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 '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 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.

Types of reasoners

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

• Accurate reasoner: An accurate reasoner never believes any false proposition. (modal axiom T)

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

• Inaccurate reasoner: An inaccurate reasoner believes at least one false proposition.

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

• Conceited reasoner: A conceited reasoner believes his or her beliefs are never inaccurate.

$\mathcal{B}[\neg\exists p ( \neg p \wedge \mathcal{B}p )]$

or

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

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

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

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

or

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

• Normal reasoner: A normal reasoner is one who, while believing p, also believes he or she believes p (modal axiom 4).

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

• Peculiar reasoner: A peculiar reasoner believes proposition p while also believing he or she does not 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.

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

• Regular reasoner: A regular reasoner is one who, while believing $p \to q$, also believes $\mathcal{B}p \to \mathcal{B}q$.

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

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

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

If a reflexive reasoner of type 4 [see below] believes $\mathcal{B}p \to p$, he or she will believe p. This is a parallelism of Löb's theorem for reasoners.

• Unstable reasoner: 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.

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

• Stable reasoner: A stable reasoner is not unstable. That is, for every p, if he or she believes Bp then he or she believes 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 p, he or she believes BBp→Bp (believing: "If I should ever believe that I believe p, then I really will believe p").

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

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

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

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

Increasing levels of rationality

• Type 1 reasoner: A type 1 reasoner has a complete knowledge of propositional logic i.e., he or she sooner or later believes every tautology (any proposition provable by truth tables). Also, his or her set of beliefs (past, present and future) is logically closed under modus ponens. If he or she ever believes p and believes p → q (p implies q) then he or she will (sooner or later) believe q.
  • $\vdash_{PC} p \Rightarrow\ \vdash \mathcal{B}p$
  • $\forall p \forall q : ( \mathcal{B}p \wedge \mathcal{B}( p \to q)) \to \mathcal{B} q )$
    This rule can also be thought of as stating that belief distributes over implication, as it's logically equivalent to $\forall p \forall q : \mathcal{B}(p \to q) \to (\mathcal{B}p \to \mathcal{B}q )$.
• Type 1* reasoner: A type 1* reasoner believes all tautologies; his or her set of beliefs (past, present and future) is logically closed under modus ponens, and for any propositions p and q, if he or she believes p→q, then he or she will believe that if he or she believes p then he or she will believe q. The type 1* reasoner has "a shade more" self awareness than a type 1 reasoner.
  • $\forall p \forall q : \mathcal{B}(p \to q) \to \mathcal{B} (\mathcal{B}p \to \mathcal{B}q )$
• Type 2 reasoner: A reasoner is of type 2 if he or she is of type 1, and if for every p and q he or she (correctly) believes: "If I should ever believe both p and p→q, then I will believe q." Being of type 1, he or she also believes the logically equivalent proposition: ℬ(p→q)→(ℬp→ℬq). A type 2 reasoner knows his or her beliefs are closed under modus ponens.
  • $\forall p \forall q : \mathcal{B}(( \mathcal{B}p \wedge \mathcal{B}( p \to q)) \to \mathcal{B} q )$
• Type 3 reasoner: A reasoner is of type 3 if he or she is a normal reasoner of type 2.
  • $\forall p: \mathcal{B} p \to \mathcal{B} \mathcal{B}p$
• Type 4 reasoner: A reasoner is of type 4 if he or she is of type 3 and also believes he or she is normal.
  • $\mathcal{B}[ \forall p ( \mathcal{B} p \to \mathcal{B} \mathcal{B}p )]$
• Type G reasoner: A reasoner of type 4 who believes he or she is modest.
  • $\mathcal{B}[ \forall p ( \mathcal{B}(\mathcal{B}p \to p) \to \mathcal{B}p ) ]$

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 him or her. There exists a statement which the reasoner must either remain forever undecided about or lose his or her accuracy. One solution is the statement:

S: "I 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, he or she 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.

Inaccuracy and peculiarity of conceited reasoners

A reasoner of type 1 is faced with the statement "I will never believe this sentence." The interesting thing now is that if the reasoner believes he or she is always accurate, then he or she will become inaccurate. Such a reasoner will reason: "The statement in question is that I won't believe the statement, so if it's false then I will believe the statement. Because I am accurate, believing the statement means it must be true. So if the statement is false then it must be true. It's tautological that if a statement being false implies the statement, then that statement is true. Therefore the statement is true."

At this point the reasoner believes the statement, which makes it false. Thus the reasoner is inaccurate in believing that the statement is true. If the reasoner hadn't assumed his or her own accuracy, he or she would never have lapsed into an inaccuracy. Formally:

$1 \ \ S \equiv \lnot \mathcal{B}S$ [definition of S]

$2 \ \ (\lnot S \to S) \to S$ [elementary tautology]

$3 \ \ (\mathcal{B}S \to S) \to S$ [because ¬S ≡ ℬS]

$4 \ \ \mathcal{B}((\mathcal{B}S \to S) \to S)$ [reasoner believes all tautologies]

$5 \ \ \mathcal{B}(\mathcal{B}S \to S) \to \mathcal{B}S$ [the reasoner is of type 1]

$6 \ \ \mathcal{B}(\mathcal{B}S \to S)$ [the reasoner is conceited]

$7 \ \ \mathcal{B}S$ [modus ponens 5 and 6]

$8 \ \ \lnot S$ [because ℬS ≡ ¬S]

Additionally, the reasoner is peculiar because he or she believes that he/she doesn't believe the statement (symbolically, ℬ(¬ℬS), which follows from ℬS because S ≡ ¬ℬS) even though he/she actually believes it.

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 he or she is stable, then he or she will become unstable. Stated otherwise, if a stable reflexive reasoner of type 4 believes that he or she is stable, then he or she will become inconsistent. Why is this? Suppose that a stable reflexive reasoner of type 4 believes that he or she is stable. We will show that he or she 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 he or she will believe ℬp (because he or she believes ℬr→r, where r is the proposition ℬp, and so he or she will believe r, which is the proposition ℬp). Being stable, he or she will then believe p.

See also

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