The Principles of Probability: From Formal Logic to Measure Theory to the Principle of Indifference
Jason Swanson
TL;DR
This work develops inductive logic as a rigorous alternative to standard probability theory, using an infinitary language and nine inductive-inference rules to generate a complete calculus with a probabilistic semantics. It shows how modern measure-theoretic probability embeds into inductive logic through models that assign probabilities to structures, while also enabling probabilistic notions—most notably, the principle of indifference—to be formulated rigorously within this framework. The framework connects syntax (derivability) and semantics (consequence) via σ-compactness and Karp-style completeness, and demonstrates that Dynkin systems arise naturally in inductive inference, clarifying fundamental aspects of probability beyond measure spaces alone. The approach yields a broad, structured theory of probability as logic, with potential applications across mathematics, philosophy of science, physics, and AI, and provides tools to analyze independence, conditional reasoning, and the role of symmetry in probabilistic statements.
Abstract
In this work, we develop a formal system of inductive logic. It uses an infinitary language that allows for countable conjunctions and disjunctions. It is based on a set of nine syntactic rules of inductive inference, and contains classical first-order logic as a special case. We also provide natural, probabilistic semantics, and prove both $σ$-compactness and completeness. We show that the whole of measure-theoretic probability theory is embedded in this system of inductive logic. The semantic models of inductive logic are probability measures on sets of structures. (Structures are the semantic models of finitary, deductive logic.) Moreover, any probability space, together with a set of its random variables, can be mapped to such a model in a way that gives each outcome, event, and random variable a logical interpretation. This embedding, however, is proper. There are scenarios that are expressible in this system of logic which cannot be formulated in a measure-theoretic probability model. The principle of indifference is an idea originating with Laplace. It says, roughly, that if we are "equally ignorant" about two possibilities, then we should assign them the same probability. The principle of indifference has no rigorous formulation in probability. It exists only as a heuristic. Moreover, its use has a problematic history and is prone to apparent paradoxes. Within inductive logic, however, we formulate it rigorously and illustrate its use through a number of examples. Many of the ideas in inductive logic have counterparts in measure theory. The principle of indifference, however, does not. Its formulation requires the structure of inductive logic, both its syntactic structure and the semantic structures embedded in its models. As such, it exemplifies the fact that inductive logic is a strictly broader theory of probability than any that is based on measure theory alone.