Doxastic logic is a type of logic concerned with reasoning about beliefs. The term doxasticderives from the ancient Greek δόξα, doxa, which means "belief". Typically, a doxastic logic uses to mean "It is believed that is the case", and the set 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:
Accurate reasoner: An accurate reasoner never believes any false proposition.
Inaccurate reasoner: An inaccurate reasoner believes at least one false proposition.
Conceited reasoner: A conceited reasoner believes their beliefs are never inaccurate.
Consistent reasoner: A consistent reasoner never simultaneously believes a proposition and its negation.
Normal reasoner: A normal reasoner is one who, while believing also believes they believe p.
Peculiar reasoner: A peculiar reasoner believes proposition p while also believing they do not believe Although a peculiar reasoner may seem like a strange psychological phenomenon, a peculiar reasoner is necessarily inaccurate but not necessarily inconsistent.
Regular reasoner: A regular reasoner is one who, while believing , also believes .
Reflexive reasoner: A reflexive reasoner is one for whom every proposition has some proposition such that the reasoner believes.
Unstable reasoner: An unstable reasoner is one who believes that they believe some proposition, but in fact does not believe it. This is just as strange a psychological phenomenon as peculiarity; however, an unstable reasoner is not necessarily inconsistent.
Stable reasoner: A stable reasoner is not unstable. That is, for every if they believe then they believe Note that stability is the converse of normality. We will say that a reasoner believes they are stable if for every proposition they believe .
Modest reasoner: A modest reasoner is one for whom for every believed proposition, only if they believe. A modest reasoner never believes unless they believe. Any reflexive reasoner of type 4 is modest.
Queer reasoner: A queer reasoner is of type G and believes they are inconsistent—but is wrong in this belief.
Timid reasoner: A timid reasoner does not believe if they believe that belief in leads to a contradictory belief.
Type 1* reasoner: A type 1* reasoner believes all tautologies; their set of beliefs is logically closed under modus ponens, and for any propositions and if they believe then they will believe that if they believe then they will believe. The type 1* reasoner has "a shade more" self awareness than a type 1 reasoner.
Type 2 reasoner: A reasoner is of type 2 if they are of type 1, and if for every and they believe: "If I should ever believe both and, then I will believe." Being of type 1, they also believe the logically equivalent proposition: A type 2 reasoner knows their beliefs are closed under modus ponens.
Type 3 reasoner: A reasoner is of type 3 if they are a normal reasoner of type 2.
Type 4 reasoner: A reasoner is of type 4 if they are of type 3 and also believe they are normal.
Type G reasoner: A reasoner of type 4 who believes they are modest.
Self-fulfilling beliefs
For systems, we define reflexivity to mean that for any there is some such that is provable in the system. Löb's theorem is that for any reflexive system of type 4, if is provable in the system, so is
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 believe every proposition . Take any proposition The reasoner believes hence by Löb's theorem they will believe . Being stable, they will then believe
