This book provides a critical examination of how the choice of what to believe is represented in the standard model of belief change. In particular the use of possible worlds and infinite remainders as objects of choice is critically examined. Descriptors are introduced as a versatile tool for expressing the success conditions of belief change, addressing both local and global descriptor revision. The book presents dynamic descriptors such as Ramsey descriptors that convey how an agent’s beliefs tend to be changed (...) in response to different inputs. It also explores sentential revision and demonstrates how local and global operations of revision by a sentence can be derived as a special case of descriptor revision. Lastly, the book examines revocation, a generalization of contraction in which a specified sentence is removed in a process that may possibly also involve the addition of some new information to the belief set. (shrink)

The paper develops an inductive extension of AGM-style belief base revision theory with the aim of formally implementing Freitag’s :254–267, 2015, Dialectica 70:185–200, 2016) solution to Goodman’s paradox. It shows that the paradox dissolves once belief revision takes place on inductively closed belief bases, rather than on belief sets.

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)

We consider the connections between belief revision, conditional logic and nonmonotonic reasoning, using as a foundation the approach to theory change developed by Alchourrón, Gärdenfors and Makinson (the AGM approach). This is first generalized to allow the iteration of theory change operations to capture the dynamics of epistemic states according to a principle of minimal change of entrenchment. The iterative operations of expansion, contraction and revision are characterized both by a set of postulates and by Grove's construction based on total (...) pre-orders on the set of complete theories of the belief logic. We present a sound and complete conditional logic whose semantics is based on our iterative revision operation, but which avoids Gärdenfors's triviality result because of a severely restricted language of beliefs and hence the weakened scope of our extended postulates. In the second part of the paper, we develop a computational approach to theory dynamics using Rott's E-bases as a representation for epistemic states. Under this approach, a ranked E-base is interpreted as standing for the most conservative entrenchment compatible with the base, reflecting a kind of foundationalism in the acceptance of evidence for a belief. Algorithms for the computation of our iterative versions of expansion, contraction and revision are presented. Finally, we consider the relationship between nonmonotonic reasoning and both conditional logic and belief revision. Adapting the approach of Delgrande, we show that the unique extension of a default theory expressed in our conditional logic of belief revision corresponds to the most conservative belief state which respects the theory: however, this correspondence is limited to propositional default theories. Considering first order default theories, we present a belief revision algorithm which incorporates the assumption of independence of default instances and propose the use of a base logic for default reasoning which incorporates uniqueness of names. We conclude with an examination of the behavior of an implemented system on some of Lifschitz's benchmark problems in nonmonotonic reasoning. (shrink)

The AGM paradigm for belief revision provides a very elegant and powerful framework for reasoning about idealized agents. The paradigm assumes that the modeled agent is a perfect reasoner with infinite memory. In this paper we propose a framework to reason about non-ideal agents that generalizes the AGM paradigm. We first introduce a structure to represent an agent's belief states that distinguishes different status of beliefs according to whether or not they are explicitly represented, whether they are currently active and (...) whether they are fully accepted or provisional. Then we define a set of basic operations that change the status of beliefs and show how these operations can be used to model agents with different capacities. We also show how different operations of belief change described in the literature can be seen as special cases of our theory. (shrink)

The AGM theory of belief revision provides a formal framework to represent the dynamics of epistemic states. In this framework, the beliefs of the agent are usually represented as logical formulas while the change operations are constrained by rationality postulates. In the original proposal, the logic underlying the reasoning was supposed to be supraclassical, among other properties. In this paper, we present some of the existing work in adapting the AGM theory for non-classical logics and discuss their interconnections and what (...) is still missing for each approach. (shrink)

In the area known as model-based diagnosis, a system is described by-means of a set of formulas together with assumptions that all the components are functioning correctly. When we observe a behavior of the system which is inconsistent with the system description, we must relax some of the assumptions. In previous work, we have presented operations of belief change which only affect the relevant part of a belief base. In this paper, we propose the application of the same strategy to (...) the problem of model-based diagnosis. We Erst isolate the subset of the system description which is relevant for a given observation. By doing this, we reduce the size of the problem and then solve the diagnosis problem for the relevant subset. (shrink)

The standard theory of belief revision was developed to describe how a rational agent should change his beliefs in the presence of new information. Many interesting tools were created, but the concept of rationality was usually assumed to be related to classical logics. In this paper, we explore the fact that the logical tools used can be extended to other sorts of logics, as proved in (Hansson and Wassermann, 2002), to describe models that are closer to the rationality of a (...) real agent. (shrink)

Both logic and philosophy of science investigate formal aspects of scientific discourse, i.e. properties of (non-monotonic) consequence operations for discursive logic. In the present paper we handle two of them: paraconsistency and enthymematycity.

ABSTRACT The well-known AGM-theory-contraction and theory-revision, due to Alchourrón, Gärdenfors and Makinson, relies heavily on the so-called postulate of recovery. This postulate is supposed to capture the requirement of “minimum mutilation”; but it does not. Recovery can be satisfied even when there is more mutilation than is necessary. Recovery also ensures that very often too little is given up in a contraction, in this paper I bring out clearly the deficiencies of the AGM-theory in these two regards, showing how it (...) is doubly off-beam. I show that some of the most serious inadequacies of the AGM-theory derive from early claims in some of its founding contributions, claims that have not been seriously questioned within the tradition since. The upshot of these investigations is that recovery cannot, and should not, be recovered. Theory contraction is hysteretic. Whether the AGM-theory can now recover is a good question. (shrink)

The Theory of theory change has contraction and revision as its central notions. Of these, contraction is the more fundamental. The best-known theory, due to Alchourrón, Gärdenfors, and Makinson, is based on a few central postulates. The most fundamental of these is the principle of recovery: if one contracts a theory with respect to a sentence, and then adds that sentence back again, one recovers the whole theory. Recovery is demonstrably false. This paper shows why, and investigates how one can (...) nevertheless characterize contraction in a theoretically fruitful way. The theory proposed lends itself to implementation, which in turn could yield new theoretical insights. The Main proposal is a ‘staining algorithm’ which identifies which sentences to reject when contracting a theory. The algorithm requires one to be clear about the structure of reasons one has for including sentences within one's theory. (shrink)

Changing the Theory of Theory Change: Towards a Computational Approach’ (Tennant [1994]; henceforth CTTC) claimed that the AGM postulate of recovery is false, and that AGM contractions of theories can be more than minimally mutilating. It also described an alternative, computational method for contracting theories, called the Staining Algorithm. Makinson [1995] and Hansson and Rott [1995] criticized CTTC's arguments against AGM-theory, and its specific proposals for an alternative, computational approach. This paper replies as comprehensively as space allows.

I reformulate the AGM-account of contraction (which would yield an account also of revision). The reformulation involves using introduction and elimination rules for relational notions. Then I investigate the extent to which the two main methods of partial meet contraction and safe contraction can be employed for theories closed under intuitionistic consequence.

We present a theory that copes with the dynamics of inconsistent information. A method is set forth to represent possibly inconsistent information by a finite state. Next, finite operations for expansion and contraction of finite states are given. No extra-logical element — a choice function or an ordering over (sets of) sentences — is presupposed in the definition of contraction. Moreover, expansion and contraction are each other's duals. AGM-style characterizations of these operations follow.

This paper presents a belief revision operator that considers time intervals for modelling norm change in the law. This approach relates techniques from belief revision formalisms and time intervals with temporalised rules for legal systems. Our goal is to formalise a temporalised belief base and corresponding timed derivation, together with a proper revision operator. This operator may remove rules when needed or adapt intervals of time when contradictory norms are added in the system. For the operator, both constructive definition and (...) an axiomatic characterisation by representation theorems are given. (shrink)

This paper generalises classical revision theory of the AGM brand to sets of norms. This is achieved substituting input/output logic for classical logic and tracking the changes. Operations of derogation and amendment—analogues of contraction and revision—are defined and characterised, and the precise relationship between contraction and derogation, on the one hand, and derogation and amendment on the other, is established. It is argued that the notion of derogation, in particular, is a very important analytical tool, and that even core deontic (...) concepts such as that of permission resists a satisfactory analysis without it. By way of illustration the last section of the paper analyses the much debated concept of positive permission, of which there turns out to be more than one kind. (shrink)

We distinguish the set of explicit beliefs of a reasoner, his "belief base", from the beliefs that are merely implicit. Syntax-based belief change governed by the structure of the belief base and the ranking ("prioritization") of its elements is reconstructed with the help of an epistemic entrenchment relation in the style of Gärdenfors and Makinson. Though priorities are essentially different from entrenchments, distinguished relations of epistemic entrenchment may be obtained from prioritized belief bases by a number of different constructions. An (...) approximate reconstruction of syntax-based belief change is possible, which can be strengthened to a strict identity in some important special cases. A competitive interpretation and a minimal change interpretation of the basic idea of entrenchment are discussed. It is argued that entrenchment essentially involves multiple belief changes, i.e., changes by sets of sentences. (shrink)

In the logic of theory change, the standard model is AGM, proposed by Alchourrón et al. (J Symb Log 50:510–530, 1985 ). This paper focuses on the extension of AGM that accounts for contractions of a theory by a set of sentences instead of only by a single sentence. Hansson (Theoria 55:114–132, 1989 ), Fuhrmann and Hansson (J Logic Lang Inf 3:39–74, 1994 ) generalized Partial Meet Contraction to the case of contractions by (possibly non-singleton) sets of sentences. In this (...) paper we present the possible worlds semantics for partial meet multiple contractions. (shrink)

The process of completing, correcting and prioritising specifications is an essential but very complex task for the maintenance and improvement of software systems. The preservation of functionalities and the ability to accommodate changes are main objectives of the software development cycle to guarantee system reliability. Logical theories able to fully model such processes are still insufficient. In this paper we propose a full formalisation of such operations on software systems inspired by the Alchourrón–Gärdenfors–Makinson paradigm for belief revision of human epistemic (...) states. We represent specifications as finite sets of formulas equipped with a priority relation that models functional entrenchment of properties. We propose to handle specification incompleteness through ordered expansion, inconsistency through ordered safe contraction and prioritisation through revision with reordering, and model all three in an algorithmic fashion. We show how the system satisfies basic properties of the AGM paradigm, including Levi’s and Harper’s identities. We offer a concrete example and complexity results for the inference and model checking problems on revision. We conclude by describing resilience and evolvability of software systems based on such revision operators. (shrink)

This paper is concerned with the construction of a base contraction (revision) operation such that the theory contraction (revision) operation generated by it will be fully AGM-rational. It is shown that the theory contraction operation generated by Fuhrmann's minimal base contraction operation, even under quite strong restrictions, fails to satisfy the "supplementary postulates" of belief contraction. Finally Fuhrmann's construction is appropriately modified so as to yield the desired properties. The new construction may be described as involving a modification of safe (...) (base) contraction so as to make it maxichoice. (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.

As lawmakers produce norms, the underlying normative system is affected showing the intrinsic dynamism of law. Through undertaken actions of legal change, the normative system is continuously modified. In a usual legislative practice, the time for an enacted legal provision to be in force may differ from that of its inclusion to the legal system, or from that in which it produces legal effects. Even more, some provisions can produce effects retroactively in time. In this article we study a simulation (...) of such process through the formalisation of a temporalised logical framework upon which a novel belief revision model tackles the dynamic nature of law. Represented through intervals, the temporalisation of sentences allows differentiating the temporal parameters of norms. In addition, a proposed revision operator allows assessing change to the legal system by including a new temporalised literal while preserving the time-based consistency. This can be achieved either by pushing out conflictive pieces of pre-existing norms or through the modification of intervals in which such norms can be either in force, or produce effects. Finally, the construction of the temporalised revision operator is axiomatically characterised and its rational behavior proved through a corresponding representation theorem. (shrink)

Although AGM theory contraction (Alchourrón et al., 1985; Alchourrón and Makinson, 1985) occupies a central position in the literature on belief change, there is one aspect about it that has created a fair amount of controversy. It involves the inclusion of the postulate known as Recovery. As a result, a number of alternatives to AGM theory contraction have been proposed that do not always satisfy the Recovery postulate (Levi, 1991, 1998; Hansson and Olsson, 1995; Fermé, 1998; Fermé and Rodriguez, 1998; (...) Rott and Pagnucco, 1999). In this paper we present a new addition, systematic withdrawal, to the family of withdrawal operations, as they have become known. We define systematic withdrawal semantically, in terms of a set of preorders, and show that it can be characterised by a set of postulates. In a comparison of withdrawal operations we show that AGM contraction, systematic withdrawal and the severe withdrawal of Rott and Pagnucco (1999) are intimately connected by virtue of their definition in terms of sets of preorders. In a future paper it will be shown that this connection can be extended to include the epistemic entrenchment orderings of Gärdenfors (1988) and Gärdenfors and Makinson (1988) and the refined entrenchment orderings of Meyer et al. (2000). (shrink)

Generalisations of theory change involving operations on arbitrary sets ofwffs instead of on belief sets (i.e., sets closed under a consequencerelation), have become known as base change. In one view, a base should bethought of as providing more structure to its generated belief set, whichmeans that it can be employed to determine the theory contraction operationassociated with a base contraction operation. In this paper we follow suchan approach as the first step in defining infobase change. We think of an infobase (...) as a finite set of wffs consisting of independently obtainedbits of information. Taking AGM theory change (Alchourrón et al. 1985) as the general framework, we present a method that uses the structure of aninfobase B to obtain an AGM theory contraction operation for contractingthe belief set Cn(B). Both the infobase and the obtained theory contraction operation then play a role in constructing a unique infobasecontraction operation. Infobase revision is defined in terms of an analogueof the Levi Identity, and it is shown that the associated theory revisionoperation satisfies the AGM postulates for revision. Because every infobaseis associated with a unique infobase contraction and revision operation, the method also allows for iterated base change. (shrink)

Generalisations of theory change involving arbitrary sets of wffs instead of belief sets have become known as base change. In one view, a base should be thought of as providing more structure to its generated belief set, and can be used to determine the theory change operation associated with a base change operation. In this paper we extend a proposal along these lines by Meyer et al. We take an infobase as a finite sequence of wffs, with each element in (...) the sequence being seen as an independently obtained bit of information, and define appropriate infobase change operations. The associated theory change operations satisfy the AGM postulates for theory change. Since an infobase change operation produces a new infobase, it allows for iterated infobase change. We measure iterated infobase change against the postulates proposed by Darwiche et al. and Lehmann. (shrink)

Scientific inquiry is represented as a process of rational hypothesis revision in the face of data. For the concept of rationality, we rely on the theory of belief dynamics as developed in [5, 9]. Among other things, it is shown that if belief states are left unclosed under deductive logic then scientific theories can be expanded in a uniform, consistent fashion that allows inquiry to proceed by any method of hypothesis revision based on "kernel" contraction. In contrast, if belief states (...) are closed under logic, then no such expansion is possible. (shrink)

Scientific inquiry is represented as a process of rational hypothesis revision in the face of data. For the concept of rationality, we rely on the theory of belief dynamics as developed in [5, 9]. Among other things, it is shown that if belief states are left unclosed under deductive logic then scientific theories can be expanded in a uniform, consistent fashion that allows inquiry to proceed by any method of hypothesis revision based on “kernel” contraction. In contrast, if belief states (...) are closed under logic, then no such expansion is possible. (shrink)

David Makinson has argued that the compelling character of counterexamples to the Recovery Condition on contraction is due to an appeal to justificational structure. In “naked theories” where such structure is ignored or is not present, Recovery does apply. This note attempts to show that Makinson is mistaken on both counts. Recovery fails when no appeal is made to justificational structure.

Formal models of theory contraction entered the philosophicalliterature with the prototype model by Alchourrón, Gärdenfors,and Makinson (Alchourrón et al. 1985). One influential modelinvolves theory contraction with respect to a relation calledepistemic entrenchment which orders the propositions of a theoryaccording to their relative degrees of theoretical importance.Various postulates have been suggested for characterizingepistemic entrenchment formally. I argue here that threesuggested postulates produce inappropriately bizarre results whenapplied to scientific theories. I argue that the postulates callednoncovering, continuing up, and continuing down, implyrespectively that, (...) (i) no scientific law is more epistemicallyentrenched than any of its instances, (ii) two distinct instances ofthe same scientific law must have different degrees of epistemicentrenchment, and (iii) any two scientific laws must have differentdegrees of epistemic entrenchment. I also argue that continuingup and continuing down each lead to incoherency. (shrink)

We give a reading of binary necessity statements of the form “ϕ is necessary for ψ” in terms of proofs. This reading is based on the idea of interpreting such statements as “Every proof of ψ uses ϕ”.

One way to construct a contraction operator for a theory (belief set) is to assign to it a base (belief base) and an operator of partial meet contraction for that base. Axiomatic characterizations are given of the theory contractions that are generated in this way by (various types of) partial meet base contractions.

Different similarity relations on sets are introduced, and their logical properties are investigated. Close relationships are shown to hold between similarity relations that are based on symmetrical difference and operators of belief contraction that are based on relational selection functions. Two new rationality criteria for minimal belief contraction, the maximizing property and the reducing property, are proposed.

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)

Two recent defences of the recovery postulate for contraction of belief sets are analyzed. It is concluded that recovery is defensible as a by-product of a formalization that is idealized in the sense of being simplified for the sake of clarity. However, recovery does not seem to be a required feature of the doxastic behaviour of ideal (perfectly rational) agents. It is reasonable to expect that there should be epistemic residues (remnants of rejected beliefs), but not that these should always (...) suffice to recover all the original beliefs if the discarded belief is reinstated. (shrink)

We prove that four theses commonly associated with coherentism are incompatible with the representation of a belief state as a logically closed set of sentences. The result is applied to the conventional coherence interpretation of the AGM theory of belief revision, which appears not to be tenable. Our argument also counts against the coherentistic acceptability of a certain form of propositional holism. We argue that the problems arise as an effect of ignoring the distinction between derived and non-derived beliefs, and (...) we suggest that the kind of coherence relevant to epistemic justification is the coherence of non-derived beliefs. (shrink)

An agent can usually hold a very large number of beliefs. However, only a small part of these beliefs is used at a time. Efficient operations for belief change should affect the beliefs of the agent locally, that is, the changes should be performed only in the relevant part of the belief state. In this paper we define a local consequence operator that only considers the relevant part of a belief base. This operator is used to define local versions of (...) the operations for belief change. Representation theorems are given for the local operators. (shrink)

Kernel contraction is a natural nonrelational generalization of safe contraction. All partial meet contractions are kernel contractions, but the converse relationship does not hold. Kernel contraction is axiomatically characterized. It is shown to be better suited than partial meet contraction for formal treatments of iterated belief change.

In the most common approaches to belief dynamics, states of belief are represented by sets that are closed under logical consequence. In an alternative approach, they are represented by non-closed belief bases. This representation has attractive properties not shared by closed representations. Most importantly, it can account for repeated belief changes that have not yet been satisfactorily accounted for in the closed approach.

A number of seminal papers on the logic of belief change by Alchourrön, Gärden-fors, and Makinson have given rise to what is now known as the AGM paradigm. The present discussion note is a response to Neil Tennant's [1994], which aims at a critical appraisal of the AGM approach and the introduction of an alternative approach. We show that important parts of Tennants's critical remarks are based on misunderstandings or on lack of information. In the course of doing this, we (...) attend to some central philosophical issues in the theory of belief change, such as the choice of a representation for belief states and the meaning of an idealized rational agent. (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 operator of contraction for a belief set (a theory) can be obtained by assigning to it a belief base and an operator of partial meet contraction for that base. It is argued that closure of the base under disjunction is an intuitively reasonable condition. Axiomatic characterizations are given of the contractions of belief sets that can be generated by (various types of) partial meet contraction on disjunctively closed bases. The corresponding revision operators are also characterized. Finally, some results are (...) reported on operations on bases that are closed under material implication. (shrink)

Traditionally, belief change is modelled as the construction of a belief set that satisfies a success condition. The success condition is usually that a specified sentence should be believed or not believed. Furthermore, most models of belief change employ a select-and-intersect strategy. This means that a selection is made among primary objects that satisfy the success condition, and the intersection of the selected objects is taken as outcome of the operation. However, the select-and-intersect method is difficult to justify, in particular (...) since the primary objects are not themselves plausible outcome candidates. Some of the most controversial features of belief change theory, such as recovery and the impossibility of Ramsey test conditionals, are closely connected with the select-and-intersect method. It is proposed that a selection mechanism should instead operate directly on the potential outcomes, and select only one of them. In this way many of the problems that are associated with the select-and-intersect method can be avoided. This model is simpler than previous models in the important Ockhamist sense of doing away with intermediate, cognitively inaccessible objects. However, the role of simplicity as a choice criterion in the direct selection among potential outcomes is left as an open issue. (shrink)

In his paper ‘Changing the Theory of Theory Change: Reply to My Critics’, N. Tennant (1997b) reacts to the critical reception of an earlier article of his. The present note rectifies some of the most serious misrepresentations in Tennant's reply.

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)