Centering with conjoined and iterated conditionals under coherence
Angelo Gilio, David E. Over, Niki Pfeifer, Giuseppe Sanfilippo · arXiv (Cornell University) · 2017
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 to extend the inference of centering for conditional events, that is inferring $B|A$ from $A$ and $B$, to compounds and iterations of both conditional events and biconditional events. Moreover, we determine the lower and upper bounds for the conclusions in the classical and extended versions of centering. We also study some other related aspects. Finally, we apply our analysis to the semantics of counterfactuals.