Can Bayesian reasoning be simplified to classical logic Classical logic is a reasonably good way of modelling rational thinking but it has limitations, as illustrated by the lottery paradox, the preface paradox, and others. One way of overcoming these limitations is by introducing a probabilistic element into logic or even going all the way to Bayesian reasoning.
I was wondering whether there has been any attempt to bridge from one to the other? Maybe under some assumptions (e.g. very high probabilities) we can simplify Bayesian reasoning in a way that the end result is classical logic?
Note: In a way, I am looking for a logical equivalent to what happens in electromagnetism (and other fields, of course) where you can take the very general Maxwell equations, simplify them for the specific case of electrical circuits, and end up with Kirchoff equations, which are not as general but are much easier to compute in the context (i.e. circuits) that grants certain assumptions. 
 A: It is a misconception that classical logic has known flaws or limitations when dealing with any so-called 'paradoxes'. The error is not at all in classical logic but in the incorrect application of it.

In particular, classical logic yields the inescapable conclusion that one cannot make nonsensical claims like "If P has at least 99% likelihood then P is true.". Also, a proper grasp of classical logic would immediately tell you where the error lies in the invalid reasoning employed in the two 'paradoxes' you cited. I shall explicitly show the mistake now. Consider the following:

(1) For every statement P in the book, the author has high confidence that P is true.
(2) The author has high confidence that, for every statement P in the book, P is true.

It is obvious that (1) and (2) are not equivalent, and that claiming their equivalence is akin to an invalid quantifier switch. Why do I use "has high confidence" in place of "believes"? Because it is indeed the case that the choice of the ambiguous word "believes" hides the fact that it is based on some threshold of confidence. Therefore the 'paradox' is just nonsense for the ordinary meaning of "believe".
If the author only believes things that he/she is 100% confident about, then (1) and (2) are actually equivalent, but this also destroys the 'paradox'; if one of the statements in the book is false, then in the first place he/she had no valid reason to believe (have 100% confidence) that it is true, unless he/she is irrational (from the viewpoint of classical logic).

Furthermore, all of (modern) probability theory can be built in ZFC set theory, which is the current conventional foundations and also a classical set theory.

That said, one can consider extensions of classical logic with generalized quantifiers. In some parts of complexity theory there have also been some attempts to simplify classifications and arguments involving atypical quantifiers such as this paper regarding "most". Still, these could be considered classical, since they are easily handled in classical set theory. The advantage of these generalized quantifiers is in deductive facility and intuitiveness and replacement of set-theoretical notions.
