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 B x {displaystyle {mathcal {B}}x} to mean 'It is believed that x {displaystyle x} is the case', and the set B {displaystyle mathbb {B} } denotes a set of beliefs. In doxastic logic, belief is treated as a modal operator. 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 B x {displaystyle {mathcal {B}}x} to mean 'It is believed that x {displaystyle x} is the case', and the set B {displaystyle mathbb {B} } 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: For systems, we define reflexivity to mean that for any p {displaystyle p} (in the language of the system) there is some q {displaystyle q} such that q ≡ B q → p {displaystyle qequiv {mathcal {B}}q o p} is provable in the system. Löb's theorem (in a general form) is that for any reflexive system of type 4, if B p → p {displaystyle {mathcal {B}}p o p} is provable in the system, so is p . {displaystyle p.} 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 p {displaystyle p} (and hence be inconsistent). Take any proposition p . {displaystyle p.} The reasoner believes B B p → B p , {displaystyle {mathcal {B}}{mathcal {B}}p o {mathcal {B}}p,} hence by Löb's theorem they will believe B p {displaystyle {mathcal {B}}p} (because they believe B r → r , {displaystyle {mathcal {B}}r o r,} where r {displaystyle r} is the proposition B p , {displaystyle {mathcal {B}}p,} and so they will believe r , {displaystyle r,} which is the proposition B p {displaystyle {mathcal {B}}p} ). Being stable, they will then believe p . {displaystyle p.}