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)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

The 1985 paper by Carlos Alchourrón, Peter Gärdenfors, and David Makinson, “On the Logic of Theory Change: Partial Meet Contraction and Revision Functions” was the starting-point of a large and rapidly growing literature that employs formal models in the investigation of changes in belief states and databases. In this review, the first twenty-five years of this development are summarized. The topics covered include equivalent characterizations of AGM operations, extended representations of the belief states, change operators not included in the original (...) framework, iterated change, applications of the model, its connections with other formal frameworks, computatibility of AGM operations, and criticism of the model. (shrink)

A descriptor is a set of sentences that are truth-functional combinations of expressions of the form \ , where \ is a metalinguistic belief predicate and p a sentence in the object language in which beliefs are expressed. Descriptor revision ) is an operation of belief change that takes us from a belief set K to a new belief set \ where \ is a descriptor representing the success condition. Previously studied operations of belief change are special cases of descriptor (...) revision, hence sentential revision can be represented as \ , contraction as \ , multiple contraction as \ , replacement as \ , etc. General models of descriptor revision are constructed and axiomatically characterized. The common selection mechanisms of AGM style belief change cannot be used, but they can be replaced by choice functions operating directly on the set of potential outcomes . The restrictions of this construction to sentential revision ) and sentential contraction give rise to operations with plausible properties that are also studied in some some detail. (shrink)

An extended review of what is known about the formal behaviour of nonmonotonic inference operations, including those generated by the principal systems in the artificial intelligence literature. Directed towards computer scientists and others with some background in logic.

In earlier papers (Lindström & Rabinowicz, 1989. 1990), we proposed a generalization of the AGM approach to belief revision. Our proposal was to view belief revision as a relation rather thanas a function on theories (or belief sets). The idea was to allow for there being several equally reasonable revisions of a theory with a given proposition. In the present paper, we show that the relational approach is the natural result of generalizing in a certain way an approach to belief (...) revision due to Adam Grove. In his (1988) paper, Grove presents two closely related modelings of functional belief revision, one in terms of a family of "spheres" around the agent's theory G and the other in terms of an epistemic entrenchment ordering of propositions. The "sphere"-terminology is natural when one looks upon theories and propositions as being represented by sets of possible worlds. Grove's spheres may be thought of as possible "fallback" theories relative to the agent's original theory: theories that he may reach by deleting propositions that are not "sufficiently" entrenched (according to standards of sufficient entrenchment of varying stringency). To put it differently, fallbacks are theories that are closed upwards under entrenchment The entrenchment ordering can be recovered from the family of fallbacks by the definition: A is at least as entrenched as B iff A belongs to every fallback to which B belongs. To revise a theory T with a proposition A, we go to the smallest sphere that contain A-worlds and intersect it with A. The relational notion of belief revision that we are interested in, results from weakening epistemic entrenchment by not assuming it to be connected. I.e., we want to allow that some propositions may be incomparable with respect to epistemic entrenchment. As a result, the family of fallbacks around a given theory will no longer have to be nested. This change opens up the possibility for several different ways of revising a theory with a given proposition. (shrink)

Prioritized bases, i.e., weakly ordered sets of sentences, have been used for specifying an agent’s ‘basic’ or ‘explicit’ beliefs, or alternatively for compactly encoding an agent’s belief state without the claim that the elements of a base are in any sense basic. This paper focuses on the second interpretation and shows how a shifting of priorities in prioritized bases can be used for a simple, constructive and intuitive way of representing a large variety of methods for the change of belief (...) states—methods that have usually been characterized semantically by a system-of-spheres modeling. Among the methods represented are ‘radical’, ‘conservative’ and ‘moderate’ revision, ‘revision by comparison’ in its raising and lowering variants, as well as various constructions for belief expansion and contraction. Importantly, none of these methods makes any use of numbers. (shrink)

This is a discussion of the problem of extending the basic AGM belief revision theory to iterated belief revision: the problem of formulating rules, not only for revising a basic belief state in response to potential new information, but also for revising one’s revision rules in response to potential new information. The emphasis in the paper is on foundational questions about the nature of and motivation for various constraints, and about the methodology of the evaluation of putative counterexamples to proposed (...) constraints. Some specific constraints that have been proposed are criticized. The paper emphasizes the importance of meta-information—information about one’s sources of information—and argues that little of substance can be said about constraints on iterated belief revision at a level of abstraction that lacks the resources for explicit representation of meta-information. (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)

ABSTRACT We consider in this work the problem of iterated belief revision. We propose a family of belief revision operators called revision with memory operators and we give a logical (both syntactical and semantical) characterization of these operators. They obey what we call the principle of strong primacy of update: when one revises his beliefs by a new evidence, then all possible worlds that satisfy this new evidence become more reliable than those that do not. We show that those operators (...) have a satisfying behaviour concerning the iteration of the revision process. Then we provide four particular operators of this family. (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 ...

Blockage contraction is an operation of belief contraction that acts directly on the outcome set, i.e. the set of logically closed subsets of the original belief set K that are potential contraction outcomes. Blocking is represented by a binary relation on the outcome set. If a potential outcome X blocks another potential outcome Y, and X does not imply the sentence p to be contracted, then Y ≠ K ÷ p. The contraction outcome K ÷ p is equal to the (...) (unique) inclusion-maximal unblocked element of the outcome set that does not imply p. Conditions on the blocking relation are specified that ensure the existence of such a unique inclusion-maximal set for all sentences p. Blockage contraction is axiomatically characterized and its relations to AGM-style operations are investigated. In a finite-based framework, every transitively relational partial meet contraction is also a blockage contraction. (shrink)

Develops a concept of revision, akin in spirit to AGM partial meet revision, but in which the postulate of 'success' may fail. The basic idea is to see such an operation as composite, with a pre-processor using a priori considerations to resolve the question of whether to revise, following which another operation revises in a manner that protects the a priori material.

A study in the logic of theory change, examining the properties of maxichoice contraction and revision operations.

Provability, Computability and Reflection.

Change, Choice and Inference develops logical theories that are necessary both for the understanding of adaptable human reasoning and for the design of intelligent systems. The book shows that reasoning processes - the drawing on inferences and changing one's beliefs - can be viewed as belonging to the realm of practical reason by embedding logical theories into the broader context of the theory of rational choice. The book unifies lively and significant strands of research in logic, philosophy, economics and artificial (...) intelligence. It elaborates on the relevant theories and provides a mathematically precise foundation for the thesis that large parts of theoretical reason can be subsumed under practical reason. (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)

Specified meet contraction is the operation defined by the identity where ∼ is full meet contraction and f is a sentential selector, a function from sentences to sentences. With suitable conditions on the sentential selector, specified meet contraction coincides with the partial meet contractions that yield a finite-based contraction outcome if the original belief set is finite-based. In terms of cognitive realism, specified meet contraction has an advantage over partial meet contraction in that the selection mechanism operates on sentences rather (...) than on temporary infinite structures (remainders) that are cognitively inaccessible. Specified meet contraction provides a versatile framework in which other types of contraction, such as severe withdrawal and base-generated contraction, can be expressed with suitably chosen properties of the sentential selector. (shrink)

The postulate of recovery is commonly regarded to be the intuitively least compelling of the six basic Gärdenfors postulates for belief contraction. We replace recovery by the seemingly much weaker postulate of core-retainment, which ensures that if x is excluded from K when p is contracted, then x plays some role for the fact that K implies p. Surprisingly enough, core-retainment together with four of the other Gärdenfors postulates implies recovery for logically closed belief sets. Reasonable contraction operators without recovery (...) do not seem to be possible for such sets. Instead, however, they can be obtained for non-closed belief bases. Some results on partial meet contractions on belief bases are given, including an axiomatic characterization and a non-vacuous extension of the AGM closure condition. (shrink)

Five types of constructions are introduced for non-prioritized belief revision, i.e., belief revision in which the input sentence is not always accepted. These constructions include generalizations of entrenchment-based and sphere-based revision. Axiomatic characterizations are provided, and close interconnections are shown to hold between the different constructions.

Epistemic conditionals have often been thought to satisfy the Ramsey test : If A, then B is acceptable in a belief state G if and only if B should be accepted upon revising G with A. But as Peter Gärdenfors has shown, RT conflicts with the intuitively plausible condition of Preservation on belief revision. We investigate what happens if RT is retained while Preservation is weakened, or vice versa. We also generalize Gärdenfors' approach by treating belief revision as a relation (...) rather than as a function.In our semantic approach, the same relation is used to model belief revision and to give truth-conditions for conditionals. The approach validates a weak version of the Ramsey Test — essentially, a restriction of RT to maximally consistent belief states. (shrink)

The outcome set of a belief change operator is the set of outcomes that can be obtained with it. Axiomatic characterizations are reported for the outcome sets of the standard AGM contraction operators and eight types of base-generated contraction. These results throw new light on the properties of some of these operators.