Doxatic logic & types of reasoners

https://en.wikipedia.org/wiki/Doxastic_logic 
Something i read long time back when i read Smullyan's books, but making a note because i love it. His types of reasoners is beautiful :)

Types of reasoners[edit]

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.

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

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

${\displaystyle {\mathcal {B}}[\neg \exists p(\neg p\wedge {\mathcal {B}}p)]\quad {\text{or}}\quad {\mathcal {B}}[\forall p({\mathcal {B}}p\to p)]}$

A conceited reasoner with rationality of at least type 1 (see below) will necessarily lapse into inaccuracy.

• 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 he or she believes p (modal axiom 4).

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

• Peculiar reasoner:^[1][4] A peculiar reasoner believes proposition p while also believing he or she does 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 [see below] believes ${\displaystyle {\mathcal {B}}p\to p}$, he or she will believe p. This is a parallelism of Löb's theorem for reasoners.

• Unstable reasoner:^[1][4] An unstable reasoner is one who believes that he or she believes 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.

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

• Stable reasoner:^[1][4] A stable reasoner is not unstable. That is, for every ${\displaystyle p,}$ if he or she believes ${\displaystyle {\mathcal {B}}p}$ then he or she believes ${\displaystyle p.}$ Note that stability is the converse of normality. We will say that a reasoner believes he or she is stable if for every proposition ${\displaystyle p,}$ he or she believes ${\displaystyle {\mathcal {B}}{\mathcal {B}}p\to {\mathcal {B}}p}$ (believing: "If I should ever believe that I believe ${\displaystyle p,}$ then I really will believe ${\displaystyle p}$").

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

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

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

• Queer reasoner:^[4] A queer reasoner is of type G and believes he or she is inconsistent—but is wrong in this belief.
• Timid reasoner:^[4] A timid reasoner does not believe ${\displaystyle p}$ [is "afraid to" believe ${\displaystyle p}$] if he or she believes ${\displaystyle {\mathcal {B}}p\to {\mathcal {B}}\bot }$