Conditional Random Quantities and Compounds of Conditionals

TL;DR

This paper develops a coherence-based framework for finite conditional random quantities, introducing iterated conditionals, conjunctions, and disjunctions of conditionals, with bounds and logical properties analyzed.

Contribution

It introduces a novel representation for conditional random quantities and extends the concept of conjunction and disjunction of conditionals within a coherence framework.

Findings

Representation of conditional random quantities as three-valued entities.

Introduction of conjunction and disjunction operations for conditionals.

Establishment of bounds and logical properties for combined conditionals.

Abstract

In this paper we consider finite conditional random quantities and conditional previsions assessments in the setting of coherence. We use a suitable representation for conditional random quantities; in particular the indicator of a conditional event $E|H$ is looked at as a three-valued quantity with values 1, or 0, or $p$, where $p$ is the probability of $E|H$. We introduce a notion of iterated conditional random quantity of the form $(X|H)|K$ defined as a suitable conditional random quantity, which coincides with $X|HK$ when $H \subseteq K$. Based on a recent paper by S. Kaufmann, we introduce a notion of conjunction of two conditional events and then we analyze it in the setting of coherence. We give a representation of the conjoined conditional and we show that this new object is a conditional random quantity. We examine some cases of logical dependencies, by also showing that the conjunction may be a conditional event; moreover, we introduce the negation of the conjunction and by De Morgan's Law the operation of disjunction. Finally, we give the lower and upper bounds for the conjunction and the disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold.