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, Gardenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourron 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 Gardenfors 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 Gardenfors investigated, with a further representation theorem established. $1. Background. The simplest and best known form of theory change is expansion, where a new proposition (axiom), hopefully consistent with a given theory A, is set-theoretically added to A, and this expanded set is then closed under logical consequence. There are, however, other kinds of theory change, the logic of which is less well understood. One form is theory contraction, where a proposition x, which was earlier in a theory A, is rejected. When A is a code of norms, this process is known among legal theorists as the derogation of x from A. The central problem is to determine which propositions should be rejected along with x so that the contracted theory will be closed under logical consequence. Another kind of change is revision, where a proposition x, inconsistent with a given theory A, is added to A under the requirement that the revised theory be consistent and closed under logical consequence. In normative contexts this kind of change is also known as amendment.
This results in the construction of transitively relational partial meet Horn contraction which is the Horn analogue of transitively relational partial meet contraction REF .
ON THE LOGIC OF THEORY CHANGE: PARTIAL MEET CONTRACTION AND REVISION FUNCTIONS
J. Symb. Log.
J. Symb. Log.
Mathematics Computer Science