The notions of fuzzy set A and B, A or B can be used for multi valued 
propositional calculus.  To wit:  p,q the true values of P,Q, 0<=p,q<=1.  
Truth value of not P is 1-p, of P & Q is min(p,q), of P or Q is max(p,q).  
Note that P or Q equivalent not(notP & notQ) and P & Q equivalent not(notP or 
notQ).  So does it fly?  Where to?  Can we take P implies Q, P -> Q, to be 
not(P & notQ) or equivalently notP or Q?
First off lets look to the finite case.  The truth values T = 
{0.1/n,2/n,...,(n-1)/n,1}.  Well this is just too much for my mind so I will 
chose n = 3 and denote three values false, maybe, true, f,m,t for 0,1/2,1.  
Does it fly?  Where do we want to fly with it?
P Q notP P&Q PorQ P->Q Q->P P=Q 
f f   t   f   f    t    t    t
f m       f   m    t    m    m
f t       f   t    t    f    f
m f   m   f   m    m    t    m
m m       m   m    m    m    m
m t       m   t    t    m    m
t f   f   f   t    f    t    f
t m       m   t    m    t    m
t t       t   t    t    t    t
Most of this seems to hang except for the last column P->Q & Q->P, P 
equivalent Q.  There's a problem here that P and Q can have the same truth 
value, 'maybe', without P=Q being 'true'.  The problem is that two statements 
aren't equivalent just because they always have the same truth values, 
they're 'maybe' equivalent.  Guess I've demonstrated my earlier claim that no 
workable multi value propositional calculus can been devised.
Got any patches for this?  Exclude the middle m? -)
... from the marsh, the world's freshest source of stagnate water.
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