Probabilities of conditionals and previsions of iterated conditionals

TL;DR

This paper examines iterated conditionals within a probabilistic framework, avoiding Lewis's triviality, and explores how different types of iterated conditionals relate to probability and latent information.

Contribution

It introduces a novel approach to analyzing iterated conditionals without triviality, formalizes various types of latent information, and investigates their probabilistic properties.

Findings

Prevision of iterated conditionals is ≥ probability of A

Bounds of Affirmation of the Consequent are analyzed

Independence of conditionals interpreted as uncorrelation

Abstract

We analyze selected iterated conditionals in the framework of conditional random quantities. We point out that it is instructive to examine Lewis's triviality result, which shows the conditions a conditional must satisfy for its probability to be the conditional probability. In our approach, however, we avoid triviality because the import-export principle is invalid. We then analyze an example of reasoning under partial knowledge where, given a conditional if $A$ then $C$ as information, the probability of $A$ should intuitively increase. We explain this intuition by making some implicit background information explicit. We consider several (generalized) iterated conditionals, which allow us to formalize different kinds of latent information. We verify that for these iterated conditionals the prevision is greater than or equal to the probability of $A$. We also investigate the lower and upper bounds of the Affirmation of the Consequent inference. We conclude our study with some remarks on the supposed ''independence'' of two conditionals, and we interpret this property as uncorrelation between two random quantities.