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'''Doxastic logic''' is a [[modal logic]] concerned with [[reasoning]] about [[belief]]s. 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 <math>\mathbb{B}</math> [[denotation|denotes]] a [[Theory (mathematical logic)|set of beliefs]]. In doxastic logic, belief is treated as a [[modal operator]].

::<math>\mathbb{B}</math>: {<math>b_{1},b_{2},...,b_{n}</math>}

There is complete parallelism between a person who believes [[proposition]]s and a [[formal system]] that [[formal proof|derives]] propositions. Using doxastic logic, one can express the [[epistemic logic|epistemic]] counterpart of [[Gödel's incompleteness theorem]] of [[metalogic]], as well as [[Löb's theorem]], and other metalogical results in terms of belief.<ref name="Logicians">[[Raymond Smullyan|Smullyan, Raymond M.]], (1986) [http://portal.acm.org/ft_gateway.cfm?id=1029818&type=pdf&coll=GUIDE&dl=GUIDE&CFID=44077077&CFTOKEN=65318791 ''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</ref>

==Types of reasoners==
To demonstrate the properties of sets of beliefs, Raymond Smullyan defines the following types of reasoners:
* '''Accurate reasoner''':<ref name="Logicians"/>{{Request quotation|date=March 2011}}<ref name="belief">http://cs.wwc.edu/KU/Logic/Book/book/node17.html Belief, Knowledge and Self-Awareness{{Dead link|date=March 2011}}
</ref>{{Dead link|date=March 2011}}<ref name= "modal">
http://moonbase.wwc.edu/~aabyan/Logic/Modal.html Modal Logics{{Dead link|date=March 2011}}
</ref>{{Dead link|date=March 2011}}<ref name="forever">
[[Raymond Smullyan|Smullyan, Raymond M.]], (1987) ''Forever Undecided'', Alfred A. Knopf Inc.
</ref>{{Request quotation|date=March 2011}} An accurate reasoner never believes any false proposition. (modal axiom '''T''')

: <math>\forall p: \mathcal{B}p \to p</math>

* '''Inaccurate reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/> An inaccurate reasoner believes at least one false proposition.

: <math>\exists p: \neg p \wedge \mathcal{B}p</math>

* '''Conceited reasoner''':<ref name="Logicians"/><ref name="forever"/> A conceited reasoner believes his or her beliefs are never inaccurate. A conceited reasoner will necessarily lapse into an inaccuracy.

: <math>\mathcal{B}[\neg\exists p: \neg p \wedge \mathcal{B}p]</math>

* '''Consistent reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/> A consistent reasoner never simultaneously believes a proposition and its negation. (modal axiom '''D''')

: <math>\neg\exists p: \mathcal{B}p \wedge \mathcal{B}\neg p</math>
: or
: <math>\forall p: \mathcal{B}p \to \neg\mathcal{B}\neg p</math>

* '''Normal reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/> A normal reasoner is one who, while believing p, also ''believes'' he or she believes p (modal axiom '''4''').

: <math>\forall p: \mathcal{B}p \to \mathcal{BB}p</math>

* '''Peculiar reasoner''':<ref name="Logicians"/><ref name="forever"/> 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.

: <math>\exists p: \mathcal{B}p \wedge \mathcal{B\neg B}p</math>

* '''Regular reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/> A regular reasoner is one who, while believing <math> p \to q </math>, also ''believes'' <math> \mathcal{B}p \to \mathcal{B}q </math>.

: <math>\forall p: \forall q: \mathcal{B}(p \to q) \to \mathcal{B} (\mathcal{B}p \to \mathcal{B}q)</math>

* '''Reflexive reasoner''':<ref name="Logicians"/><ref name="forever"/> A reflexive reasoner is one for whom every proposition p has some q such that the reasoner believes <math> q \equiv ( \mathcal{B}q \to p) </math>.

: <math>\forall p:\exists q:\mathcal{B}(q \equiv ( \mathcal{B}q \to p)) </math>

:If a reflexive reasoner of type 4 [see [[#Increasing levels of rationality|below]]] believes <math> \mathcal{B}p \to p </math>, he or she will believe p. This is a parallelism of [[Löb's theorem]] for reasoners.

* '''Unstable reasoner''':<ref name="Logicians"/><ref name="forever"/> 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.

: <math>\exists p: \mathcal{B}\mathcal{B}p \wedge \neg\mathcal{B}p </math>

* '''Stable reasoner''':<ref name="Logicians"/><ref name="forever"/> 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").

: <math>\forall p: \mathcal{BB}p\to\mathcal{B}p</math>

* '''Modest reasoner''':<ref name="Logicians"/><ref name="forever"/> A modest reasoner is one for whom every believed proposition p, <math> \mathcal{B}p \to p </math> 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]])

: <math>\forall p: \mathcal{B}(\mathcal{B}p \to p) \to \mathcal{B}p</math>

* '''Queer reasoner''':<ref name="forever"/> A queer reasoner is of type G and believes he or she is inconsistent—but is wrong in this belief.

* '''Timid reasoner''':<ref name="forever"/> A timid reasoner is afraid to believe p [i.e., he or she does not believe p] if he or she believes <math> \mathcal{B}p \to \mathcal{B}\bot </math>

==Increasing levels of rationality==
* '''Type 1 reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/><ref name="possible">Rod Girle, ''Possible Worlds'', McGill-Queen's University Press (2003) ISBN 0-7735-2668-4 ISBN 978-0773526686</ref> A type 1 reasoner has a complete knowledge of [[propositional logic]] i.e., he or she sooner or later believes every [[tautology (logic)|tautology]] (any proposition provable by [[truth tables]]). Also, his or her set of beliefs (past, present and future) is [[Deductive closure|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 .

**<math> \vdash_{PC} p \Rightarrow \vdash \mathcal{B}p</math>
**<math>\forall p: \forall q: \mathcal{B}(p \to q) \to (\mathcal{B}p \to \mathcal{B}q )</math>

* '''Type 1* reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/> 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.

**<math>\forall p: \forall q: \mathcal{B}(p \to q) \to \mathcal{B} (\mathcal{B}p \to \mathcal{B}q )</math>

* '''Type 2 reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/> 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 [[logical equivalence|logically equivalent]] proposition: B(p→q)→(Bp→Bq). A type 2 reasoner knows his or her beliefs are closed under modus ponens.

**<math>\forall p: \forall q: \mathcal{B}(( \mathcal{B}p \wedge \mathcal{B}( p \to q)) \to \mathcal{B} q )</math>

* '''Type 3 reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/> A reasoner is of type 3 if he or she is a normal reasoner of type 2.

* '''Type 4 reasoner''':<ref name="Logicians"/><ref name="belief"/><ref name="modal"/><ref name="forever"/><ref name="possible"/> A reasoner is of type 4 if he or she is of type 3 and also believes he or she is normal.

* '''Type G reasoner''':<ref name="Logicians"/><ref name="forever"/> A reasoner of type 4 who believes he or she is modest.

== [[Godel incompleteness|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 [[contradiction|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.<ref name="Logicians"/><ref name="forever"/>

==Inconsistency 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: "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 his or her own accuracy, he or she would never have lapsed into an inaccuracy.

It can also be shown that a conceited reasoner is peculiar.<ref name="Logicians"/><ref name="forever"/>

==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.<ref name="Logicians"/><ref name="forever"/>

==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.<ref name="Logicians"/><ref name="forever"/>

==See also==
{{Portal|Logic}}
* [[Modal logic]]
* [[Raymond Smullyan]]
* [[Jaakko Hintikka]]
* [[George Boolos]]
* [[Belief revision]]
* [[Common knowledge (logic)]]

==References==
{{Reflist}}

== Further reading ==
*{{cite journal |last=Lindström |first=St. |first2=Wl. |last2=Rabinowicz |title=DDL Unlimited. Dynamic Doxastic Logic for Introspective Agents |journal=[[Erkenntnis]] |volume=51 |year=1999 |issue=2–3 |pages=353–385 |doi=10.1023/A:1005577906029 }}
*{{cite journal |last=Linski |first=L. |title=On Interpreting Doxastic Logic |journal=[[Journal of Philosophy]] |volume=65 |year=1968 |issue=17 |pages=500–502 |jstor=2024352 }}
*{{cite journal |last=Segerberg |first=Kr. |title=Default Logic as Dynamic Doxastic Logic |journal=Erkenntnis |volume=50 |issue=2–3 |year=1999 |pages=333–352 |doi=10.1023/A:1005546526502 }}
*{{cite journal |last=Wansing |first=H. |title=A Reduction of Doxastic Logic to Action Logic |journal=Erkenntnis |volume=53 |issue=1–2 |year=2000 |pages=267–283 |doi=10.1023/A:1005666218871 }}

{{Logic}}

[[Category:Belief]]
[[Category:Belief revision]]
[[Category:Modal logic]]
[[Category:Reasoning]]

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