On 01-13-98 William Elliot wrote to John Boone... 
 
        Hello William and thanks for writing, 
  
 WE> The notions of fuzzy set A and B, A or B can be used for 
 WE> multi valued propositional calculus.  To wit:  p,q the true  
 WE> values of P,Q, 0<=p,q<=1.  Truth value of not P is 1-p, of  
 WE> P & Q is min(p,q), of P or Q is max(p,q).  Note that P or Q  
 WE> equivalent not(notP & notQ) and P & Q equivalent not(notP  
 WE> or notQ).  So does it fly?  Where to?  Can we take P  
 WE> implies Q, P -> Q, to be not(P & notQ) or equivalently notP  
 WE> or Q? 
   
 WE> First off lets look to the finite case.  The truth values T =  
 WE> {0.1/n,2/n,...,(n-1)/n,1}.  Well this is just too much for  
 WE> my mind so I will chose n = 3 and denote three values  
 WE> false, maybe, true, f,m,t for 0,1/2,1.  Does it fly?  Where  
 WE> do we want to fly with it? 
 >  
 WE> P Q notP P&Q PorQ P->Q Q->P P=Q  
 WE> f f   t   f   f    t    t    t 
 WE> f m       f   m    t    m    m 
 WE> f t       f   t    t    f    f 
 WE> m f   m   f   m    m    t    m 
 WE> m m       m   m    m    m    m 
 WE> m t       m   t    t    m    m 
 WE> t f   f   f   t    f    t    f 
 WE> t m       m   t    m    t    m 
 WE> t t       t   t    t    t    t 
 WE> Most of this seems to hang except for the last column P->Q  
 WE> & Q->P, P equivalent Q.  There's a problem here that P and  
 WE> Q can have the same truth value, 'maybe', without P=Q being  
 WE> 'true'.  The problem is that two statements aren't  
 WE> equivalent just because they always have the same truth  
 WE> values, they're 'maybe' equivalent.  Guess I've  
 WE> demonstrated my earlier claim that no workable multi value  
 WE> propositional calculus can been devised. 
 
  I don't have time to respond (as I need to finish a book 
I have been reading by Schering, for work which is more 
pressing at the present time), but I am not ignoring this; 
so, I printed it out to ponder when I have time. 
  I did have a chance of "look at" your other response 
I printed, and yes it is interesting. 
 
Take care, 
John 
 
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