Propositional Measure Logic
Francisco Aragão
TL;DR
This work introduces Propositional Measure Logic (PML), a Hilbert-style extension of propositional logic where each formula is assigned a degree of truth in $[0,1]$ via a probabilistic semantics built on well-formed sequences of truth valuations and a weight-based measure. A key construct is the Logical Measure $\mu(\alpha)=\sum_{s\in J_{\alpha}} p(s)$, which couples justification sequences with a weight function, enabling formal notions of probabilistic entailment and soundness. The authors prove that classical tautologies have positive probability (probabilistic soundness) and discuss the non-equivalence of completeness between classical and probabilistic semantics, outlining rich avenues for future work including first-order generalization, algorithmic applications, and axiomatization. The framework aims to support robust reasoning under uncertainty and has potential applications in Bayesian nets and related AI reasoning systems, while maintaining deductive structure from classical logic. $\mu$, $k$, and sets like $J_{\alpha}$ and $A_{\alpha}$ provide a concrete, measure-theoretic foundation for logical truth under uncertainty.
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.