Doxastic logic is a type of logic concerned with reasoning about beliefs.

The term derives from the Ancient Greek (doxa, "opinion, belief"), from which the English term doxa denotes the 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.

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)

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

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

  Normal reasoner: also believes (modal axiom 4).

• also believes (modal axiom 5).

• Peculiar reasoner: 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.

  Regular reasoner:, also believes.

  Reflexive reasoner:.

• If a reflexive reasoner of type 4 [see below. This is a parallelism of Löb's theorem for reasoners.

• Conceited reasoner: A conceited reasoner believes their beliefs are never inaccurate.

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

• This implies that:

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

• Unstable reasoner: 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.

  Stable reasoner:"). This corresponds to having a dense accessibility relation in Kripke semantics, and any accurate reasoner is always stable.

  Modest reasoner:. Any reflexive reasoner of type 4 is modest. (Löb's Theorem)

  Queer reasoner: A queer reasoner is of type G (see below) and believes they are inconsistent—but is wrong in this belief.

• Timid reasoner: leads to a contradictory belief.

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/theorem (any proposition provable by truth tables):

• is a tautology/theorem provable in Propositional Calculus. Also, their set of beliefs (past, present and future) is logically closed under modus ponens:

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

  1. .

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

• Type 1* reasoner:. The type 1* reasoner has "a shade more" self awareness than a type 1 reasoner.

  Type 2 reasoner:." Being of type 1, they also believe the logically equivalent A type 2 reasoner knows their beliefs are closed under modus ponens.

  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 believe they are normal.

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

Self-fulfilling beliefs

is provable in the system. Löb's theorem

Inconsistency of the belief in one's stability

See also

• Belief revision

• Common knowledge (logic)

• Epistemic modal logic

• George Boolos

• Jaakko Hintikka

• Modal logic

• Raymond Smullyan

References

Further reading