Possibilities of logically equivalent expressions

If A is logically equivalent to B, it is not necessary that #(A) = p(B). This technical note proves, however, that if A is in CNF, there exists some logically equivalent DNF, B, such that p(A) = #(B). It is well known that this admits anomalies, such as #(P/x ~P) > 0 and #(P v ~P) < 1. Further, two logically equivalent expressions can have different possibility values. For example, P v Q-~(P ^ ~Q), but for #(P) ¼ and p(Q)= 1 = 2, p(P v Q) = Max{/~(P), p(Q)} = ' ~, and p(,-~ (P/x ,-~Q)) = 1-3 Min{/~(P), 1-~(Q)} = ~. Consider the possibility value for a CNF wff. Proof. We shall proceed by induction. Suppose S has only one disjunct-that is, S = L1 v L2 v ... v Lk. Then this is also DNF with k conjuncts (each literal is a conjunct by itself). Define S' = S (not merely S'-S), so #(S) = p(S'). Now suppose the theorem is true for all wff's having m-1 or fewer disjuncts in CNF. Let S-D1/x D 2 A '-A D,,_ 1 /x Din, and let Dm = Lml v L,,2 v ... v Link, where k = k(m) (dropping dependence on m for notational convenience). From the induction hypothesis, let R = KI v K2 v ... v K. be a DNF equivalent of D1 ^ D2 ^ ... A D m_ 1 with the same possibility value, and note S = R ^ Din. Applying distributiv-ity, we obtain S' = C1 v C2 v ..