Doxastic logic

Doxastic logic is a modal logic that is concerned with reasoning about beliefs. The term doxastic is derived from the ancient Greek doxa which means 'belief.' Typically, a doxastic logic uses 'Bx' to mean "It is believed that x is the case" and the set $\mathbb {B}$ denotes a set of beliefs. In doxastic logic, belief is treated as a modal operator.

$\mathbb {B}$: {$b_{1},b_{2},...,b_{n}$}

There is a 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 mathematical results in terms of belief.

Types of reasoners

In order to demonstrate the properties of sets of beliefs, Raymond Smullyan defines the following types of reasoners:

• Accurate reasoner: A reasoner is accurate if they never believe any false proposition. (modal axiom T)

$\forall$p(Bp$\to$p)

• Inaccurate reasonerA reasoner is inaccurate if there exists a proposition which they believe and which is not true.

$\exists$p(Bp$\wedge$$\neg$p)

• Conceited reasoner: A reasoner is conceited, if they believe they are never inaccurate. A conceited reasoner will necessarily lapse into an inaccuracy.

B($\neg$$\exists$p(Bp$\wedge$$\neg$p))

• Consistent reasoner: A consistent reasoner never believes any proposition and its negation. (modal axiom D)

$\neg$$\exists$p((Bp$\wedge$B$\neg$p))

• Normal reasoner: A reasoner is normal if whenever they believe p, they also believe that they believe p. (modal axiom 4)

$\forall$p(Bp$\to$BBp)

• Peculiar reasoner: A reasoner is peculiar if there is some proposition p such that they believe p and also believe they don't believe 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.

$\exists$p(Bp$\wedge$B$\neg$Bp)

• Regular reasoner: A reasoner is regular if all their their beliefs are distributive over logical operations. (modal axiom K)

$\forall$p($\forall$q(B(p$\to$q)$\to$(Bp$\to$Bq)))

• Reflexive reasoner: A reasoner is reflexive if for every proposition p there is some q such that the reasoner believes q≡(Bq→p). And so if a reflexive reasoner of type 4 believes Bp→p, they will believe p. This is a parallelism of Löb's theorem for reasoners.

• Unstable reasoner: A reasoner is unstable if there is some proposition p they believe that they believe p, but don't really believe p. This is just as strange a psychological phenomenon as peculiarity, however, an unstable reasoner is not necessarily inconsistent.

• Stable reasoner: A reasoner is stable if they are not unstable. That is, for every p, if they believe Bp then they believe p. Note that stability is the converse of normality. We will say that a reasoner believes they are stable if for every proposition p, they believe BBp→Bp (they believe: "If I should ever believe that I believe p, then I really will believe p).

BBp$\to$Bp

• Modest reasoner: A reasoner is modest if for every proposition p they believe $Bp\to p$ only if they believe p. A modest reasoner never believes Bp→p unless they believe p. Any reflexive reasoner of type 4 is modest. (Löb's Theorem)

B(Bp→p)→Bp

• Queer reasoner: A reasoner is a queer reasoner if they are of type G and believe they are inconsistent—but they are wrong in this belief!

• Timid reasoner: A timid reasoner is afraid to believe p if they believe $Bp\to B\bot$

Increasing levels of rationality

• Type 1 reasoner: A type 1 reasoner has a complete knowledge of propositional logic i.e, they sooner or later believe every tautology (any proposition provable by truth tables) (modal axiom N). Also, their set of beliefs (past, present and future) is logically closed under modus ponens. If they ever believe p and believes p→q (p implies q) then they will (sooner or later) believe q (modal axiom K). This is equivalent to modal system K.

• $p\models Bp$
• (Bp$\wedge$B(p$\to$q))$\to$Bq

• Type 1* reasoner: 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 p and q, if they believe p→q, then they will believe that if they believe p then they will believe q. The type 1* reasoner has a shade more self awareness than a type 1 reasoner.

• B(p$\to$q)$\to$B(Bp$\to$Bq)

• Type 2 reasoner: A reasoner is of type 2 if they are of type 1, and if for every p and q they (correctly) believe: "If I should ever believe both p and p→q, then I will believe q." Being of type 1, they also believe the logically equivalent proposition: B(p→q)→(Bp→Bq). A type 2 reasoner knows their beliefs are closed under modus ponens.

• B((Bp$\wedge$B(p$\to$q))$\to$Bq)

• Type 3 reasoner: A reasoner is of type 3 if they are a normal reasoner of type 2.

• Type 4 reasoner: A reasoner is of type 4 if they are of type 3 and also believes they are normal.

• Type G reasoner : A reasoner of type 4 who believes they are modest.

Gödel incompleteness and doxastic undecidability

Let us say, an accurate reasoner is faced with the task of assigning a truth value to a statement posed to them. There exists a statement which the reasoner must either remain forever undecided or lose their accuracy. One solution is the statement:

S: "You will never believe this statement."

If the reasoner ever believes the statement S, it becomes falsified by that fact, making S an untrue belief and hence making the reasoner inaccurate in believing S.

Therefore, since the reasoner is accurate, they will never believe S. Hence the statement was true, because that is exactly what it claimed. It further follows that the reasoner will never have the false belief that S is true. The reasoner cannot believe either that the statement is true or false without becoming inconsistent (i.e. holding two contradictory beliefs). And so the reasoner must remain forever undecided as to whether the statement S is true or false.

The equivalent theorem is that for any formal system F, there exists a mathematical statement which can be interpreted as "This statement is not provable in formal system F". If the system F is consistent, neither the statement nor its opposite will be provable in it

Inconsistency and peculiarity of conceited reasoners

A reasoner of type 1 is faced with the statement "You will never believe this sentence." The interesting thing now is that if the reasoner believes they are always accurate, then they will become inaccurate. Such a reasoner will reason: "If I believe the statement then it will be made false by that fact, which means that I will be inaccurate. This is impossible, since I'm always accurate. Therefore I can't believe the statement, it must be false."

At this point the reasoner believes that the statement is false, which makes the statement true. Thus the reasoner is inaccurate in believing that the statement is false. If the reasoner hadn't assumed their own accuracy, they would never have lapsed into an inaccuracy.

It can also be shown that a conceited reasoner is peculiar.

Self fulfilling beliefs

For systems, we define reflexivity to mean that for any p (in the language of the system) there is some q such that q≡(Bq→p) is provable in the system. Löb's theorem (in a general form) is that for any reflexive system of type 4, if Bp→p is provable in the system, so is p.

Inconsistency of the belief in one's stability

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 (and hence be inconsistent). Take any proposition p. The reasoner believes BBp→Bp, hence by Löb's theorem they will believe Bp (because they believe Br→r, where r is the proposition Bp, and so they will believe r, which is the proposition Bp). Being stable, they will then believe p.

See also

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