Latent Belief Theory and Belief Dependencies: A Solution to the Recovery
  Problem in the Belief Set Theories
  The AGM recovery postulate says: assume a set of propositions X; assume that
it is consistent and that it is closed under logical consequences; remove a
belief P from the set minimally, but make sure that the resultant set is again
some set of propositions X' which is closed under the logical consequences; now
add P again and close the set under the logical consequences; and we should get
a set of propositions that contains all the propositions that were in X. This
postulate has since met objections; many have observed that it could bear
counter-intuitive results. Nevertheless, the attempts that have been made so
far to amend it either recovered the postulate in full, had to relinquish the
assumption of the logical closure altogether, or else had to introduce fresh
controversies of their own. We provide a solution to the recovery paradox in
this work. Our theoretical basis is the recently proposed belief theory with
latent beliefs (simply the latent belief theory for short). Firstly, through
examples, we will illustrate that the vanilla latent belief theory can be made
more expressive. We will identify that a latent belief, when it becomes
visible, may remain visible only while the beliefs that triggered it into the
agent's consciousness are in the agent's belief set. In order that such
situations can be also handled, we will enrich the latent belief theory with
belief dependencies among attributive beliefs, recording the information as to
which belief is supported of its existence by which beliefs. We will show that
the enriched latent belief theory does not possess the recovery property. The
closure by logical consequences is maintained in the theory, however. Hence it
serves as a solution to the open problem in the belief set theories.
