Doxastic logic

Doxastic logic is a type of logic concerned with reasoning about beliefs. The term "doxastic" derives from the ancient Greek δόξα, "doxa", which means "belief". Typically, a doxastic logic uses formula_1 to mean "It is believed that formula_2 is the case", and the set formula_3 denotes a set of beliefs. In doxastic logic, belief is treated as a modal operator.
There is complete parallelism between a person who believes propositions and a formal system that derives propositions. Using doxastic logic, one can express the epistemic counterpart of Gödel's incompleteness theorem of metalogic, as well as Löb's theorem, and other metalogical results in terms of belief.
Types of reasoners.
To demonstrate the properties of sets of beliefs, Raymond Smullyan defines the following types of reasoners:
Self-fulfilling beliefs.
For systems, we define reflexivity to mean that for any formula_16 (in the language of the system) there is some formula_17 such that formula_64 is provable in the system. Löb's theorem (in a general form) is that for any reflexive system of type 4, if formula_33 is provable in the system, so is formula_11
Inconsistency of the belief in one's stability.
If a consistent reflexive reasoner of type 4 believes that they are stable, then they will become unstable. Stated otherwise, if a stable reflexive reasoner of type 4 believes that they are stable, then they will become inconsistent. Why is this? Suppose that a stable reflexive reasoner of type 4 believes that they are stable. We will show that they will (sooner or later) believe every proposition formula_16 (and hence be inconsistent). Take any proposition formula_11 The reasoner believes formula_69 hence by Löb's theorem they will believe formula_23 (because they believe formula_71 where formula_72 is the proposition formula_73 and so they will believe formula_74 which is the proposition formula_23). Being stable, they will then believe formula_11