Doxastic logic

Doxastic logic is a type of logic concerned with reasoning about beliefs. The term derives from the Ancient Greek (doxa, "opinion, belief"), from which the English term doxa ("popular opinion or belief") is also borrowed. Typically, a doxastic logic uses the notation 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. 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. (modal axiom T) Inaccurate reasoner: An inaccurate reasoner believes at least one false proposition. Consistent reasoner: A consistent reasoner never simultaneously believes a proposition and its negation. (modal axiom D) Normal reasoner: A normal reasoner is one who, while believing also believes they believe p (modal axiom 4). A variation on this would be someone who, while not believing also believes they don't believe p (modal axiom 5). 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 (see Moore's paradox), 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 . If a reflexive reasoner of type 4 [see below] believes , they will believe p. This is a parallelism of Löb's theorem for reasoners. Conceited reasoner: A conceited reasoner believes their beliefs are never inaccurate.