On 01-06-98 William Elliot wrote to John Boone... 
 
        Hello William and thanks for writing, 
  
 WE>  JB> I'll give you -his-, Bart Kosko's, words. 
 WE>  JB> At first I worked with symbols on abstract math theorums."  
 
 WE> So far this is gab, where's the math? 
   
  "Fuzzy Logic", he wrote wasn't a "math" book; as, he said  
there were other books that dealt with the math. 
  
 WE>  JB> In Fuzzy logic set theroy, a "thing" may belong to a set 
 WE>  JB> (A) -AND- (not A) at the same time giving truth values  
 WE>  JB> between and inclusive of zero and one.   
 
 WE> Gab, gab, gab.  What's the definitions?  What I understand is that in 
 
   I am unsure. 
  
 WE> fuzzy 
 WE> set theory an element x belongs to a set A with degree d, 0  
 WE> <= d <= 1, x member A (d).  For example, A = {rocks in  
 
  This is my understanding of it. 
  
 WE> river}, r a rock. r member A (1) if r is completely 
 WE> submerged.  r member A (1/2) if r is half submerged, r  
 WE> member A (1/3) if r is 1/3 submerged, r member A (0) if r  
 WE> is completely out of the river. 
  
 WE> So a half submerged rock is in A with degree 1/2 and in not  
 WE> A with degree 1/2, while a fully submerged rock is in A  
 WE> with degree 1 and in not A with degree 0.  In general x  
 WE> member notA (d) when x member A (1-d), the definition of  
 WE> not A. 
   
 WE> Now probability is different.  An infinite multi valued  
 WE> propositional calculus that is successful is probability  
 WE> theory.  If two independent events, P and Q have  
 WE> probability p and q, then probability P and Q is pq, not P  
 WE> is 1 - p, and P or Q is, of course, from PorQ equivalent  
 WE> not(notP and notQ) is 1 - (1-p)(1-q) = p + q - pq.  This  
 WE> method however is not applicable for a finite valued  
 WE> propositional calculus.   
   
 WE> Of interest is P implies Q, not(P and notQ) witch has  
 WE> probability 1 - p(1-q) = 1 - p + pq.  So if p = 1 then  
 WE> probability of P implies Q is q and if p = 0 it's 1.  Which  
 WE> is to be expected.  Now if p = 1/2, it's 1/2 + q/2.  Brain  
 WE> teaser, is it not? 
 
  Yes, it is, I am going to print it out to ponder it a bit. 
 
Take care, 
John 
 
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