Doxastic logic

 ( Belief ) 

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

The term derives from the Ancient Greek ( doxa, "opinion, belief"), from which the English term doxa ("popular opinion or belief") is also borrowed. Typically, a doxastic logic uses the notation $\backslash mathcal\{B\}\_c\; x$ to mean "reasoner $c$ believes that $x$ is true", and the set $\backslash mathbb\{B\}\_c\; :\; \backslash left\; \backslash \{\; b\_1,\; \backslash ldots\; ,b\_n\; \backslash right\; \backslash \}$ denotation the set of beliefs of $c$. In doxastic logic, belief is treated as a modal operator.

There is complete parallelism between a person who believes and a formal system that formal proof propositions. Using doxastic logic, one can express the epistemic logic counterpart of Gödel's incompleteness theorem of metalogic, as well as Löb's theorem, and other metalogical results in terms of belief.Raymond Smullyan, (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

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

• Accurate reasoner: Belief, Knowledge and Self-Awareness

https://web.archive.org/web/20070213054220/http://moonbase.wwc.edu/~aabyan/Logic/Modal.html Modal Logics

Raymond Smullyan, (1987) Forever Undecided, Alfred A. Knopf Inc. An accurate reasoner never believes any false proposition. (modal axiom T)

:$\backslash forall\; p:\; \backslash mathcal\{B\}\_c\; p\; \backslash to\; p$

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

: $\backslash exists\; p:\; \backslash neg\; p\; \backslash wedge\; \backslash mathcal\{B\}\_c\; p$

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

:$\backslash neg\backslash exists\; p:\; \backslash mathcal\{B\}\_c\; p\; \backslash wedge\; \backslash mathcal\{B\}\_c\; \backslash neg\; p\; \backslash quad\; \backslash text\{or\}\; \backslash quad\; \backslash forall\; p:\; \backslash mathcal\{B\}\_c\; p\; \backslash to\; \backslash neg\backslash mathcal\{B\}\_c\; \backslash neg\; p$

• Normal reasoner: A normal reasoner is one who, while believing $p,$ also believes they believe $p$ (modal axiom 4).

: $\backslash forall\; p:\; \backslash mathcal\{B\}\_c\; p\; \backslash to\; \backslash mathcal\{BB\}p$

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

: $\backslash forall\; p:\; \backslash neg\backslash mathcal\{B\}\_c\; p\; \backslash to\; \backslash mathcal\{B\}(\backslash neg\; \backslash mathcal\{B\}\_c\; p)$

• Peculiar reasoner: A peculiar reasoner believes proposition $p$ while also believing they do 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.

: $\backslash exists\; p:\; \backslash mathcal\{B\}\_c\; p\; \backslash wedge\; \backslash mathcal\{B\backslash neg\; B\}p$

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

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

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

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

If a reflexive reasoner of type 4 see believes $\backslash mathcal\{B\}\_c\; p\; \backslash to\; p$, they will believe $p$. This is a parallelism of Löb's theorem for reasoners.

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

: $\backslash mathcal\{B\}\backslash neg\backslash exists\; \backslash quad\; \backslash text\{or\}\; \backslash quad\; \backslash mathcal\{B\}\backslash forall$

:

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

: $\backslash forall\; p\backslash mathcal\{B\}$

This implies that:

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

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.

:$\backslash exists\; p:\; \backslash mathcal\{B\}\backslash mathcal\{B\}\_c\; p\; \backslash wedge\; \backslash neg\backslash mathcal\{B\}\_c\; p$

• Stable reasoner: A stable reasoner is not unstable. That is, for every $p,$ if they believe $\backslash mathcal\{B\}\_c\; p$ 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 $\backslash mathcal\{B\}\backslash mathcal\{B\}\_c\; p\; \backslash to\; \backslash mathcal\{B\}\_c\; p$ (believing: "If I should ever believe that I believe $p,$ then I really will believe $p$"). This corresponds to having a dense accessibility relation in Kripke semantics, and any accurate reasoner is always stable.

:$\backslash forall\; p:\; \backslash mathcal\{BB\}p\backslash to\backslash mathcal\{B\}\_c\; p$

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

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

• Queer reasoner: A queer reasoner is of type G (see below) and believes they are inconsistent—but is wrong in this belief.
• Timid reasoner: A timid reasoner does not believe $p$ is if they believe that belief in $p$ leads to a contradictory belief.

:$\backslash forall\; p:\; \backslash mathcal\{B\}(\backslash mathcal\{B\}\_c\; p\; \backslash to\; \backslash mathcal\{B\}\backslash bot)\; \backslash to\; \backslash neg\backslash mathcal\{B\}\_c\; p$

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Increasing levels of rationality

• Type 1 reasoner:Rod Girle, Possible Worlds, McGill-Queen's University Press (2003) 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):

:$\backslash vdash\_\{PC\}\; p\; \backslash Rightarrow\backslash \; \backslash vdash\; \backslash mathcal\{B\}\_c\; p$

The symbol $\backslash vdash\_\{PC\}p$ means $p$ 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 $p$ and $p\; \backslash to\; q$ then they will (sooner or later) believe $q$:

:$\backslash forall\; p\; \backslash forall\; q\; :\; (\; \backslash mathcal\{B\}\_c\; p\; \backslash wedge\; \backslash mathcal\{B\}(\; p\; \backslash to\; q))\; \backslash to\; \backslash mathcal\{B\}\; q$

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

:$\backslash forall\; p\; \backslash forall\; q\; :\; \backslash mathcal\{B\}(p\; \backslash to\; q)\; \backslash to\; (\backslash mathcal\{B\}\_c\; p\; \backslash to\; \backslash mathcal\{B\}q\; )$.

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

• Type 1* reasoner: 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 $p\; \backslash to\; q,$ 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.

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

• 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 $p\; \backslash to\; q,$, then I will believe $q$." Being of type 1, they also believe the logically equivalent proposition: $\backslash mathcal\{B\}(p\; \backslash to\; q)\; \backslash to\; (\backslash mathcal\{B\}\_c\; p\; \backslash to\; \backslash mathcal\{B\}q).$ A type 2 reasoner knows their beliefs are closed under modus ponens.

:$\backslash forall\; p\; \backslash forall\; q\; :\; \backslash mathcal\{B\}((\; \backslash mathcal\{B\}\_c\; p\; \backslash wedge\; \backslash mathcal\{B\}(\; p\; \backslash to\; q))\; \backslash to\; \backslash mathcal\{B\}\; q\; )$

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

:$\backslash forall\; p:\; \backslash mathcal\{B\}\; p\; \backslash to\; \backslash mathcal\{B\}\; \backslash mathcal\{B\}\_c\; p$

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

:$\backslash mathcal\{B\}$

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

:$\backslash mathcal\{B\}$

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Self-fulfilling beliefs For systems, logicians define reflexivity to mean that for any $p$ (in the language of the system) there is some $q$ such that $q\; \backslash equiv\; \backslash mathcal\{B\}q\; \backslash to\; p$ is provable in the system. Löb's theorem (in a general form) is that for any reflexive system of type 4, if $\backslash mathcal\{B\}\_c\; p\; \backslash to\; p$ is provable in the system, so is $p.$

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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 $\backslash mathcal\{B\}\backslash mathcal\{B\}\_c\; p\; \backslash to\; \backslash mathcal\{B\}\_c\; p,$ hence by Löb's theorem they will believe $\backslash mathcal\{B\}\_c\; p$ (because they believe $\backslash mathcal\{B\}r\; \backslash to\; r,$ where $r$ is the proposition $\backslash mathcal\{B\}\_c\; p,$ and so they will believe $r,$ which is the proposition $\backslash mathcal\{B\}\_c\; p$). Being stable, they will then believe $p.$

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See also

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

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Further reading

Categories: Belief, Belief Revision, Epistemic Logic, Reasoning

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