Bounded Revision: Two-Dimensional Belief Change Between Conservative and Moderate Revision

In this paper I present the model of ‘bounded revision’ that is based on two-dimensional revision functions taking as arguments pairs consisting of an input sentence and a reference sentence. The key idea is that the input sentence is accepted as long as (and just a little longer than) the reference sentence is ‘cotenable’ with it. Bounded revision satisfies the AGM axioms as well as the Same Beliefs Condition (SBC) saying that the set of beliefs accepted after the revision do not depend on the reference sentence (although the posterior belief state does depend on it). Bounded revision satisfies the Darwiche-Pearl (DP) axioms. If the reference sentence is fixed to be a tautology or a contradiction, two well-known one-dimensional revision operations result; bounded revision thus naturally fills the space between conservative revision (also known as natural revision) and moderate revision (also known as lexicographic revision). I compare this approach to the two-dimensional model of ‘revision by comparison’ investigated by Fermé and Rott (Artificial Intelligence 157, 2004) that satisfies neither SBC nor the DP axioms. I conclude that two-dimensional revision operations add substantially to the expressive power of qualitative approaches that do not make use of numbers as measures of degrees of belief.