Probabilistic inferences from conjoined to iterated conditionals

Abstract

There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, $P(\text{if }A\text{ then }B)$, is the conditional probability of $B$ given $A$, $P(B|A)$. We identify a conditional which is such that $P(\text{if }A\text{ then }B)=P(B|A)$ 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.

References in corpus (6)

• Conditional Random Quantities and Compounds of Conditionals
• Quasi Conjunction, Quasi Disjunction, T-norms and T-conorms: Probabilistic Aspects
• Probabilistic squares and hexagons of opposition under coherence
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Cited by in corpus (7)

• 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
• On Trivalent Logics, Compound Conditionals, and Probabilistic Deduction Theorems