Impossibility in Belief Merging

TL;DR

This paper extends the framework of belief merging to complex epistemic states and profiles, introduces epistemic social postulates, and establishes a general version of Arrow's Impossibility Theorem applicable to various epistemic representations.

Contribution

It introduces a novel, generalized framework for belief merging with epistemic states and profiles, and proves an epistemic version of Arrow's Impossibility Theorem.

Findings

Established representation theorems for belief merging operators.

Proved an epistemic version of Arrow's Impossibility Theorem.

Applicable to multiple representations of epistemic states.

Abstract

With the aim of studying social properties of belief merging and having a better understanding of impossibility, we extend in three ways the framework of logic-based merging introduced by Konieczny and Pino P\'erez. First, at the level of representation of the information, we pass from belief bases to complex epistemic states. Second, the profiles are represented as functions of finite societies to the set of epistemic states (a sort of vectors) and not as multisets of epistemic states. Third, we extend the set of rational postulates in order to consider the epistemic versions of the classical postulates of Social Choice Theory: Standard Domain, Pareto Property, Independence of Irrelevant Alternatives and Absence of Dictator. These epistemic versions of social postulates are given, essentially, in terms of the finite propositional logic. We state some representation theorems for these operators. These extensions and representation theorems allow us to establish an epistemic and very general version of Arrow's Impossibility Theorem. One of the interesting features of our result, is that it holds for different representations of epistemic states; for instance conditionals, Ordinal Conditional functions and, of course, total preorders.