I have no idea what you mean when you say that MIN (I presume you mean MAX,
since MIN is the aAND operator) "is quite crude". Also, any goo book on fuzzy
theory will tell you that the "bounded sum" OR is min(a+b, 1). It is only this
operator which will produce the desired result in the example I gave, for good
theoretical reasons.
>
>>Actually, the problem is not inherent in "any multivalued logic system. The
>>principle here is that IF we want to preserve excluded middle and
>>>contradiction, and IF we have semantically inconsistent operands, all we
>>have to do is switch to bounded sum for OR and bounded difference for
>>AND and we are home free. In the example given just above, using the
>>bounded sum OR makes NOT(Small OR Medium) precisely equal to Large.
>Well, that's a very common technique in fuzzy logic..
Actually, it's not at all common. Most fuzzy people don't give a damn about
preserving excluded middle and contradiction.
>> but you're still violating the law of the excluded middle - you already did
>>that when you defined triangular membership functions. Fuzzy logic is by
>>definition a violation of the law of the excluded middle, as it permits
values to
>>belong to both set A and not(set A).
I don't know where you got your definition, but the definitions of excluded
middle for multivalued logics are A AND NOT A = 0, and A OR NOTA = 1. The
failure of max-min fuzzy logic to obey these laws is why conventional fuzzy
logic does not obey excluded middle and contradiction.
If the bounded sum and bounded difference operators are used when both A and
NOT A appear in the same proposition, then these laws are obeyed. If A dnd NOT
A do not appear in the same proposition, then excluded middle and contradiction
don't arise, and you can use any multivalued logic you please for whatever
reason.
Maurits, I appreciate you interest and willingness to engage in a discussion of
what I consider to be important points. However, be careful about dogmatic
statements; the theory here is not all that simple, and there are a whole bunch
of misconceptions floating around that are (unfortunately) accepted as gospel
truth by many. Jim Buckley and I have a paper coming out in Fuzzy Sets and
Systems which deals in detail with a family of multivalued logics which do obey
both excluded middle and contradiction, but the paper is pretty technical. It
should be out very soon now.
William Siler
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