Conditionals Based on Selection Functions, Modal Operators and Probabilities
Tommaso Flaminio, Lluis Godo, Gluliano Rosella
TL;DR
This work develops a general, selection-function-based framework linking conditionals to probability updates on finite Boolean algebras. It proves that the probability of a selection-function conditional equals an imaged belief function (Bel_a) and, under suitable constraints, coincides with a normal modal update (□^f_a). It further shows that Bayesian (λ) updates are not generally representable as the probability of a conditional unless the closest-world set is a singleton, highlighting a fundamental limit in representing updating procedures via conditionals. The results unify various conditional families (variably strict, Stalnaker, preferential) within a common algebraic and modal-theoretic perspective and point toward modal representations while leaving open questions about broader operator characterizations of updates.
Abstract
Methods for probability updating, of which Bayesian conditionalization is the most well-known and widely used, are modeling tools that aim to represent the process of modifying an initial epistemic state, typically represented by a prior probability function P, which is adjusted in light of new information. Notably, updating methods and conditional sentences seem to intuitively share a deep connection, as is evident in the case of conditionalization. The present work contributes to this line of research and aims at shedding new light on the relationship between updating methods and conditional connectives. Departing from previous literature that often focused on a specific type of conditional or a particular updating method, our goal is to prove general results concerning the connection between conditionals and their probabilities. This will allow us to characterize the probabilities of certain conditional connectives and to understand what class of updating procedures can be represented using specific conditional connectives. Broadly, we adopt a general perspective that encompasses a large class of conditionals and a wide range of updating methods, enabling us to prove some general results concerning their interrelation.