Ranking Theory

Introduction

Epistemic or doxastic attitudes 1 representing how the world is like come in degrees, whether you call them degrees of belief, uncertainty, plausibility, etc.

There are various accounts of those degrees, amply presented in this handbook. 2

The interests in those accounts are manifold. Philosophers are concerned with the rational nature of those degrees, AI researchers are interested in their computational feasibility, psychologists deal with their actual manifestations, and all sides argue about how well they are suited to model human reasoning.

However, we also have the notion of belief simpliciter. Related notions are those of acceptance or judgment. These are indeed the more basic notions when it comes to truth, to truly representing the world. Beliefs can be true, but degrees of belief cannot. The latter rather relate to action (see chapter 8.2 by Peterson in this volume). Accounts of degrees of belief invariably have great difficulties in doing 1 Strictly speaking, "epistemic" only refers to knowledge, although it is often used more widely. Because we will talk only about belief, we prefer to use "doxastic" throughout. See also chapter 5.1 by van Ditmarsch (in this volume). justice to this fundamental point. There is a questionable tendency to take degrees of belief as basic and to belittle those difficulties.

So we need to theoretically account for belief simpliciter. The first attempt was doxastic logic (see chapter 5.1 by van Ditmarsch in this volume). However, it is static and misses a dynamic perspective. This has been unfolded in belief revision theory (see chapter 5.2 by Rott in this volume). However, it has problems with iterated belief revision required for a complete dynamic account.

Ranking theory promises both to represent belief and degrees of belief and to provide a complete dynamics for both. These features give it a prominent place in the spectrum of possible theories. It was first presented in English in Spohn (1988) and fully developed in Spohn (2012). Easy access is provided in Spohn (2009). Its far-reaching applications in philosophy of science, epistemology, and even to normative reasoning may be found, e.g., in Spohn (2012Spohn ( , 2015Spohn ( , 2019.

There is no place here to go into any of them.

Below we present the basics of the theory in Section 2 and its dynamic aspects in Section 3. Section 4 is comparative. Section 5 gives a short introduction to its relevance for Artificial Intelligence, and Section 6 explains how it can be put to use in psychology.

The Basics of Ranking Theory

Grammatically, "believe" is a transitive verb. In the phrase "a believes that p", "a" refers to a (human) subject and "that p" seems to be the object. What does "that p"

stand for, what are the objects of belief? This is a difficult and most confusing issue extensively discussed in philosophy (under the rubric "propositions"; see, e.g., McGrath 2012). Here, we cut short the issue, as usual in formal epistemology, by saying in a non-committal way that "that p" stands for the proposition expressed by "p", where that proposition is its truth condition, the set of possibilities or possible worlds in which p obtains or "p" is true.

Hence, we simply assume a set W of (mutually exclusive and jointly exhaustive) possibilities. These may be coarse-grained and refer only to a few things of interest; they need not consist of entire possible worlds. Each subset of W, i.e. each element of the power set P(W) of W, is a proposition. Now, the basic representation of a belief state is simply as the set of propositions believed or taken to be true in that state, its belief set. Traditionally, a belief set B Í P(W) has to satisfy two rationality requirements: B must be consistent, i.e., ⋂B ≠ AE, and B must be deductively closed, i.e., if ⋂B Í A, then A Î B.

5

These two rationality requirements may seem entirely obvious. The rationale of deductive logic is to check what we must not believe and what we are committed to believe. Note, however, that deductive closure is lost when we identify belief with probability above a certain threshold; it easily happens that the probabilities of two propositions is above the threshold, while that of their conjunction is below. Thus, the lottery and the preface paradox and the general desire to stick to a probabilistic representation of belief have led to a contestation of these requirements (see, e.g., Christensen 2005 Let us start with some brief formal explanations.

Definition 1: k is a negative ranking function for W iff k is a function from P(W)

into the set of natural numbers plus infinity ¥ such that for all A, B Í W:

(1) k(W) = 0 and k(AE) = ¥,

The basic interpretation is that k expresses degrees of disbelief (whence the qualification 'negative'). If k(A) = 0, A is not disbelieved at all. This allows that (1) says that the tautology W is not disbelieved, and hence the contradiction AE is not believed. This entails that beliefs are consistent according to k.

(1) moreover says that the contradiction is indeed maximally disbelieved. And (2) states that you cannot disbelieve a disjunction less strongly than its disjuncts. This entails in particular that if you believe two conjuncts, you also believe their conjunction. Hence, beliefs are deductively closed according to k. In other words, the belief set B = {A | k(~A) > 0} associated with k satisfies the two basic rationality requirements. Note, moreover, that (1) and (2) entail:

(3) either k(A) = 0 or k(~A) = 0 (or both) (the law of negation),

i.e., you cannot (dis)believe A and ~A at once.

For an illustration, consider Tweetie. Tweetie has, or fails to have, each of three properties: being a bird (B), being a penguin (P), and being able to fly (F). We will see the reasons for starting with negative ranks. But, of course, we can also introduce the positive counterpart by defining b to be a positive ranking function iff there is a negative ranking function k such that b(A) = k(~A) for all propositions A. b represents degrees of belief. Of course, (1) and (2) translate directly into axioms for b.

We may as well represent degrees of belief and degrees of disbelief in a single function by defining t to be a two-sided ranking function iff there is a negative ranking function k and the corresponding positive ranking function b such that

< 0, or = 0 according to whether A is believed, disbelieved, or neither in t.

Therefore this is perhaps the most intuitive notion. However, the mathematics is best done in terms of negative ranking functions. It is clear, though, that the three functions are interdefinable.

There is an important interpretational degree of freedom that we have not yet noticed. So far, we said that belief in A is represented by

However, we may often find it useful to raise the threshold for belief, as we do informally in asking: "Do you really believe A?" That is, we may as well say that belief in A is only represented by k(~A) = b(A) = t(A) > z for some z ≥ 0. This seems to be a natural move. Belief is vague. Where does it commence, when does it cease? And this vagueness seems well represented by that parameter z. This move at the same time enlarges the range of suspension of judgment to the interval from -z to z. The remarkable point about axioms (1) and (2) is that they guarantee belief sets to be consistent and deductively closed, however we choose the threshold z. They are indeed equivalent to this general guarantee.

In fact, we need no more than twofold non-vacuous contractions. Raidl & Skovgaard-Olsen 2016). Therefore, our attitude has always been to independently develop ranking theory as a theory of belief.

Conditional Ranks, Reasons, and the Dynamics of Ranks

So far, we have sketched only the static part of ranking theory. However, we mentioned that the numeric ranks are essentially used to account for the dynamics of belief; they are not just to represent greater and lesser firmness of (dis)belief.

To achieve this, the crucial notion is that of conditional ranks.

Definition 2: Let k be a negative ranking function for W and k(A) < ¥. Then the

We might rewrite this definition as:

However, if A is true, B must be false; and this adds k(B | A).

It immediately follows for all propositions A and B with k(A) < ¥:

This law says that even conditional belief must be consistent. If both, k(B | A) and k(~B | A), were > 0, both, B and ~B, would be (dis-)believed given A, and this must be excluded, as long as the condition A itself is considered possible. Indeed, given definition 2 and axiom (1), we could axiomatize ranking theory also by (5) instead of (2). Hence, the only substantial assumption written into ranking functions is conditional consistency.

Axioms (1) and (2) did not refer to any cardinal properties of ranking functions.

However, the definition of conditional ranks involves arithmetical operations and thus presupposes a cardinal understanding of ranks. We will see below how this may be justified. We hasten to add that one could as well define positive

As an illustration, consider again Table 1 and the conditional beliefs contained therein. We can see that precisely the (material) if-then propositions nonvacuously held true correspond to conditional beliefs. According to the k specified, you believe, e.g., that Tweetie can fly given it is a bird (since k(~F | B)

= 1) and also given it is a bird, but not a penguin (since k(~F | B Ç ~P) = 2), and that Tweetie cannot fly given it is a penguin (since k(F | P) = 3). Hence, your vacuous belief in the material implication "if Tweety is a penguin, it can fly" does not amount to a corresponding conditional belief. In other words: "if, then" expresses conditional belief rather than material implication (see also chapter 6.1 by Starr in this volume).

A first fundamental application of conditional ranks lies in the notion of an Hartmann in this volume) works equally well in ranking theory (see Goldszmidt & Pearl, 1996).

A second fundamental application of conditional ranks lies in the dynamics of beliefs and ranks. As in probability theory, we may say that we should simply move to the degrees of belief conditional on the evidence E learned. Thereby, though, the evidence E acquires maximal certainty, either probability 1 or positive rank ¥. This seems too restrictive. In general, evidence may be (slightly) uncertain, and our rules for doxastic change through evidence or learning-we do not attend to changes caused in other ways like forgetting-should take account of this. In ranking theory, it is achieved by two principles: first, conditional ranks given the evidence E and given its negation ~E are not changed by the evidence itself-how could it change them?-and second, the evidence E does not become maximally certain, but improves its position by n ranks, where n is a free parameter characterizing the specific information process. 4 These two assumptions suffice to uniquely determine the kinematics of ranking functions,

i.e., ranking-theoretic conditionalization.

In order to see how this works look again at our Tweety example. Suppose you learn in some way and accept with firmness 2 that Tweetie is a bird. Thus you shift up ~B-possibilities by 2 and keep constant the rank differences within B and within ~B. This results in the posterior ranking function k':

~F 2 1 2 10 (Table 2) In k' you believe that Tweetie is a bird able to fly, but not a penguin; you still neglect this possiblity. So, in k' you believe more than in k; in belief revision 4 This is completely analogous to Jeffrey conditionalization in probability theory (see Jeffrey 1983, ch. 11, and chapter Next, to your surprise, you tentatively learn and accept, say with firmness 1, that Tweetie is indeed a penguin. This results in another ranking function k'', which shifts all P-possibilities down by 1 and all ~P-possibilities up by 1, so that P is indeed believed with firmness 1 (i.e., k''(~P) = 1): Table 3) So, you have changed your mind and believe in k'' that Tweetie is a penguin bird that cannot fly. In belief revision theory this would be called a belief revision.

Obviously, belief contraction (cf. Chapter 5.2 of Rott in this volume), where you simply give up a belief previously held without replacing it by a new one, can also be modeled by ranking-theoretic conditionalization. The example already demonstrates that this rule of belief change can be iteratively applied ad libitum.

An important application of ranking-theoretic conditionalization is that it delivers a measurement procedure for ranks that justifies the cardinality of ranks.

This procedure refers to iterated belief contraction. Its point is this: if your iterated contractions behave as prescribed by ranking theory 5 , then that behavior uniquely determines your ranking function up to a multiplicative constant. That is, your ranks can thereby be measured on a ratio scale (see Hild & Spohn 2008). The consequences of the fact that ranks are measured only on a ratio scale await investigation. They imply, e.g., a problem analogous to the problem of the interpersonal comparison of utilities.

Comparisons

The formal structure defined by axioms (1) and (2) has been called Baconian probability by Cohen (1980). Its first clear appearance is in the functions of potential surprise developed by Shackle (1949). The structure is also hidden in Rescher (1964) and is clearly found in Cohen's own work in Cohen (1970Cohen ( , 1977.

The crucial formal advance of ranking theory lies in the definition of conditional ranks, which is nowhere found in these works and which makes the theory a properly cardinal one.

Ranking Functions in Artificial Intelligence

Besides probability theory and logic, ranking functions are among the most popular formalisms used for knowledge representation 6 and reasoning (KRR), and their popularity is still increasing because they provide a very versatile framework for many central operations in KRR, as already sections 2-4 pointed out. Most importantly, ranking functions are a convenient common basic tool for 6

In AI, the distinction between knowledge and belief is usually quite vague.

nonmonotonic reasoning and belief revision. Belief revision has been already explained in more detail in section 3, and nonmonotonic reasoning also deals with belief dynamics in that conclusions may be given up when new information arrives (so, the consequence relation is not monotonic, as in classical logic). Both fields emerged in the 1980's (partly) as a reaction to the incapability of classical logic to handle problems in everyday life that intelligent systems like robots were expected to tackle. Knowledge, or belief about the world is usually uncertain, and the world is always changing. Therefore, AI systems built upon classical logics failed. So-called preferential models (see Makinson 1989) provide an important semantics for nonmonotonic logics, their basic idea is to order worlds according to normality and focus on the minimal, i.e., the most plausible ones for reasoning.

Likewise, AGM belief revision theory (see chapter 5.2 by Rott in this volume) needs orderings of worlds to become effective. For both fields, ranking functions offer quite a perfect technical tool that also complies nicely with the intuitions behind the techniques. Moreover, they can also evaluate conditionals and are an attractive qualitative counterpart to probabilities (see section 3).

Judea Pearl was probably the first renowned AI scientist to make use of ranking functions; his famous system Z (Pearl 1990) is based on them. He has continuously emphasized the structural commonsense qualities of probabilities and developed ranking functions as an interesting qualitative counterpart to probabilities. He set up his system Z as an "ultimate system of nonmonotonic reasoning" in terms of ranking functions. To date, it is one of the best and most convenient approaches to implement high-quality nonmonotonic reasoning.

Consequently, ranking functions are deeply connected with nonmonotonic and uncertain reasoning and with belief change, which are core topics in KRR. Many researchers make use of them in one way or another even if they rely on more general frameworks. Darwiche & Pearl (1997) presented general postulates for the iterated revision of general epistemic states, but illustrated their account with ranking functions. So did Jin & Thielscher (2007) and Delgrande & Jin (2012) when they devised novel postulates for iterated and multiple revision.

Interestingly, the independence properties for advanced belief revision which were proposed in those papers can be related to independence with respect to ranking functions (see Spohn 2012, ch. 7) in analogy with probabilistic independence (see Kern-Isberner & Huvermann, 2017).

Indeed, as suggested in section 3, ranking functions are particularly well suited for iterated belief change because they can easily be changed in accordance with AGM theory, returning new ranking functions which are readily available for a successive change operation. The main AGM operations are revision (adopting a belief) and contraction (giving up a belief), related by Levi and Harper identities (see chapter 5.2). In ranking theory, the connections between these operations are even deeper, since (iterated) contraction is just a special kind of (iterated) ranking conditionalization. Indeed, the results of (Kern- Isberner et al., 2017) show that iterated revision and contractions can be performed by a common methodology.

Continuing on that, and beyond practicality and diversity of ranking functions, it is crucial to understand that they are not just a pragmatically good choice but indeed allow for deep theoretical foundations of approaches to reasoning. It is the ease and naturalness with which they can handle conditionals-very similar to probabilities-that make them an excellent formal tool for modeling reasoning.

Given that conditionals are, on the one hand, crucial entities for nonmonotonic and commonsense reasoning and belief change, and, on the other hand, formal entities fully accessible to conditional logics, this capability provides a key feature for logic-based approaches connecting nonmonotonic logics and belief change theories with commonsense and general human reasoning. More precisely, conditional ranks give meaning to differences between degrees in belief when from conditional belief bases. Ranking theory is one of the few formal frameworks that is rich and expressive enough to allow such a precise formalization of conditional preservation which supports both belief change and inductive reasoning as a common methodology, probability theory is another.