📝 SummarXiv

Abstract

There has not been an established mathematical measure of evidence. Some Bayesians have argued that probability can be an objectively correct measure of ``rational degrees of belief,'' which we do not distinguish from evidence. However, support for the objectivist view has been limited due to the lack of a general method for assigning probabilities to evidence (belief) de novo. The standard axioms of Kolmogorov and Cox solve only the calculation problem, specifying how probabilities can be calculated from other probabilities. They do not solve the measurement problem of how to determine the uniquely correct value of P(A) given only A. The prototypical solution has always been Laplace's principle of indifference, which assigns equal (uniform) probabilities to all possibilities. However, uniformity is well known to be inconsistent with the standard axioms when there are infinite possibilities. Here we introduce new axioms that resolve this inconsistency. We first show that all measures must ultimately be based on uniformity, and that uniformity is inevitable if all propositions are adequately defined. We then reconcile uniformity with infinite possibilities by using hyperrational numbers, so that an infinite sum of infinitesimal probabilities can equal one. Our axioms thereby provide a general and relatively simple solution to the measurement problem. We discuss a variety of conceptual obstacles that have made this solution difficult to recognize.