Conditionals Based on Selection Functions, Modal Operators and Probabilities

TL;DR

This paper explores the deep connection between probability updating methods, especially Bayesian conditionalization, and conditional connectives, aiming to characterize their probabilities and the updating procedures they can represent.

Contribution

It provides general results linking conditionals and updating methods, broadening understanding beyond specific cases and encompassing various conditionals and updates.

Findings

Characterizes probabilities of certain conditional connectives.

Identifies classes of updating procedures representable by specific conditionals.

Establishes a general framework connecting conditionals with probability updating methods.

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.