Conditioning and AGM-like belief change in the Desirability-Indifference framework

TL;DR

This work extends the AGM belief-change program to conditioning within the Desirability-Indifference (DI) framework, modeling beliefs as abstract statement models over an option space and events as call-off operations. It develops both expansion and revision notions for conditioning, showing that conditioning generally behaves as belief revision rather than expansion, and provides a DI-specific revision operator that yields updated background models $V_e$ and revised desirability sets $D\Vert e$. The theory specializes to concrete classical and quantum probabilistic inferences, recovering Bayes-like conditioning in the classical case and Lüders-type updates in the quantum case, via explicit correspondences between events, kernels, and projections. The framework thus unifies conditioning across probabilistic paradigms under an AGM-inspired, abstract belief-change lens and points to future work on broader background models and Accept-Desirability formalisms.

Abstract

We show how the AGM framework for belief change (expansion, revision, contraction) can be extended to deal with conditioning in the so-called Desirability-Indifference framework, based on abstract notions of accepting and rejecting options, as well as on abstract notions of events. This level of abstraction allows us to deal simultaneously with classical and quantum probability theory.

Theorems & Definitions (25)

• Definition 2.1
• Theorem 2.1
• Proposition 3.1
• Proposition 5.1
• Proposition 5.2
• Proposition 5.3: Harper's identity
• Proposition 5.4: Levi's identity
• proof : Proof of \ref{['eq:D:compatibility']}
• proof : Proof of \ref{['eq:least:resolved:model']}
• proof : Proof of \ref{['eq:event:extra:properties']}