Doxastic logic
Doxastic logic is a type of logic concerned with reasoning about beliefs.
The term derives from the Ancient Greek, from which the English term doxa is also borrowed. Typically, a doxastic logic uses the notation to mean "reasoner believes that is true", and the set denotes the set of beliefs of. 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.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 .Peculiar reasoner: A peculiar reasoner believes proposition 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.- Conceited reasoner: A conceited reasoner believes their beliefs are never inaccurate.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 . This corresponds to having a dense accessibility relation in Kripke semantics, and any accurate reasoner is always stable.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.