Eduardo Fermé

Selective revision operators are well-known non-prioritized revision operators that allow for the acceptance of only part of the new information while rejecting the rest. In this paper, we introduce their contraction counterparts, which we refer to as selective contraction operators. These operators enable the removal of only part of the input information. As such, selective contractions can be viewed as a method for removing some of the beliefs that support the input belief without eliminating the belief itself. In this sense, selective contraction operators allow for a weakening of the input belief’s strength without discarding it. We provide representation theorems for various classes of selective contraction operators.

The purpose of this chapter is to introduce the AGM account of belief change, originally developed by Alchourrón, Gärdenfors and Makinson [4]. In Sections 3.1–3.3 we introduce the formal apparatus of belief sets and in Section 3.4 the operations of change. In Sections 3.5–3.6 we introduce the axioms of the AGM model. In Section 3.7 the relations between contraction and revision are specified and in Section 3.8 we introduce the basic construtive method of the AGM model, partial meet contraction and revision functions.

It was understood from the beginning that the use of logically closed sets of sentences to represent belief states is not cognitively realistic. In an article published in 1985 Makinson pointed out that “in real life, when we perform a contraction or derogation, we never do it to the theory itself (in the sense of a set of propositions closed under consequence) but rather on some finite or recursive or at least recursively enumerable base for the theory” [238, p. 357]. The use of belief bases rather than (logically closed) belief sets has turned out to increase the expressive power of the belief change framework in important ways.

This chapter introduces a new approach to belief revision that was recently presented in [182] and has been further developed in [190, 183, 185, 189, 187, 184, 341]. It provides us with a mechanism for belief change that is in important respects more general than previous approaches. Descriptor revision is based on two new formal constructions: A generalized notation for success conditions and the application of choice mechanisms directly to the set of potential outcomes of the belief change operation.

This thesis consists in six articles and a comprehensive summary. • The pourpose of the summary is to introduce the AGM theory of belief change and to exemplify the diversity and significance of the research that has been inspired by the AGM article in the last 25 years. The research areas associated with AGM was divided in three parts: criticisms, where we discussed some of the more common criticisms of AGM. Extensions where the most common extensions and variations of AGM are presented and applications where we provided an overview of applications and connections with other areas of research. • Article I elaborates on the connection between partial meet contractions [AGM85] and kernel contractions [Han94a] in belief change theory. Also both functions are equivalent in belief sets, there are notequivalent in belief bases. A way to define incision functions from selection functions and vice versa is presented. It is explained under which conditions there are exact correspondences between selection and incision functions so that the same contraction operations can be obtained by using either of them. • Article II proposes an axiomatic characterization for ensconcement-based contraction functions, belief base functions proposed byWilliams and relates this function with other kinds of base contraction functions. • Article III adapts the Fermé and Hansson model of Shielded Contraction [FH01] as well as Hansson et all Credibility-Limited Revision [HFCF01] for belief bases, to join two of the many variations of the AGM model [AGM85], i.e. those in which knowledge is represented through belief bases instead of logic theories, and those in which the object of the epistemic change does not get the priority over the existing information as it is the case in the AGM model. • Article IV introduces revision by comparison a refined method for changing beliefs by specifying constraints on the relative plausibility of propositions. Like the earlier belief revision models, the method proposed is a qualitative one, in the sense that no numbers are needed in order to specify the posterior plausibility of the new information. The method uses reference beliefs in order to determine the degree of entrenchment of the newly accepted piece of information. Two kinds of semantics for this idea are proposed and a logical characterization of the new model is given. • Article V focuses on the extension of AGM that allows change for a belief base by a set of sentences instead of a single sentence. In [FH94], Fuhrmann and Hansson presented an axiomatic for Multiple Contraction and a construction based on the AGM Partial Meet Contraction. This essay proposes for their model another way to construct functions: Multiple Kernel Contraction, that is a modification of Kernel Contraction,proposed by Hansson [Han94a] to construct classical AGM contractions and belief base contractions. • Article VI relates AGM model with the DFT model proposed by Carlos Alchourrón [Alc93]. Alchourrón devoted his last years to the analysis of the notion of defeasible conditionalization. His definition of the defeasible conditional is given in terms of strict implication operator and a modal operator f which is interpreted as a revision function at the language level. This essay points out that this underlying revision function is more general than AGM revision. In addition, a complete characterization of that more general kind of revision that permits to unify models of revision given by other authors is given.

This paper aims at comparing and relating belief revision and argumentation as approaches to model reasoning processes. Referring to some prominent literature references in both fields, we will discuss their (implicit or explicit) assumptions on the modeled processes and hence commonalities and differences in the forms of reasoning they are suitable to deal with. The intended contribution is on one hand assessing the (not fully explored yet) relationships between two lively research fields in the broad area of defeasible reasoning and on the other hand pointing out open issues and potential directions for future research.

Revision operation is the consistent expansion of a theory by a new belief-representing sentence. We consider that in a paraconsistent setting this desideratum can be accomplished in at least three distinct ways: the output of a revision operation should be either non-trivial or non-contradictory (in general or relative to the new belief). In this paper those distinctions will be explored in the constructive level by showing how the remainder sets could be refined, capturing the key concepts of paraconsistency in a dynamical scenario. These are preliminaries results of a wider project on Paraconsistent Belief Change conduced by the authors.