# A note on Darwiche and Pearl

It is shown that Darwiche and Pearl's postulates imply an interesting property, not noticed by the authors.

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## 1 A short remark

In [DarwPearl:AIJ], Darwiche and Pearl propose postulates for iterated revisions, noted (R*1) to (R*6) and (C1) to (C4). In particular, the postulate (C3) reads:

 (C3)   If Ψ∘α⊨μ, then (Ψ∘μ)∘α⊨μ.

It will be shown that, in the presence (R*1) to (R*6), (C1) and (C3) imply:

 (∗∗)    If Ψ∘α⊨μ, then (Ψ∘μ)∘α≡Ψ∘α.

First, a lemma.

###### Lemma 1

Assuming (R*1) to (R*6), if , then .

Proof: Since , . By (R*4), . Therefore .

If is satisfiable, then, since , is satisfiable and, by (R*5), and therefore .

If is not satisfiable, then, by (R*3), is not satisfiable, and is not satisfiable. By (R*1), then, .

###### Lemma 2

Assuming (R*1) to (R*6), (C1) and (C3), if , then .

Proof: Suppose . By Lemma 1, . By (C1), . But, by (C3), and, by Lemma 1, .

We conclude that .