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]

Types of reasoners

To demonstrate the properties of sets of beliefs, Raymond Smullyan defines the following types of reasoners:

• Accurate reasoner:^[1][2][3][4] An accurate reasoner never believes any false proposition. (modal axiom T)

${\displaystyle \forall p:{\mathcal {B}}p\to p}$

• Inaccurate reasoner:^[1][2][3][4] An inaccurate reasoner believes at least one false proposition.

${\displaystyle \exists p:\neg p\wedge {\mathcal {B}}p}$

• Consistent reasoner:^[1][2][3][4] A consistent reasoner never simultaneously believes a proposition and its negation. (modal axiom D)

${\displaystyle \neg \exists p:{\mathcal {B}}p\wedge {\mathcal {B}}\neg p\quad {\text{or}}\quad \forall p:{\mathcal {B}}p\to \neg {\mathcal {B}}\neg p}$

• Normal reasoner:^[1][2][3][4] A normal reasoner is one who, while believing ${\displaystyle p,}$ also believes they believe p (modal axiom 4).

${\displaystyle \forall p:{\mathcal {B}}p\to {\mathcal {BB}}p}$

A variation on this would be someone who, while not believing ${\displaystyle p,}$ also believes they don't believe p (modal axiom 5).

${\displaystyle \forall p:\neg {\mathcal {B}}p\to {\mathcal {B}}(\neg {\mathcal {B}}p)}$

• Peculiar reasoner:^[1][4] A peculiar reasoner believes proposition p while also believing they do not believe ${\displaystyle p.}$ 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.

${\displaystyle \exists p:{\mathcal {B}}p\wedge {\mathcal {B\neg B}}p}$

• Regular reasoner:^[1][2][3][4] A regular reasoner is one who, while believing ${\displaystyle p\to q}$, also believes ${\displaystyle {\mathcal {B}}p\to {\mathcal {B}}q}$.

${\displaystyle \forall p\forall q:{\mathcal {B}}(p\to q)\to {\mathcal {B}}({\mathcal {B}}p\to {\mathcal {B}}q)}$

• Reflexive reasoner:^[1][4] A reflexive reasoner is one for whom every proposition ${\displaystyle p}$ has some proposition ${\displaystyle q}$ such that the reasoner believes ${\displaystyle q\equiv ({\mathcal {B}}q\to p)}$.

${\displaystyle \forall p\exists q:{\mathcal {B}}(q\equiv ({\mathcal {B}}q\to p))}$

If a reflexive reasoner of type 4 believes ${\displaystyle {\mathcal {B}}p\to p}$, they will believe p. This is a parallelism of Löb's theorem for reasoners.

• Conceited reasoner:^[1][4] A conceited reasoner believes their beliefs are never inaccurate.

${\displaystyle {\mathcal {B}}\quad {\text{or}}\quad {\mathcal {B}}}$

Rewritten in de re form, this is logically equivalent to:

${\displaystyle \forall p}$

This implies that:

${\displaystyle \forall p({\mathcal {B}}{\mathcal {B}}p\to {\mathcal {B}}p)}$

This shows that a conceited reasoner is always a stable reasoner (see below).

• Unstable reasoner:^[1][4] An unstable reasoner is one who believes that they believe some proposition, but in fact do not believe it. This is just as strange a psychological phenomenon as peculiarity; however, an unstable reasoner is not necessarily inconsistent.

${\displaystyle \exists p:{\mathcal {B}}{\mathcal {B}}p\wedge \neg {\mathcal {B}}p}$

• Stable reasoner:^[1][4] A stable reasoner is not unstable. That is, for every ${\displaystyle p,}$ if they believe ${\displaystyle {\mathcal {B}}p}$ then they believe ${\displaystyle p.}$ Note that stability is the converse of normality. We will say that a reasoner believes they are stable if for every proposition ${\displaystyle p,}$ they believe ${\displaystyle {\mathcal {B}}{\mathcal {B}}p\to {\mathcal {B}}p}$ (believing: "If I should ever believe that I believe ${\displaystyle p,}$ then I really will believe ${\displaystyle p}$"). This corresponds to having a dense accessibility relation in Kripke semantics, and any accurate reasoner is always stable.

${\displaystyle \forall p:{\mathcal {BB}}p\to {\mathcal {B}}p}$

• Modest reasoner:^[1][4] A modest reasoner is one for whom for every believed proposition ${\displaystyle p}$, ${\displaystyle {\mathcal {B}}p\to p}$ only if they believe ${\displaystyle p}$. A modest reasoner never believes ${\displaystyle {\mathcal {B}}p\to p}$ unless they believe ${\displaystyle p}$. Any reflexive reasoner of type 4 is modest. (Löb's Theorem)

${\displaystyle \mathrm{\forall <!--\; \forall \; -->}p:\mathcal{B}(\mathcal{B}p}$Zdroj:https://en.wikipedia.org?pojem=Doxastic
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