Compound conditionals as random quantities and Boolean algebras

TL;DR

This paper develops a method to assign conditional random quantities to compound conditionals, revealing that they form a Boolean algebra, thus linking three-valued conditionals with Boolean algebraic structures.

Contribution

It introduces a natural procedure to attach conditional random quantities to arbitrary compound conditionals and demonstrates their Boolean algebraic structure.

Findings

Compound conditionals can be explicitly associated with random quantities.

The set of compound conditionals forms a Boolean algebra.

This bridges three-valued conditionals with Boolean algebra frameworks.

Abstract

Conditionals play a key role in different areas of logic and probabilistic reasoning, and they have been studied and formalized from different angles. In this paper we focus on the de Finetti's notion of conditional as a three-valued object, with betting-based semantics, and its related approach as random quantity as mainly developed by two of the authors. Compound conditionals have been studied in the literature, but not in full generality. In this paper we provide a natural procedure to explicitly attach conditional random quantities to arbitrary compound conditionals that also allows us to compute their previsions. By studying the properties of these random quantities, we show that, in fact, the set of compound conditionals can be endowed with a Boolean algebraic structure. In doing so, we pave the way to build a bridge between the long standing tradition of three-valued conditionals and a more recent proposal of looking at conditionals as elements from suitable Boolean algebras.