Newsgroups: comp.ai.fuzzy
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From: kpm@netcom.com (Keith Morgan)
Subject: Re: Kosko
In-Reply-To: gerla@bridge.dia.unisa.it's message of 30 Jun 1995 13:02:23 GMT
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Date: Mon, 3 Jul 1995 13:36:05 GMT
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gerla@bridge.dia.unisa.it writes:
>
>Kosko's position is that probability is contained in fuzzy logic. On the
>basis of such a claim is the interpretation of the quantity 
>                     S(A,B)=M(A\intB)/M(A) 
>as the inclusion degree of the set A in B (M(X) denotes the number of
>elements of X, \int the intersection operator). Now, I think that such an
>interpretation is misleading in account of the following paradoxes.
>
>*** PARADOX 1 *** Let A, B and C sets and consider the multivalued
>translation of the implication
>  A contained in B  AND  B contained in C  INPLIES  A contained in C.
>We obtain an inequality like
>             S(A,B)*S(B,C) <= S(A,C)
>where * is a suitable interpretation of the connective AND in the lattice
>[0,1].  Now, assume that M(C)=n (n a very big number), a is an element not
>in C, A={a} and B=A\unionC.  Then it is immediate that S(A,B)=1,
>S(B,C)=n/(n+1) and S(A,C)=0.  So, we have the absurdity
>                 1*(n/(n+1))=0.
>In words, A is contained in B, B is almost completely contained in C but A
>is not contained in C.
>
>*** PARADOX 2 ***  Consider the multivalued translation of the implication
>        B contained in C  IMPLIES  B-C' contained in C-C'.
>We obtain
>              S(B,C) <= S(B-C',C-C'). 
>Set C=C'\union{c}, B=C\union{a}, a different
>from c and a and c not in C'. Then it is immediate that 
>S(B,C)=(n-1)/n and S(B-C',C-C')=1/2. 
>If n is very big, this contraddicts the above inequality. 
>

Interesting post. I hadn't thought of it before, but the subsethood
measure Kosko proposes is *much * weaker than the "traditional" fuzzy
set one (e.g. A < B iff ua(x) < ub(x) for all x). Kosko's measure is
nonzer whenever A *overlaps* B to any degree. It would be interesting
to see which classical set theorems could be meaningfully generalized
to use the new definition.Or, perhaps the definition itself would need
rework to yield meaningful results.

--
Keith Morgan | kpm@netcom.com
-- 
Keith Morgan | kpm@netcom.com
