View source for Philosophy:Doxastic logic

{{Short description|Type of logic regarding reasoning about beliefs}}
'''Doxastic logic''' is a type of logic concerned with reasoning about [[Philosophy:Belief|belief]]s.

The term ''{{wt|en|doxastic}}'' derives from the [[Social:Ancient Greek|Ancient Greek]] {{wt|grc|δόξα}} (''doxa'', "opinion, belief"), from which the English term ''[[Philosophy:Doxa|doxa]]'' ("popular opinion or belief") is also borrowed. Typically, a doxastic logic uses the notation <math>\mathcal{B}x</math> to mean "It is believed that <math>x</math> is the case", and the set <math>\mathbb{B} : \left \{ b_1, \ldots ,b_n \right \}</math> [[Philosophy:Denotation|denotes]] a [[Theory (mathematical logic)|set of beliefs]]. In doxastic logic, belief is treated as a [[Modal operator|modal operator]].

There is complete parallelism between a person who believes [[Proposition|proposition]]s and a [[Formal system|formal system]] that [[Formal proof|derives]] propositions. Using doxastic logic, one can express the epistemic counterpart of Gödel's incompleteness theorem of [[Metalogic|metalogic]], as well as [[Löb's theorem]], and other metalogical results in terms of belief.<ref name="Logicians">[[Biography: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, [[Biography:Raymond Smullyan|Raymond Smullyan]] defines the following types of reasoners:

* '''Accurate reasoner''':<ref name="Logicians"/><ref name="belief">https://web.archive.org/web/20070930165226/http://cs.wwc.edu/KU/Logic/Book/book/node17.html Belief, Knowledge and Self-Awareness
 </ref><ref name="modal">
https://web.archive.org/web/20070213054220/http://moonbase.wwc.edu/~aabyan/Logic/Modal.html Modal Logics
 </ref><ref name="forever">
[[Biography:Raymond Smullyan|Smullyan, Raymond M.]], (1987) ''Forever Undecided'', Alfred A. Knopf Inc.
</ref> 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>

* '''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  \quad \text{or} \quad  \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 <math>p,</math> also ''believes'' they believe p (modal axiom '''4''').
:: <math>\forall p: \mathcal{B}p \to \mathcal{BB}p</math> 
:A variation on this would be someone who, while not believing <math>p,</math> also ''believes'' they don't believe p (modal axiom '''5''').
:: <math>\forall p: \neg\mathcal{B}p \to \mathcal{B}(\neg \mathcal{B}p)</math>
* '''Peculiar reasoner''':<ref name="Logicians" /><ref name="forever" /> A peculiar reasoner believes proposition p while also believing they do not believe <math>p.</math> Although a peculiar reasoner may seem like a strange psychological phenomenon (see [[Philosophy:Moore's paradox|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 <math>p</math> has some proposition <math>q</math> 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 below] believes <math> \mathcal{B}p \to p </math>, they will believe p. This is a parallelism of [[Löb's theorem]] for reasoners.

*'''Conceited reasoner''':<ref name="Logicians" /><ref name="forever" /> A conceited reasoner believes their beliefs are never inaccurate.
:: <math>\mathcal{B}[\neg\exists p ( \neg p \wedge \mathcal{B}p )] \quad \text{or} \quad \mathcal{B}[\forall p( \mathcal{B}p \to p) ]</math> 
::
:[[Rewriting|Rewritten]] in ''[[De dicto and de re#Representing de dicto and de re in modal logic|de re]]'' form, this is [[Logical equivalence|logically equivalent]] to:
:: <math>\forall p[\mathcal{B} ( \mathcal{B}p \to p) ]</math> 
:This implies that:
:: <math>\forall p(\mathcal{B} \mathcal{B}p \to \mathcal{B}p )</math>
:This shows that a conceited reasoner is always a stable reasoner (see below).

* '''Unstable reasoner''':<ref name="Logicians"/><ref name="forever"/> An unstable reasoner is one who believes that they believe some proposition, but in fact do 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 <math>p,</math> if they believe <math>\mathcal{B}p</math> then they believe <math>p.</math> Note that stability is the converse of normality. We will say that a reasoner believes they are stable if for every proposition <math>p,</math> they believe <math>\mathcal{B}\mathcal{B}p \to \mathcal{B}p</math> (believing: "If I should ever believe that I believe <math>p,</math> then I really will believe <math>p</math>"). This corresponds to having a [[Dense order#Generalizations|dense]] accessibility relation in [[Kripke semantics#Common modal axiom schemata|Kripke semantics]], and any accurate reasoner is always stable.
::<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 for every believed proposition <math>p</math>, <math> \mathcal{B}p \to p </math> only if they believe <math>p</math>. A modest reasoner never believes <math>\mathcal{B}p \to p</math> unless they believe <math>p</math>. 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 they are inconsistent—but is wrong in this belief.
* '''Timid reasoner''':<ref name="forever"/> A timid reasoner does not believe <math>p</math> [is "afraid to" believe <math>p</math>] if they believe that belief in <math>p</math> leads to a contradictory belief.
::<math>\forall p: \mathcal{B}(\mathcal{B}p \to \mathcal{B}\bot) \to \neg\mathcal{B}p </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., they sooner or later believe every [[Philosophy:Tautology (logic)|tautology]]/theorem (any proposition provable by truth tables):
::<math> \vdash_{PC} p \Rightarrow\  \vdash \mathcal{B}p</math>   
:The symbol <math> \vdash_{PC}p</math> means <math> p</math> is a tautology/theorem provable in Propositional Calculus. Also, their set of beliefs (past, present and future) is [[Deductive closure|logically closed]] under [[Modus ponens|modus ponens]]. If they ever believe <math>p</math> and <math>p \to q</math> then they will (sooner or later) believe <math>q</math>:
::<math>\forall p \forall q : ( \mathcal{B}p \wedge  \mathcal{B}( p \to q)) \to \mathcal{B} q )</math>
:This rule can also be thought of as stating that belief distributes over implication, as it's logically equivalent to 
::<math>\forall p \forall q : \mathcal{B}(p \to q) \to (\mathcal{B}p \to \mathcal{B}q )</math>.
:Note that, in reality, even the assumption of type 1 reasoner may be too strong for some cases (see [[Lottery paradox]]).
* '''Type 1* reasoner''':<ref name="Logicians" /><ref name="belief" /><ref name="modal" /><ref name="forever" /> 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 <math>p</math> and <math>q,</math> if they believe <math>p \to q,</math> then they will believe that if they believe <math>p</math> then they will believe <math>q</math>. 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 they are of type 1, and if for every <math>p</math> and <math>q</math> they (correctly) believe: "If I should ever believe both <math>p</math> and <math>p \to q,</math>, then I will believe <math>q</math>." Being of type 1, they also believe the [[Logical equivalence|logically equivalent]] proposition: <math>\mathcal{B}(p \to q) \to (\mathcal{B}p \to \mathcal{B}q).</math> A type 2 reasoner knows their 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 they are a normal reasoner of type 2.
::<math>\forall p: \mathcal{B} p \to \mathcal{B} \mathcal{B}p </math>
* '''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 they are of type 3 and also believe they are normal.
::<math>\mathcal{B}[ \forall p ( \mathcal{B} p \to \mathcal{B} \mathcal{B}p )]</math>
* '''Type G reasoner''':<ref name="Logicians"/><ref name="forever"/> A reasoner of type 4 who believes they are modest.
::<math>\mathcal{B}[ \forall p ( \mathcal{B}(\mathcal{B}p \to p) \to \mathcal{B}p ) ]</math>

==Self-fulfilling beliefs==
For systems, we define reflexivity to mean that for any <math>p</math> (in the language of the system) there is some <math>q</math> such that <math>q \equiv \mathcal{B}q \to p</math> is provable in the system. [[Löb's theorem]] (in a general form) is that for any reflexive system of type 4, if <math>\mathcal{B}p \to p</math> is provable in the system, so is <math>p.</math><ref name="Logicians"/><ref name="forever"/>

==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 <math>p</math> (and hence be inconsistent). Take any proposition <math>p.</math> The reasoner believes <math>\mathcal{B}\mathcal{B}p \to \mathcal{B}p,</math> hence by Löb's theorem they will believe <math>\mathcal{B}p</math> (because they believe <math>\mathcal{B}r \to r,</math> where <math>r</math> is the proposition <math>\mathcal{B}p,</math> and so they will believe <math>r,</math> which is the proposition <math>\mathcal{B}p</math>). Being stable, they will then believe <math>p.</math><ref name="Logicians"/><ref name="forever"/>

==See also==
* [[Epistemic modal logic]]
* [[Belief revision]]
* [[Philosophy:Common knowledge (logic)|Common knowledge (logic)]]
* [[Biography:George Boolos|George Boolos]]
* [[Biography:Jaakko Hintikka|Jaakko Hintikka]]
* [[Modal logic]]
* [[Biography:Raymond Smullyan|Raymond Smullyan]]

==References==
{{Reflist}}

==Further reading==
*{{cite journal |last1=Lindström |first1=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 |s2cid=116984078 }}
*{{cite journal |last=Linski |first=L. |title=On Interpreting Doxastic Logic |journal=Journal of Philosophy |volume=65 |year=1968 |issue=17 |pages=500–502 |doi=10.2307/2024352 |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 |s2cid=118747031 }}
*{{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 |s2cid=58939606 }}

{{Non-classical logic}}

[[Category:Belief]]
[[Category:Belief revision]]
[[Category:Reasoning]]

{{Sourceattribution|Doxastic logic}}