Nuances of Probability Theory - Minka, Tom (1998)
The probability statement P(A | B) = p has a very different meaning from the logical statement "B implies A with certainty p". The logical statement means that whenever B is true then A is true with certainty p. This applies regardless of any other information we may have. In other words, it is modular. But the probability statement is not modular: it applies when the only thing we know is B. If anything else is known, e.g. C, than we must refer to P(A | B, C) instead. The only exception is when we can prove that C is conditionally independent of A given B, so that P(A | B, C) = P(A | B).