B.1. Probability Raising
Claim. The following conditions for probabilistic causality are equivalent in non-deterministic cases:
Proof. Assume that 1 > P(C) > 0 (and thus 1 > P(¬C) > 0). By definition:
Then, if P(E|¬C) > P(E|C), it must be that P(E|C) < P(E) to maintain the equality, and if P(E|C) > P(E|¬C), then by the same reason P(E|C) > P(E). Thus, if (B.2) is satisfied, (B.1) is satisfied. Conversely, if P(E|C) > P(E), then we must have P(E|C) > P(E|¬C). Thus, if (B.1) is satisfied (B.2) is satisfied and finally we conclude ...