Introduction: Belief vs Degrees of Belief
Hannes Leitgeb
LMU Munich
October 2014
My three lectures will be devoted to answering this question:
How does rational (all-or-nothing) belief relate to degrees of belief?
But first I will prepare the ground by considering some preliminary questions:
What is belief?
How can we talk about belief?
What should belief be like?
Plan:
1
The Nature of Belief
2
Concepts of Belief
3
Emptiness, Reduction/Elimination, Independence
4
Norms on Belief
5
Bridge Principles for Rational Belief and Degrees of Belief
The Nature of Belief
Assumption 1: Belief is a propositional attitude of cognitive agents:
an agent’s belief that X is a mental state that has as its content the
proposition that X .
The Nature of Belief
Assumption 1: Belief is a propositional attitude of cognitive agents:
an agent’s belief that X is a mental state that has as its content the
proposition that X .
But what kind of propositional attitude is it?
Belief is usually explained functionally and with the help of normative terms:
the propositional attitude the function of which is to reach the goal. . . and
to satisfy the norms. . . and to realize the valuable state. . .,
. . . or to achieve all that at least to great extent and in normal circumstances.
(cf. Lewis 1966, 1972)
The epistemic functional role of belief:
Assumption 2: Belief is an agent’s representation of what the world is like;
it aims at the truth.
,→ Lecture 2!
The epistemic functional role of belief:
Assumption 2: Belief is an agent’s representation of what the world is like;
it aims at the truth.
,→ Lecture 2!
The pragmatic functional role of belief:
Assumption 3: In combination with an agent’s desires (and subject to a
ceteris paribus clause), belief should commit an agent to rational action.
,→ Lecture 3!
The functional role of beliefs vis-à-vis beliefs:
Assumption 4: “An agent’s beliefs are subject to an ideal of integration.
Other things equal one should be able to agglomerate one’s various
beliefs into a larger, overall view; and this larger view should satisfy
demands for consistency and coherence” (Bratman 1999, p. 17).
,→ Lectures 1–3!
The functional role of beliefs vis-à-vis beliefs:
Assumption 4: “An agent’s beliefs are subject to an ideal of integration.
Other things equal one should be able to agglomerate one’s various
beliefs into a larger, overall view; and this larger view should satisfy
demands for consistency and coherence” (Bratman 1999, p. 17).
,→ Lectures 1–3!
Another, especially salient, pragmatic functional role of beliefs:
Assumption 5: If an agent is capable of linguistic discourse, then what is
expressed by the agent’s assertions should be her beliefs:
an agent ought to assert that X only if she believes that X .
,→ Lecture 3!
In the literature on belief vs degrees of belief, there is a constant temptation
to drop some of the above as being constitutive of belief.
E.g.: Acceptance vs belief ,→ Lecture 3!
In the literature on belief vs degrees of belief, there is a constant temptation
to drop some of the above as being constitutive of belief.
E.g.: Acceptance vs belief ,→ Lecture 3!
For me, none of the above will be negotiable.
This said, my theory will also have to give up on certain assumptions on belief:
Assumption∗ [NOT SATISFIED]: “Reasonable belief is, in an important
way, context independent: at any one time a reasonable agent normally
either believes something (to degree n) or does not believe it (to that
degree). She does not at the same time believe that p relative to one
context but not relative to another” (Bratman 1999, p. 18)
Concepts of Belief
Assumption 6: Belief can be ascribed by means of different concepts,
including a categorical (classificatory ) concept of belief and a
numerical (degree of ) belief concept.
(There are more options: e.g, belief on an interval scale—cf. Spohn 2012.)
These two concepts of belief occupy different scales of measurement and
belong to different intellectual traditions.
All-or-nothing belief of agent a:
a believes that X , or
a disbelieves that X (that is, believes that ¬X ), or
a suspends judgment on X (that is, neither believes nor disbelieves it).
,→ Epistemology, philosophy of mind. Cognitive psychology. AI. Logic.
Degree of belief of agent a:
a believes that X to a degree of P (X ) (i.e., with P (X ) · 100 percent).
,→ Subjective probability theory. Decision theory. Economics.
Bayesian epistemology / philosophy of science / psychology. . .
E.g., a subjective probability measure P:
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E.g., a subjective probability measure P:
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P conditionalized
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Bayesianism is omnipresent in many areas these days, and often it competes
with the more traditional (logic-based) approaches:
Our Assumption 1–5 from before can be understood as assumptions on
categorical belief or as assumptions on numerical belief.
The two of them are in the same “business”.
If the two concepts of belief are so similar in terms of their defining features:
Do they also denote the same propositional attitude?
Emptiness, Reduction/Elimination, Independence
Actually, there are three options here:
OPTION (i): At least one of the two concepts of belief is empty.
Belief is the propositional attitude the function of which is. . .
Will the uniqueness condition fail? Unlikely (cf. Lewis 1970).
Will the existence condition fail? Unlikely (contra Churchland 1981).
Emptiness, Reduction/Elimination, Independence
Actually, there are three options here:
OPTION (i): At least one of the two concepts of belief is empty.
Belief is the propositional attitude the function of which is. . .
Will the uniqueness condition fail? Unlikely (cf. Lewis 1970).
Will the existence condition fail? Unlikely (contra Churchland 1981).
Assumption 7: Both the categorical concept of belief and the degree of
belief concept refer.
OPTION (ii): Both concepts of belief refer, and to the same phenomenon.
Carnap (1950) on classificatory vs quantitative concepts:
Among the kinds of concept used in science, three are of special
importance. We call them classificatory, comparative, and
quantitative concepts. . . In prescientific thinking classificatory
concepts are used most frequently. In the course of the development
of science they are replaced in scientific formulations more and more
by concepts of the other two kinds, although they remain always
useful for the formulation of observational results. (pp. 8f)
Classificatory concepts are the simplest and least effective kind of
concept. Comparative concepts are more powerful, and quantitative
concepts still more; that is to say, they enable us to give a more
precise description of a concrete situation and, more important, to
formulate more comprehensive general laws. (p. 12)
In many cases a quantitative concept corresponds to a
classificatory concept. Thus temperature corresponds to the property
Warm; and the concept of a distance of less than five miles
corresponds to the relation of proximity. (p. 9)
This suggests a reduction (not an elimination) of the classificatory concept:
proximity is nothing but distance of less than five miles
belief is nothing but high enough degree of belief (or the like)
This suggests a reduction (not an elimination) of the classificatory concept:
proximity is nothing but distance of less than five miles
belief is nothing but high enough degree of belief (or the like)
Richard Jeffrey (1970, pp. 171f) took this one step further:
By ‘belief’ I mean the thing that goes along with valuation in
decision-making: degree-of-belief, or subjective probability, or
personal probability, or grade of credence. I do not care what you call
it because I can tell you what it is, and how to measure it, within
limits. . . Nor am I disturbed by the fact that our ordinary notion of
belief is only vestigially present in the notion of degree of belief. I am
inclined to think Ramsey sucked the marrow out of the ordinary
notion, and used it to nourish a more adequate view.
This eliminativist position is much more radical and much less plausible than
the reductionist one. (Heavy burden of proof!)
OPTION (iii): Both concepts of belief refer, but not to the same phenomenon.
This is certainly conceivable (see, e.g., Ross and Schroeder, forthcoming).
Here is my own “dual process” take on this ontological independence option:
!
!!!!!Perception!
Belief:!
Conscious!
!
Simple!
!
Linguistic!
X"
Y"
X"∧!Y"
Maintenance!
C!
o!
h!
e!
r!
e!
n!
c!
e!
Desire!
!!!!!!!!!Action!
Degrees!of!Belief:!
P(X)!=!0.7!
P(¬X)!=!0.3!
P(Y)!=!0.5!
"
Maintenance!
Unconscious!
!
Complex!
!
Only!Partially!Linguistic!
Norms on Belief
Epistemic norms guide belief to truth.
Pragmatic norms guide belief (and desire) to rational action.
Coherence norms: beliefs ought to relate to each other so that they aim at
the truth and facilitate rational action.
It is the coherence norms that will take center stage in my three lectures.
They tell us what one ought to believe given certain belief circumstances.
Equivalently: what a perfectly rational agent must believe given certain belief
circumstances.
Assumption 4’: An agent’s all-or-nothing beliefs are subject to an ideal of
integration. Other things equal one should be able to agglomerate one’s
various all-or-nothing beliefs into a larger, overall view; and this larger
view should satisfy demands for consistency and coherence.
Assumption 4”: An agent’s degrees of beliefs are subject to an ideal of
integration. Other things equal one should be able to agglomerate one’s
various degrees of beliefs into a larger, overall view; and this larger view
should satisfy demands for consistency and coherence.
Assumption 4”’: An agent’s beliefs and degrees of belief are subject to an
ideal of integration. Other things equal one should be able to agglomerate
one’s various all-or-nothing beliefs and degrees of belief into a larger,
overall view; and this larger view should satisfy demands for consistency
and coherence.
With OPTION (ii), Assumption 4”’ is about coherence between concepts.
With OPTION (iii), Assumption 4”’ is about coherence between mental states.
I will specify Assumptions 4’ on coherence for beliefs and Assumption 4” on
coherence for degrees of belief conservatively:
Assumption 8: The coherence norms on all-or-nothing belief are precisely
what the canonical literature on the logic of belief takes them to be:
(i) Synchronically, the set of beliefs of a perfectly rational agent is closed
under logic (cf. Hintikka 1962).
(ii) Diachronically, belief change of a perfectly rational agent is governed
by the axioms of belief revision (cf. AGM 1985).
(In parts of the lectures, I will actually be able to derive Assumption 8.)
Assumption 9: The coherence norms on degrees of belief are precisely
what the canonical Bayesian literature takes them to be:
(i) Synchronically, the degree of belief assignment of a perfectly rational
agent satisfies the axioms of probability.
(ii) Diachronically, degree of belief change of a perfectly rational agent is
given by conditionalization.
Remark:
Without logical closure, pressure rises for belief to be reduced or eliminated.
For without logical closure,
belief is no longer that simple (so why not go for the numerical concept
from the start?),
OPTION (iii) of an independent belief system becomes implausible.
On the next slide I will give you an example: belief revision without closure
under conjunction is hard!
P1 If the degrees of belief that the agent assigns to two propositions are
identical, then either the agent believes both of them or neither of them.
For all X , Y : if Pt (X ) = Pt (Y ) then
Belt (X ) iff Belt (Y ).
P2 If the agent already believes X , then updating on the piece of evidence X
does not change her system of (all-or-nothing) beliefs at all.
For all X : if the evidence that the agent obtains between t and t 0 > t is the
proposition X , but it holds already that Belt (X ), then for all Y :
Belt 0 (Y ) iff Belt (Y ).
P3 The agent’s learning is captured probabilistically by conditionalization.
For all X (with Pt (X ) > 0): if the evidence that the agent obtains between
t and t 0 > t is the proposition X , then for all Y :
Pt 0 (Y ) = Pt (Y | X ).
P1 If the degrees of belief that the agent assigns to two propositions are
identical, then either the agent believes both of them or neither of them.
For all X , Y : if Pt (X ) = Pt (Y ) then
Belt (X ) iff Belt (Y ).
P2 If the agent already believes X , then updating on the piece of evidence X
does not change her system of (all-or-nothing) beliefs at all.
For all X : if the evidence that the agent obtains between t and t 0 > t is the
proposition X , but it holds already that Belt (X ), then for all Y :
Belt 0 (Y ) iff Belt (Y ).
P3 The agent’s learning is captured probabilistically by conditionalization.
For all X (with Pt (X ) > 0): if the evidence that the agent obtains between
t and t 0 > t is the proposition X , then for all Y :
Pt 0 (Y ) = Pt (Y | X ).
C The agent’s all-or-nothing beliefs are closed under conjunction:
If Belt (X ) and Belt (Y ), then Belt (X ∩ Y ).
Bridge Principles for Rational Belief and Degrees of Belief
How shall we specify Assumption 4”’ on the coherence between belief and
degrees of belief?
Most of the more traditional proposals belong to one of the following categories:
(1) The Certainty Proposal (e.g., Gärdenfors 1986):
Bel (X ) iff P (X ) = 1.
(contra Ass. 3?)
(2) The Lockean Thesis (e.g., Kyburg 1961, Foley 1993):
Bel (X ) iff P (X ) > r .
(contra Ass. 4?)
(3) Decision-Theoretic Accounts (e.g., Hempel 1962, Levi 1967, Kaplan
1996):
Bel (X ) iff
∑
P ({w }) · u (bel X , w ) is so-and-so. (contra Ass. x?)
w ∈W
(4) The Nihilistic Proposal (e.g., Wolfgang Spohn?)
Many people are working on this topic, or on related ones, right now,
and they are having new ideas:
Hanti Lin and Kevin Kelly (2012a, 2012b),
Branden Fitelson (monograph!),
Richard Pettigrew (see e.g. his M-Phi blog entry)
the Amsterdam Group (a lot in the making!),
..
.
References:
“The Review Paradox. A Note on the Diachronic Costs of Not Closing Rational
Belief Under Conjunction”, to appear in Nous.
I used parts of the monograph on The Stability of Belief that I am writing.
Soon a draft will appear at
https://lmu-munich.academia.edu/HannesLeitgeb