If one believes that they believe that p, then do they believe that 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 . Since they omit this axiom (commonly called the T axiom), 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 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
In his contribution on partial logic to the Handbook of Philosophical Logic, Stephen Blamey introduces a ‘value gap introducing’ connective named ‘transplication’ to the standard 3-valued partial logic, the Strong Kleene logic. Blamey suggests the possibility of reading the transplication connective as a type of conditional. I was interested to see how the transplication connective fares as a conditional by testing it against a list of inferences concerning conditionals.
Here is the investigation: Transplication as Implication
I have not come across much material concerning transplication. Does anyone else have any other references or ideas?