# From Propositional Logic to Plausible Reasoning: A Uniqueness Theorem

## TL;DR

This paper demonstrates that extending propositional logic to a plausible reasoning framework under certain natural requirements necessarily leads to probability theory, providing a foundational justification for classical probability as a unique extension.

## Contribution

It establishes a set of simple, natural conditions that uniquely characterize probability theory as the extension of propositional logic for plausible reasoning.

## Key findings

- Extended logic is isomorphic to finite-set probability theory
- Classical probability emerges as a unique solution under the specified conditions
- Probabilities are derived as a theorem from logical premises

## Abstract

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."