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.