On probability and logic

TL;DR

This paper explores the relationship between probability and classical propositional logic, introducing a new probabilistic entailment concept and a decidable extension of propositional logic with a semantics for constraining probabilities.

Contribution

It proposes a novel probabilistic entailment that aligns with classical logic and introduces a decidable enriched language for probabilistic constraints.

Findings

Probabilistic entailment collapses to classical entailment in unchanged languages.

A decidable conservative extension of propositional logic is developed.

A sound and weakly complete axiomatization is provided using real closed ordered fields.

Abstract

Within classical propositional logic, assigning probabilities to formulas is shown to be equivalent to assigning probabilities to valuations. A novel notion of probabilistic entailment enjoying desirable properties of logical consequence is proposed and shown to collapse into the classical entailment when the language is left unchanged. Motivated by this result, a decidable conservative enrichment of propositional logic is proposed by giving the appropriate semantics to a new language construct that allows the constraining of the probability of a formula. A sound and weakly complete axiomatization is provided using the decidability of the theory of real closed ordered fields.