Plausibility and probability in deductive reasoning

TL;DR

This paper develops a normative Bayesian-inspired model for rational uncertainty in deductive reasoning, integrating probability, algorithms, and computational resource limitations to understand unproven mathematical statements.

Contribution

It introduces a novel model combining Bayesian probability and algorithmic theory to address deductive uncertainty and explores its implications for semi-rigorous proofs and arbitrage.

Findings

The model captures subjective uncertainty in unproven statements.

Connections between deductive reasoning and semi-rigorous proofs are discussed.

Implications for financial models with limited computational resources are analyzed.

Abstract

We consider the problem of rational uncertainty about unproven mathematical statements, remarked on by G\"odel and others. Using Bayesian-inspired arguments we build a normative model of fair bets under deductive uncertainty which draws from both probability and the theory of algorithms. We comment on connections to Zeilberger's notion of "semi-rigorous proofs", particularly that inherent subjectivity would be present. We also discuss a financial view with models of arbitrage where traders have limited computational resources.