Two-Dimensional Belief Change

The idea of two-dimensional belief change operators is that a belief state is transformed by an input sentence $A$ in such a way that $A$ gets accepted with at least the strength or certainty of a sentence $B$ (the reference sentence). The input of such a transformation may alternatively be conceived as `$B leq A$' [`$B$ less-than-or-equal-to $A$']. This notation makes explicit that the process induced is basically one of doxastic preference change. The principal case of two-dimensional belief change obtains when $B$ is a prior belief which is more strongly accepted than both $A$ and $ eg A$, but the non-principal cases are interesting in their own right. Various two-dimensional revision operators were studied by Cantwell (1997, `raising' and `lowering'), Fermé and Rott (2003, `revision by comparison'), and Rott (2007, `bounded revision'). Special choices of a fixed input sentence $A$ or a fixed reference sentence $B$ lead to some well-known unary oparators of belief change: `irrevocable' (aka `radical') revision, `severe withdrawal' (aka `mild contraction'), `natural' (aka `conservative') and `lexicographic' (aka `moderate') revision. The talk gives a survey of several variants of two-dimensional belief change and their representations. I argue that two-dimensional belief change operators offer an interesting qualitative model with an expressive power between (all too poor) unary operators and (all too demanding) quantitative models of belief change.