Algorithmic Randomness, Exchangeability, and the Principal Principle

Abstract

We introduce a framework uniting algorithmic randomness with exchangeable credences to address foundational questions in philosophy of probability and philosophy of science. To demonstrate its power, we show how one might use the framework to derive the Principal Principle -- the norm that rational credence should match known objective chance -- without circularity. The derivation brings together de Finetti's exchangeability, Martin-L\"of randomness, Lewis's and Skyrms's chance-credence norms, and statistical constraining laws (arXiv:2303.01411). Laws that constrain histories to algorithmically random sequences naturally pair with exchangeable credences encoding inductive symmetries. Using the de Finetti representation theorem, we show that this pairing directly entails the Principal Principle of this framework. We extend the proof to partial exchangeability and provide finite-history bounds that vanish in the infinite limit. The Principal Principle thus emerges as a mathematical consequence of the alignment between nomological constraints and inductive learning. This reveals how algorithmic randomness and exchangeability can illuminate foundational questions about chance, frequency, and rational belief.