# How much does Mathematical Logic relate to Statistics?

1. To wit, to what extent did statistics professors study Mathematical Logic (“ML” henceforth)?

2. To what extent does Statistics use ML? How relevant is ML?

I mean Mathematical Logic in books like Hils and Loeser’s A First Journey Through Logic

The book starts with a presentation of naive set theory, the theory of sets that mathematicians use on a daily basis. Each subsequent chapter presents one of the main areas of mathematical logic: first order logic and formal proofs, model theory, recursion theory, Gödel’s incompleteness theorem, and, finally, the axiomatic set theory.

Chiswell and Hodges’s Mathematical Logic

Alongside the practical examples, readers learn what can and can’t be calculated; for example the correctness of a derivation proving a given sequent can be tested mechanically, but
there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich’s theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and
completeness proofs of the relevant logics are provided, with careful attention to the languages involved. Optional sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch.

and Walicki’s Introduction to Mathematical Logic

Starting with the basics of set theory, induction and computability, it covers propositional and first order logic — their syntax, reasoning systems and semantics. Soundness and completeness results for Hilbert’s and Gentzen’s systems are presented, along with simple decidability arguments. The general applicability of various concepts and techniques is demonstrated by highlighting their consistent reuse in different contexts.

You can derive Bayesian statistics from mathematical logic. See its axiomatization in logic at

Cox, R. T. (1961). The Algebra of Probable Inference. Baltimore, MD: Johns Hopkins University Press.

I cannot answer how many statistics professors are required to study logic at a deep level.

You can find its extension at

Edwin Thompson Jaynes, Probability Theory: The Logic of Science, Cambridge University Press (2003).

You can also pick up anything on decision theory and you are back in mathematical logic combined with a utility function.