[cscurmudgeon]

Deductive logic is sort of like the machine code for reasoning. It encompasses everything, but it is hard to see in daily usage, just like machine code. Bayes Rule is just one part of it.

If you look at the Mizar mathematical library you will find a formalization of Bayes Rule using plain FOL.

In fact, one of my Logic II (an intermediate logic class) exercises was formalizing a version of Bayes' rule in classic first-order logic.