A complete characterization of a notion of contraction based on information-value.
Horacio Arló‐Costa, Isaac Levi · 2004
We present a decision-theoretically motivated notion of contraction which, we claim, encodes the principles of minimal change and entrenchment. Contraction is seen as an operation whose goal is to minimize losses of informational value. The operation is also compatible with the principle that in contracting A one should preserve the sentences better entrenched than A (when the belief set contains A). Even when the principle of minimal change and the latter motivation for entrenchment figure prominently among the basic intuitions in the works of, among others, Quine (Quine 1951), Levi (Levi 1980) (Levi 1991), Harman (Harman 1999), and Gardenfors (Gärdenfors 1988), recent formal accounts of belief change (like AGM- see (Gärdenfors 1988)) have abandoned both principles (see (Rott 2000)). We argue for the principles and we show how to construct a contraction operation which obeys both. An axiom system is proposed. We also prove that the decision-theoretic notion of contraction can be completely characterized in terms of the given axioms. Proving this type of completeness result is a well-known open problem in the field, whose solution requires employing both decision-theoretical techniques and logical methods recently used in belief change. Principles for Belief Change The following principle has guided research in belief revision at least since it was sketched in (Quine 1951) and (van O. Quine & Ullian 1978) (we use here a recent formulation of it presented by Hans Rott and Maurice Pagnucco in (Rott & Pagnucco 1999).) The principle of Economy Keep loss to a minimum in contraction. Of course one needs to specify what is the index that is minimized in changing beliefs. A usual way of making this explicit is done as follows (see (Gärdenfors & Rott 1988), p.