Studia Logica

, Volume 102, Issue 4, pp 709–729

# Conditional Random Quantities and Compounds of Conditionals

• Angelo Gilio
• Giuseppe Sanfilippo
Article

## Abstract

In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as $${XH + \mathbb{P}(X|H)H^c}$$ ; in particular (the indicator of) a conditional event E|H is looked at as EHP(E|H)H c . This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give a meaning to the sum X|H + Y|K and we can define the iterated c.r.q. (X|H)|K. We analyze the conjunction of two conditional events, introduced by the authors in a recent work, in the setting of coherence. We show that the conjoined conditional is a conditional random quantity, which may be a conditional event when there are logical dependencies. Moreover, we introduce the negation of the conjunction and by applying De Morgan’s Law we obtain the disjoined conditional. Finally, we give the lower and upper bounds for the conjunction and disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold.

## Keywords

Conditional events Conditional random quantities Coherence Iterated conditioning Import–Export principle Conjunction Disjunction Lower/upper prevision bounds

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