A probabilistic analysis of selected notions of iterated conditioning under coherence

TL;DR

This paper analyzes different notions of iterated conditioning within coherence, showing that many basic probabilistic properties are not preserved, and introduces a new iterated conditional that satisfies key probabilistic and logical properties.

Contribution

It compares existing iterated conditioning notions in three-valued logics and proposes a new approach that maintains essential probabilistic properties.

Findings

Many existing iterated conditionals do not preserve the compound probability theorem.

A new iterated conditional is introduced that satisfies the compound prevision theorem.

The proposed iterated conditional aligns with recent developments by Gilio and Sanfilippo.

Abstract

It is well know that basic conditionals satisfy some desirable basic logical and probabilistic properties, such as the compound probability theorem, but checking the validity of these becomes trickier when we switch to compound and iterated conditionals. We consider de Finetti's notion of conditional as a three-valued object and as a conditional random quantity in the betting framework. We recall the notions of conjunction and disjunction among conditionals in selected trivalent logics. First, in the framework of specific three-valued logics we analyze the notions of iterated conditioning introduced by Cooper-Calabrese, de Finetti and Farrell, respectively. We show that the compound probability theorem and other basic properties are not preserved by these objects, by also computing some probability propagation rules. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem and some of the desirable properties. We also check the validity of two generalized versions of Bayes' Rule for iterated conditionals. We study the p-validity of generalized versions of Modus Ponens and two-premise centering for iterated conditionals. Finally, we observe that all the basic properties are satisfied only by the iterated conditional mainly developed in recent papers by Gilio and Sanfilippo in the setting of conditional random quantities.