From Propositional Logic to Plausible Reasoning: A Uniqueness Theorem
This provides a foundational result for AI and logic, offering a rigorous basis for plausible reasoning, though it is incremental relative to Cox's Theorem.
The paper tackles the problem of extending propositional logic to plausible reasoning by proposing four simpler requirements than Cox's Theorem, and proves that any such extension must be isomorphic to finite-set probability theory, recovering the classical definition of probability as a theorem.
We consider the question of extending propositional logic to a logic of plausible reasoning, and posit four requirements that any such extension should satisfy. Each is a requirement that some property of classical propositional logic be preserved in the extended logic; as such, the requirements are simpler and less problematic than those used in Cox's Theorem and its variants. As with Cox's Theorem, our requirements imply that the extended logic must be isomorphic to (finite-set) probability theory. We also obtain specific numerical values for the probabilities, recovering the classical definition of probability as a theorem, with truth assignments that satisfy the premise playing the role of the "possible cases."