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)

This paper extends the AGM theory of belief revision to accommodate infinitary belief change. We generalize both axiomatization and modeling of the AGM theory. We show that most properties of the AGM belief change operations are preserved by the generalized operations whereas the infinitary belief change operations have their special properties. We prove that the extended axiomatic system for the generalized belief change operators with a Limit Postulate properly specifies infinite belief change. This (...) framework provides a basis for first-order belief revision and the theory of revising a belief state by a belief state. (shrink)

We present a general framework for representing belief-revision rules and use it to characterize Bayes's rule as a classical example and Jeffrey's rule as a non-classical one. In Jeffrey's rule, the input to a belief revision is not simply the information that some event has occurred, as in Bayes's rule, but a new assignment of probabilities to some events. Despite their differences, Bayes's and Jeffrey's rules can be characterized in terms of the same axioms: "responsiveness", which (...) requires that revised beliefs incorporate what has been learnt, and "conservativeness", which requires that beliefs on which the learnt input is "silent" do not change. To illustrate the use of non-Bayesian belief revision in economic theory, we sketch a simple decision-theoretic application. (shrink)

I develop a dynamic logic for reasoning about "interrogative belief revision", a new branch of belief revision theory that has been developed in a small number of papers, beginning with E. J. Olsson and D. Westlund's paper "On the role of the research agenda in epistemic change" [12]. In interrogative belief revision, epistemic states are taken to include a research agenda, consisting of questions the agent seeks to answer. I present a logic for (...) class='Hi'>revision of such epistemic states based on the notion of an epistemic strategy, a stable plan of action that determines changes in the agent's research agenda. This idea is a further development of an idea put forward in [6], that changes in the research agenda of an agent should be determined by stable, "long term" research interests. I provide complete axioms and a decidability result for the logic. (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)

We propose a dynamic hyperintensional logic of belief revision for non-omniscient agents, reducing the logical omniscience phenomena affecting standard doxastic/epistemic logic as well as AGM belief revision theory. Our agents don’t know all a priori truths; their belief states are not closed under classical logical consequence; and their belief update policies are such that logically or necessarily equivalent contents can lead to different revisions. We model both plain and conditional belief, then focus on (...) dynamic belief revision. The key idea we exploit to achieve non-omniscience focuses on topic- or subject matter-sensitivity: a feature of belief states which is gaining growing attention in the recent literature. (shrink)

We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge (...) and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws. (shrink)

In this paper two systems of AGM-like Paraconsistent Belief Revision are overviewed, both defined over Logics of Formal Inconsistency (LFIs) due to the possibility of defining a formal consistency operator within these logics. The AGM° system is strongly based on this operator and internalize the notion of formal consistency in the explicit constructions and postulates. Alternatively, the AGMp system uses the AGM-compliance of LFIs and thus assumes a wider notion of paraconsistency - not necessarily related to the notion (...) of formal consistency. (shrink)

We use the normality framework of Goodman and Salow (2018, 2021, 2023b) to investigate of dynamics of rational belief. The guiding idea is that people are entitled to believe that their circumstances aren’t especially abnormal. More precisely, a rational agent’s beliefs rule out all and only those possibilities that are either (i) ruled out by their evidence or (ii) sufficiently less normal than some other possibility not ruled out by their evidence. Working within this framework, we argue that the (...) logic of rational belief revision is much weaker than is usually supposed. We do so by isolating a natural family of orthodox principles about belief revision, describing realistic cases in which these principles seem to fail, and showing how these counterexamples are predicted by independently motivated models of the cases in question. In these models, whether one evidential possibility counts as sufficiently less normal than another is determined by underlying probabilities (together with a contextually determined question). We argue that the resulting probabilistic account of belief compares favorably with other such accounts, including Lockeanism (Foley, 1993), a ‘stability’ account inspired by Leitgeb (2017), the ‘tracking theory’ of Lin and Kelly (2012), and the influential precursor of Levi (1967). We show that all of these accounts yield subtly different but similarly heterodox logics of belief revision. (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)

The well known AGM framework for belief revision has recently been extended to include a model of the research agenda of the agent, i.e. a set of questions to which the agent wishes to find answers (Olsson & Westlund in Erkenntnis , 65 , 165–183, 2006 ). The resulting model has later come to be called interrogative belief revision . While belief revision has been studied extensively from the point of view of modal logic, (...) so far interrogative belief revision has only been dealt with in the metalanguage approach in which AGM was originally presented. In this paper, I show how to model interrogative belief revision in a modal object language using a class of operators for questions. In particular, the solution I propose will be shown to capture the notion of K-truncation , a method for agenda update in the case of expansion constructed by Olsson & Westlund. Two case studies are conducted: first, an interrogative extension of Krister Segerberg’s system DDL, and then a similar extension of Giacomo Bonanno’s modal logic for belief revision. Sound and complete axioms will be provided for both of the resulting logics. (shrink)

Postulational approaches attempt to understand the dynamics of belief revision by appealing to no more than the set of beliefs held by an agent and the logical relations between them. It is argued there that such an approach cannot work. A proper account of belief revision must also appeal to the arguments supporting beliefs, and recognize that those arguments can be defeasible. If we begin with a mature epistemological theory that accommodates this, it can be seen (...) that the belief revision operators on which the postulational theories are based are ill-defined. It is further argued that there is no way to repair the definitions so as to retain the spirit of those theory. Belief revision is better studied from within an independently motivated epistemological theory. (shrink)

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. (shrink)

Belief revision theories standardly endorse a principle of intensionality to the effect that ideal doxastic agents do not discriminate between pieces of information that are equivalent within classical logic. I argue that this principle should be rejected. Its failure, on my view, does not require failures of logical omniscience on the part of the agent, but results from a view of the update as _mighty_: as encoding what the agent learns might be the case, as well as what (...) must be. The view is motivated by consideration of a puzzle case, obtained by transposing into the context of belief revision a kind of scenario that Kit Fine has used to argue against intensionalism about counterfactuals. Employing the framework of truthmaker semantics, I go on to develop a novel account of belief revision, based on a conception of the update as mighty, which validates natural hyperintensional counterparts of the usual AGM postulates. (shrink)

Krister Segerberg proposed irrevocable belief revision, to be contrasted with standard belief revision, in a setting wherein belief of propositional formulas is modelled explicitly. This suggests that in standard belief revision is revocable: one should be able to unmake (‘revoke’) the fresh belief in the revision formula, given yet further information that contradicts it. In a dynamic epistemic logical setting for belief revision, for multiple agents, we investigate what the (...) requirements are for revocable belief revision. By this we not merely mean recovering belief in non-modal propositions, as in the recovery principle for belief contraction, but recovering belief in modal propositions: beliefs about beliefs. These requirements are almost never met, a surprising result. (shrink)

In this article, we propose a belief revision approach for families of (non-classical) logics whose semantics are first-order axiomatisable. Given any such (non-classical) logic , the approach enables the definition of belief revision operators for , in terms of a belief revision operation satisfying the postulates for revision theory proposed by Alchourrrdenfors and Makinson (AGM revision, Alchourrukasiewicz's many-valued logic. In addition, we present a general methodology to translate algebraic logics into classical logic. (...) For the examples provided, we analyse in what circumstances the properties of the AGM revision are preserved and discuss the advantages of the approach from both theoretical and practical viewpoints. (shrink)

This chapter discusses an account of belief-revision that is compatible with the rational belief of contradictions. In the process, a formal account of the model of rationality of the preceding chapter is provided. The account of belief-revision is contrasted with the familiar AGM account.

According to the cognitive model of psychopathology, maladaptive beliefs about oneself, others, and the world are the main factors contributing to the development and persistence of various forms of mental suffering. Therefore, the key therapeutic process of Cognitive Behavioral Therapy (CBT)—a therapeutic approach rooted in the cognitive model—is cognitive restructuring, i.e., a process of revision of such maladaptive beliefs. In this paper, I examine the philosophical assumptions underlying CBT and offer theoretical reasons to think that the effectiveness of (...) class='Hi'>belief revision in psychotherapy is very limited. This is the case, I argue, because the cognitive model wrongly assumes that our body of beliefs is unified, while it is in fact fragmented. (shrink)

In this rather technical paper we establish a useful combination of belief revision and verisimilitude according to which better theories provide better predictions, and revising with more verisimilar data results in theories that are closer to the truth. Moreover, this paper presents two alternative definitions of refined verisimilitude, which are more perspicuous than the algebraic version used in previous publications.

In this study, we examine the problem of belief revision, defined as deciding whic h of several initially-accepted sentences to disbelieve, when new information presents a l ogical inconsistency with the initial set. In the first three experiments, the initial sentence set included a conditional sentence, a non-conditional sentence, and an inferred conclusi on drawn from the first two. The new information contradicted the inferred conclusion. Results indicated that the conditional sentences were more readily abandoned than non-c onditional (...) sentences, even when either choice would lead to a consistent belief state, and that this preference was more pronounced when problems used natural language cover sto ries rather than symbols. The pattern of belief revision choices differed depending on whe ther the contradicted conclusion from the initial belief set had been a modus ponens or m odus tollens inference. Two additional experiments examined alternative model-theoretic definitions of minimal change to a belief state, using problems that contained multiple mo dels of the initial belief state and of the new information that provided the contradiction. The results indicated that people did not follow any of four formal definitions of minimal change on these problems. The new information and the contradiction it offered was not, for example, used to select a particular model of the initial belief state as a way of reconci ling the contradiction. The preferred revision was to retain only those initial sentences th at had the same, unambiguous truth value within and across both the initial and new info rmation sets. The study and results are presented in the context of certain logic-based for malizations of belief revision, syntactic and model-theoretic representations of belief stat es, and performance models of human deduction. Principles by which some types of sent ences might be more "entrenched" than others in the face of contradiction are also discuss ed from the perspective of induction and theory revision. (shrink)

This paper presents a probabilist paraconsistent theory of belief revision. This theory is based on a very general theory of probability, that fits with a wide range of classical and nonclassical logics. The theory incorporates a version of Jeffrey conditionalisation as its method of updating. A Dutch book argument is given, and the theory is applied to the problem of choosing a logical system.

In the literature of belief revision, it is widely accepted that: there is only one revision phase in belief revision which is well characterized by the Bayes’ Rule, Jeffrey’s Rule, etc.. However, as I argue in this article, there are at least four successive phases in belief revision, namely first/second order evaluation and first/second order revision. To characterize these phases, I propose mainly four rules of belief revision based on agent’s (...) criteria, and make one composition rule to characterize belief revision as a whole. (shrink)

This thesis deals with the theory of interrogative belief revision, a recent development in the subfield of philosophical logic known as belief revision theory. While classical belief revision theory deals with the dynamics of an agent’s beliefs, interrogative belief revision aims to describe the dynamics of the agent’s questions about the world, or research agenda, as well. Great emphasis is put on the interaction between belief change and changes in the research (...) agenda. I develop both the technical framework for this theory by constructing and investigating modal logics for reasoning about the dynamics of research interests, as well as pursuing further the philosophical foundations and applications of the theory. Among the contributions of the thesis are: - a novel interpretation of the theory founded on a notion of epistemic strategies - a solution to the previously open problem of contraction and revision in this context - several systems of modal logic that can be used to reason about interrogative belief revision - completeness results and decidability results for these logics - a characterization of a class of relevant conditionals based on a notion of agenda relevance. (shrink)

In this paper we present a new modeling for belief revision that is what we term irrevocable. This modeling is of philosophical interest since it captures some features of suppositional reasoning, and of formal interest since it is closely connected with AGM, yet provides for iterated belief revision. The analysis is couched in terms of dynamic doxastic logic.

The present paper aims at a synthesis of belief revision theory with the Sneed formalism known as the structuralist theory of science. This synthesis is brought about by a dynamisation of classical structuralism, with an abductive inference rule and base generated revisions in the style of Rott (2001). The formalism of prioritised default logic (PDL) serves as the medium of the synthesis. Why seek to integrate the Sneed formalism into belief revision theory? With the hybrid system (...) of the present investigation, a substantial simplification of the ranking information that is necessary to define revisions and contractions uniquely is achieved. This system is, furthermore, expressive enough to capture complex and non-trivial scientific examples. It is thus closely related to a novel research area within belief revision theory which addresses the dynamics of scientific knowledge. (shrink)

A syntactic formalism for the modeling of belief revision in perfect information games is presented that allows to define the rationality of a player's choice of moves relative to the beliefs he holds as his respective decision nodes have been reached. In this setting, true common belief in the structure of the game and rationality held before the start of the game does not imply that backward induction will be played. To derive backward induction, a “forward (...) class='Hi'>belief” condition is formulated in terms of revised rather than initial beliefs. Alternative notions of rationality as well as the use of knowledge instead of belief are also studied within this framework. Footnotes1 I would like to thank Wlodek Rabinowicz and three anonymous referees for very helpful comments. (shrink)

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)

In an earlier paper [Rational choice and AGM belief revision, _Artificial Intelligence_, 2009] a correspondence was established between the set-theoretic structures of revealed-preference theory (developed in economics) and the syntactic belief revision functions of the AGM theory (developed in philosophy and computer science). In this paper we extend the re-interpretation of those structures in terms of one-shot belief revision by relating them to the trichotomous attitude towards information studied in Garapa (Rev Symb Logic, 1–21, (...) 2020) where information may be either (1) fully accepted or (2) rejected or (3) taken seriously but not fully accepted. We begin by introducing the syntactic notion of _filtered belief revision_ and providing a characterization of it in terms of a mixture of both AGM revision and contraction. We then establish a correspondence between the proposed notion of filtered belief revision and the above-mentioned set-theoretic structures, interpreted as semantic _partial_ belief revision structures. We also provide an interpretation of the trichotomous attitude towards information in terms of the degree of implausibility of the information. (shrink)

We present different constructions for nonprioritized belief revision, that is, belief changes in which the input sentences are not always accepted. First, we present the concept of explanation in a deductive way. Second, we define multiple revision operators with respect to sets of sentences (representing explanations), giving representation theorems. Finally, we relate the formulated operators with argumentative systems and default reasoning frameworks.

The Egli-Milner power-ordering is used to define verisimilitude orderings on theories from preference orderings on models. The effects of the definitions on constraints such as stopperedness and soundness are explored. Orderings on theories are seen to contain more information than orderings on models. Belief revision is defined in terms of both types of orderings, and conditions are given which make the two notions coincide.

The paper attacks the widely held view that belief revision theories, as they have been studied in the past two decades, are founded on the Principle of Informational Economy. The principle comes in two versions. According to the first, an agent should, when accepting a new piece of information, aim at a minimal change of his previously held beliefs. If there are different ways to effect the belief change, then the agent should, according to he second version, (...) give up those beliefs that are least entrenched. Although widely proclaimed by belief revision theorists, I argue that both versions of the principle are dogmas that are not (and perhaps should not be) adhered to. I substantiate this claim with two simple mathematical observations, and I defend it against four possible objections that involve contractions, reconstructions, dispositions, and truths. (shrink)

A general criterion for the theory of belief revision is that when we revise a state of belief by a sentence A, as much of the old information as possible should be retained in the revised state of belief. The motivating idea in this paper is that if a belief B is irrelevant to A, then B should still be believed in the revised state. The problem is that the traditional definition of statistical relevance suffers (...) from some serious shortcomings and cannot be used as a tool for defining belief revision processes. In particular, the traditional definition violates the requirement that if A is irrelevant to C and B is irrelevant to C, then A&B is irrelevant to C. In order to circumvent these drawbacks, I develop an amended notion of relevance which has the desired properties. On the basis of the new definition, I outline how it can be used to simplify a construction of a belief revision method. (shrink)

The theory of belief revision deals with (rational) changes in beliefs in response to new information. In the literature a distinction has been drawn between belief revision and belief update (see [6]). The former deals with situations where the objective facts describing the world do not change (so that only the beliefs of the agent change over time), while the letter allows for situations where both the facts and the doxastic state of the agent change (...) over time. We focus on belief revision and propose a temporal framework that allows for iterated revision. We model the notion of “minimal” or “conservative” belief revision by considering logics of increasing strength. We move from one logic to the next by adding one or more axioms and show that the corresponding logic captures more stringent notions of minimal belief revision. The strongest logic that we propose provides a full axiomatization of the well-known AGM theory of belief revision. (shrink)

We show how belief revision can be treated systematically in the format of dynamicepistemic logic, when operators of conditional belief are added. The core engine consists of definable update rules for changing plausibility relations between worlds, which have been proposed independently in the dynamic-epistemic literature on preference change. Our analysis yields two new types of modal result. First, we obtain complete logics for concrete mechanisms of belief revision, based on compositional reduction axioms. Next, we show (...) how various abstract postulates for belief revision can be analyzed by standard modal frame correspondences for model-changing operations. (shrink)

We examine carefully the rationale underlying the approaches to belief change taken in the literature, and highlight what we view as methodological problems. We argue that to study belief change carefully, we must be quite explicit about the ontology or scenario underlying the belief change process. This is something that has been missing in previous work, with its focus on postulates. Our analysis shows that we must pay particular attention to two issues that have often been taken (...) for granted: the first is how we model the agent's epistemic state. (Do we use a set of beliefs, or a richer structure, such as an ordering on worlds? And if we use a set of beliefs, in what language are these beliefs are expressed?) We show that even postulates that have been called beyond controversy are unreasonable when the agent's beliefs include beliefs about her own epistemic state as well as the external world. The second is the status of observations. (Are observations known to be true, or just believed? In the latter case, how firm is the belief?) Issues regarding the status of observations arise particularly when we consider iterated belief revision, and we must confront the possibility of revising by and then by ¬. (shrink)

The Bayesian maxim for rational learning could be described as conservative change from one probabilistic belief or credence function to another in response to newinformation. Roughly: ‘Hold fixed any credences that are not directly affected by the learning experience.’ This is precisely articulated for the case when we learn that some proposition that we had previously entertained is indeed true (the rule of conditionalisation). But can this conservative-change maxim be extended to revising one’s credences in response to entertaining propositions (...) or concepts of which one was previously unaware? The economists Karni and Vierø (2013, 2015) make a proposal in this spirit. Philosophers have adopted effectively the same rule: revision in response to growing awareness should not affect the relative probabilities of propositions in one’s ‘old’ epistemic state. The rule is compelling, but only under the assumptions that its advocates introduce. It is not a general requirement of rationality, or so we argue. We provide informal counterexamples. And we show that, when awareness grows, the boundary between one’s ‘old’ and ‘new’ epistemic commitments is blurred. Accordingly, there is no general notion of conservative change in this setting. (shrink)

Belief revision theory studies how an ideal doxastic agent should revise her beliefs when she receives new information. In part I I will first present the AGM theory of belief revision (Alchourrón & Gärdenfors & Makinson 1985). Then I will focus on the problem of iterated belief revisions.

This paper offers a logico-algebraic investigation of AGM belief revision based on the logic of paradox ( \(\mathrm {LP}\) ). First, we define a concrete belief revision operator for \(\mathrm {LP}\), proving that it satisfies a generalised version of the traditional AGM postulates. Moreover, we investigate to what extent the Levi and Harper identities, in their classical formulation, can be applied to a paraconsistent account of revision. We show that a generalised Levi-type identity still yields (...) paraconsistent-based revisions that are fully compatible with the AGM postulates. The main outcome is that, once the classical AGM framework is lifted up to an appropriate level of generality, it still appears as a regulative ideal for treating of paraconsistent-based epistemic operators. (shrink)

Classical modal logics, based on the neighborhood semantics of Scott and Montague, provide a generalization of the familiar normal systems based on Kripke semantics. This paper defines AGM revision operators on several first-order monotonic modal correspondents, where each first-order correspondence language is defined by Marc Pauly’s version of the van Benthem characterization theorem for monotone modal logic. A revision problem expressed in a monotone modal system is translated into first-order logic, the revision is performed, and the new (...) belief set is translated back to the original modal system. An example is provided for the logic of Risky Knowledge that uses modal AGM contraction to construct counter-factual evidence sets in order to investigate robustness of a probability assignment given some evidence set. A proof of correctness is given. (shrink)

Belief Revision studies how rational agents change their beliefs in response to new information. The main objective of the works in this area is modelling these dynamics by defining some formal operators of change. This paper is an overview on the AGM model for revision, introducing some of the major questions that are addressed in this area of study.

We introduce a new operator, called rational evaluation, in belief change. The operator evaluates new information according to the agent’s core beliefs, and then exports the plausible part of the new information. It belongs to the decision module in belief change. We characterize rational evaluation by axiomatic postulates and propose two functional constructions for it, based on the well-known constructions of kernel sets and remainder sets, respectively. The main results of the paper are two representation theorems with respect (...) to the two functional constructions. We also compare and connect the evaluation operator with some related operators in belief revision, including contraction, set contraction, and prioritized multiple revision. (shrink)

The paper introduces ten open problems in belief revision theory, related to the representation of the belief state, to different notions of degrees of belief, and to the nature of change operations. It is argued that these problems are all issues in philosopical logic, in the strong sense of requiring inputs from both logic and philosophy for their solution.

‘Incongruity is never superfluous’Thea Arbee (2016) Is the superfluous a (new) modality?11. Even if much has been written about ingredients that trigger laughter, researchers are still far from hav...

This is a short version of Lindström & Rabinowicz 1991.In earlier papers, we proposed a generalization of the AGM approach to belief revision. The proposal was to view belief revision as a relation rather than as a function on theories (or belief sets). Going relational means that one allows for 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 presented 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 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). 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 if one allows 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)