Polyvalued vs Probabilistic Logic

Under construction Name Probabilistic notation Polyvalued notation the product rule P(AB|C) = P(A|C) * P(B|AC) (p=>(q&r)) (p=>q) * ((p&q)=>r) the sum rule P(A+B|C) = P(A|C) + P(B|C) + P(AB|C) (p=>(qvr)) (p=>q) + (p=>r) - (p=>(q&r)) Bayes' theorem P(A|BC) = P(A|C) * P(B|AC) / P(B|C) ((p&q)=>r) (p=>r) * ((p&r)=>q) / (p=>q) Proof of the product rule: (p=>(q&r)) (p=>q) * ((p&q)=>r)     |- |-     (p&q&r) / p ((p&q)/p) * ((p&q&r)/(p&q)) |-     (p&q&r)*p*(p&q) (p&q)*(p&q&r)*p     # Proof of the sum rule: (p=>(qvr)) (p=>q) + (p=>r) - (p=>(q&r))     |- |-     (p&(qvr))/p (p&q)/p + (p&r)/p - (p&q&r)/p |-     (p&(qvr))/p ((p&q) + (p&r) - (p&q&r))/p |-     ((p&q&r) + (p&q&~r) + (p&~q&r))/p ((p&q) + (p&r) - (p&q&r))/p |-     ((p&q&r) + (p&q&~r) + (p&~q&r))/p ((p&q&r) + (p&q&~r) + (p&r) - (p&q&r))/p |-     ((p&q&r) + (p&q&~r) + (p&~q&r))/p ((p&q&r) + (p&q&~r) + (p&q&r) + (p&~q&r) - (p&q&r))/p |-     ((p&q&r) + (p&q&~r) + (p&~q&r))/p ((p&q&r) + (p&q&~r) + (p&~q&r))/p     # Proof of Bayes' theorem: ((p&q)=>r) (p=>r) * ((p&r)=>q) / (p=>q)     |- |-     (p&q&r) / (p&q) ((p&r)/p) * ((p&q&r)/(p&r)) / ((p&q)/p) |-     (p&q&r)*p*(p&r)*(p&q) (p&r)*(p&q&r)*p*(p&q)     #