The Belief Model

Chapter 5 is an introduction to the AGM model of belief revision. It begins with an explanation of belief and how sets of sentences are used to model psychological states of an agent. The model’s norms are shown to flow directly from Jamesian ideas that inquiry targets truth and the avoidance of error. Suspended judgement and disbelief are discussed, and the model’s treatment of them is used to spell out its transition rules. Slowly and carefully the main technical idea in the model’s transition theory—known as ‘partial meet contraction’—is explained for the beginner. The model’s postulates are then listed, its revision theorem is explained, and its approach to conditional belief is spelled out. The chapter closes by describing a notionally possible agent, Bella, whose psychology matches the Belief Model.

Kernel contraction

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.

Belief contraction in the context of the general theory of rational choice

This paper reorganizes and further develops the theory of partial meet contraction which was introduced in a classic paper by Alchourrón, Gärdenfors, and Makinson. Our purpose is threefold. First, we put the theory in a broader perspective by decomposing it into two layers which can respectively be treated by the general theory of choice and preference and elementary model theory. Second, we reprove the two main representation theorems of AGM and present two more representation results for the finite case that “lie between” the former, thereby partially answering an open question of AGM. Our method of proof is uniform insofar as it uses only one form of “revealed preference”, and it explains where and why the finiteness assumption is needed. Third, as an application, we explore the logic characterizing theory contractions in the finite case which are governed by the structure of simple and prioritized belief bases.

Theory contraction and base contraction unified

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.

On the logic of theory change: Partial meet contraction and revision functions

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, Gärdenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourrón 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 Gärdenfors 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 Gàrdenfors investigated, with a further representation theorem established.