Algebraic aspects and coherence conditions for conjoined and disjoined conditionals

TL;DR

This paper explores the algebraic structure and coherence conditions of conjoined and disjoined conditional events within a framework of conditional random quantities, extending classical properties and providing new coherence criteria.

Contribution

It introduces a novel algebraic approach to conditional events, establishing properties like decomposition, inclusion-exclusion, and coherence conditions under independence.

Findings

Proves a decomposition formula for conditional events.

Derives a generalized inclusion-exclusion formula.

Provides necessary and sufficient coherence conditions under independence.

Abstract

We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula and we prove a suitable distributivity property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions. The first condition gives a geometrical characterization for the coherence of prevision assessments on a family $\mathscr{F}$ constituted by $n$ conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $\mathscr{F}\cup \mathscr{K}$, where $\mathscr{K}$ is the set of conditional constituents associated with the conditional events in $\mathscr{F}$. Then, we give a further theoretical result and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications.