[spacehacker]

I've been surprised to not find anything for ctrl-F "Bayes". It turns out that traditional deductive logic is simply a special case of applying Bayes rule in order to find out the state of some binary variables given the other variables in which the probabilities are only 0 or 1. See for example these examples from Barber's "Bayesian Reasoning and Machine Learning" (p. 39):

https://i.imgur.com/W1b0Wem.png

https://i.imgur.com/JOFvd9C.png

[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.

[GregBuchholz]

I think you might enjoy E.T. Jaynes

http://bayes.wustl.edu/

https://www.amazon.com/Probability-Theory-E-T-Jaynes/dp/0521...