3 Ranking Theory

Subjective probability theory as well as the theory of DS belief functions take the objects of belief to be propositions. Possibility theory does so only indirectly, although possibility measures on a field of propositions [ltr]AA[/ltr] can also be defined without recourse to a possibility distribution on the underlying set [ltr]WW[/ltr] of possibilities.
A possibility [ltr]ww[/ltr] in [ltr]WW[/ltr] is a complete and consistent description of the world relative to the expressive power of [ltr]WW[/ltr]. [ltr]WW[/ltr] may contain just two possibilities: according to [ltr]w[size=13]1w1[/ltr] tomorrow it will be sunny in Vienna, according to [ltr]w2w2[/ltr] it will not. On the other end of the spectrum, [ltr]WW[/ltr] may comprise all metaphysically possible, or even all logically possible worlds (for more see the entry on possible worlds.)[/size]
Usually we are not certain which of the possibilities in [ltr]WW[/ltr] corresponds to the actual world. Otherwise these possibilities would not be genuine possibilities for us, and our degree of belief function would collapse into a truth value assignment. However, to say that we are not certain which possibility it is that corresponds to the actual world does not mean that all possibilities are on a par. Some of them will be really far-fetched, while others will seem to be more reasonable candidates for the actual possibility.
This gives rise to the following consideration. We can partition the set of possibilities, that is, form sets of possibilities that are mutually exclusive and jointly exhaustive. Then we can orderthe cells of this partition according to their plausibility. The first cell in this ordering contains the possibilities that we take to be the most reasonable candidates for the actual possibility. The second cell contains the possibilities which we take to be the second most reasonable candidates. And so on.
If you are still equipped with your possibility distribution from the preceding section you can use your degrees of possibility for the various possibilities to obtain such an ordered partition. Note, though, that an ordered partition — in contrast to your possibility distribution — contains no more than ordinal information. While your possibility distribution enables you to say how possible you take a particular possibility to be, an ordered partition only allows you to say that one possibility [ltr]w[size=13]1w1[/ltr] is more plausible than another possibility [ltr]w2w2[/ltr]. In fact, an ordered partition does not even enable you to express that the difference between your plausibility for [ltr]w1w1[/ltr] (say, tomorrow the temperature in Vienna will be between 70°F and 75°F) and for [ltr]w2w2[/ltr] (say, tomorrow the temperature in Vienna will be between 75°F and 80°F) is smaller than the difference between your plausibility for [ltr]w2w2[/ltr] and for the far-fetched [ltr]w3w3[/ltr] (say, tomorrow the temperature in Vienna will be between 120°F and 125°F).[/size]
This takes us directly to ranking theory (Spohn 1988 and 1990 and, especially, 2012), which goes one step further. Rather than merely ordering the possibilities in [ltr]WW[/ltr], a pointwise ranking function [ltr]κ:W→N∪{∞}κ:W→N∪{∞}[/ltr] additionally assigns natural numbers from [ltr]N∪{∞}N∪{∞}[/ltr] to the cells of possibilities. These numbers represent the grades of disbelief you assign to the various (cells of) possibilities in [ltr]WW[/ltr]. The result is a numbered partition of [ltr]WW[/ltr],

[ltr]κ[size=13]−1(0),κ−1(1),κ−1(2),…,κ−1(n)={w∈W:κ(w)=n},…κ−1(∞).κ−1(0),κ−1(1),κ−1(2),…,κ−1(n)={w∈W:κ(w)=n},…κ−1(∞).[/ltr][/size]

The first cell [ltr]κ[size=13]−1(0)κ−1(0)[/ltr] contains the possibilities which are not disbelieved (which does not mean that they are believed). The second cell [ltr]κ−1(1)κ−1(1)[/ltr] is the set of possibilities which are disbelieved to degree 1. And so on. It is important to note that, except for [ltr]κ−1(0)κ−1(0)[/ltr], the cells [ltr]κ−1(n)κ−1(n)[/ltr] may be empty, and so would not appear at all in the corresponding ordered partition. [ltr]κ−1(0)κ−1(0)[/ltr] must not be empty, though. The reason is that one cannot consistently disbelieve everything.[/size]
More precisely, a function [ltr]κ:W→N∪{∞}κ:W→N∪{∞}[/ltr] from a set of possibilities [ltr]WW[/ltr] into the set of natural numbers extended by [ltr]∞∞[/ltr], [ltr]N∪{∞}N∪{∞}[/ltr], is a pointwise ranking function just in case [ltr]κ(w)=0κ(w)=0[/ltr] for at least one [ltr]ww[/ltr] in [ltr]WW[/ltr], i.e., just in case [ltr]κ[size=13]−1(0)≠∅κ−1(0)≠∅[/ltr]. The latter requirement says that you should not disbelieve every possibility. It is justified, because you know for sure that one possibility is actual. A pointwise ranking function [ltr]κ:W→N∪{∞}κ:W→N∪{∞}[/ltr] on [ltr]WW[/ltr] induces a ranking function [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] on a field [ltr]AA[/ltr] of propositions over [ltr]WW[/ltr] by defining for each [ltr]AA[/ltr] in [ltr]AA[/ltr],[/size]

[ltr]ϱ(A)=min{κ(w):w∈A} (=∞ if A=∅).ϱ(A)=min{κ(w):w∈A} (=∞ if A=∅).[/ltr]

This entails that ranking functions [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] are (finitely) minimitive (and hence super-additive), i.e., for all [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr],

[ltr]ϱ(A∪B)=min{ϱ(A),ϱ(B)}.(9)(9)ϱ(A∪B)=min{ϱ(A),ϱ(B)}.[/ltr]

As in the case of possibility theory, (finitely minimitive and non-conditional) ranking functions can be directly defined on a field [ltr]AA[/ltr] of propositions over a set of possibilities [ltr]WW[/ltr] as functions [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] such that for all [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr]:

[ltr]=∞,[/ltr]

The triple [ltr]⟨W,A,ϱ⟩⟨W,A,ϱ⟩[/ltr] is a (finitely minimitive) ranking space. Suppose [ltr]AA[/ltr] is closed under countable/complete intersections (and thus a [ltr]σσ[/ltr]-/[ltr]γγ[/ltr]-field). Suppose further that [ltr]ϱϱ[/ltr] additionally satisfies, for all countable/arbitrary [ltr]B⊆AB⊆A[/ltr],

[ltr]ϱ(∪B)=min{ϱ(A):A∈B}.ϱ(∪B)=min{ϱ(A):A∈B}.[/ltr]

Then [ltr]ϱϱ[/ltr] is a countably/completely minimitive ranking function, and [ltr]⟨W,A,ϱ⟩⟨W,A,ϱ⟩[/ltr] is a [ltr]σσ[/ltr]- or countably/[ltr]γγ[/ltr]- or completely minimitive ranking space. Finally, a ranking function [ltr]ϱϱ[/ltr] on [ltr]AA[/ltr] isregular just in case [ltr]ϱ(A)<∞ϱ(A)<∞[/ltr] for every non-empty or consistent proposition [ltr]AA[/ltr] in [ltr]AA[/ltr]. For more see Huber (2006), which discusses under which conditions ranking functions on fields of propositions induce pointwise ranking functions on the underlying set of possibilities.
Let us pause for a moment. The previous paragraphs introduce a lot of terminology for something that seems to add only little to what we have already discussed. Let the necessity measures of possibility theory assign natural instead of real numbers in the unit interval to the various propositions so that [ltr]∞∞[/ltr] instead of 1 represents maximal necessity/possibility. The axioms for necessity measures then become

[ltr]N(∅)N(W)N(A∩B)=0,=∞ (instead of 1),=min{N(A),N(B)}.N(∅)=0,N(W)=∞ (instead of 1),N(A∩B)=min{N(A),N(B)}.[/ltr]

Now think of the rank of a proposition [ltr]AA[/ltr] as the degree of necessity of its negation [ltr]W∖A,ϱ(A)=N(W∖A)W∖A,ϱ(A)=N(W∖A)[/ltr]. Seen this way, finitely minimitive ranking functions are a mere terminological variant of necessity measures, for

[ltr]ϱ(∅)ϱ(W)ϱ(A∪B)=N(W)=∞=N(∅)=0=N((W∖A)∩(W∖B))=min{N(W∖A),N(W∖B)}=min{ϱ(A),ϱ(B)}.ϱ(∅)=N(W)=∞ϱ(W)=N(∅)=0ϱ(A∪B)=N((W∖A)∩(W∖B))=min{N(W∖A),N(W∖B)}=min{ϱ(A),ϱ(B)}.[/ltr]

(If we take necessity measures as primitive rather than letting them be induced by possibility measures, and if we continue to follow the rank-theoretic policy of adopting a well-ordered range, we can obviously also define countably and completely minimitive necessity measures.) Of course, the fact that (finitely minimitive and non-conditional) ranking functions and necessity measures are formally alike does not mean that their interpretations are the same.
The latter is the case, though, when we compare ranking functions and Shackle’s degrees ofpotential surprise (Shackle 1949; 1969). (These degrees of potential surprise have made their way into philosophy mainly through the work of Isaac Levi. See Levi 1967a; 1978.) So what justifies devoting a whole section to ranking functions?
Shackle’s theory lacks a notion of conditional potential surprise. (Shackle 1969, 79ff, seems to assume a notion of conditional potential surprise as primitive that appears in his axiom 7. This axiom further relies on a connective that behaves like conjunction except that it is not commutative and is best interpreted as “[ltr]AA[/ltr] followed by [ltr]BB[/ltr]”. Axiom 7, in its stronger version from p. 83, seems to say that the degree of potential surprise of “A followed by [ltr]BB[/ltr]” is the greater of the degree of potential surprise of [ltr]AA[/ltr] and the degree of potential surprise of [ltr]BB[/ltr] given [ltr]AA[/ltr], i.e., [ltr]ς(A followed by B)=max{ς(A),ς(B∣A)}ς(A followed by B)=max{ς(A),ς(B∣A)}[/ltr] where [ltr]ςς[/ltr] is the measure of potential surprise. Spohn 2009, sct. 4.1, discusses Shackle’s struggle with the notion of conditional potential surprise.)
Possibility theory, on the other hand, offers two notions of conditional possibility (Dubois & Prade 1988). The first notion of conditional possibility is obtained by the equation

[ltr]Π(A∩B)=min{Π(A),Π(B∣A)}.Π(A∩B)=min{Π(A),Π(B∣A)}.[/ltr]

It is mainly motivated by the desire to have a notion of conditional possibility that also makes sense if possibility does not admit of degrees, but is a merely comparative notion. The second notion of conditional possibility is obtained by the equation

[ltr]Π(A∩B)=Π(A)⋅Π(B∣A).Π(A∩B)=Π(A)⋅Π(B∣A).[/ltr]

The inspiration for this notion seems to come from probability theory. While none of these two notions is the one we have in ranking theory, Spohn (2009), relying on Halpern (2003), shows that by adopting the second notion of conditional possibility one can render possibility theory isomorphic to a real-valued version of ranking theory.
For standard ranking functions with a well-ordered range conditional ranks are defined as follows. The conditional ranking function [ltr]ϱ(⋅∣⋅):A×A→N∪{∞}ϱ(⋅∣⋅):A×A→N∪{∞}[/ltr] on [ltr]AA[/ltr] (based on the non-conditional ranking function [ltr]ϱϱ[/ltr] on [ltr]A)A)[/ltr] is defined for all pairs of propositions [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr] with [ltr]A≠∅A≠∅[/ltr] by

[ltr]ϱ(A∣B)=ϱ(A∩B)−ϱ(B),ϱ(A∣B)=ϱ(A∩B)−ϱ(B),[/ltr]

where [ltr]∞−∞=0∞−∞=0[/ltr]. Further stipulating [ltr]ϱ(∅∣B)=∞ϱ(∅∣B)=∞[/ltr] for all [ltr]BB[/ltr] in [ltr]AA[/ltr] guarantees that [ltr]ϱ(⋅∣B):A→N∪{∞}ϱ(⋅∣B):A→N∪{∞}[/ltr] is a ranking function, for every [ltr]BB[/ltr] in [ltr]AA[/ltr]. It would, of course, also be possible to take conditional ranking functions [ltr]ϱ(⋅,given ⋅):A×A→N∪{∞}ϱ(⋅,given ⋅):A×A→N∪{∞}[/ltr] as primitive and define (non-conditional) ranking functions in terms of them as [ltr]ϱ(A)=ϱ(A,given W)ϱ(A)=ϱ(A,given W)[/ltr] for all propositions [ltr]AA[/ltr] in [ltr]AA[/ltr].
The number [ltr]ϱ(A)ϱ(A)[/ltr] represents the agent’s degree of disbelief for the proposition [ltr]AA[/ltr]. If [ltr]ϱ(A)>0ϱ(A)>0[/ltr], the agent disbelieves [ltr]AA[/ltr] to a positive degree. Therefore, on pain of inconsistency, she cannot also disbelieve [ltr]W∖AW∖A[/ltr] to a positive degree. In other words, for every proposition [ltr]AA[/ltr] in [ltr]AA[/ltr], at least one of [ltr]A,W∖AA,W∖A[/ltr] has to be assigned rank 0. If [ltr]ϱ(A)=0ϱ(A)=0[/ltr], the agent does not disbelieve [ltr]AA[/ltr] to a positive degree. However, this does not mean that she believes [ltr]AA[/ltr] to a positive degree [ltr]−−[/ltr] the agent may suspend judgment and assign rank 0 to both [ltr]AA[/ltr] and [ltr]W∖AW∖A[/ltr]. So belief in a proposition is characterized by disbelief in its negation.
For each ranking function [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] we can define a corresponding belief function [ltr]β:A→Z∪{∞}∪{−∞}β:A→Z∪{∞}∪{−∞}[/ltr] that assigns positive numbers to those propositions that are believed, negative numbers to those propositions that are disbelieved, and 0 to those propositions with respect to which the agent suspends judgment:

[ltr]β(A)=ϱ(W∖A)−ϱ(A)β(A)=ϱ(W∖A)−ϱ(A)[/ltr]

Each ranking function [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] induces a belief set

[ltr]B={A∈A:ϱ(W∖A)>0}={A∈A:ϱ(W∖A)>ϱ(A)}={A∈A:β(A)>0}.B={A∈A:ϱ(W∖A)>0}={A∈A:ϱ(W∖A)>ϱ(A)}={A∈A:β(A)>0}.[/ltr]

[ltr]BB[/ltr] is the set of all propositions the agent believes to some positive degree, or equivalently, whose complements she disbelieves to a positive degree. The belief set [ltr]BB[/ltr] induced by a ranking function[ltr]ϱϱ[/ltr] is consistent and deductively closed (in the finite sense). The same is true for the belief set induced by a possibility measure [ltr]Π:A→RΠ:A→ℜ[/ltr],

If [ltr]ϱϱ[/ltr] is a countably/completely minimitive ranking function, the belief set [ltr]BB[/ltr] induced by [ltr]ϱϱ[/ltr] is consistent and deductively closed in the following countable/complete sense: [ltr]∩C≠∅∩C≠∅[/ltr] for every countable/arbitrary [ltr]C⊆BC⊆B[/ltr]; and [ltr]A∈BA∈B[/ltr] whenever [ltr]∩C⊆A∩C⊆A[/ltr] for some countable/arbitrary [ltr]C⊆BC⊆B[/ltr] and any [ltr]A∈AA∈A[/ltr]. Ranking theory thus offers a link between belief and degrees of belief that is preserved when we move from the finite to the countably or uncountably infinite case. As shown by the example in Section 3.2, this is not the case for possibility theory. (Of course, as indicated above, the possibility theorist can copy ranking theory by taking necessity measures as primitive and by adopting a well-ordered range).
Much as for subjective probabilities, there are rules for updating one’s doxastic state represented by a ranking function. In case the new information comes in form of a certainty, ranking theory’s counterpart to probability theory’s strict conditionalization is
Plain Conditionalization.
If evidence comes only in form of certainties, if [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] is your ranking function at time [ltr]tt[/ltr], and if between [ltr]tt[/ltr] and [ltr]t[size=13]′t′[/ltr] you become certain of [ltr]A∈AA∈A[/ltr] and no logically stronger proposition in the sense that your new rank for [ltr]W∖AW∖A[/ltr], but no logically weaker proposition, is [ltr]∞∞[/ltr] (and your ranks are not directly affected in any other way such as forgetting etc.), then your ranking function at time [ltr]t′t′[/ltr] should be [ltr]ϱ(⋅∣A)ϱ(⋅∣A)[/ltr].[/size]
If the new information merely changes your ranks for various propositions, ranking theory’s counterpart to probability theory’s Jeffrey conditionalization is
Spohn Conditionalization.
If evidence comes only in form of new grades of disbelief for the elements of a partition, if [ltr]ϱ:A→N∪{∞}ϱ:A→N∪{∞}[/ltr] is your ranking function at time [ltr]tt[/ltr], and between [ltr]tt[/ltr] and [ltr]t[size=13]′t′[/ltr] your ranks in the mutually exclusive and jointly exhaustive propositions [ltr]Ai∈AAi∈A[/ltr] are directly affected and change to [ltr]ni∈N∪{∞}ni∈N∪{∞}[/ltr] with [ltr]minini=0minini=0[/ltr], and the finite part of your ranking function does not change on any superset of the partition [ltr]{Ai}{Ai}[/ltr] (and your ranks are not directly affected in any other way such as forgetting etc.), then your ranking function at time [ltr]t′t′[/ltr] should be [ltr]ϱ′(⋅)=mini{ϱ(⋅∣Ai)+ni}ϱ′(⋅)=mini{ϱ(⋅∣Ai)+ni}[/ltr].[/size]
As the reader will have noticed by now, whenever we substitute 0 for 1, [ltr]∞∞[/ltr] for 0, [ltr]minmin[/ltr] for [ltr]∑∑[/ltr], [ltr]∑∑[/ltr]for [ltr]∏∏[/ltr], and [ltr]>>[/ltr] for [ltr]<<[/ltr], a true statement about probabilities almost always turns into a true statement about ranking functions. (There are but a few known exceptions to this transformation. Spohn 1994 mentions one.) For a comparison of probability theory and ranking theory see Spohn (2009, sct. 3).
Three complaints about Jeffrey conditionalization carry over to Spohn conditionalization. First, Jeffrey conditionalization is not commutative (Levi 1976b). The same is true of Spohn conditionalization. Second, any two regular probability measures can be related to each other via Jeffrey conditionalization (by letting the evidential partition consist of the set of singletons [ltr]{w}{w}[/ltr]containing the possibilities [ltr]ww[/ltr] in [ltr]W)W)[/ltr]. The same is true of any two regular ranking functions and Spohn conditionalization. Therefore, so the complaint goes, these rules are empty as normative constraints. Third, Weisberg (2015) argues that Spohn conditionalization cannot handle perceptual undermining either.
The first complaint misfires, because both Jeffrey and Spohn conditionalization are result- ratherevidence-oriented: the parameters [ltr]p[size=13]ipi[/ltr] and [ltr]nini[/ltr] characterize the resulting degree of (dis)belief in [ltr]EiEi[/ltr] rather than the amount by which the evidence received between [ltr]tt[/ltr] and [ltr]t′t′[/ltr] boosts or lowers the degree of (dis)belief in [ltr]EiEi[/ltr]. Therefore these parameters depend on both the prior doxastic states Pr and [ltr]ϱϱ[/ltr], respectively, and the evidence received between [ltr]tt[/ltr] and [ltr]t′t′[/ltr]. Evidence first shifting [ltr]EE[/ltr] from [ltr]pp[/ltr] to [ltr]p′p′[/ltr] and then to [ltr]p′′p″[/ltr] is not a rearrangement of evidence first shifting [ltr]EE[/ltr] from [ltr]pp[/ltr] to [ltr]p′′p″[/ltr] and then to [ltr]p′p′[/ltr]. Field (1978) presents a probabilistic update rule that is evidence-oriented in the sense of characterizing the evidence as such, independently of the prior doxastic state. Shenoy (1991) presents a rank-theoretic update rule that is evidence-oriented in this sense. These two update rules are commutative.[/size]
The second complaint misfires, because it confuses input and output: Jeffrey conditionalization does not rule out any evidential input of the appropriate format, just as it does not rule out any prior epistemic state not already ruled out by the probability calculus. The same is true of Spohn conditionalization and the ranking calculus. That does not mean that these rules are empty as normative constraints, though. On the contrary, for each admissible prior doxastic state and each admissible evidential input there is only one posterior doxastic state not ruled by Jeffrey (Spohn) conditionalization. Huber (2014) defends Jeffrey and Spohn conditionalization against Weisberg’s charge.

One reason why an ideal doxastic agent’s degrees of belief should obey the probability calculus is that otherwise she is vulnerable to a Dutch Book (standard version) or an inconsistent evaluation of the fairness of bets (depragmatized version). For similar reasons she should update her subjective probability according to strict or Jeffrey conditionalization, depending on the format of the new information. Why should grades of disbelief obey the ranking calculus? And why should an ideal doxastic agent update her ranking function according to plain or Spohn Conditionalization?
The answers to these questions require a bit of terminology. An ideal doxastic agent’s degree ofentrenchment for a proposition [ltr]AA[/ltr] is the number of “independent and minimally positively reliable” information sources saying [ltr]AA[/ltr] that it takes for the agent to give up her disbelief that [ltr]AA[/ltr]. If the agent does not disbelieve [ltr]AA[/ltr] to begin with, her degree of entrenchment for [ltr]AA[/ltr] is 0. If no finite number of information sources is able to make the agent give up her disbelief that [ltr]AA[/ltr], her degree of entrenchment for [ltr]AA[/ltr] is [ltr]∞∞[/ltr]. Suppose we want to determine Sophia’s degree of entrenchment for the proposition that Vienna is the capital of Austria. This can be done by putting her on, say, the Stephansplatz, a popular place in the old town of Vienna, and by counting the number of people passing by and telling her that Vienna is the capital of Austria. Her degree of entrenchment for the proposition that Vienna is the capital of Austria equals [ltr]nn[/ltr] precisely if she stops disbelieving that Vienna is the capital of Austria after [ltr]nn[/ltr] people have passed by and told her it is. The relation between these operationally defined degrees of entrenchment and the theoretical grades of disbelief is similar to the relation between betting ratios and degrees of belief: under suitable conditions (when the information sources are independent and minimally positively reliable) the former can be used to measure the latter. Most of the time the conditions are not suitable, though. In section 2.2 primitivism seemed to be the only plausible game in town. In the present case “going hypothetical” (Eriksson & Hájek 2007) is more promising: the agent’sgrade of disbelief in [ltr]AA[/ltr] is the number of information sources saying [ltr]AA[/ltr] that it would take for her to give up her qualitative disbelief that [ltr]AA[/ltr], if those sources were independent and minimally positively reliable.
Now we are in the position to say why degrees of disbelief should obey the ranking calculus. They should do so, because an agent’s belief set is and will always be consistent and deductively closed in the finite/countable/complete sense just in case her entrenchment function is a finitely/countably/completely minimitive ranking function and, depending on the format of the evidence, the agent updates according to plain or Spohn conditionalization (Huber 2007b). This theorem can be used to establish the thesis that an ideal doxastic agent’s beliefs should obey the synchronic and diachronic rules of the ranking calculus. It can be used to provide a means-ends justification for this thesis in the spirit of epistemic consequentialism (Percival 2002, Stalnaker 2002). The idea is that obeying the normative constraints of the ranking calculus is a (necessary and sufficient) means to attaining the end of being “eternally consistent and deductively closed.” The latter end in turn is a (necessary, but insufficient) means to attaining the end of always having only true beliefs, and as many thereof as possible. Brössel, Eder & Huber (2013) discuss the importance of this result as well as its Bayesian role-model, Joyce’s (1998; 2009) “non-pragmatic vindication of probabilism” discussed above, for means-ends epistemology in general.
It follows that the above notion of conditional ranks is the only good notion for standard ranking functions with a well-ordered domain: plain and Spohn conditionalization depend on the notion of conditional ranks, and the theorem does not hold if we replace this notion by another one. Furthermore, one reason for adopting standard ranking functions with a well-ordered domain is that the notion of degree of entrenchement makes sense only for natural (or ordinal) numbers, because one has to count the independent and minimally positively reliable information sources. The seemingly small differences between possibility theory and ranking theory thus turn out to be crucial.
With the possible exception of decision making (see, however, Giang & Shenoy 2000), it seems that we can do everything with ranking functions that we can do with probability measures. Ranking theory also has a notion of qualitative belief that is vital if we want to stay in tune with traditional epistemology. This allows for rank-theoretic theories of belief revision and of nonmonotonic reasoning, which are the topic of the final two sections.

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3 Ranking Theory

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3 Ranking Theory

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