Doxastic logic

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)

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

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

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

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

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

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

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

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

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

• Peculiar reasoner:^[1][4] A peculiar reasoner believes proposition ${\displaystyle p}$ 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.

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

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

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

• 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 ${\displaystyle {\mathcal {B}}_{c}p\to p}$, they will believe ${\displaystyle p}$. This is a parallelism of Löb's theorem for reasoners.

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

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

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

${\displaystyle \forall p[{\mathcal {B}}({\mathcal {B}}_{c}p\to p)]}$

This implies that:

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

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 does not believe it. This is just as strange a psychological phenomenon as peculiarity; however, an unstable reasoner is not necessarily inconsistent.

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

• Stable reasoner:^[1][4] A stable reasoner is not unstable. That is, for every ${\displaystyle p,}$ if they believe ${\displaystyle {\mathcal {B}}_{c}p}$ 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 ${\displaystyle {\mathcal {B}}{\mathcal {B}}_{c}p\to {\mathcal {B}}_{c}p}$ (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.

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

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

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

• Queer reasoner:^[4] A queer reasoner is of type G (see below) and believes they are 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 they believe that belief in ${\displaystyle p}$ leads to a contradictory belief.

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

• Type 1 reasoner:^{[1][2][3][4][5]} A type 1 reasoner has a complete knowledge of propositional logic i.e., they sooner or later believe every tautology/theorem (any proposition provable by truth tables):

${\displaystyle \vdash _{PC}p\Rightarrow \ \vdash {\mathcal {B}}_{c}p}$

The symbol ${\displaystyle \vdash _{PC}p}$ means ${\displaystyle p}$ is a tautology/theorem provable in Propositional Calculus. Also, their set of beliefs (past, present and future) is logically closed under modus ponens. If they ever believe ${\displaystyle p}$ and ${\displaystyle p\to q}$ then they will (sooner or later) believe ${\displaystyle q}$:

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

This rule can also be thought of as stating that belief distributes over implication, as it's logically equivalent to

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

Note that, in reality, even the assumption of type 1 reasoner may be too strong for some cases (see Lottery paradox).

• Type 1* reasoner:^[1][2][3][4] A type 1* reasoner believes all tautologies; their set of beliefs (past, present and future) is logically closed under modus ponens, and for any propositions ${\displaystyle p}$ and ${\displaystyle q,}$ if they believe ${\displaystyle p\to q,}$ then they will believe that if they believe ${\displaystyle p}$ then they will believe ${\displaystyle q}$. The type 1* reasoner has "a shade more" self awareness than a type 1 reasoner.

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

• Type 2 reasoner:^[1][2][3][4] A reasoner is of type 2 if they are of type 1, and if for every ${\displaystyle p}$ and ${\displaystyle q}$ they (correctly) believe: "If I should ever believe both ${\displaystyle p}$ and ${\displaystyle p\to q}$, then I will believe ${\displaystyle q}$." Being of type 1, they also believe the logically equivalent proposition: ${\displaystyle {\mathcal {B}}(p\to q)\to ({\mathcal {B}}_{c}p\to {\mathcal {B}}q).}$ A type 2 reasoner knows their beliefs are closed under modus ponens.

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

• Type 3 reasoner:^[1][2][3][4] A reasoner is of type 3 if they are a normal reasoner of type 2.

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

• Type 4 reasoner:^{[1][2][3][4][5]} A reasoner is of type 4 if they are of type 3 and also believe they are normal.

${\displaystyle {\mathcal {B}}[\forall p({\mathcal {B}}p\to {\mathcal {B}}{\mathcal {B}}_{c}p)]}$

• Type G reasoner:^[1][4] A reasoner of type 4 who believes they are modest.

${\displaystyle {\mathcal {B}}[\forall p({\mathcal {B}}({\mathcal {B}}_{c}p\to p)\to {\mathcal {B}}_{c}p)]}$

For systems, logicians define reflexivity to mean that for any ${\displaystyle p}$ (in the language of the system) there is some ${\displaystyle q}$ such that ${\displaystyle q\equiv {\mathcal {B}}q\to p}$ is provable in the system. Löb's theorem (in a general form) is that for any reflexive system of type 4, if ${\displaystyle {\mathcal {B}}_{c}p\to p}$ is provable in the system, so is ${\displaystyle p.}$^[1][4]

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 ${\displaystyle p}$ (and hence be inconsistent). Take any proposition ${\displaystyle p.}$ The reasoner believes ${\displaystyle {\mathcal {B}}{\mathcal {B}}_{c}p\to {\mathcal {B}}_{c}p,}$ hence by Löb's theorem they will believe ${\displaystyle {\mathcal {B}}_{c}p}$ (because they believe ${\displaystyle {\mathcal {B}}r\to r,}$ where ${\displaystyle r}$ is the proposition ${\displaystyle {\mathcal {B}}_{c}p,}$ and so they will believe ${\displaystyle r,}$ which is the proposition ${\displaystyle {\mathcal {B}}_{c}p}$). Being stable, they will then believe ${\displaystyle p.}$^[1][4]

• Epistemic modal logic
• Belief revision
• Common knowledge (logic)
• George Boolos
• Jaakko Hintikka
• Modal logic
• Raymond Smullyan