Propositional Measure Logic

TL;DR

This paper introduces a probabilistic propositional logic where formulas are assigned real measures representing degrees of truth, enabling reasoning under uncertainty and applied to Bayesian Networks.

Contribution

It proposes a novel probabilistic semantics for propositional logic that maintains deductive structure while handling uncertainty, with a soundness proof and applications.

Findings

The logic assigns real measures to formulas, representing degrees of truth.

The system is proven sound for reasoning under uncertainty.

Applied to Bayesian Networks, demonstrating practical relevance.

Abstract

We present a propositional logic with fundamental probabilistic semantics, in which each formula is given a real measure in the interval $[0,1]$ that represents its degree of truth. This semantics replaces the binarity of classical logic, while preserving its deductive structure. We demonstrate the soundness theorem, establishing that the proposed system is sound and suitable for reasoning under uncertainty. We discuss potential applications and avenues for future extensions of the theory. We apply probabilistic logic to a still refractory problem in Bayesian Networks.