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

TLDR

This paper extends earlier work on formal aspects of theory contraction and revision, building on Gärdenfors’ postulates and Alchourrón and Makinson’s study of maximal contraction functions. The study explores a broader class of contraction functions, particularly partial meet contraction functions that may be less than maximal. They define partial meet contraction functions as the intersection of nonempty families of maximal subsets that fail to imply the eliminated proposition, and analyze special relational subclasses, linking them to supplementary postulates and proving representation theorems. They show that partial meet contraction functions satisfy Gärdenfors’ postulates and admit a representation theorem, and that relational subclasses further satisfy supplementary postulates with additional representation results.

Abstract

Abstract 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.