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Re: [ontolog-forum] Ontology of Rough Sets

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Sat, 15 Jan 2011 13:08:41 -0600
Message-id: <4B4568B9-0A0A-4E57-BD5F-9D54D9658758@xxxxxxxx>
On Jan 15, 2011, at 10:27 AM, Rich Cooper wrote:
Hi Chris,
A Boolean variable is bivalent – it can take on True or False as its value.  

So I've heard. ;-)

A multi-valent variable can be True False or Unknown, as you define it.

Actually, three valued logics are a very special case of multivalent, or many-valued, logics.  Multivalent logics can have arbitrarily many "truth" values, even an uncountable infinity (notably, in fuzzy logic).  

Rough sets are multivalent because their boundaries are not crisply defined.

And thus my puzzlement. "Multivalent" in logic generally refers to the number of values that a salient (typically, semantic) function can take.  To call a *set* multivalent seems like a category mistake.  I suppose, however, that what you might have in mind is that, in rough set theory, as in fuzzy set theory, the characteristic function for a set (i.e., the function that says whether a given object is a member of the set) can take on many values (representing the degree, or probability, of membership) rather than just "in" or "out".

Actually, that doesn't seem to be what you have in mind:

They can be in the set, out of the set, or in the bracketing upper and lower boundaries.

I don't believe that rather coarse-grained fact is the sense in which multivalence is relevant to rough set theory.  Rather, again, as I understand things, it comes into the picture insofar as the rough membership function can in general take on infinitely many values, where the value of the function on a given object x relative to a given set S indicates the probability that x is a member of S (all relative to a given "indiscernibility" relation R on the universe that Pawlak uses to represent imperfect knowledge).


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