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)

We show in this paper that the AGM postulates are too weak to ensure the rational preservation of conditional beliefs during belief revision, thus permitting improper responses to sequences of observations. We remedy this weakness by proposing four additional postulates, which are sound relative to a qualitative version of probabilistic conditioning. Contrary to the AGM framework, the proposed postulates characterize belief revision as a process which may depend on elements of an epistemic state that are not necessarily captured by a (...) belief set. We also show that a simple modification to the AGM framework can allow belief revision to be a function of epistemic states. We establish a model-based representation theorem which characterizes the proposed postulates and constrains, in turn, the way in which entrenchment orderings may be transformed under iterated belief revision. (shrink)

This paper presents a formalism that combines useful properties of both logic and probabilities. Like logic, the formalism admits qualitative sentences and provides symbolic machinery for deriving deductively closed beliefs and, like probability, it permits us to express if-then rules with different levels of firmness and to retract beliefs in response to changing observations. Rules are interpreted as order-of-magnitude approximations of conditional probabilities which impose constraints over the rankings of worlds. Inferences are supported by a unique priority ordering on rules (...) which is syntactically derived from the knowledge base. This ordering accounts for rule interactions, respects specificity considerations and facilitates the construction of coherent states of beliefs. Practical algorithms are developed and analyzed for testing consistency, computing rule ordering, and answering queries. Imprecise observations are incorporated using qualitative versions of Jeffrey's rule and Bayesian updating, with the result that coherent belief revision is embodied naturally and tractably. Finally, causal rules are interpreted as imposing Markovian conditions that further constrain world rankings to reflect the modularity of causal organizations. These constraints are shown to facilitate reasoning about causal projections, explanations, actions and change. (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 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, α is an acceptable theory for evidence β if and only if α 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.

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)

Belief revision theory concerns methods for reformulating an agent's epistemic state when the agent's beliefs are refuted by new information. The usual guiding principle in the design of such methods is to preserve as much of the agent's epistemic state as possible when the state is revised. Learning theoretic research focuses, instead, on a learning method's reliability or ability to converge to true, informative beliefs over a wide range of possible environments. This paper bridges the two perspectives by assessing the (...) reliability of several proposed belief revision operators. Stringent conceptions of minimal change are shown to occasion a limitation called inductive amnesia: they can predict the future only if they cannot remember the past. Avoidance of inductive amnesia can therefore function as a plausible and hitherto unrecognized constraint on the design of belief revision operators. (shrink)

In this paper it is argued that, in order to solve the problem of iterated belief change, both the belief state and its input should be represented as epistemic entrenchment (EE) relations. A belief revision operation is constructed that updates a given EE relation to a new one in light of an evidential EE relation. It is shown that the operation in question satisfies generalized versions of the Gärdenfors revision postulates. The account offered is motivated by Spohn's ordinal conditionalization functions, (...) and can be seen as the Jeffrization of a proposal considered by Rott. (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)

"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 paper compares the epistemological conception of Isaac Levi with mine. We are joined in both giving a constructive answer to the relation of belief and probability, without reducing one to the other. However, our constructions differ in at least nine more or less important ways, all discussed in the paper. In particular, the paper explains the similarities and differences of Shackle's functions of potential surprise, as used by Levi, and my ranking functions in formal as well as in philosophical (...) respects. The appendix explains how ranking and probability theory can be combined in the notion of a ranked probability measure (or probabilified ranking function). (shrink)

The paper builds on the basically Humean idea that A is a cause of B iff A and B both occur, A precedes B, and A raises the metaphysical or epistemic status of B given the obtaining circumstances. It argues that in pursuit of a theory of deterministic causation this ‘status raising’ is best explicated not in regularity or counterfactual terms, but in terms of ranking functions. On this basis, it constructs a rigorous theory of deterministic causation that successfully deals (...) with cases of overdetermination and pre-emption. It finally indicates how the account's profound epistemic relativization induced by ranking theory can be undone. Introduction Variables, propositions, time Induction first Causation Redundant causation Objectivization. (shrink)

The paper is based on ranking theory, a theory of degrees of disbelief (and hence belief). On this basis, it explains enumerative induction, the confirmation of a law by its positive instances, which may indeed take various schemes. It gives a ranking theoretic explication of a possible law or a nomological hypothesis. It proves, then, that such schemes of enumerative induction uniquely correspond to mixtures of such nomological hypotheses. Thus, it shows that de Finetti's probabilistic representation theorems may be transformed (...) into an account of confirmation of possible laws and that enumerative induction is equivalent to such an account. The paper concludes with some remarks about the apriority of lawfulness or the uniformity of nature. (shrink)

The characteristic difference between laws and accidental generalizations lies in our epistemic or inductive attitude towards them. This idea has taken various forms and dominated the discussion about lawlikeness in the last decades. Hence, ranking theory with its resources of formalizing defeasible reasoning or inductive schemes seems ideally suited to explicate the idea in a formal way. This is what the paper attempts to do. Thus it will turn out that a law is simply the deterministic analogue of a sequence (...) of independent, identically distributed random variables. This entails that de Finetti‘s representation theorems can be directly transformed into an account of confirmation of laws thus conceived. (shrink)

The characteristic difference between laws and accidental generalizations lies in our epistemic or inductive attitude towards them. This idea has taken various forms and dominated the discussion about lawlikeness in the last decades. Likewise, the issue about ceteris paribus conditions is essentially about how we epistemically deal with exceptions. Hence, ranking theory with its resources of defeasible reasoning seems ideally suited to explicate these points in a formal way. This is what the paper attempts to do. Thus it will turn (...) out that a law is simply the deterministic analogue of a sequence of independent, identically distributed random variables. This entails that de Finetti's representation theorems can be directly transformed into an account of confirmation of laws thus conceived. (shrink)

In this paper two theories of defeasible reasoning, Pollock's account and my theory of ranking functions, are compared, on a strategic level, since a strictly formal comparison would have been unfeasible. A brief summary of the accounts shows their basic difference: Pollock's is a strictly computational one, whereas ranking functions provide a regulative theory. Consequently, I argue that Pollock's theory is normatively defective, unable to provide a theoretical justification for its basic inference rules and thus an independent notion of admissible (...) rules. Conversely, I explain how quite a number of achievements of Pollock's account can be adequately duplicated within ranking theory. The main purpose of the paper, though, is not to settle a dispute with formal epistemology, but rather to emphasize the importance of formal methods to the whole of epistemology. (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)

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.

As the paper explains, it is crucial to epistemology in general and to the theory of causation in particular to investigate the properties of conditional independence as completely as possible. The paper summarizes the most important results concerning conditional independence with respect to two important representations of epistemic states, namely (strictly positive) probability measures and natural conditional (or disbelief or ranking) functions. It finally adds some new observations.

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.

Sometimes appearances provide prima facie support that is later undercut. In (Weisberg 2009) I argued that this phenomenon is incompatible with Bayesianism because its update rules have a formal property known as “rigidity”. Here I generalize that argument to cover Dempster-Shafer the-ory and Ranking Theory. The update rules of these models, in virtue of their rigidity, are similarly incompatible with the undermining of perceptual support. I then critique three solutions that have been proposed to the problem. I conclude that perceptual (...) undermining poses a more pervasive and more severe problem than is currently thought. (shrink)

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)