To the extent that there is a difference between Bayesians and Frequentists, it's a disagreement about interpretations, not math. It's not like Frequentists disagree with P(H|E) = P(E|H)*P(H)/P(E), or have sworn not to use it!

There are at least two meanings to the Bayesian/frequentist debate; one is a disagreement about methods (or at least a different set of tools), and the other is a disagreement about the deeper meaning of probability. This is an article about methods, not meaning. The major difference is that Bayesian methods make the prior explicit. The p-value is, perhaps, the quintessential frequentist statistic. Here, we can easily imagine Solomon publishing his paper in the Ancient Journal of Statistical Law and citing a p-value < 0.001 -- but without knowing the actual P(defendant), we don't know how many times he made the correct decision (in terms of the facts; as noted in another thread, from a child-welfare perspective, the decision was probably correct regardless).

What I would call the three main components of Bayes are explicitly considering hypotheses, explicitly searching for tests with high likelihood ratios, rather than just high likelihoods, and explicitly incorporating prior information. I'm content with examples that show off only one of those components.

Frequentists also care about high likelihood ratios.