**Appendix B**

Proofs

**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 ...

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