The Force of Proof by Which Any Argument Prevails

TL;DR

This paper introduces a new axiomatic framework for quantifying the uncertainty inherent in the inference process itself, extending probability theory to better model arguments and their strength.

Contribution

It develops axioms analogous to Kolmogorov's for uncertainty in inference, generalizes to arguments spanning Boolean algebras, and connects to evidence theory, advancing AI applications.

Findings

Axioms specify uncertainty in inference rather than premises.

Framework generalizes probability to arguments between Boolean algebras.

Application expands Shafer's evidence theory for broader AI use.

Abstract

Jakob Bernoulli, working in the late 17th century, identified a gap in contemporary probability theory. He cautioned that it was inadequate to specify force of proof (probability of provability) for some kinds of uncertain arguments. After 300 years, this gap remains in present-day probability theory. We present axioms analogous to Kolmogorov's axioms for probability, specifying uncertainty that lies in an argument's inference/implication itself rather than in its premise and conclusion. The axioms focus on arguments spanning two Boolean algebras, but generalize the obligatory: "force of proof of A implies B is the probability of B or not A" in the case that the Boolean algebras are identical. We propose a categorical framework that relies on generalized probabilities (objects) to express uncertainty in premises, to mix with arguments (morphisms) to express uncertainty embedded directly in inference/implication. There is a direct application to Shafer's evidence theory (Dempster-Shafer theory), greatly expanding its scope for applications. Therefore, we can offer this framework not only as an optimal solution to a difficult historical puzzle, but also to advance the frontiers of contemporary artificial intelligence. Keywords: force of proof, probability of provability, Ars Conjectandi, non additive probabilities, evidence theory.