Ranking Functions

Recent work has considered the problem of extending to the case of iterated belief change the so-called `Harper Identity' (HI), which defines single-shot contraction in terms of single-shot revision. The present paper considers the prospects of providing a similar extension of the Levi Identity (LI), in which the direction of definition runs the other way. We restrict our attention here to the three classic iterated revision operators--natural, restrained and lexicographic, for which we provide here the first collective characterisation in the (...) literature, under the appellation of `elementary' operators. We consider two prima facie plausible ways of extending (LI). The first proposal involves the use of the rational closure operator to offer a `reductive' account of iterated revision in terms of iterated contraction. The second, which doesn't commit to reductionism, was put forward some years ago by Nayak et al. We establish that, for elementary revision operators and under mild assumptions regarding contraction, Nayak's proposal is equivalent to a new set of postulates formalising the claim that contraction by ¬A should be considered to be a kind of `mild' revision by A. We then show that these, in turn, under slightly weaker assumptions, jointly amount to the conjunction of a pair of constraints on the extension of (HI) that were recently proposed in the literature. Finally, we consider the consequences of endorsing both suggestions and show that this would yield an identification of rational revision with natural revision. We close the paper by discussing the general prospects for defining iterated revision in terms of iterated contraction. (shrink)

In considering what we ought to say, we can evaluate a proposition both for whether it is assertable and for how assertable it is. The latter notion, that of comparative assertability, has an important role to play, both in our epistemic evaluations of speech and in our pragmatic reasoning. Yet, despite this, it has received little prior discussion. This paper takes up the investigation of comparative assertability. §§1–2 provide a preliminary, informal overview of the topic and an operationalization of the (...) target notion. §3 introduces Probabilism, the thesis that a proposition's degree of assertability is determined by its probability. Probabilism has been assumed in much of what prior discussion on comparative assertability there is. In §4 I present two kinds of problem for Probabilism—problems which, I suggest, when taken in combination, should lead us to look for alternatives. In §5, I formulate and defend one such alternative. Under this proposal, comparative assertability is a matter, not of comparative probability, but of comparative normality. I conclude by demonstrating how adopting this approach allows us to avoid both kinds of problem which beset Probabilism. (shrink)

In this paper, we argue that in a possible world there is a determinate ranking of teams. Our argument rests on the premise: In theory, nothing prevents a determinate better than ranking. This premise in turn rests on assumptions with regard to stipulations regarding ‘better than’ and nature of a competition as well as a right answer theory of interpretation. We then speculate that in some actual leagues in some years, there were determinate rankings. We consider objections that focus on (...) ties, noise, intransitivity, meaning of ‘better than’, and league recognition of the best team. We finish by noting that the issues involved in sport also apply to ranking candidates to occupy roles in our lives. (shrink)

Theories of epistemic justification are commonly assessed by exploring their predictions about particular hypothetical cases – predictions as to whether justification is present or absent in this or that case. With a few exceptions, it is much less common for theories of epistemic justification to be assessed by exploring their predictions about logical principles. The exceptions are a handful of ‘closure’ principles, which have received a lot of attention, and which certain theories of justification are well known to invalidate. But (...) these closure principles are only a small sample of the logical principles that we might consider. In this paper, I will outline four further logical principles that plausibly hold for justification and two which plausibly do not. While my primary aim is just to put these principles forward, I will use them to evaluate some different approaches to justification and (tentatively) conclude that a ‘normic’ theory of justification best captures its logic. (shrink)

This note is a sequel to Huber. It is shown that obeying a normative principle relating counterfactual conditionals and conditional beliefs, viz. the royal rule, is a necessary and sufficient means to attaining a cognitive end that relates true beliefs in purely factual, non-modal propositions and true beliefs in purely modal propositions. Along the way I will sketch my idealism about alethic or metaphysical modality.

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

Wolfgang Spohn's theory of ranking functions is an elegant and powerful theory of the structure and dynamics of doxastic states. In two recent papers, Spohn has applied it to the analysis of conditionals, claiming to have presented a unified account of indicative and subjunctive (counterfactual) conditionals. I argue that his analysis fails to account for counterfactuals that refer to indirect causes. The strategy of taking the transitive closure that Spohn employs in the theory of causation is not available for counterfactuals. (...) I have a close look at Spohn's treatment of the famous Oswald-Kennedy case in order to illustrate my points. I sketch an alternative view that seems to avoid the problems. (shrink)

In this chapter, the normic support condition on justification is approached via the widely accepted _safety_ condition on knowledge. It is argued that the normic support condition can be portrayed as a modification of the safety condition, with the notion of world normalcy substituted for the more familiar notion of world similarity. This idea is used to generate a contextualist theory of justified belief attributions that mirrors a safety-based contextualist theory of knowledge attributions. The relative _modal robustness_ of world normalcy, (...) as opposed to world similarity, is used to explore anew the long-standing debate between internalism and externalism about justification. (shrink)

According to the principle of multiple premise closure, if one has justification for believing each of a series of propositions, one has justification for believing their joint deductive consequences. A powerful objection to this principle arises from the phenomenon of risk aggregation, made vivid by the preface paradox. In this chapter two theories of justification are outlined—a pure normic theory that respects multiple premise closure and a hybrid theory that violates it. It is shown that multiple premise closure is derivable (...) from a range of further formal principles for justification and, as such, the formal consequences of rejecting multiple premise closure are difficult to anticipate. A case is made for the possibility of justifiably believing propositions that run a high risk of falsity, and some consequences for the psychology of human reasoning are explored. (shrink)

Lotteries, and our intuitions about them, lie at the heart of a number of persistent puzzles in epistemology. These include the lottery paradox, the lottery driven sceptical puzzle, Harman’s puzzle about statistical vs testimonial evidence concerning lottery outcomes and Vogel’s puzzle about iterated lotteries. In this chapter, these puzzles are considered anew in light of the suggestion that justification requires normic support.

In this chapter, the risk minimisation conception of justification is criticised for its predictions about the force of statistical evidence, and it is argued that justification cannot be understood solely in terms of probability. A non-probabilistic support relation between evidence and propositions—termed _normic support_—is introduced and put forward as a necessary condition for justification. The distinction between normic support and probabilistic support turns out to be akin to the more widely recognised distinction between ceteris paribus laws and brute statistical generalisations. (...) A formal model of normic support, involving a normalcy ordering of possible worlds, is proposed. (shrink)

This chapter develops a formal theory of degrees of safety and of normic support. The theory predicts that the degrees of normic support imposed upon propositions by a body of evidence will meet the conditions for a _ranking function_. Degrees of normic support are used to develop an alternative epistemology of degrees of belief that contrasts sharply with the more standard Bayesian approach. A defence of what Lewis in _Counterfactuals_ (1973) calls the ‘limit assumption’ is outlined.

Say that two goals are _normatively coincident_ just in case one cannot aim for one goal without automatically aiming for the other. While knowledge and justification are distinct epistemic goals, this chapter begins from the suggestion that they are nevertheless normatively coincident—aiming for knowledge and aiming for justification are one and the same activity. A number of consequences are derived from this, including limitations on how we can ascribe justification and knowledge in lottery cases. Ultimately, the suggestion is shown to (...) be in tension with the widespread _risk minimisation conception_ of justification, according to which one has justification for believing a proposition just when it is highly probable, given one’s evidence. (shrink)

In this chapter it is argued that the hybrid theory of justification has difficulty in accounting for justification _comparisons_—a problem shared by many ‘knowledge-first’ approaches to justification. The formal model of normic support is enriched to accommodate _degrees_ of normic support, enabling us to distinguish two versions of the normic theory of justification; a simple normic theory and a _threshold_ normic theory, on which justification requires a level of normic support that exceeds a given threshold value. One option explored here (...) is to let the value of the threshold be fixed as a function of one’s current practical interests. (shrink)

This book explores a question central to philosophy--namely, what does it take for a belief to be justified or rational? According to a widespread view, whether one has justification for believing a proposition is determined by how probable that proposition is, given one's evidence. In this book this view is rejected and replaced with another: in order for one to have justification for believing a proposition, one's evidence must normically support it--roughly, one's evidence must make the falsity of that proposition (...) abnormal in the sense of calling for special, independent explanation. This conception of justification bears upon a range of topics in epistemology and beyond. Ultimately, this way of looking at justification guides us to a new, unfamiliar picture of how we should respond to our evidence and manage our own fallibility. This picture is developed here. (shrink)

In this chapter the formal features of safety and of normic support are explored using sphere models for conditional logic. The formal structure of normic support is contrasted with the formal structure of probabilistic support, and it is argued that the former corresponds more closely with the formal structure of epistemic justification. It is argued, in particular, that normic support exhibits a number of crucial formal properties that probabilistic support lacks—properties such as agglomeration, amalgamation, cautious monotonicity, cumulative transitivity, and rational (...) monotonicity. (shrink)

A number of epistemologists have attempted to refine the risk minimisation conception of justification, proposing more complex probabilistic rules for justification that enable one to avoid the lottery paradox without giving up multiple premise closure. These attempts to refine the risk minimisation conception often appear ad hoc, and are also beset by formal difficulties. An important result, proved by Douven and Williamson (in 2006), comes close to showing that the ambition behind these refinements cannot be realised. In this chapter it (...) is shown that a significant limitation of Douven and Williamson’s proof—namely, its assumption of a finite, uniform probability space—can be overcome, and the result is available for infinite probability spaces as well. While a normic theory of justification is able to accommodate multiple premise closure within a fallibilist framework, it is concluded that no purely probabilistic theory of justification can do so. (shrink)

Sometimes appearances provide epistemic support that gets undercut later. In an earlier paper I argued that standard Bayesian update rules are at odds with this phenomenon because they are ‘rigid’. Here I generalize and bolster that argument. I first show that the update rules of Dempster–Shafer theory and ranking theory are rigid too, hence also at odds with the defeasibility of appearances. I then rebut three Bayesian attempts to solve the problem. I conclude that defeasible appearances pose a more difficult (...) and pervasive challenge for formal epistemology than is currently thought. 1 The Challenge for Bayesianism1.1 Updating and experience1.2 The problem1.3 Objections2 The Challenge for Dempster–Shafer Theory2.1 Background on Dempster–Shafer theory2.2 The problem for Dempster–Shafer theory3 The Challenge for Ranking Theory4 The Appeal to Metacognition5 The Appeal to Richer Inputs6 The Appeal to a Generic Underminer7 Conclusion. (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)

This is a short discussion review of Wolfgang Spohn, The Laws of Belief, Oxford UP 2012. I argue, first, that it is important to account for incomparabilities in the plausibilities of possible worlds or propositions, and second, that the meaning of input parameters specifying the degree to which a proposition is to be accepted should be independent of the agent's belief state.

As the ongoing literature on the paradoxes of the Lottery and the Preface reminds us, the nature of the relation between probability and rational acceptability remains far from settled. This article provides a novel perspective on the matter by exploiting a recently noted structural parallel with the problem of judgment aggregation. After offering a number of general desiderata on the relation between finite probability models and sets of accepted sentences in a Boolean sentential language, it is noted that a number (...) of these constraints will be satisfied if and only if acceptable sentences are true under all valuations in a distinguished non-empty set W. Drawing inspiration from distance-based aggregation procedures, various scoring rule based membership conditions for W are discussed and a possible point of contact with ranking theory is considered. The paper closes with various suggestions for further research. (shrink)

In this paper we investigate three approaches to iterated contraction, namely: the Moderate (or Priority) contraction, the Natural (or Conservative) contraction, and the Lexicographic contraction. We characterise these three contraction functions using certain, arguably plausible, properties of an iterated contraction function. While we provide the characterisation of the first two contraction operations using rationality postulates of the standard variety for iterated contraction, we found doing the same for the Lexicographic contraction more challenging. We provide its characterisation using a variation of (...) Epistemic ranking function instead. (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.

The above account of natural modalities is forbiddingly subjectivistic; these modalities are explained only relative to a doxastic state, a ranking function. This chapter explores the extent to which this relativization can be undone; it thus carries a heavy philosophical weight. The issue is addressed in a fully general way. First it is defined what it means to call a feature of a ranking function objectifiable, i.e., what it means to uniquely reconstruct a ranking function and the way in which (...) it realizes that feature from an objectively true or false proposition associated with that realization (so that the function can also be called true or false). Then it is proved that belief is objectifiable and that reasons are not. The main part provides two objectification methods for causation, one that works only for sufficient causes and another that even works for causal overdetermination. Subjective laws are objectifiable if and only if their assumptions about the single case are so. These are substantial results that, as is finally explained, complete the projectivistic strategy pursued in this book. (shrink)

The final chapter intends to revive reason or rationality as a source of the a priori in a roughly Kantian spirit. Therefore, the two notions of apriority introduced in Chapter 6 and their dynamic character are more thoroughly explained. This helps grasping the conceptual a priori and its three forms of appearance. However, concepts are not the only source of the a priori. The rational structure of reasons is so as well, as is unfolded in a series of principles. The (...) starting point is the Basic Empiricist principle that is the core of all empiricist attitudes and that is shown to entail two coherence principles expressing something like the unity of science. Then the chapter proceeds to the fundamental issue of the truth-conduciveness of reasons and argues that the Basic Empiricist principle should be strengthened to the Basic Belief-Truth and Basic Reason-Truth Connection, which are further amended by certain stability assumptions. On the basis of Chapter 14, these stronger principles provably entail a weak principle of causality. The moral of all this is: there is substantial and rigorous theorizing about the a priori beyond the merely conceptual a priori. (shrink)

There is a large spectrum of representations of belief states besides probabilities and ranks. The chapter cannot give a complete survey, but it presents comparisons with a number of important accounts. It first discusses the predecessors of ranking theory: Shackle’s functions of potential surprise, Rescher’s hypothetical reasoning, and Cohen’s account of inductive support. It saves AGM belief revision theory from various critical examples. It carefully discusses Levi’s epistemology and finds discrepancies everywhere. It is critical of Lehrer’s account of knowledge and (...) undefeated justification. Pollock’s theory of defeasible reasoning looks similar, but is quite different. This leads to a discussion of the nature of normative epistemology. The chapter closes with brief comparative remarks on formal learning theory, possibility theory, Dempster-Shafer belief functions, and non-monotonic reasoning. (shrink)

Like conditional probability, conditional belief is not easy to understand. The chapter argues that conditional belief is best interpreted as suppositional belief. This raises the question of how supposition relates to updating and conditionalization as presented in chapter 5. The question leads to a ranking-theoretic treatment of auto-epistemology. Hence, the reflection principle is discussed and then shown to be equivalent to an auto-epistemological conditionalization rule that subsumes ordinary conditionalization. In this perspective, updating turns out to be an auto-epistemic version of (...) supposing. (shrink)

This is by far the longest chapter of the book. It deals with the most important and mysterious natural modality: causation. It proceeds from the basic idea underlying all deterministic and probabilistic theories of causation that a cause raises the metaphysical or epistemic status of the effect given the obtaining circumstances. For direct causes the circumstances can be explained in a non-circular way. Thus, the basic idea immediately yields a precise explication of deterministic direct causes in ranking-theoretic terms. The explication (...) is extended to direct causal dependence between variables and allows proving all basic principles such as the common cause principle and the causal Markov condition. Hence, causal theorizing in terms of Bayesian nets works here equally well. These definitions of causal relations are frame-relative, but they allow inferring what causation is in absolute terms. Moreover, a way is shown of how not to deny, but to accommodate so-called interactive forks, which prima facie violate the common cause principle. Finally, the analysis is extended to indirect causation by a detailed argument for the transitivity of causation, which, however, entails the transitivity of causal dependence between variables only under additional assumptions. All this is done with a continuous look at many examples and many other accounts of causation. An appendix briefly extends the analysis to causal explanation. And a final appendix argues that inference to the best explanation is basically the ranking analogue of Bayes’ theorem and, moreover, that there is no problem of the catch-all hypothesis within the ranking-theoretic framework. (shrink)

This is the central chapter of this book. It criticizes belief revision theory for providing only a non-iterable account of belief change and thus only an incomplete dynamics of belief. An analysis of the incompleteness motivates the basic notion of this book: ranking functions. The chapter introduces some variants of this notion and develops its basic theory. It explains how ranks represent belief, and by defining conditional ranks, it is able to adequately explicate conditional belief. The latter notion finally allows (...) stating a complete dynamics of belief by specifying various rules of conditionalization. Expansion, revision, and contraction turn out to be special cases of those rules. An appendix discusses the existing insufficient attempts at iterated belief change, and a further appendix explicates so-called multiple contraction in ranking-theoretic terms. (shrink)

This account of laws is naturally extended to an account of ceteris paribus laws or conditions. In the main, “ceteris paribus” is here interpreted as “other things being normal”. A brief review displays how unsatisfactory current accounts of ceteris paribus conditions are. Relative to a ranking function, however, normal and exceptional conditions can be precisely explicated in a way that avoids the inadequacies of other accounts. In particular, this explication makes sense of double and higher degrees of exceptionality. Starting from (...) the old observation that reduction sentences for dispositions hold only ceteris paribus, the chapter then proceeds to an analysis of dispositions. It is argued that those reduction sentences are defeasibly a priori and, if qualified by normal conditions, even unrevisably a priori; this is what we a priori know about dispositions. The chapter concludes with a comparative discussion of the metaphysical and epistemological aspects of dispositions, confirming the account given in this chapter. (shrink)

This chapter introduces and discusses the standard static laws of belief, consistency and deductive closure, as well as the first steps towards a dynamics of belief. The latter amounts to an introduction into AGM belief revision theory and some variant formulations. The chapter thus explains the three kinds of belief change commonly taken as basic: expansions, revisions, and contractions.

The remaining chapters are devoted to various philosophical applications of ranking theory. The first goal is to understand natural modalities as covertly epistemological modalities by a kind of projectivistic strategy on the basis of ranking theory, which has already proved how well it accounts for epistemological modalities. The first of those natural modalities, lawlikeness, is treated in this chapter. A review of the difficulties with that notion concludes that laws are best understood via their inductive behavior. The study of that (...) behavior arrives, after thorough argument, at the notion of a subjective (deterministic) law that is simply the ranking-theoretic analogue of the notion of a Bernoulli measure or statistical law. It turns out then that de Finetti’s philosophy of probability, incarnated in his representation theorem, can be fully carried over to the deterministic side. Thus, the confirmation of deterministic laws works in essentially the same way as that of statistical laws. The chapter concludes with some observations about the apriority of the belief in lawfulness, i.e., in the uniformity of nature. (shrink)

Probability theory will provide the all-important guide-line and standard of comparison of ranking theory, as it will be developed in this book. Therefore, this chapter provides a short introduction into Bayesian epistemology. However, as the lottery paradox shows, probability theory is unable to capture the notion of belief. This point initiates the subsequent exploration of how to construct a theory that is able to represent belief while preserving the epistemological virtues of probability theory.

The previous chapters display a close similarity between probability and ranking theory. This calls for in-depth analysis of their relation. Various ways of formally integrating ranks and probabilities are discussed. It is argued that non-standard probabilities are disproportionate, that Popper measures are unsatisfactory (for the same reasons as AGM belief revision theory), and that the best unification is provided by so-called probabilified ranking functions. Still, the preferred attitude is so-called methodological separatism. The substantial convergences and divergences of ranks and probabilities (...) are then discussed under the experiential, the inferential, the behavioral, the truth, the reality, and the objectivity aspect. Since the behavioral aspect is not further dealt with in the book, an appendix presents the so far best account of decision theory in ranking-theoretic terms. (shrink)

The antonym of (conditional) relevance is (conditional) irrelevance or independence. The chapter introduces this notion and develops its theory. It thus turns out that the full theory of Bayesian nets can be carried over to ranking theory. This results in a most illuminating (and extensively exemplified) algorithmic theory of the propagation of evidential impact within ranking nets and thus of inductive reasoning. An appendix shows how this theory extends to an account of (independent) testimonies (that is much simpler than its (...) probabilistic counterpart). (shrink)

Ranks are cardinal degrees of belief (that also represent belief). This calls for a theory of measurement of those numerical ranks. The chapter provides no less than three different methods of measurement. The basic theory is that of difference measurement. The main measurement method uniquely reconstructs a ranking function from its iterated contraction behavior on a ratio scale. Thereby, it offers a complete set of laws of iterated contraction, thus solving an old problem of belief revision theory. The chapter should (...) be understood as providing additional justification of ranking theory. At the end, two possible further lines of justification, a Cox-type of justification and an analogy to the Dutch book argument, are presented and discussed. (shrink)

Conditional ranks also allow explicating a subjective notion of (deductive or non-deductive) reason, i.e., positive relevance, which is at the base of all current notions of (theoretical) reasons. The chapter introduces and defends this explication; it distinguishes sufficient, necessary, insufficient, and supererogatory reasons (in an unconditional and a conditional version); it states the formal behavior of reasons thus explicated; it explains how such reasons can be weighed; and it shows that it is reasons precisely in this sense that drive the (...) dynamics of belief. Moreover, it explains what maximal certainty and unrevisability mean in ranking-theoretic terms and thus provides a dynamic approach to two notions of apriority: (traditional) unrevisable apriority and so-called defeasible apriority. (shrink)

This chapter deals with the Agrippan justification trilemma. First, the account of justification that goes with the ranking-theoretic account of reasons is made explicit and related to various other accounts of justification. Thereby it becomes clear how various forms of externalism and contextualism can be accommodated in the ranking-theoretic picture. Then the chapter proceeds to the issue of the experiential belief base. Referring to perception or even direct perception does not really get at this base, since both notions are frame-relative. (...) Hence, the chapter goes on to discuss impressions, appearances, or sense data. They come in two forms. Appearances as if _p_ are argued to be defeasibly a priori reasons for _p_ itself, and vice versa; this is the so-called Schein-Sein principle, which suggests a coherentist construal of the belief base. This form is to be distinguished from the demonstrative form “it appears _thus_ to me”. The latter leads to an investigation of contents of consciousness, which are characterized by the so-called Conscious Essence principle, and of the epistemological role of those contents. The chapter arrives at a peculiar mixture of minimal externalism, strong foundationalism, and coherentism, which, however, is a strict consequence of those two principles within the ranking-theoretic account. (shrink)

This chapter provides a philosophical introduction on skepticim and the perennial problem of induction. It transforms the latter into the issue of the dynamics of belief that drives this book and emphasizes the normative character of this issue. Finally, it gives an overview of the contents of this book.

This chapter introduces and discusses the nature of the objects of belief as well as belief itself, in a non-committal way that everybody can share and in a philosophically opinionated way. It thus lays the algebraic as well as the philosophical foundations of this book.

In the context of a general framework for belief dynamics which interprets revision as doxastic constraint satisfaction, we discuss a proposal for revising quasi-probabilistic belief measures with finite sets of graded conditionals. The belief states are ranking measures with divisible values (generalizing Spohn’s epistemology), and the conditionals are interpreted as ranking constraints. The approach is inspired by the minimal information paradigm and based on the principle-guided canonical construction of a ranking model of the input conditionals. This is achieved by extending (...) techniques known from conditional default reasoning. We give an overview of how it handles different principles for conditional and parallel revision and compare it with similar accounts. (shrink)

Ranking functions have been introduced under the name of ordinal conditional functions in Spohn (1988; 1990). They are representations of epistemic states and their dynamics. The most comprehensive and up to date presentation is Spohn (manuscript).

Degrees of belief are familiar to all of us. Our confidence in the truth of some propositions is higher than our confidence in the truth of other propositions. We are pretty confident that our computers will boot when we push their power button, but we are much more confident that the sun will rise tomorrow. Degrees of belief formally represent the strength with which we believe the truth of various propositions. The higher an agent’s degree of belief for a particular (...) proposition, the higher her confidence in the truth of that proposition. For instance, Sophia’s degree of belief that it will be sunny in Vienna tomorrow might be .52, whereas her degree of belief that the train will leave on time might be .23. The precise meaning of these statements depends, of course, on the underlying theory of degrees of belief. These theories offer a formal tool to measure degrees of belief, to investigate the relations between various degrees of belief in different propositions, and to normatively evaluate degrees of belief. (shrink)

This paper combines various structures representing degrees of belief, degrees of disbelief, and degrees of non-belief (degrees of expectations) into a unified whole. The representation uses relations of comparative necessity and possibility, as well as non-probabilistic functions assigning numerical values of necessity and possibility. We define all-encompassing necessity structures which have weak expectations (mere hypotheses, guesses, conjectures, etc.) occupying the lowest ranks and very strong, ineradicable ('a priori') beliefs occupying the highest ranks. Structurally, there are no differences from the top (...) to the bottom. I argue that belief is a vague notion, and that thresholds for belief, if there are any, are context-dependent. (shrink)

"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)

This is a survey paper. Contents: 1 Introduction ― 2 Preliminary remarks on information, truth and mind - 2.1 Remarks on information and truth - 2.2 Some clues from the philosophy of mind - 2.3 Functionalism as applied to belief revision - 2.4 Filling in the parameters ― 3 Belief change = revision + reflection - 3.1 Foundationalism - 3.2 Coherentism ― 4 Inference operations for simple change operations: three examples - 4.1 Example 1: flat data bases - 4.2 Example (...) 2: up-sets in prioritized data bases - 4.3 Example 3: maximal-consistent sets in prioritized data bases ― 5 Representing coherentist belief change by operations on prioritized data bases - 5.1 Moderate revision - 5.2 Revision by comparison ― 6 A very brief history of belief revision - 6.1 The 1980s: AGM's classical model - 6.2 The 1990s: iteration - 6.3 The 2000s: merging ― 7 Conclusion. (shrink)

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.

This paper starts by indicating the analysis of Hempel's conditions of adequacy for any relation of confirmation (Hempel, 1945) as presented in Huber (submitted). There I argue contra Carnap (1962, Section 87) that Hempel felt the need for two concepts of confirmation: one aiming at plausible theories and another aiming at informative theories. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative theories. The main part of the paper (...) consists in working out the claim that one can have Hempel's cake and eat it too - in the sense that there is a logic of theory assessment that takes into account both of the two conflicting aspects of plausibility and informativeness. According to the semantics of this logic, a is an acceptable theory for evidence β if and only if a is both sufficiently plausible given β and sufficiently informative about β. This is spelt out in terms of ranking functions (Spohn, 1988) and shown to represent the syntactically specified notion of an assessment relation. The paper then compares these acceptability relations to explanatory and confirmatory consequence relations (Flach, 2000) as well as to nonmonotonic consequence relations (Kraus et al., 1990). It concludes by relating the plausibility-informativeness approach to Carnap's positive relevance account, thereby shedding new light on Carnap's analysis as well as solving another problem of confirmation theory. (shrink)

The paper provides an argument for the thesis that an agent’s degrees of disbelief should obey the ranking calculus. This Consistency Argument is based on the Consistency Theorem. The latter says that an agent’s belief set is and will always be consistent and deductively closed iff her degrees of entrenchment satisfy the ranking axioms and are updated according to the ranktheoretic update rules.