Boolean algebras of conditionals, probability and logic

TL;DR

This paper explores the structure of conditional events within Boolean algebras, linking probability measures and logic, and introduces a construction that models conditional probabilities as one-place functions.

Contribution

It introduces a finite Boolean algebra of conditionals derived from any Boolean algebra of events, distinguishing probabilistic and logical properties of conditionals.

Findings

Standard two-place conditional probabilities can be viewed as one-place probability functions.

A logical counterpart of Boolean algebras of conditionals is developed, connecting to non-monotonic reasoning.

The framework offers a new perspective on the interplay between logic and probability in conditional knowledge.

Abstract

This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in this context. In particular we introduce a construction which defines a (finite) {\em Boolean algebra of conditionals} from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on conditional events. We also consider a logical counterpart of our Boolean algebras of conditionals with links to preferential consequence relations for non-monotonic reasoning. The overall framework of this paper provides a novel perspective on the rich interplay between logic and probability in the representation of conditional knowledge.