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Doxastic logic is a modal logic concerned with reasoning about beliefs. 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 
denotes a set of beliefs. In doxastic
logic, belief is treated as a modal operator.
-
- :
{b1,b2,...,
bn}
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 mathematical results in terms of belief.[1]
Types of
reasoners
To demonstrate the properties of sets of beliefs, Raymond
Smullyan defines the following types of reasoners:
- Accurate reasoner[1][2][3][4]:
An accurate reasoner never believes any false proposition. (modal
axiom T)
p(Bp
p)
- Inaccurate reasoner[1][2][3][4]
An inaccurate reasoner believes at least one false
proposition.
p(Bp&¬p)
- Conceited reasoner[1][4]:
A conceited reasoner believes its beliefs are never inaccurate. A
conceited reasoner will necessarily lapse into an inaccuracy.
- B(¬p(Bp&¬p))
- Consistent reasoner[1][2][3][4]:
A consistent reasoner never simultaneously believes a proposition
and its negation. (modal axiom D)
- ¬p((Bp&B¬p))
- Normal reasoner[1][2][3][4]:
A normal reasoner is one who, while believing p, also
believes it believes p (modal axiom
4).
- p(BpBBp)
- Peculiar reasoner[1][4]:
A peculiar reasoner believes proposition p while also believing it
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.
- p(Bp&B¬Bp)
- Regular reasoner[1][2][3][4]:
A regular reasoner is one for whom all beliefs are distributive
over logical operations. (modal axiom K)
- p(q(B(pq)(BpBq)))
- Reflexive reasoner[1][4]:
A reflexive reasoner is one for whom every proposition p has some q
such that the reasoner believes q≡(Bq→p). So if a reflexive
reasoner of type 4 [see below] believes Bp→p,
it will believe p. This is a parallelism of Löb's
theorem for reasoners.
- Unstable reasoner[1][4]:
An unstable reasoner is one for whom there is some proposition p
such that it believes it believes p, but who does not really
believe p. This is just as strange a psychological phenomenon as
peculiarity; however, an unstable reasoner is not necessarily
inconsistent.
- Stable reasoner[1][4]:
A stable reasoner is not unstable. That is, for every p, if it
believes Bp then it believes p. Note that stability is the converse
of normality. We will say that a reasoner believes it is stable if
for every proposition p, it believes BBp→Bp (believing: "If I
should ever believe that I believe p, then I really will believe
p").
- BBpBp
- Modest reasoner[1][4]:
A modest reasoner is one for whom every believed proposition p,
only if it believes p. A modest reasoner never believes Bp→p unless
it believes p. Any reflexive reasoner of type 4 is modest. (Löb's Theorem)
- B(Bp→p)→Bp
- Queer reasoner[4]:
A queer reasoner is of type G and believes it is inconsistent—but
is wrong in this belief.
- Timid reasoner[4]:
A timid reasoner is afraid to believe p [i.e., it does not believe
p] if it believes
Increasing levels of
rationality
- Type 1 reasoner[1][2][3][4][5]:
A type 1 reasoner has a complete knowledge of propositional logic i.e, it sooner or later
believes every tautology (any proposition provable
by truth tables) (modal axiom
N). Also, its set of beliefs (past, present and
future) is logically closed under modus ponens. If it
ever believes p and believes p→q (p implies q) then it will (sooner
or later) believe q (modal axiom K). This is
equivalent to modal system K.
-
- Type 1* reasoner[1][2][3][4]:
A type 1* reasoner believes all tautologies; its set of beliefs
(past, present and future) is logically closed under modus ponens,
and for any propositions p and q, if it believes p→q, then it will
believe that if it believes p then it will believe q. The type 1*
reasoner has a shade more self awareness than
a type 1 reasoner.
-
- Type 2 reasoner[1][2][3][4]:
A reasoner is of type 2 if it is of type 1, and if for every p and
q it (correctly) believes: "If I should ever believe both p and
p→q, then I will believe q." Being of type 1, it also believes the
logically equivalent proposition:
B(p→q)→(Bp→Bq). A type 2 reasoner knows its beliefs are closed
under modus ponens.
-
- B((Bp&B(pq))Bq)
- Type 3 reasoner[1][2][3][4]:
A reasoner is of type 3 if it is a normal reasoner of type 2.
- Type 4 reasoner[1][2][3][4][5]:
A reasoner is of type 4 if it is of type 3 and also believes it is
normal.
- Type G reasoner[1][4] :
A reasoner of type 4 who believes it is modest.
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 it. There exists a statement
which the reasoner must either remain forever undecided about or
lose its 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, it 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 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.[1][4]
Inconsistency
and peculiarity of conceited reasoners
A reasoner of type 1 is faced with the statement "You will never
believe this sentence." The interesting thing now is that if the
reasoner believes it is always accurate, then it 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 its own accuracy, it would never have lapsed into an
inaccuracy.
It can also be shown that a conceited reasoner is peculiar.[1][4]
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.[1][4]
Inconsistency
of the belief in one's stability
If a consistent reflexive reasoner of type 4 believes that it is
stable, then it will become unstable. Stated otherwise, if a stable
reflexive reasoner of type 4 believes that it is stable, then it
will become inconsistent. Why is this? Suppose that a stable
reflexive reasoner of type 4 believes that it is stable. We will
show that it 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 it will believe Bp (because
it believes Br→r, where r is the proposition Bp, and so it will
believe r, which is the proposition Bp). Being stable, it will then
believe p.[1][4]
See also
References
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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
- ^ a
b
c
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http://cs.wwc.edu/KU/Logic/Book/book/node17.html
Belief, Knowledge and Self-Awareness
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http://moonbase.wwc.edu/~aabyan/Logic/Modal.html
Modal Logics
- ^ a
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Smullyan,
Raymond M., (1987) Forever Undecided, Alfred A. Knopf
Inc.
- ^ a
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Rod Girle, Possible Worlds, McGill-Queen's University
Press (2003) ISBN 0773526684 ISBN 978-0773526686
Bp
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