Conditional Probabilities (CP) (P|A/B) and the Deductive Closure Principle (DCP) (If S knows P and P-> Q, P knows Q) are tightly knit concepts which is helpful for considering probabilities in axiomatic theories, especially in science and religion. Suppose that we evaluate the truth of some proposition P. Given axioms, say, A_{1}…A_{n, }a person S believes P on the basis of the presumed axioms. In this case, with respect to conditional probabilities, P is probable—more probable than not, that is—given the axioms (and so S accepts P). Suppose, further, that P->Q (Q being a logical entailment of P). It is, then, appropriate to say that given the axioms, Q is more probable than not. I find that this is very significant for understanding many issues found in conceptual problems which arise in science. Here is an example from quantum physics:

- Jones, having reviewed the evidence for the Many Worlds Interpretation E
_{1}…E_{n}, believes that the Many Worlds Interpretation to Quantum Mechanics is true.

Jones, so far as we know, does not rely on any previous axioms in holding that the Many Worlds Interpretation is true. Indeed, Jones can be said to merely make his judgement on the evidence presented to him; however, this might become problematic when other factors come into consideration. For instance, suppose that the evidence in question was inconclusive, that is, didn’t amount to a proof and that, being persuaded by other considerations i.e., some other epistemological hypothesis, Jones decided that the epistemological hypothesis—since it was, let us say, inconsistent with the Many Worlds Interpretation—amounted to rejecting the Many Worlds Interpretation (since Jones had better grounds for thinking that the epistemological hypothesis was more probable than his evidence for the Many Worlds Interpretation. Jones has applied conditional probability in the following way: Suppose A is “Many Worlds Interpretation” and B “Epistemological Hypothesis”, it would mean that for Jones (P|A/B) = 0. Two inconsistent propositions have a 0 probability when taken conjunctively. (To see this consider the proposition “at some time T some object O exists and simultaneously does not exist”, a proposition with 0 probability since it is impossible—logically and metaphysically). So, since the evidence in favor or B is higher than A, he sees it as (P|A/B) < (P|not-A/B) (in the latter case, the probability amounts to 1. Inasmuch as Jones is not being *ad hoc* here (in denying A on the grounds that he does not like it (and so happens to accept B), Jones is perfectly rational in his conditional probability. Things are not much different when we consider the following case:

2. Sharon, a theist, believes that the Many Worlds Interpretation to Quantum Mechanics is false.

In a nutshell, Jones is pro-Many Worlds, and Sharon is against Many-Worlds. Let us apply CP and DCP here to see what is at work.

For Sharon, she believes that, where A is the proposition “that God exists” and B “the relevant evidence”, that (P|A/B) > .5. She, then, claims to *know *A (since, on conditional probability, B is more probable than not). But, since theism—some would argue (and I shall just presume it here)—that the Many Worlds Interpretation is logically inconsistent with A, it follows that A->Q (where Q is the proposition that “the Many Worlds Interpretation is false.” So, given the conditional probability of A on B, applying the DCP, Sharon reasonably believes that A and therefore that the Many Worlds Interpretation is false. The question now arises though: What is the conceptual difference between the case of Jones and the case of Sharon? My answer: nothing at all.

Whether it is a theological hypothesis or an epistemic one, it makes no difference regarding the probability of such statements. If S accepts P on the basis that it is more probable than not that P is true given Q, it does not matter if P is theological or epistemological. Even if one accepts P on the basis that they know some other proposition P* which implies P, if P* is a theological hypothesis or historical or scientific or whatever, it has no bearing on the truth of P itself. If one denies this, it can be for no other reason than *begging the question* (and I suspect that this goes on all the time). [1]

[1] The latter half of this post could have been “Issues in Philosophy of Mind/Religion, Metaphysics, Logic” et cetera. I just used the quantum mechanical example as an example of a possible area that my speculations could have been relevant.