# Probabilistic inferences from conjoined to iterated conditionals

## TL;DR

This paper explores the probabilistic interpretation of natural language conditionals, demonstrating how compounds and iterations of conditionals can be coherently analyzed as conditional random quantities, extending inference rules and addressing counterfactuals.

## Contribution

It provides a coherence-based probabilistic framework for compounds and iterations of conditionals, linking them to de Finetti's conditional event and extending inference rules.

## Key findings

- Conditional probability of 'if A then B' equals de Finetti's conditional event B|A.
- Probabilistic analysis of compounds and iterations as conditional random quantities.
- Extended inference rules and uncertainty propagation for complex conditional structures.

## Abstract

There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, $P(\textit{if } A \textit{ then } B)$, is the conditional probability of $B$ given $A$, $P(B|A)$. We identify a conditional which is such that $P(\textit{if } A \textit{ then } B)= P(B|A)$ with de Finetti's conditional event, $B|A$. An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of conditional events. In this paper, we illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities which, given some logical dependencies, may reduce to conditional events. We show how the inference to $B|A$ from $A$ and $B$ can be extended to compounds and iterations of both conditional events and biconditional events. Moreover, we determine the respective uncertainty propagation rules. Finally, we make some comments on extending our analysis to counterfactuals.