PAC Learning Theory: How Many Examples Suffice for Knowledge?
Computational learning theory transforms the problem of induction into precise, solvable sample complexity bounds
Kolmogorov Complexity and Induction: A Computational Theory of Simplicity
How algorithmic information theory turns Occam's razor into a provable theorem about optimal prediction
Cromwell's Rule: Why Certainty Is Epistemically Forbidden
The mathematics of why rational agents must never assign probability zero or one to contingent claims
The Maximum Entropy Principle: Objective Bayesianism's Best Hope?
Can information theory rescue Bayesian epistemology from the arbitrariness of priors?
Epistemic Logic and Common Knowledge: What Formal Systems Reveal About Mutual Belief
The infinite hierarchy from shared belief to common knowledge, and why that gap determines what rational coordination can achieve
The Dutch Book Argument Reconsidered: Does Coherence Really Imply Rationality?
The classic betting argument proves less about rational belief than Bayesians typically assume
The Preface Paradox and Rational Inconsistency
When believing each claim rationally entails believing you're wrong somewhere—formal epistemology confronts irreducible inconsistency.
The Problem of Old Evidence: Why Bayesianism Struggles With Scientific History
Bayesian confirmation theory's elegant mathematics cannot explain why old facts confirm new theories—a problem still unresolved.
Truthlikeness and Verisimilitude: Measuring Distance from Truth
Why Popper's elegant definition of verisimilitude collapsed and how possible worlds rescued the concept of approximate truth
Defeasible Reasoning: Non-Monotonic Logic for Uncertain Inference
Why rational inference requires conclusions that can be withdrawn
Carnap's Inductive Logic: The Dream of Algorithmic Rationality
Why Carnap's quest to make inductive reasoning purely logical revealed the irreducible role of judgment in rational inference.
The Ramsey Test: Conditionals Through Belief Revision
How Ramsey's elegant test for conditionals encountered impossibility theorems and forced formal epistemologists to choose which intuitions to preserve
Higher-Order Probability: Can You Have Beliefs About Your Own Uncertainty?
The formal epistemology of uncertainty about uncertainty—when and why rational agents doubt their own beliefs.
The Lottery Paradox: Why Knowledge Isn't Just High Probability
Kyburg's paradox proves that no probability threshold can ground rational belief without violating basic logical closure—formal epistemology must choose its casualties.
Dempster-Shafer Theory: When Probability Isn't Enough for Uncertainty
Beyond probability distributions lies a formal framework for genuine ignorance—when your evidence supports possibilities without choosing between them.
AGM Theory: The Logic of Belief Revision
How rationality postulates constrain the logic of changing your mind—and why probability alone cannot capture belief revision.
Imprecise Probabilities: Embracing Indeterminacy in Rational Belief
When evidence fails to determine exact probabilities, credal sets formalize rational humility about belief—with surprising consequences for learning and decision.
Why Probability Can't Escape Philosophy: The Hidden Assumptions Behind Every Bayesian Model
Every Bayesian calculation inherits philosophical commitments about probability's meaning, prior selection, and rationality that mathematics presupposes but cannot prove.
Conditionalization and Its Rivals: When Should You Update Your Beliefs?
Master the formal criteria for choosing between Bayesian conditionalization, Jeffrey updating, and imaging based on your evidential context.
The Conjunction Fallacy Revisited: What Formal Epistemology Says About Human Reasoning
The famous proof of human irrationality may reveal more about experimental interpretation than cognitive limitation.
Scoring Rules and Epistemic Utility: A Mathematical Theory of Accuracy
How mathematical scoring rules reveal that probabilistic coherence is the only path to guaranteed accuracy—no betting required.