A Question About Belief

If one believes that they believe that p, then do they believe that p (\text{B} \text{B} p \supset \text{B} p)?

I was recently thinking about this property and its absence from standard systems of doxastic logic. Systems of doxastic logic rightly do not validate the property \text{B} p \supset p. Since they omit this axiom (commonly called the T axiom), \text{B} \text{B} p \supset \text{B} p cannot be simply derived. But although the T axiom should not be valid in a doxastic logic, it is fair to say that the axiom \text{B} \text{B} p \supset \text{B} p should be valid; if one believes that they believe that p, then they do believe that p.

This type of agent is apparently termed a stable reasoner by Raymond Smullyan:

Stable reasoner: A stable reasoner is not unstable. That is, for every p, if it believes Bp then it believes p.

A list of doxastic reasoner types can be found here

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