://www.jstor.org/stable/2586541 published at Journal of Symbolic Logic (see also http://www.logicmatters.net/2010/05/curry-lukasiewicz-and-field ).

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I'm not sure what you mean by "deep". However, I believe you will think it reasonable to postulate that if something leads to the creation or expansion of new mathematics, then it seems reasonable to think of it as deep in some sense. Well, fuzzy set theory can get said to have lead to the creation of new branches of group theory (and other branches of abstract algebra), topology, differential calculus, arithmetic, geometry, trigonometry, graph theory, etc., see here and here for potential sources. On top of this, there exists possibility theory, and generalized measure theory where fuzzy sets usually come up at some point one way or another. This, of course, is not an argument that you should like fuzzy set theory, nor an argument that you should dislike fuzzy set theory, just as if you read enough philosophers you'll almost surely find someone extremely deep who you intensely dislike.

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This is what yours truly answered on a related question on MathOverflow:

Fuzzy measure theory has applications in pure measure theory. The Choquet capacity theorem is a standard tool for showing the universal measurability of analytic sets. The theory of capacities (or fuzzy measures) is fairly well developed and strongly related to "normal" analysis.

The theory of capacities was not created in the context of fuzzy mathematics, but M. Sugeno developed a form of fuzzy integration in his PhD thesis that shares many formal similarities with the Choquet integral and some work on the Sugeno integral carried over to the Choquet integral.

A rather extensive introduction to these topics is given in the book Generalized Measure Theory by Wang and Klir.

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As I understand it, the Wang and Klir book basically indicates that fuzzy measures don't use the term "fuzzy" in the same way when talks about fuzzy sets or fuzzy logic, including Sugeno's integral. That's what I recall Wang and Klir writing, and Klir is definitely a strong advocate of fuzzy theory. There doesn't exist anything about imprecise boundaries (fuzzy sets) or degrees of truth (fuzzy logic in the narrow sense/purely logical sense) in the concept of a fuzzy measure, AFAIK. So, does this answer really apply to the question here? –  Doug Spoonwood Feb 7 '12 at 0:32
    
I think it applies "half". It belongs to fuzzy mathematics, but not fuzzy set theory or logic proper. The imprecision comes, for example, from working with a functional of family of measures instead of a measure. –  Michael Greinecker Feb 7 '12 at 0:44
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