Transmutations of Knowledge Systems

Within the AGM paradigm revision and contraction operators are constrained by a set of rationality postulates. The logical properties of a set of knowledge are not strong enough to uniquely determine a revision or contraction operation, therefore constructions for these operators rely on some form of underlying preference relation, such as a systems of spheres, or an epistemic entrenchment ordering. The problem of iterated revision is determining how the underlying preference relation should change in response to the acceptance or contraction of information. We call this process a transmutation. Generalizing Spohn’s approach we define a transmutation of a well-ordered system of spheres using ordinal conditional functions. Similarly, we define the transmutation of a well-ordered epistemic entrenchment using ordinal epistemic entrenchment functions. We provide several conditions which capture the relationship between an ordinal conditional function and an ordinal epistemic entrenchment function, and their corresponding transmutations. These conditions allow an ordinal epistemic entrenchment function to be explicitly constructed from an ordinal conditional function, and vice versa, in such a way that the epistemic state and its dynamic properties are preserved.