# Good resources on the intersection of probability theory and logic from a foundations/philosophical perspective?

What are some good books, courses, or online resources for probability theory that highlights differences between classical, frequentist, Bayesian, epistemic etc.? I majored in philosophy and am now transitioning to Data Science. As an undergrad, I focused on logic, epistemology, philosophy of mind, and the philosophy of science. As I delve deeper into statistics, I think reading a book from the perspective of mathematical logic and the foundations of mathematics would help me understand stats and probability at a level that connects with my previous education. I may not have "mathematical maturity", but I have had to write rigorous logical proofs about very similar subjects. So I would say I have at least an intermediate level of philosophical maturity. Does anyone know of any books by mathematicians that focus on this topic in a logical, abstract, and conceptual way?

• I don't think you want a mathematical logic book. You want a straight philosophy book, albeit one written by someone familiar with modern probability theory. Commented Dec 11, 2017 at 0:34
• I am more interested into looking into propositional logic from a mathematical standpoint because mathematicians do not focus as much on the meaning of the words within a proposition. This is because they are more concerned with the relationships between symbols and not the logical validity of a sentence. Of course, once you get to higher levels, formal logic in both fields begins to look more similar, as mathematicians become more concerned with philosophy/semantics and philosophers become more concerned with mathematics/symbols. Commented Dec 11, 2017 at 2:33
• But none of that has anything to do with the epistemology of probability theory or its interpretation. Commented Dec 11, 2017 at 8:37
• Mathematicians not concerned with "the logical validity of a sentence". Eh? You mean they don't care about what validly follows from what?? Really??? Commented Dec 11, 2017 at 10:42
• "mathematicians do not focus as much on the meaning of the words within a proposition" Um, what? Commented Dec 11, 2017 at 19:20

I'm not too sure that a "a book from the perspective of mathematical logic and the foundations of mathematics" is going to give you what you want, if you want to know more about interpretations of probability or (rather differently) foundations of statistics. Indeed, I suspect you probably need to get clearer about what you are seeking by doing some introductory reading around. Perhaps useful places to start would be with some of the relevant articles at that rich resource, the Stanford Encyclopedia of Philosophy, e.g. on

Interpretations of Probability

Logic and Probability

Philosophy of Statistics

These all have numerous pointers to further literature.

It's by no means comprehensive with respect to probability theory, but the Schaum's outline of Logic, written by three professors of philosophy, has a section on probability theory.

Peter Smith's answer is exactly what I would recommend EXCEPT there is (at least) one book directly on the topic of probability logic: A Primer of Probability Logic, Earnest W. Adams. 1998 but being a primer that shouldn't be an issue; I haven't looked for a more recent book on this topic (yet--but now I will).