Defending Bayes, Part 6
Credence and Logical Induction
In our previous exploration of credence, we distinguished empirical credence—credence that directly corresponds to objective probabilities or Measures within our Quantum Branching Universe framework—from conceptual or logical credence, which represents purely epistemic uncertainty. While empirical credence reflects measurable uncertainty about states of the physical world, conceptual or logical credence addresses uncertainty about truths that are not empirically determined.
This distinction raises a challenging question: how can credence meaningfully adhere to the laws of probability when there is no underlying objective probability—particularly in cases involving unresolved logical or mathematical statements? To address this, we turn to Logical Induction.
Logical Induction, introduced by Garrabrant, Benson-Tilsen, Critch, Soares, and Taylor, rigorously formalizes epistemic uncertainty about logical and mathematical statements. It systematically assigns probabilistic credences to logical propositions in a manner that ensures rational coherence and consistency. Logical Induction generates a sequence of credences that become progressively more accurate and precise as logical evidence—such as proofs, partial results, computational checks, or algorithmic verifications—accumulates over time. Crucially, these credences rigorously satisfy the axioms and laws of probability, even though they do not correspond to any empirical probability.
To better understand how Logical Induction accomplishes this, consider its operational analogy to market dynamics. Logical uncertainty is conceptualized as a "market," wherein hypothetical "traders" represent distinct algorithmic strategies or computational heuristics that "bet" on logical statements. Each trader makes predictions or wagers on whether particular logical propositions are true or false. As new information (proofs, computational outcomes, or heuristic evaluations) is revealed, these traders update their beliefs accordingly. This dynamic interaction gradually drives the "market" credences toward accurate logical beliefs. The analogy ensures internal coherence, rational updating, and robust epistemic management of uncertainty.
Through this approach, Logical Induction resolves a significant philosophical tension:
Credences must obey probabilistic laws to maintain rational consistency.
Logical credences, however, are epistemically grounded rather than empirically objective.
By employing a structured, market-like mechanism, Logical Induction illustrates how rational, probabilistic consistency is maintained without assuming objective probabilities. Thus, it provides a clear epistemic foundation for credences in logical or mathematical contexts—credences that can be systematically updated, are free from contradiction, and avoid irrational vulnerability such as Dutch-book scenarios.
In essence, Logical Induction allows us to rigorously embrace epistemic probabilities. It clarifies how and why rational, probabilistic credences can be meaningfully assigned even in contexts devoid of empirical probabilities, thereby significantly enriching our philosophical and practical understanding of credence itself.