Ranking theory is one of the salient formal representations of doxastic states. It differs from others in being able to represent belief in a proposition (= taking it to be true), to also represent degrees of belief (i.e. beliefs as more or less firm), and thus to generally account for the dynamics of these beliefs. It does so on the basis of fundamental and compelling rationality postulates and is hence one way of explicating the rational structure of doxastic (...) states. Thereby it provides foundations for accounts of defeasible or nonmonotonic reasoning. It has widespread applications in philosophy, it proves to be most useful in Artificial Intelligence, and it has started to find applications as a model of reasoning in psychology. (shrink)

An organizing theme of the dissertation is the issue of how to make philosophical theories useful for scientific purposes. An argument for the contention is presented that it doesn’t suffice merely to theoretically motivate one’s theories, and make them compatible with existing data, but that philosophers having this aim should ideally contribute to identifying unique and hard to vary predictions of their theories. This methodological recommendation is applied to the ranking-theoretic approach to conditionals, which emphasizes the epistemic relevance and (...) the expression of reason relations as part of the semantics of the natural language conditional. As a first step, this approach is theoretically motivated in a comparative discussion of other alternatives in psychology of reasoning, like the suppositional theory of conditionals, and novel approaches to the problems of compositionality and accounting for the objective purport of indicative conditionals are presented. In a second step, a formal model is formulated, which allows us to derive quantitative predictions from the ranking-theoretic approach, and it is investigated which novel avenues of empirical research that this model opens up for. Finally, a treatment is given of the problem of logical omniscience as it concerns the issue of whether ranking theory (and other similar approaches) makes too idealized assumptions about rationality to allow for interesting applications in psychology of reasoning. Building on the work of Robert Brandom, a novel solution to this problem is presented, which both opens up for new perspectives in psychology of reasoning and appears to be capable of satisfying a range of constraints on bridge principles between logic and norms of reasoning, which would otherwise stand in a tension. (shrink)

Ranking theory is a formal epistemology that has been developed in over 600 pages in Spohn's recent book The Laws of Belief, which aims to provide a normative account of the dynamics of beliefs that presents an alternative to current probabilistic approaches. It has long been received in the AI community, but it has not yet found application in experimental psychology. The purpose of this paper is to derive clear, quantitative predictions by exploiting a parallel between ranking (...) theory and a statistical model called logistic regression. This approach is illustrated by the development of a model for the conditional inference task using Spohn's ranking theoretic approach to conditionals. (shrink)

In this paper we compare Leitgeb’s stability theory of belief and Spohn’s ranking-theoretic account of belief. We discuss the two theories as solutions to the lottery paradox. To compare the two theories, we introduce a novel translation between ranking functions and probability functions. We draw some crucial consequences from this translation, in particular a new probabilistic belief notion. Based on this, we explore the logical relation between the two belief theories, showing that models of Leitgeb’s theory (...) correspond to certain models of Spohn’s theory. The reverse is not true. Finally, we discuss how these results raise new questions in belief theory. In particular, we raise the question whether stability is rightly thought of as a property pertaining to belief. (shrink)

Wolfgang Spohn presents the first full account of the dynamic laws of belief, by means of ranking theory. This book is his long-awaited presentation of ranking theory and its ramifications.

Belief revision theory studies how an ideal doxastic agent should revise her beliefs when she receives new information. In part I, I have first presented the AGM theory of belief revision. Then I have focused on the problem of iterated belief revisions. In part II, I will first present ranking theory (Spohn 1988). Then I will show how it solves the problem of iterated belief revisions. I will conclude by sketching two areas of future research.

"A Survey of Ranking Theory": The paper gives an up-to-date survey of ranking theory. It carefully explains the basics. It elaborates on the ranking theoretic explication of reasons and their balance. It explains the dynamics of belief statable in ranking terms and indicates how the ranks can thereby be measured. It suggests how the theory of Bayesian nets can be carried over to ranking theory. It indicates what it might mean to (...) objectify ranks. It discusses the formal and the philosophical aspects of the tight relation and the complementarity of ranks and probabilities. It closes with comparative remarks on predecessors and other philosophical proposals as well as formal models developed in AI. (shrink)

Meets what? Ranking theory is, as far as I know, the only existing theory suited for underpinning Keith Lehrer’s account of knowledge and justification. If this is true, it’s high time to bring both together. This is what I shall do in this paper. However, the result of defining Lehrer’s primitive notions in terms of ranking theory will be disappointing: justified acceptance will, depending on the interpretation, either have an unintelligible structure or reduce to mere (...) acceptance, and in the latter interpretation knowledge will reduce to true belief. Of course, this result will require a discussion of who should be disappointed. So, the plan of the paper is simple: In section 1 I shall briefly state what is required for underpinning Lehrer’s account and why most of the familiar theories fail to do so. In section 2 I shall briefly motivate and introduce ranking theory. Basing Lehrer’s account on it will be entirely straightforward. Section 3 proves the above-mentioned results. Section 4, finally, discusses the possible conclusions. (shrink)

The Laws of Belief: Ranking Theory & its Philosophical Applications, by SpohnWolfgang. New York: Oxford University Press, 2012. Pp. xv + 598.

Fitelson and McCarthy have proposed an accuracy measure for confidence orders which favors probability measures and Dempster-Shafer belief functions as accounts of degrees of belief and excludes ranking functions. Their accuracy measure only penalizes mistakes in confidence comparisons. We propose an alternative accuracy measure that also rewards correct confidence comparisons. Thus we conform to both of William James’ maxims: “Believe truth! Shun error!” We combine the two maxims, penalties and rewards, into one criterion that we call prioritized accuracy optimization. (...) That is, PAO punishes wrong comparisons and rewards right comparisons. And it requires to prioritize being right und avoiding to be wrong in a specific way. Thus PAO is both, a scoring rule and a decision rule. It turns out that precisely confidence orders representable by two-sided ranking functions satisfy PAO. The point is not to argue that PAO is the better accuracy goal. The point is only that ranking theory can also be supported by accuracy considerations. Thus, those considerations by themselves cannot decide about rational formats for degrees of belief, but are part and parcel of an overall normative assessment of those formats. (shrink)

I want to look at recent developments of representing AGM-style belief revision in dynamic epistemic logics and the options for doing something similar for ranking theory. Formally, my aim will be modest: I will define a version of basic dynamic doxastic logic using ranking functions as the semantics. I will show why formalizing ranking theory this way is useful for the ranking theorist first by showing how it enables one to compare ranking (...) class='Hi'>theory more easily with other approaches to belief revision. I will then use the logic to state an argument for defining ranking functions on larger sets of ordinals than is customary. Secondly, I will argue that the only way to extend the account of belief revision given by ranking theory to higher-order beliefs and revisions is by continuing the approach taken by me and defining ranking theoretical equivalents of dynamic epistemic logics. For proponents of dynamic epistemic logic, such logics will naturally be of interest provided they are convinced of the revision operator defined by ranking theory. (shrink)

We introduce a family of local ranks $D_Q$ depending on a finite set Q of pairs of the form $(\varphi (x,y),q(y)),$ where $\varphi (x,y)$ is a formula and $q(y)$ is a global type. We prove that in any NSOP $_1$ theory these ranks satisfy some desirable properties; in particular, $D_Q(x=x)<\omega $ for any finite tuple of variables x and any Q, if $q\supseteq p$ is a Kim-forking extension of types, then $D_Q(q) (...) a Kim-non-forking extension, then $D_Q(q)=D_Q(p)$ for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP $_1$ theories. We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory $T_\infty $ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in $T_\infty $, strengthening an earlier observation that $T_\infty $ satisfies the existence axiom for forking independence. Finally, we slightly modify our definitions and go beyond NSOP $_1$ to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP $_1$ and NTP $_2$ theories. (shrink)

I argue that fitting resentment tracks unacceptable ‘ecological’ imbalances in relative social strength between victims and perpetrators that arise from violations of legitimate moral expectations. It does not respond purely, or even primarily, to offenders’ attitudes, and its proper targets need not be fully developed moral agents. It characteristically involves a wish for the restoration of social equilibrium rather than a demand for moral recognition or good will. To illuminate these contentions, I focus on cases that I believe demonstrate a (...) corollary thesis, namely, that strength, broadly construed, is a necessary condition of resentment-worthiness. I argue that weakness can make resentment unfitting in two ways. First, weakness may prevent a wrongdoer from shifting the balance of social power. Second, a weak wrongdoer may do social damage but be so lowly that resentment, which would represent him as excessively strong, would be inapt. Finally, I consider how accepting the ecological view might affect our theorizing about moral responsibility and the ethics of blame. (shrink)

A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non-standard models of (...) the fragment \ + \-Separation of \; and Gaifman’s technique of iterated ultrapowers is employed to show that any countable model of \ can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of “strong rank-cut” is characterized in terms of the theory \, and in terms of fixed-point sets of self-embeddings. (shrink)

We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP $_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP $_{1}$ theories.

We prove that it is not possible to define an appropriate notion of rank in set theories without the axiom of foundation.

Conditionals somehow express conditional beliefs. However, conditional belief is a bi-propositional attitude that is generally not truth-evaluable, in contrast to unconditional belief. Therefore, this article opts for an expressivistic semantics for conditionals, grounds this semantics in the arguably most adequate account of conditional belief, that is, ranking theory, and dismisses probability theory for that purpose, because probabilities cannot represent belief. Various expressive options are then explained in terms of ranking theory, with the intention to set (...) out a general interpretive scheme that is able to account for the most variegated usage of conditionals. The Ramsey test is only the first option. Relevance is another, familiar, but little understood item, which comes in several versions. This article adds a further family of expressive options, which is able to subsume also counterfactuals and causal conditionals, and indicates at the end how this family allows for partial recovery of truth conditions for conditionals. (shrink)

Starting from the definition of `amorphous set' in set theory without the axiom of choice, we propose a notion of rank (which will only make sense for, at most, the class of Dedekind finite sets), which is intended to be an analogue in this situation of Morley rank in model theory.

Empirical evidence from both utility and psychophysical experiments suggests that people respond quite differently—perhaps discontinuously—to stimulus pairs when one consequence or signal is set to `zero.' Such stimuli are called unitary. The author's earlier theories assumed otherwise. In particular, the key property of segregation relating gambles and joint receipts (or presentations) involves unitary stimuli. Also, the representation of unitary stimuli was assumed to be separable (i.e., multiplicative). The theories developed here do not invoke separability. Four general cases based on two (...) distinctions are explored. The first distinction is between commutative joint receipts, which are relevant to utility, and the non-commutative ones, which are relevant to psychophysics. The second distinction concerns how stimuli of the form (x, C; y) and the operation of joint receipt are linked: by segregation, which mixes stimuli and unitary ones, and by distributivity, which does not involve any unitary stimuli. A class of representations more general than rank-dependent utility (RDU) is found in which monotonic functions of increments U(x)-U(y), where U is an order preseving representation of gambles, and joint receipt play a role. This form and its natural generalization to gambles with n > 2 consequences, which is also axiomatized, appear to encompass models of configural weights and decision affect. When joint receipts are not commutative, somewhat similar representations of stimuli arise, and joint receipts are shown to have a conjoint additive representation and in some cases a constant bias independent of signal intensity is predicted. (shrink)

We are able to incorporate an antisymmetric second-rank tensor into null aesthetic field theory. There are some changes in the solutions due to the introduction of this antisymmetric second-rank tensor, which we discuss. We are not able to find an acceptable bounded particle system in four space-time dimensions.

An example is given of a complete theory with minimal models of arbitrarily large minimality rank.

Ranking theory delivers an account of iterated contraction; each ranking function induces a specific iterated contraction behavior. The paper shows how to reconstruct a ranking function from its iterated contraction behavior uniquely up to multiplicative constant and thus how to measure ranks on a ratio scale. Thereby, it also shows how to completely axiomatize that behavior. The complete set of laws of iterated contraction it specifies amend the laws hitherto discussed in the literature.

Dual-ranking act-consequentialism (DRAC) is a rather peculiar version of act-consequentialism. Unlike more traditional forms of act-consequentialism, DRAC doesn’t take the deontic status of an action to be a function of some evaluative ranking of outcomes. Rather, it takes the deontic status of an action to be a function of some non-evaluative ranking that is in turn a function of two auxiliary rankings that are evaluative. I argue that DRAC is promising in that it can accommodate certain features (...) of commonsense morality that no single-ranking version of act-consequentialism can: supererogation, agent-centered options, and the self-other asymmetry. I also defend DRAC against three objections: (1) that its dual-ranking structure is ad hoc, (2) that it denies (putatively implausibly) that it is always permissible to make self-sacrifices that don’t make things worse for others, and (3) that it violates certain axioms of expected utility theory, viz., transitivity and independence. (shrink)

We introduce a ranking of multidimensional alternatives, including uncertain prospects as a particular case, when these objects can be given a matrix form. This ranking is separable in terms of rows and columns, and continuous and monotonic in the basic quantities. Owing to the theory of additive separability developed here, we derive very precise numerical representations over a large class of domains (i.e., typically notof the Cartesian product form). We apply these representationsto (1)streams of commodity baskets through (...) time, (2)uncertain social prospects, (3)uncertain individual prospects. Concerning(1), we propose a finite horizon variant of Koopmans’s (1960) axiomatization of infinite discounted utility sums. The main results concern(2). We push the classic comparison between the exanteand expostsocial welfare criteria one step further by avoiding any expected utility assumptions, and as a consequence obtain what appears to be the strongest existing form of Harsanyi’s (1955) Aggregation Theorem. Concerning(3), we derive a subjective probability for Anscombe and Aumann’s (1963) finite case by merely assuming that there are two epistemically independent sources of uncertainty. (shrink)

st of these lectures, I present a conception of dignity that preserves its ancient association with rank and station, and a conception of human dignity that amounts to a generalization of high status across all human beings. The lectures argue that this provides a better understanding of human dignity and of the work it does in theories of rights than the better-known Kantian conception. The second lecture focuses particularly on the importance of dignity - understood in this way - as (...) a status defining a person's relation to law: their presentation as persons capable of self-applying the law, capable of presenting and arguing a point of view, and capable of responding to law's demands without brute coercion. Together the two lectures also illuminate the relation between dignity conceived as the ground of rights and dignity conceived as the content of rights; they also illuminate important ideas about dignity as noble bearing and dignity as the subject of a right against degrading treatment; and they help us understand the sense in which dignity is better conceived as a status than as a kind of value. (shrink)

Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. Previously, the author proved that this theory is supersimple. In supersimple theories there is a notion of rank defined in analogy with Lascar U-rank for superstable theories. It is also possible to define a notion of dimension for types in DCFA based on transcendence degree of realization of the types. In this paper we compute the rank of a model (...) of DCFA, give some properties regarding rank and dimension, and give an example of a definable set with finite rank but infinite dimension. Finally we prove that for the case of definable subgroup of the additive group being finite-dimensional and having finite rank are equivalent. (shrink)

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any$\Pi (...) ^1_1$sound extension of$\mathsf {ACA}_0$. We prove that for any$\Pi ^1_1$sound theoryTextending$\mathsf {ACA}_0^+$, the reflection rank ofTequals the$\Pi ^1_1$proof-theoretic ordinal ofT. We also prove that the$\Pi ^1_1$proof-theoretic ordinal of$\alpha $iterated$\Pi ^1_1$reflection is$\varepsilon _\alpha $. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles. (shrink)

First, ranking functions are argued to be superior to AGM belief revision theory in two crucial respects. Second, it is shown how ranking functions are uniquely reflected in iterated belief change. More precisely, conditions on threefold contractions are specified which suffice for representing contractions by a ranking function uniquely up to multiplication by a positive integer. Thus, an important advantage AGM theory seemed to have over ranking functions proves to be spurious.

We show that if T is a trivial uncountably categorical theory of Morley Rank 1 then T is model complete after naming constants for a model.

The theory of ranked partial structures allows a reinterpretation of several of the standard results of model theory and first-order logic and is intended to provide a proof-theoretic method which allows for the intuitions of model theory. A version of the downward Löwenheim-Skolem theorem is central to our development. In this paper we will present the basic theory of ranked partial structures and their logic including an appropriate version of the completeness theorem.

The Spohnian paradigm of ranking functions is in many respects like an order-of-magnitude reverse of subjective probability theory. Unlike probabilities, however, ranking functions are only indirectly—via a pointwise ranking function on the underlying set of possibilities W —defined on a field of propositions A over W. This research note shows under which conditions ranking functions on a field of propositions A over W and rankings on a language L are induced by pointwise ranking functions (...) on W and the set of models for L, ModL, respectively. (shrink)

Partial order optimality theory is a conservative generalization of classical optimality theory that makes possible the modeling of free variation and quantitative regularities without any numerical parameters. Solving the ranking problem for PoOT has so far remained an outstanding problem: allowing for free variation, given a finite set of input/output pairs, i.e., a dataset, \ that a speaker S knows to be part of some language L, how can S learn the set of all grammars G under (...) some constraint set C compatible with \?. Here, allowing for free variation, given the set of all PoOT grammars GPoOT over a constraint set C, for an arbitrary \, I provide set-theoretic means for constructing the actual set G compatible with \. Specifically, I determine the set of all STRICT ORDERS of C that are compatible with \. As every strict total order is a strict order, our solution is applicable in both PoOT and COT, showing that the ranking problem in COT is a special instance of a more general one in PoOT. (shrink)

In [R. Vaught, Denumerable models of complete theories, in: Infinitistic Methods, Pregamon, London, 1961, pp. 303–321] Vaught conjectured that a countable first order theory has countably many or 20 many countable models. Here, the following special case is proved.

The paper investigates the prospects of Spohn’s ranking theory with respect to Goodman’s New Riddle of Induction. Based on a novel analysis of the riddle (Freitag [manuscript]), we show it to be an inductive extension of Hansson’s puzzle (Hansson 1992, 1999). As a consequence, a solution needs to take into account the dependence relations between evidential beliefs: “grue” is unprojectible because it depends on evidence whose projection is defeated. It will be suggested that this solution can be implemented (...) into Spohn’s ranking theory, but that ranking theory is unable to provide a proper explanation of the required conditional ranks. (shrink)

In traditional consequentialism the good is position-neutral. A single evaluative ranking of states of affairs is correct for everyone, everywhere regardless of their positions. Recently, position-relative forms of consequentialism have been developed. These allow for the correct rankings of states to depend on connections that hold between the state being evaluated and the position of the evaluator. For example, perhaps being an agent who acts in a certain state requires me to rank that state differently from someone else who (...) lacks this connection. In this chapter several different kinds of position-relative rankings related to agents, times, physical locations, and possible worlds are explored. Arguments for and against adopting a position-relative axiology are examined, and it is suggested that position-relative consequentialism is a promising moral theory that has been underestimated. (shrink)

We develop a model of Disappointment in which disappointment and elation arise from comparing the outcome received, not with an expected value as in previous models, but rather with the other individual outcomes of the lottery. This approach may better reflect the way individuals are liable to experience disappointment. The model obtained accounts for classic behavioral deviations from the normative theory, offers a richer structure than previous disappointment models, and leads to a Rank-Dependent Utility formulation in a transparent way. (...) Thus, our disappointment model may provide a clear psychological rationale for the subjective transformation of probabilities. (shrink)

Ranking finite subsets of a given set X of elements is the formal object of analysis in this article. This problem has found a wide range of economic interpretations in the literature. The focus of the article is on the family of rankings that are additively representable. Existing characterizations are too complex and hard to grasp in decisional contexts. Furthermore, Fishburn (1996), Journal of Mathematical Psychology 40, 64–77 showed that the number of sufficient and necessary conditions that are needed (...) to characterize such a family has no upper bound as the cardinality of X increases. In turn, this article proposes a way to overcome these difficulties and allows for the characterization of a meaningful (sub)family of additively representable rankings of sets by means of a few simple axioms. Pattanaik and Xu’s (1990), Recherches Economiques de Louvain 56, 383–390) characterization of the cardinality-based rule will be derived from our main result, and other new rules that stem from our general proposal are discussed and characterized in even simpler terms. In particular, we analyze restricted-cardinality based rules, where the set of “focal” elements is not given ex-ante; but brought out by the axioms. (shrink)