Bayesian Inference On 1st Order Logic

February 21, 2021

David Chapman’s blog post titled Probability theory does not extend logic has stirred up some controversy. In it, Chapman argues that so-called Bayesian logic, as it currently understood, is limited to propositional logic (0th order logic), but cannot generalize to higher order logics (e.g. predicate logic a.k.a. 1st order logic), and thus cannot be a general foundation for inference from data under uncertainty.
Chapman provides a few counter-examples that supposedly demonstrate that doing Bayesian inference on statements in 1st order logic is incoherent. I think there is a lot of confusion surrounding this point because Chapman does not use proper probability notation. In the following article I show how Chapman’s examples can be properly written and made sense of using random variables. Hopefully this clarifies some things.
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