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Fodor and M. Jin, Y. Li and C. Li, Robustness of fuzzy reasoning via locally equivalence measure. Information Sciences, No. Li, D. Li, W. Pedrycz and J. Wu, An approach to measure the robustness of fuzzy reasoning, Int. Journal of Intelligent Systems, 20 No. Li, Approximation and robustness of fuzzy finite automata, Int.
Journal of Approximate Reasoning, 47 No. Reiser and B. Santiago, B. Bedregal, and B. Zhang and K. Cai, Optimal fuzzy reasoning and its robustness analysis, Int. Journal of Intelligent Systems, 19 No. Journal Help. User Username Password Remember me. Notifications View Subscribe. Article Tools Print this article.
How to cite item. Email this article Login required. Sleep Disorder Syndrome elemento triangular geometria complexa gynecological examination software BlenderTM virtual reality von Neumann's algorithm. 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. 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.
Fuzzy Logic | Set 2 (Classical and Fuzzy Sets)
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. 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.
This is what yours truly answered on a related question on MathOverflow:.
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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. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question.
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Asked 7 years, 7 months ago. Active 2 years, 1 month ago. Viewed 2k times. 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. Clark Feb 6 '12 at 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.
The internal logic of a topos is always intuitionistic, which turns out to be more or less incompatible with fuzzy logic. That doesn't mean that fuzzy logic isn't effective for tasks, only that it wouldn't be interesting in a foundation way. If the logic is finite, they just use "multi-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. There is certainly a lot of deep work in that area. 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 Doug Spoonwood 8, 1 1 gold badge 23 23 silver badges 44 44 bronze badges. That's what I recall Wang and Klir writing, and Klir is definitely a strong advocate of fuzzy theory. So, does this answer really apply to the question here? It belongs to fuzzy mathematics, but not fuzzy set theory or logic proper.