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Why doesn't one develop fuzzy logic by extending topos theory, by simply extending the subobject classifier $\Omega$ to the unit interval [0,1]? Have people done that?

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    The topos of sheaves on $[0, 1]$ has as its subobject classifier the lattice of all open subsets of $[0, 1]$. But as far as I know this is not what is meant by fuzzy logic. Can you please elaborate on your idea?2011-08-06
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    Also, see this [MathOverflow thread](http://mathoverflow.net/questions/8225/encoding-fuzzy-logic-with-the-topos-of-set-valued-sheaves).2011-08-06
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    Zhen: Thanks for your reply. I'll need some time to learn all this. I don't really have a detailed idea, except that I use a new form of fuzzy logic different from Zadeh's in my A.I. research. Basically my category of fuzziness is what people would call "degrees" -- ie, a degree is a number in [0,1]. And probabilities can be distributed over degrees, resulting in fuzzy-probabilistic logic. I'm just wondering, could the concept of "degree" coincide with the topos with \Omega = [0,1]?2011-08-08
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    PS: I think a topos with $\Omega$ = [0,1] can be regarded as a fuzzy set (or predicate). For example, "tallness" is a fuzzy predicate, mapping objects like "john" to, say, 0.7. This is exactly Zadeh's definition of fuzzy sets. But my interpretation of it is different -- it is regarded as a fuzzy predicate returning a *degree* of something, NOT set membership. Thus, Barr's worry about fuzzy set equality need not bother us -- we're just talking about the equality of 2 degrees, which can be crisp. You get my idea?2011-08-08
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    Oh, I realize there is a conceptual problem... topos is a generalization of set, but if I take the interpretation of fuzziness as "degree", then fuzziness is no longer a set or similar to sets. So my idea of making topos fuzzy may be unsound. I just think that fuzzy sets are not the right way to formulate fuzzy logic(s).2011-08-08
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    If you're only thinking about fuzzy *propositional logic* there is no need to invoke the machinery of categories or toposes. The point of these studies, as I understand it, was to investigate the possibility of developing fuzzy *higher-order logic* following the pattern of toposes, which have a intuitionistic higher-order logic as their internal logic.2011-08-08
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    To see why the subobject classifier route probably wouldn't get taken, I'd suggest seeing Zadeh's 1975 paper "The Concept of a Linguistic Variable and Its Applications to Approximate Reasoning." http://www.cs.berkeley.edu/~zadeh/papers/index.htm Truth values in fuzzy logic in the narrow sense can be "unknown" or "undefined", and, as I understand it, entries of truth-tables in fuzzy logic become questions. Can this happen via subobject classifiers?2011-08-26

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