Unified Inductive Logic: From Formal Learning to Statistical Inference to Supervised Learning

TL;DR

The paper proposes a Peircean, unified inductive logic that connects formal learning theory, statistics, and a substantial portion of supervised learning under the principle 'Strive for the Highest Achievable.' It formalizes empirical problems as quadruples $(\mathsf{H}, \mathsf{E}, \mathsf{W}, \mathsf{Loss})$ and analyzes three convergence modes: (i) nonstochastic identification, (ii) stochastic identification, and (iii) stochastic approximation, showing how these standards unify evaluation across disciplines. Through paradigmatic examples—easy raven, fair coin, coin bias, and binary classification—it shows how these modes are achievable or not and argues that higher standards logically imply lower ones via a uniform framework. The discussion situates this unification within the history of Peirce and discusses connections to Bayesian approaches, outlining potential extensions to reinforcement learning while acknowledging challenges for unsupervised learning. The result is a conceptual and mathematical framework that clarifies inductive logic across disciplines and suggests a common basis for evaluating non-deductive inference methods.

Abstract

While the traditional conception of inductive logic is Carnapian, I develop a Peircean alternative and use it to unify formal learning theory, statistics, and a significant part of machine learning: supervised learning. Some crucial standards for evaluating non-deductive inferences have been assumed separately in those areas, but can actually be justified by a unifying principle.