It's probable logic.
Definition 1: The probability of B given A is the probability that A -> B given A occurs. If A occurs, then if B lies in the space, A->B, but the probability of this happening is obviously contingent on the probability of A itself, by the usual modus ponens argument.
This definition says the probability that B happens given A happens is greater than the probability that B happens without A. This is why i use implication in our notation, since A -> B means if A then B, but B is certainly plausible without A, but not B given A is impossible.
Definition 2,3: a is A's negation, b is B's negation.
The rest of the argument is thus:
The probability of B happening given A is equivalent to its set-theoretic complement (since F is a sigma-algebra, each case study in F on the probability space Omega is subject to closure under complements, arbitrary unions and intersections). Of course, this is equal to saying the complement of B happening given A is equal to the probability that B does not happen given A being less likely than B not happening given A not happening
This is because of our implication defined above. Then whoever wrote this proceeds to play with negations using DeMorgan's laws for negation and substitute the unnecessary definition 2 and 3.
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Summary: Whoever wrote this shows an incredibly naive and incomplete understanding of what probabilities actually mean, and they should go die.
