Conditional Random Quantities and Compounds of 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.

References in corpus (3)

• Quasi Conjunction, Quasi Disjunction, T-norms and T-conorms: Probabilistic Aspects
• Probabilistic entailment in the setting of coherence: The role of quasi conjunction and inclusion relation
• Reassessing Accuracy Rates of Median Decisions

Cited by in corpus (10)

• Probabilistic squares and hexagons of opposition under coherence
• Probabilistic inferences from conjoined to iterated conditionals
• Probabilities of conditionals and previsions of iterated conditionals
• Generalized Logical Operations among Conditional Events
• Compound conditionals, Fréchet-Hoeffding bounds, and Frank t-norms
• Algebraic aspects and coherence conditions for conjoined and disjoined conditionals
• Conjunction of Conditional Events and T-norms
• Iterated Conditionals and Characterization of P-entailment
• Interpreting connexive principles in coherence-based probability logic
• On Trivalent Logics, Compound Conditionals, and Probabilistic Deduction Theorems