Doxastic logic is a
type of logic concerned with
reasoning
Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...
about
belief
A belief is an attitude that something is the case, or that some proposition is true. In epistemology, philosophers use the term "belief" to refer to attitudes about the world which can be either true or false. To believe something is to take i ...
s.
The term ' derives from the
Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
(''doxa'', "opinion, belief"), from which the English term ''
doxa
Doxa (; from verb ) Liddell, Henry George, and Robert Scott. 1940.δοκέω" In ''A Greek-English Lexicon'', edited by H. S. Jones and R. McKenzie. Oxford. Clarendon Press. – via Perseus Project. is a common belief or popular opinion. In cla ...
'' ("popular opinion or belief") is also borrowed. Typically, a doxastic logic uses the notation
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
A modal connective (or modal operator) is a logical connective for modal logic. It is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non- truth-functional in the following se ...
.
There is complete parallelism between a person who believes
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s and a
formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
that
derives propositions. Using doxastic logic, one can express the
epistemic counterpart of
Gödel's incompleteness theorem of
metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
, as well as
Löb's theorem
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula ''P'', if it is provable in PA that "if ''P'' is provable in PA then ''P'' is true", then ''P'' is provable in PA. If Pr ...
, and other metalogical results in terms of belief.
[ 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
Types of reasoners
To demonstrate the properties of sets of beliefs,
Raymond Smullyan
Raymond Merrill Smullyan (; May 25, 1919 – February 6, 2017) was an American mathematician, magician, concert pianist, logician, Taoist, and philosopher.
Born in Far Rockaway, New York, his first career was stage magic. He earned a BSc from th ...
defines the following types of reasoners:
* Accurate reasoner:
[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.
] An accurate reasoner never believes any false proposition. (modal axiom T)
::
* Inaccurate reasoner:
An inaccurate reasoner believes at least one false proposition.
::
* 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).
::
:A variation on this would be someone who, while not believing
also ''believes'' they don't believe p (modal axiom 5).
::
* 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
Moore's paradox concerns the apparent absurdity involved in asserting a first-person present-tense sentence such as "It is raining, but I do not believe that it is raining" or "It is raining, but I believe that it is not raining." The first author ...
), a peculiar reasoner is necessarily inaccurate but not necessarily inconsistent.
::
* Regular reasoner:
A regular reasoner is one who, while believing
, also ''believes''
.
::
* 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
below.html" ;"title="Increasing_levels_of_rationality.html" ;"title="ee #Increasing levels of rationality">below">Increasing_levels_of_rationality.html" ;"title="ee #Increasing levels of rationality">belowbelieves
, they will believe p. This is a parallelism of
Löb's theorem
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula ''P'', if it is provable in PA that "if ''P'' is provable in PA then ''P'' is true", then ''P'' is provable in PA. If Pr ...
for reasoners.
*Conceited reasoner:
A conceited reasoner believes their beliefs are never inaccurate.
::
::
: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:
::
.html" ;"title="s "afraid to" believe
p">s "afraid to" believe
pif they believe that belief in
p leads to a contradictory belief.
::
\forall p: \mathcal(\mathcalp \to \mathcal\bot) \to \neg\mathcalp
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
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
i.e., they sooner or later believe every
tautology/theorem (any proposition provable by
truth tables
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argume ...
):
::
\vdash_ p \Rightarrow\ \vdash \mathcalp
:The symbol
\vdash_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
In propositional logic, ''modus ponens'' (; MP), also known as ''modus ponendo ponens'' (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. ...
. If they ever believe
p and
p \to q then they will (sooner or later) believe
q:
::
\forall p \forall q : ( \mathcalp \wedge \mathcal( p \to q)) \to \mathcal q )
:This rule can also be thought of as stating that belief distributes over implication, as it's logically equivalent to
::
\forall p \forall q : \mathcal(p \to q) \to (\mathcalp \to \mathcalq ).
:Note that, in reality, even the assumption of type 1 reasoner may be too strong for some cases (see
Lottery paradox The lottery paradoxKyburg, H. E. (1961). ''Probability and the Logic of Rational Belief'', Middletown, CT: Wesleyan University Press, p. 197. arises from Henry E. Kyburg Jr. considering a fair 1,000-ticket lottery that has exactly one winning ti ...
).
* 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 \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
In philosophy of self, self-awareness is the experience of one's own personality or individuality. It is not to be confused with consciousness in the sense of qualia. While consciousness is being aware of one's environment and body and lifesty ...
than a type 1 reasoner.
::
\forall p \forall q : \mathcal(p \to q) \to \mathcal (\mathcalp \to \mathcalq )
* 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 \to q,, then I will believe
q." Being of type 1, they also believe the
logically equivalent
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
proposition:
\mathcal(p \to q) \to (\mathcalp \to \mathcalq). A type 2 reasoner knows their beliefs are closed under modus ponens.
::
\forall p \forall q : \mathcal(( \mathcalp \wedge \mathcal( p \to q)) \to \mathcal q )
* Type 3 reasoner:
A reasoner is of type 3 if they are a normal reasoner of type 2.
::
\forall p: \mathcal p \to \mathcal \mathcalp
* Type 4 reasoner:
A reasoner is of type 4 if they are of type 3 and also believe they are normal.
::
\mathcal \forall p ( \mathcal p \to \mathcal \mathcalp )/math>
* Type G reasoner: A reasoner of type 4 who believes they are modest.
::\mathcal \forall p ( \mathcal(\mathcalp \to p) \to \mathcalp ) /math>
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 \equiv \mathcalq \to p is provable in the system. Löb's theorem
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula ''P'', if it is provable in PA that "if ''P'' is provable in PA then ''P'' is true", then ''P'' is provable in PA. If Pr ...
(in a general form) is that for any reflexive system of type 4, if \mathcalp \to p is provable in the system, so is p.
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 \mathcal\mathcalp \to \mathcalp, hence by Löb's theorem they will believe \mathcalp (because they believe \mathcalr \to r, where r is the proposition \mathcalp, and so they will believe r, which is the proposition \mathcalp). Being stable, they will then believe p.
See also
* Epistemic modal logic Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applic ...
* Belief revision Belief revision is the process of changing beliefs to take into account a new piece of information. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational age ...
* Common knowledge (logic)
Common knowledge is a special kind of knowledge for a group of agents. There is ''common knowledge'' of ''p'' in a group of agents ''G'' when all the agents in ''G'' know ''p'', they all know that they know ''p'', they all know that they all know ...
* George Boolos
George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
Life
Boolos is of Greek-Jewish descent. He graduated with an A.B. i ...
* Jaakko Hintikka
Kaarlo Jaakko Juhani Hintikka (12 January 1929 – 12 August 2015) was a Finnish philosopher and logician.
Life and career
Hintikka was born in Helsingin maalaiskunta (now Vantaa).
In 1953, he received his doctorate from the University of Helsin ...
* Modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
* Raymond Smullyan
Raymond Merrill Smullyan (; May 25, 1919 – February 6, 2017) was an American mathematician, magician, concert pianist, logician, Taoist, and philosopher.
Born in Far Rockaway, New York, his first career was stage magic. He earned a BSc from th ...
References
Further reading
*
*
*
*
{{Non-classical logic
Belief
Belief revision
Modal logic
Reasoning