The logic of KM belief update is contained in the logic of AGM belief revision

TL;DR

This paper demonstrates that the logic underlying KM belief update is encompassed within the logic of AGM belief revision, showing that AGM revision is a special case of KM update through modal logic axioms.

Contribution

It establishes a formal relationship between KM belief update and AGM belief revision by translating their axioms into modal logic and comparing their logical strength.

Findings

Every axiom of KM belief update logic is a theorem of AGM belief revision logic.

AGM belief revision can be viewed as a special case of KM belief update.

The difference between the two logics is characterized by a single axiom related to unsurprising information.

Abstract

For each axiom of KM belief update we provide a corresponding axiom in a modal logic containing three modal operators: a unimodal belief operator $B$, a bimodal conditional operator $>$ and the unimodal necessity operator $\square$. We then compare the resulting logic to the similar logic obtained from converting the AGM axioms of belief revision into modal axioms and show that the latter contains the former. Denoting the latter by $\mathcal L_{AGM}$ and the former by $\mathcal L_{KM}$ we show that every axiom of $\mathcal L_{KM}$ is a theorem of $\mathcal L_{AGM}$. Thus AGM belief revision can be seen as a special case of KM belief update. For the strong version of KM belief update we show that the difference between $\mathcal L_{KM}$ and $\mathcal L_{AGM}$ can be narrowed down to a single axiom, which deals exclusively with unsurprising information, that is, with formulas that were not initially disbelieved.