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

Giacomo Bonanno

cs.AIFeb 26, 2026

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

Authors: Giacomo Bonanno

Organizations: University of California, Davis, USA

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.

Figures & tables

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

	$\neg\square\neg(\phi\wedge\psi)\,\rightarrow\,\neg\square\neg\phi$	$(C_{\neg\square\neg})$
		(see Remark 3 )
	$\big(\neg\square\neg(\phi\wedge\psi)\wedge B(\phi>\psi)\big)\,\rightarrow\,\left(\neg\square\neg\phi\wedge B(\phi>\psi)\right)$	1, PL
	$\left(\neg\square\neg\phi\wedge B(\phi>\psi)\right)\rightarrow\neg B(\phi>\neg\psi)$	$\text{axiom }(A^{*5b}_{\diamond 3b})$
	$\big(\neg\square\neg(\phi\wedge\psi)\wedge B(\phi>\psi)\big)\,\rightarrow\,\neg B(\phi>\neg\psi)$	2, 3, PL
	$\chi\rightarrow(\psi\rightarrow\chi)$	tautology
	$B(\phi>\chi)\rightarrow B\left(\phi>(\psi\rightarrow\chi)\right)$	$\text{5, rule }(RM_{B>})$

Appendix

Table 1

	$\neg\square\neg(\phi\wedge\psi)\wedge B\left((\phi\wedge\psi)>\chi\right)$
	$\rightarrow\,B\big(\phi>(\psi\rightarrow\chi)\big)$	$\text{axiom }(A*7)$
	$\neg\square\neg(\phi\wedge\psi)\wedge B\left((\phi\wedge\psi)>\chi\right)\wedge B(\phi>\psi)$
	$\rightarrow\,\big(B(\phi>\psi)\wedge B\big(\phi>(\psi\rightarrow\chi)\big)\big)$	15, PL
	$\big(B(\phi>\psi)\wedge B(\phi>(\psi\rightarrow\chi)\big)\,\rightarrow\,B(\phi>\chi)$	$\text{axiom }(A^{*1}_{\diamond 0})$
	$\big(\neg\square\neg(\phi\wedge\psi)\wedge B(\phi>\psi)\wedge B\left((\phi\wedge\psi)>\chi\right)\big)\,\rightarrow\,B(\phi>\chi)$	16, 17, PL
	$\big(\neg\square\neg(\phi\wedge\psi)\wedge B(\phi>\psi)\big)\,\rightarrow\,\big(B\left((\phi\wedge\psi)>\chi\right)\rightarrow B(\phi>\chi)\big)$	18, PL

Appendix

Table 2

	$\phi\leftrightarrow\big((\phi\vee\psi)\wedge\phi\big)$	tautology
	$B(\phi>\chi)\leftrightarrow B\big(\left((\phi\vee\psi)\wedge\phi\right)>\chi\big)$	$\text{1, rule }(R^{*6}_{\diamond 4})$
	$B(\phi>\chi)\rightarrow B\big(\left((\phi\vee\psi)\wedge\phi\right)>\chi\big)$	2, PL
	$\phi\rightarrow\big((\phi\vee\psi)\wedge\phi\big)$	tautology
	$\neg\square\neg\phi\rightarrow\neg\square\neg\left((\phi\vee\psi)\wedge\phi\right)$	$\text{1, rule }(RM_{\neg\square\neg})$
		$(\text{see Remark \ref{REM:derivedAxRul}})$
	$\neg\square\neg\phi\wedge B(\phi>\chi)$

Appendix

Table 3

	$\neg\square\neg\phi\wedge B(\phi>\chi)\wedge\neg\square\neg\psi\wedge B(\psi>\chi)$
	$\rightarrow\,B\big((\phi\vee\psi)>(\phi\rightarrow\chi)\big)\wedge B\big((\phi\vee\psi)>(\psi\rightarrow\chi)\big)$	8, 9, PL
	$B\big((\phi\vee\psi)>(\phi\rightarrow\chi)\big)\wedge B\big((\phi\vee\psi)>(\psi\rightarrow\chi)\big)$
	$\rightarrow B\big((\phi\vee\psi)>(\phi\rightarrow\chi\wedge\psi\rightarrow\chi)\big)$	$\text{axiom }(C_{B})$
	$(\phi\rightarrow\chi)\wedge(\psi\rightarrow\chi)\,\rightarrow\left((\phi\vee\psi)\rightarrow\chi\right)$	tautology
	$B\big((\phi\vee\psi)>(\phi\rightarrow\chi\wedge\psi\rightarrow\chi)\big)$
	$\rightarrow B\big((\phi\vee\psi)>\left((\phi\vee\psi)\rightarrow\chi\right)\big)$	$\text{12, rule }(RM_{B>})$

Appendix

Table 4

	$B(\phi\vee\neg\phi)\,\rightarrow\,\big(B(\phi\rightarrow\psi)\leftrightarrow B\left((\phi\vee\neg\phi)>(\phi\rightarrow\psi)\right)\big)$	$\text{axiom }(A\diamond 2)$
	$\phi\vee\neg\phi$	tautology
	$B(\phi\vee\neg\phi)$	$\text{2, rule }(N_{B})$
	$B(\phi\rightarrow\psi)\leftrightarrow B\left((\phi\vee\neg\phi)>(\phi\rightarrow\psi)\right)$	$\text{1, 3, rule }(MP)$
	$\neg\square\neg((\phi\vee\neg\phi)\wedge\phi)\wedge B\left((\phi\vee\neg\phi)\wedge\phi)>\psi\right)$
	$\rightarrow B((\phi\wedge\neg\phi)>(\phi\rightarrow\psi))$	$\text{axiom }(A\diamond 5)$
	$\phi\rightarrow(\phi\vee\neg\phi)\wedge\phi$	tautology

Appendix

Table 5

Belief Revision

Belief revision studies how agents change, retain, or retract beliefs when new evidence conflicts with what they already accept.

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Jun 18, 2026J/K move · Enter open · S save

Study on Quantitative Dynamic Epistemic Logic for Belief Revision

Felipe Nunes de Souza Camargo

State University of Campinas (UNICAMP), Campinas-SP, Brasil

Belief Revision Dynamic Epistemic Logic Epistemic Logic

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Belief revision is a process in which an agent begins to believe in something she previously did not. I begin the paper by presenting, based on (Gärdenfors, 1998; Hansson, 1999), postulates for belief revision that constitute the basis of the AGM theory. I will then briefly show the semantics of a modal logic introduced in (van Ditmarsch, 2005), which I call

$P$'. This logic formalizes static epistemic states and has greater expressive power than AGM in doing so because it captures the quantitative notion of "degrees of conviction". The third step is to introduce revision operators on $P$ and, mostly following (van Ditmarsch, 2005), obtain the Dynamic Epistemic Logic (DEL) I call

P*

'. It models processes of belief revision in several ways. Original results are presented in the following two sections. The first one of these sections revolves around a formalization of AGM postulates within

P*

by proving some theorems related to the satisfaction of those postulates by revisions defined in

P*

. The last section features an analysis of

P*

's revisions that go beyond the mere satisfaction of postulates. I compare their formal behavior with respect to some philosophical criteria. At last, I conclude that the functions presented in (van Ditmarsch, 2005) are not good formalizations of the philosophical intuition behind AGM. Instead, it is captured by the function

*^0

originally defined in this paper (but highly inspired by (van Benthem, 2007)). An implementation of this function is also provided.

Abstract

Belief Revision

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