Given that A1, ... An are completely incompatible hypotheses, from the fact that the conditional probability of A1 given evidence E is greater than the conditional probability of any other Ai given E, we can infer that:

A.
The argument A1; therefore E is inductively valid.

B.
Given the evidence E, A1 is the most cogent out of the set of rival hypotheses, A1,...,An.

C.
The argument E; therefore A1 is inductively valid.

D.
The belief that E is strongly inductively supported by the evidence that A1, and critics of that belief should just shut up!

E.
The argument Not-A1, therefore, Not-E is inductively valid.

The correct answer is B. Given the information provided, we can infer that A1 is the most cogent (plausible or reasonable) hypothesis out of the set of rival hypotheses A1, ..., An, when considering the evidence E.

To understand why this is the case, we need to consider the concept of conditional probability. Conditional probability measures the likelihood of an event A given that event B has occurred. In this scenario, we are comparing the conditional probabilities of each hypothesis (A1, ..., An) given the evidence E.

The fact that the conditional probability of A1 given E is greater than the conditional probability of any other Ai given E signifies that A1, when considering the evidence E, is more likely or reasonable than the other hypotheses.

Therefore, we can conclude that A1 is the most cogent hypothesis among A1, ..., An in light of the evidence E based on the given information.