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Iterated belief revision: from postulates to abilities

The belief revision field is opulent in new proposals and indigent in analyses of existing approaches. Much work hinge on postulates, employed as syntactic characterizations: some revision mechanism is equivalent to some properties. Postulates constraint specific revision instances: certain revisions update certain beliefs in a certain way. As an example, if the revision is consistent with the current beliefs, it is incorporated with no other change. A postulate like this tells what revisions must do and neglect what they can do. Can they reach a certain state of beliefs? Can they reach all possible states of beliefs? Can they reach all possible states of beliefs from no previous belief? Can they reach a dogmatic state of beliefs, where everything not believed is impossible? Can they make two conditions equally believed? An application where every possible state of beliefs is sensible requires each state of beliefs to be reachable. An application where conditions may be equally believed requires such a belief state to be reachable. An application where beliefs may become dogmatic requires a way to make them dogmatic. Such doxastic states need to be reached in a way or another. Not in specific way, as dictated by a typical belief revision postulate. This is an ability, not a constraint: the ability of being plastic, equating, dogmatic. Amnesic, correcting, believer, damascan, learnable are other abilities. Each revision mechanism owns some of these abilities and lacks the others: lexicographic, natural, restrained, very radical, full meet, radical, severe, moderate severe, deep severe, plain severe and deep severe revisions, each of these revisions is proved to possess certain abilities.

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