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 to mean "It is believed that is the case", and the set denotes a 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:
- Consistent reasoner: A consistent reasoner never simultaneously believes a proposition and its negation. (modal axiom D)
- Normal reasoner: A normal reasoner is one who, while believing also believes they believe p (modal axiom 4).
- Peculiar reasoner: A peculiar reasoner believes proposition p while also believing they do not believe 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.
- Reflexive reasoner: A reflexive reasoner is one for whom every proposition has some proposition such that the reasoner believes .
- If a reflexive reasoner of type 4 [see below] believes , they will believe p. This is a parallelism of Löb's theorem for reasoners.
- 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: A stable reasoner is not unstable. That is, for every if they believe then they believe Note that stability is the converse of normality. We will say that a reasoner believes they are stable if for every proposition they believe (believing: "If I should ever believe that I believe then I really will believe ").
- Modest reasoner: A modest reasoner is one for whom every believed proposition , only if they believe . A modest reasoner never believes unless they believe . Any reflexive reasoner of type 4 is modest. (Löb's Theorem)
- Queer reasoner: A queer reasoner is of type G and believes they are inconsistent—but is wrong in this belief.
- Timid reasoner: A timid reasoner does not believe [is "afraid to" believe ] if they believe that belief in 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 (any proposition provable by truth tables). Also, their set of beliefs (past, present and future) is logically closed under modus ponens. If they ever believe and then they will (sooner or later) believe .
- This rule can also be thought of as stating that belief distributes over implication, as it's logically equivalent to
- 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 and if they believe then they will believe that if they believe then they will believe . The type 1* reasoner has "a shade more" self awareness than a type 1 reasoner.
- Type 2 reasoner: A reasoner is of type 2 if they are of type 1, and if for every and they (correctly) believe: "If I should ever believe both and , then I will believe ." Being of type 1, they also believe the logically equivalent proposition: A type 2 reasoner knows their beliefs are closed under modus ponens.
For systems, we define reflexivity to mean that for any (in the language of the system) there is some such that is provable in the system. Löb's theorem (in a general form) is that for any reflexive system of type 4, if is provable in the system, so is
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 (and hence be inconsistent). Take any proposition The reasoner believes hence by Löb's theorem they will believe (because they believe where is the proposition and so they will believe which is the proposition ). Being stable, they will then believe
- Smullyan, Raymond M., (1986) Logicians who reason about themselves, Proceedings of the 1986 conference on Theoretical aspects of reasoning about knowledge, Monterey (CA), Morgan Kaufmann Publishers Inc., San Francisco (CA), pp. 341–352
- https://web.archive.org/web/20070930165226/http://cs.wwc.edu/KU/Logic/Book/book/node17.html Belief, Knowledge and Self-Awareness
- https://web.archive.org/web/20070213054220/http://moonbase.wwc.edu/~aabyan/Logic/Modal.html Modal Logics
- Smullyan, Raymond M., (1987) Forever Undecided, Alfred A. Knopf Inc.
- Rod Girle, Possible Worlds, McGill-Queen's University Press (2003) ISBN 0-7735-2668-4 ISBN 978-0773526686
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- Segerberg, Kr. (1999). "Default Logic as Dynamic Doxastic Logic". Erkenntnis. 50 (2–3): 333–352. doi:10.1023/A:1005546526502.
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