The book can also be used as an excellent text for graduate-level courses in AI, operations research, or applied probability.

Degrees of belief; Dempster's rule of combination; Simple and separable support functions; The weights of evidence; Compatible frames of discernment; Support functions; The discernment of evidence; Quasi support functions; Consonance; Statistical evidence; The dual nature of probable reasoning.

The AGM (Alchourrón-Gärdenfors-Makinson) model of belief change is extended to cover changes on sets of beliefs that are not closed under logical consequence (belief bases). Three major types of change operations, namely contraction, internal revision, and external revision are axiomatically characterized, and their interrelations are studied. In external revision, the Levi identity is reversed in the sense that one first adds the new belief to the belief base, and afterwards contracts its negation. It is argued that external revision represents an (...) intuitively plausible way of revising one's beliefs. Since it typically involves the temporary acceptance of an inconsistent set of beliefs, it can only be used in belief representations that distinguish between different inconsistent sets of belief. (shrink)

This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gardenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourron and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of (...) its axiomatic bases), that fails to imply the proposition being eliminated. In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate "partial meet contraction functions", which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gardenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are "relational" and "transitively relational", are studied in detail, and their connections with certain "supplementary postulates" of Gardenfors investigated, with a further representation theorem established. (shrink)

In the middle of the 1980s, logical tools were discovered that make it possible to model changes in belief and knowledge in entirely new ways. These logical tools turned out to be applicable both to human beliefs and to the contents of databases. This is the first textbook in this new area. It contains both discursive chapters with a minimum of formalism and formal chapters in which proofs and proof methods are presented. By using different selections from the formal section (...) the book can be used on all levels of University education. (shrink)

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.

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)

The main ingredients of Spohn's theory of epistemic beliefs are (1) a functional representation of an epistemic state called a disbelief function and (2) a rule for revising this function in light of new information. The main contribution of this paper is as follows. First, we provide a new axiomatic definition of an epistemic state and study some of its properties. Second, we study some properties of an alternative functional representation of an epistemic state called a Spohnian belief function. Third, (...) we state a rule for combining disbelief functions that is mathematically equivalent to Spohn's belief revision rule. Whereas Spohn's rule is defined in terms of the initial epistemic state and some features of the final epistemic state, the rule of combination is defined in terms of the initial epistemic state and the incremental epistemic state representing the information gained. Fourth, we state a rule of subtraction that allows one to recover the addendum epistemic state from the initial and final epistemic states. Fifth, we study some properties of our rule of combination. One distinct advantage of our rule of combination is that besides belief revision, it can be used to describe an initial epistemic state for many variables when this information is given as several independent epistemic states each involving few variables. Another advantage of our reformulation is that we can show that Spohn's theory of epistemic beliefs shares the essential abstract features of probability theory and the Dempster-Shafer theory of belief functions. One implication of this is that we have a ready-made algorithm for propagating disbelief functions using only local computation. (shrink)

The standard use of the propositional calculus ('P.C.?) in analyzing the validity of inferences involving conditionals leads to fallacies, and the problem is to determine where P.C. may be ?safely? used. An alternative analysis of criteria of reasonableness of inferences in terms of conditions of justification rather than truth of statements is proposed. It is argued, under certain restrictions, that P. C. may be safely used, except in inferences whose conclusions are conditionals whose antecedents are incompatible with the premises in (...) the sense that if the antecedent became known, some of the previously asserted premises would have to be withdrawn. (shrink)

The book was planned and written as a single, sustained argument. But earlier versions of a few parts of it have appeared separately. The object of this book is both to establish the existence of the paradoxes, and also to describe a non-Pascalian concept of probability in terms of which one can analyse the structure of forensic proof without giving rise to such typical signs of theoretical misfit. Neither the complementational principle for negation nor the multiplicative principle for conjunction applies (...) to the central core of any forensic proof in the Anglo-American legal system. There are four parts included in this book. Accordingly, these parts have been written in such a way that they may be read in different orders by different kinds of reader. (shrink)

I argue that the analysis most capable of systematising our intuitions about coherence as a relation is one according to which a set of beliefs, A, coheres with another set, B, if and only if the set-theoretical union of A and B is a coherent set. The second problem I consider is the role of coherence in epistemic justification. I submit that there are severe problems pertaining to the idea, defended most prominently by Keith Lehrer, that justification amounts to coherence (...) with an acceptance system. Instead I advance a more dynamic approach according to which the problem of justification is the problem of how to merge new information with old coherently, a process which is seen to be closely connected with relational coherence. (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, it is argued that 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, and an axiomatic characterization of this operation is given. Unlike most belief revision operations, the one developed here can handle both multiple belief revision and iterated belief revision.

The AGM theory of belief contraction is extended tomultiple contraction, i.e. to contraction by a set of sentences rather than by a single sentence. There are two major variants: Inpackage contraction all the sentences must be removed from the belief set, whereas inchoice contraction it is sufficient that at least one of them is removed. Constructions of both types of multiple contraction are offered and axiomatically characterized. Neither package nor choice contraction can in general be reduced to contractions by single (...) sentences; in the finite case choice contraction allows for reduction. (shrink)

We describe a model of iterated belief revision that extends the AGM theory of revision to account for the effect of a revision on the conditional beliefs of an agent. In particular, this model ensures that an agent makes as few changes as possible to the conditional component of its belief set. Adopting the Ramsey test, minimal conditional revision provides acceptance conditions for arbitrary right-nested conditionals. We show that problem of determining acceptance of any such nested conditional can be reduced (...) to acceptance tests for unnested conditionals. Thus, iterated revision can be accomplished in a "virtual" manner, using uniterated revision. (shrink)

This paper studies the idea of conservatism with respect to belief change strategies in the setting of unary, iterated belief revision functions (based on the conclusions of Rott, ‘Coherence and Conservatism in the Dynamics of Belief, Part I: Finding the Right Framework’, Erkenntnis 50, 1999, 387–412). Special attention is paid to the case of ‘basic belief change’ where neither the (weak) AGM postulates concerning conservatism with respect to beliefs nor the (stong) supplementary AGM postulates concerning dispositional coherence need to be (...) satisfied. One‐step belief change generated by ‘basic entrenchment’ is combined with a natural conservative method of revising entrenchment relations. A logical characterization of this method is presented, and it is compared with three other methods known from the literature which I call ‘external’, ‘radical’ and ‘moderate’ belief revision. While conservative belief change turns out to be incoherent in its treatment of the recency of information, moderate belief change is more satisfactory in this respect. (shrink)

SUGGESTED COURSES Introductory level A (Requires very little background in logic .): 4: -9 - - -7 -2 Introductory level B: -9,:+-+ -,2:+,2: -,3:20+-22+ -7 -2 ...

This paper is concerned with formal aspects of the logic of theory change, and in particular with the process of shrinking or contracting a theory to eliminate a proposition. It continues work in the area by the authors and Peter Gärdenfors. The paper defines a notion of safe contraction of a set of propositions, shows that it satisfies the Gärdenfors postulates for contraction and thus can be represented as a partial meet contraction, and studies its properties both in general and (...) under various natural constraints. (shrink)

In this paper I discuss the foundations of a formal theory of coherent and conservative belief change that is (a) suitable to be used as a method for constructing iterated changes of belief, (b) sensitive to the history of earlier belief changes, and (c) independent of any form of dispositional coherence. I review various ways to conceive the relationship between the beliefs actually held by an agent and her belief change strategies (that also deal with potential belief sets), show the (...) problems they suffer from, and suggest that belief states should be represented by unary revision functions that take sequences of inputs. Three concepts of coherence implicit in current theories of belief change are distinguished: synchronic, diachronic and dispositional coherence. Diachronic coherence is essentially identified with what is known as conservatism in epistemology. The present paper elaborates on the philosophical motivation of the general framework; formal details and results are provided in a companion paper. (shrink)

The central claim of the paper is, roughly, that the fact that it looks to somebody as if p is a defeasibly a priori reason for assuming that p (and vice versa), for any person, even for the perceiver himself. As a preparation, it outlines a doxastic conception suitable to explicate this claim and explains how to analyse dispositions within this conception. Since an observable p has the disposition to look as if p, this analysis generalizes to the central claim (...) which is then argued to be at the bottom of coherentism. Thus, the defense of the claim supports coherentism as opposed to foundationalism and at the same time provides an answer to skepticism about the external world. (shrink)

In this book Pollock deals with the subject of probabilistic reasoning, making general philosophical sense of objective probabilities and exploring their ...

The paper proposes two principles of coherence (thus taking up work started in Spohn (1991) "A Reason for Explanation: Explanations Provide Stable Reasons"). The latter indeed serves as a weak, but precise explication of the notion of coherence as it is used in the current epistemological discussion. After discussing their epistemological setting, the paper considers four ways of establishing these principles. They may be inferred neither from enumerative induction, nor from the nature of propositions as objects of belief, nor in (...) a Kantian way from self-consciousness. Rather, I propose a fairly rigorous way to infer them from an even more fundamental rationality principle of non-dogmatism and an elementary theory of perception. (shrink)

This is a survey paper. Contents: 1 Introduction -- 2 The representation of belief -- 3 Kinds of belief change -- 4 Coherence constraints for belief revision -- 5 Different modes of belief change -- 6 Two strategies for characterizing rational changes of belief - 6.1 The postulates strategy - 6.2 The constructive strategy -- 7 An abstract view of the elements of belief change -- 8 Iterated changes of belief -- 9 Further developments - 9.1 Variants and extensions of (...) belief revision - 9.2 Updates - 9.3 Default inferences as expectation revision - 9.4 Belief merging -- 10 Concluding remarks. (shrink)