Given the notion of a contraction we can now state what the agent’s new belief set, i.e., her old belief set [ltr]BB[/ltr] revised by the new information [ltr]αα[/ltr], [ltr]B∗αB∗α[/ltr], should look like. First, the agent should clear [ltr]BB[/ltr] to make it consistent with [ltr]αα[/ltr]. That is, first the agent should contract [ltr]BB[/ltr] by [ltr]¬α¬α[/ltr]. Then she should simply add [ltr]αα[/ltr] and close under (classical) logical consequence. This gives us the agent’s new belief set [ltr]B∗αB∗α[/ltr], her old belief set [ltr]BB[/ltr] revised by [ltr]αα[/ltr]. The recipe just described is known as theLevi identity:
Revision [ltr]∗∗[/ltr] defined in this way satisfies a corresponding list of properties. For every set of sentences [ltr]B⊆LB⊆L[/ltr] and any sentences [ltr]α,β∈Lα,β∈L[/ltr] (where the contradictory sentence [ltr]⊥⊥[/ltr] can be defined as the negation of the tautological sentence [ltr]⊤⊤[/ltr], i.e., [ltr]¬⊤)¬⊤)[/ltr]:

[ltr](∗1)(∗1)[/ltr]	[ltr]B∗α=Cn(B∗α)B∗α=Cn(B∗α)[/ltr]
[ltr](∗2)(∗2)[/ltr]	[ltr]α∈B∗αα∈B∗α[/ltr]
[ltr](∗3)(∗3)[/ltr]	If [ltr]¬α∉Cn(B)¬α∉Cn(B)[/ltr], then [ltr]B∗α=Cn(B∪{α})B∗α=Cn(B∪{α})[/ltr]
[ltr](∗4)(∗4)[/ltr]	If [ltr]¬α∉Cn(∅)¬α∉Cn(∅)[/ltr], then [ltr]⊥∉B∗α⊥∉B∗α[/ltr]
[ltr](∗5)(∗5)[/ltr]	If [ltr]Cn({α})=Cn({α})Cn({α})=Cn({α})[/ltr], then [ltr]B∗α=B∗βB∗α=B∗β[/ltr]
[ltr](∗6)(∗6)[/ltr]	If [ltr]B=Cn(B)B=Cn(B)[/ltr], then [ltr](B∗α)∩B=B÷¬α(B∗α)∩B=B÷¬α[/ltr]
[ltr](∗7)(∗7)[/ltr]	If [ltr]B=Cn(B)B=Cn(B)[/ltr], then [ltr]B∗(α∧β)⊆Cn(B∗α∪{β})B∗(α∧β)⊆Cn(B∗α∪{β})[/ltr]
[ltr](∗8)(∗8)[/ltr]	If [ltr]B=Cn(B)B=Cn(B)[/ltr] and [ltr]¬β∉B∗α¬β∉B∗α[/ltr], then [ltr]Cn(B∗α∪{β})⊆B∗(α∧β)Cn(B∗α∪{β})⊆B∗(α∧β)[/ltr]

In standard belief revision theory the new belief set is always deductively closed, as required by [ltr]∗1∗1[/ltr]. This requirement can be dropped by using belief bases instead of belief sets (Hansson 1999). In standard belief revision theory the new information is always part of the new belief set, as required by [ltr]∗2∗2[/ltr]. Non-prioritized belief revision relaxes this requirement (Hansson 1999). The idea is that the idealdoxastic agent might consider the new information to be too implausible to be added and decide to reject it; or she might add only a sufficiently plausible part of the new information; or she might add the new information and then check for consistency, which makes her give up part or all of the new information again, because her old beliefs turn out to be moreentrenched.
The notion of entrenchment provides the connection to degrees of belief. In order to decide which part of her belief set she wants to give up, belief revision theory equips the ideal doxastic agent agent with an entrenchment ordering. Technically, this is a relation [ltr]≼≼[/ltr] on [ltr]LL[/ltr] (i.e., [ltr]≼⊆L)≼⊆L)[/ltr]such that for all [ltr]α,β,γα,β,γ[/ltr] in [ltr]LL[/ltr]:

[ltr](≼1)(≼1)[/ltr]	If [ltr]α≼βα≼β[/ltr] and [ltr]β≼γβ≼γ[/ltr], then [ltr]α≼γα≼γ[/ltr]	Transitivity
[ltr](≼2)(≼2)[/ltr]	If [ltr]α⊢βα⊢β[/ltr], then [ltr]α≼βα≼β[/ltr]	Dominance
[ltr](≼3)(≼3)[/ltr]	[ltr]α≼α∧βα≼α∧β[/ltr] or [ltr]β≼α∧ββ≼α∧β[/ltr]	Conjunctivity
[ltr](≼4)(≼4)[/ltr]	If [ltr]⊥∉Cn(B)⊥∉Cn(B)[/ltr], then [ltr][α∉B[α∉B[/ltr] if and only if for all [ltr]ββ[/ltr] in [ltr]L:α≼β]L:α≼β][/ltr]	Minimality
[ltr](≼5)(≼5)[/ltr]	If for all [ltr]αα[/ltr] in [ltr]L:α≼βL:α≼β[/ltr], then [ltr]β∈Cn(∅)β∈Cn(∅)[/ltr]	Maximality

[ltr]BB[/ltr] is a fixed set of background beliefs. Given an entrenchment ordering [ltr]≼≼[/ltr] on [ltr]LL[/ltr] and letting [ltr]α[size=16]⪱βα⪱β[/ltr]hold just in case [ltr]α≼βα≼β[/ltr] and [ltr]β⋠αβ⋠α[/ltr], we can define a revision operator [ltr]∗∗[/ltr] as follows:[/size]
Then one can prove the following representation theorem:

Theorem 1: 
Let [ltr][size=18]LL[/ltr] be a language, let [ltr]B⊆LB⊆L[/ltr] be a set of sentences, and let [ltr]α∈Lα∈L[/ltr] be a sentence. Each entrenchment ordering [ltr]≼≼[/ltr] on [ltr]LL[/ltr] induces a revision operator [ltr]∗∗[/ltr] on [ltr]LL[/ltr] satisfying [ltr]∗1∗1[/ltr]–[ltr]∗8∗8[/ltr] by defining

[/size][size]
For each revision operator [ltr]∗∗[/ltr] on [ltr]LL[/ltr] satisfying [ltr]∗1∗1[/ltr]–[ltr]∗8∗8[/ltr] there is an entrenchment ordering [ltr]≼≼[/ltr] on [ltr]LL[/ltr]that induces [ltr]∗∗[/ltr] in exactly this way.[/size]

Grove (1988) proves an analogous representation theorem for a systems of spheres semantics that generalizes Lewis’ (1973) semantics for counterfactuals. Segerberg (1995) formulates the AGM approach in the framework of dynamic doxastix logic. Lindström & Rabinowicz (1999) extend this to iterated belief revision.
It is, however, fair to say that belief revision theorists distinguish between degrees of belief and entrenchment. Entrenchment, so they say, characterizes the agent’s unwillingness to give up a particular qualitative belief, which may be different from her degree of belief for the respective sentence or proposition. Although this distinction might violate Occam’s razor by introducing an additional doxastic level, it corresponds to Spohn’s parallelism (Spohn 2009, sct. 3) between subjective probabilities and ranking functions as well as Stalnaker’s stance in his (1996, sct. 3). Weisberg (2011, sct. 7) offers a similar distinction.
If the agent’s doxastic state is represented by a regular ranking function [ltr]ϱϱ[/ltr] (on a field of propositions over the set of models [ltr]Mod[size=13]LModL[/ltr] for the language [ltr]LL[/ltr], as explained in section 1.3) the ordering [ltr]≼ϱ≼ϱ[/ltr] that is defined for all [ltr]α,βα,β[/ltr] in [ltr]LL[/ltr] by[/size]
is an entrenchment ordering for [ltr]B={α∈L:ϱ(Mod(¬α))>0}B={α∈L:ϱ(Mod(¬α))>0}[/ltr]. Ranking theory thus covers AGM belief revision theory as a special case (Rott 2009a defines, among others, entrenchment orderings and ranking functions for beliefs as well as for disbeliefs and non-beliefs). It is important to see how ranking theory goes beyond AGM belief revision theory. In the latter theory the agent’s prior doxastic state is characterized by a belief set [ltr]BB[/ltr] together with an entrenchment ordering [ltr]≼≼[/ltr]. If the agent receives new information in the form of a proposition [ltr]AA[/ltr], the entrenchment ordering is used to turn the old belief set into a new one, viz. [ltr]B∗AB∗A[/ltr]. The agent’s posterior doxastic state is thus characterized by a belief set only. The entrenchment ordering itself is not updated. Therefore AGM belief revision theory cannot handle iterated belief changes. To the extent that belief revision is not simply a one step process, AGM belief revision theory is thus no theory of belief revision at all. (The analogous situation in terms of subjective probabilities would be to characterize the agent’s prior doxastic state by a set of propositions together with a subjective probability measure, and to use that measure to update the set of propositions without ever updating the probability measure itself.)
In ranking theory the agent’s prior doxastic state is characterized by a ranking function [ltr]ϱϱ[/ltr] (on a field over [ltr]Mod[size=13]L)ModL)[/ltr]. This function determines the agent’s prior belief set [ltr]BB[/ltr], and so there is no need to specify [ltr]BB[/ltr] in addition to [ltr]ϱϱ[/ltr]. If the agent receives new information in form of a proposition[ltr]AA[/ltr], as AGM belief revision theory has it, there are infinitely many ways to update her ranking function that all give rise to the same new belief set [ltr]B∗AB∗A[/ltr]. Let [ltr]nn[/ltr] be an arbitrary positive number in [ltr]N∪{∞}N∪{∞}[/ltr]. Then Spohn conditionalization on the partition [ltr]{A,W∖A}{A,W∖A}[/ltr] with [ltr]n>0n>0[/ltr] as new rank for [ltr]W∖AW∖A[/ltr] (and consequently 0 as new rank for [ltr]A),ϱ′n(W∖A)=nA),ϱn′(W∖A)=n[/ltr], determines a new ranking function [ltr]ϱ′nϱn′[/ltr] that induces a belief set [ltr]B′nBn′[/ltr]. We have for any two positive numbers [ltr]m,nm,n[/ltr] in [ltr]N∪{∞}:B′m=B′n=B∗AN∪{∞}:Bm′=Bn′=B∗A[/ltr], where the latter is the belief set described two paragraphs ago.[/size]

Plain conditionalization is the special case of Spohn conditionalization with [ltr]∞∞[/ltr] as new rank for [ltr]W∖AW∖A[/ltr]. The new ranking function obtained in this way is [ltr]ϱ[size=13]′∞=ϱ(⋅∣A)ϱ∞′=ϱ(⋅∣A)[/ltr], and the belief set it induces is the same [ltr]B∗AB∗A[/ltr] as before. However, once the ideal doxastic agent assigns rank [ltr]∞∞[/ltr] to [ltr]W∖AW∖A[/ltr], she can never get rid of [ltr]AA[/ltr] again (in the sense that the only information that would allow her to give up her belief that [ltr]AA[/ltr] is to become certain that [ltr]AA[/ltr] is false, i.e., assign rank [ltr]∞∞[/ltr] to [ltr]AA[/ltr]; that in turn would make her doxastic state collapse in the sense of turning it into the tabula rasa ranking that is agnostic with respect to all consistent propositions and so assigns rank 0 to all of them). Just as in probabilism you are stuck with [ltr]AA[/ltr] once you assign it probability 1, so you are basically stuck with [ltr]AA[/ltr] once you assign its negation rank [ltr]∞∞[/ltr]. As we have seen, AGM belief revision theory is compatible with always updating in this way. That is one way to see why it cannot handle iterated belief revision. To rule out this behavior one has to impose further constraints on entrenchment orderings. Nayak (1994) and Boutilier (1996) and Darwiche & Pearl (1997) and others do so by postulating constraints compatible with, but not yet implying ranking theory (see Rott 2009b, who provides an excellent overview of qualitative and comparative approaches to iterated belief revision extending the AGM approach). Hild & Spohn (2008) argue that one really has to go all the way to ranking functions in order to adequately deal with iterated belief revision. Stalnaker (2009) critically discusses these approaches and argues that one needs to distinguish different kinds of information, including meta-information about the agent’s own beliefs and revision policies as well as about the sources of her information. For more on AGM belief revision theory, iterated belief revisions, and ranking functions see Huber (2013a, 2013b). For a discussion of belief revision theory in the setting of possibility theory see Dubois & Prade (2009).[/size]