Doxastic logic

Doxastic logic is a type of logic concerned with reasoning about beliefs.

The term doxastic derives from the Ancient Greek δόξα (doxa, "opinion, belief"), from which the English term doxa ("popular opinion or belief") is also borrowed. Typically, a doxastic logic uses the notation ${\displaystyle {\mathcal {B}}x}$ to mean "It is believed that ${\displaystyle x}$ is the case", and the set ${\displaystyle \mathbb {B} :\left\{b_{1},\ldots ,b_{n}\right\}}$ 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.^[1]