Logic of Probability and Conjecture

Abstract

I introduce a formalization of probability which takes the concept of 'evidence' as primitive. In parallel to the intuitionistic conception of truth, in which 'proof' is primitive and an assertion A is judged to be true just in case there is a proof witnessing it, here 'evidence' is primitive and A is judged to be probable just in case there is evidence supporting it. I formalize this outlook by representing propositions as types in Martin-Lof type theory (MLTT) and defining a 'probability type' on top of the existing machinery of MLTT, whose inhabitants represent pieces of evidence in favor of a proposition. One upshot of this approach is the potential for a mathematical formalism which treats 'conjectures' as mathematical objects in their own right. Other intuitive properties of evidence occur as theorems in this formalism