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

Giacomo Bonanno

TL;DR

It is shown that the difference between KM belief update and AGM belief revision can be narrowed down to a single axiom, which deals exclusively with unsurprising information, that is, with formulas that were not initially disbelieved.

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.

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

TL;DR

It is shown that the difference between KM belief update and AGM belief revision can be narrowed down to a single axiom, which deals exclusively with unsurprising information, that is, with formulas that were not initially disbelieved.

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 , a bimodal conditional operator and the unimodal necessity operator . 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 and the former by we show that every axiom of is a theorem of . 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 and can be narrowed down to a single axiom, which deals exclusively with unsurprising information, that is, with formulas that were not initially disbelieved.
Paper Structure (9 sections, 14 theorems, 50 equations, 4 figures)

This paper contains 9 sections, 14 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The KM theory of belief update
  3. The KM theory of belief update
  4. Kripke-Lewis semantics
  5. A modal logic for belief update
  6. Frame correspondence
  7. Relating KM logic to AGM logic
  8. Conclusion
  9. Proofs

Key Result

Lemma 1

$(K\diamond 7s)$ follows from $(K\diamond 7)$ and $(K\diamond 8)$

Figures (4)

  • Figure 1: Semantic characterization of the KM axioms.
  • Figure 2: The frame properties of Figure \ref{['Fig_1']} and the corresponding modal formulas
  • Figure 3: The KM axioms and the corresponding modal axioms/rules of inference
  • Figure 4: The correspondence between AGM axioms and their modal counterparts.

Theorems & Definitions (39)

  • Remark 1
  • Lemma 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Definition 4
  • Definition 5
  • Remark 3
  • Definition 6
  • ...and 29 more