Lecture 1: The Humean Thesis on Belief
Hannes Leitgeb
LMU Munich
October 2014
What do a perfectly rational agent’s beliefs and degrees of belief have to be like
in order for them to cohere with each other?
Plan: Give an answer in terms of a joint theory of belief and degrees of belief.
1
The Humean Thesis on Belief
2
Consequence 1: The Logic of Belief
3
Consequence 2: The Lockean Thesis
4
Consequence 3: Decision Theory
5
Conclusions
In all of this I will focus on (inferentially) perfectly rational agents only.
(There is related literature in computer science:
Benferhat et al. 1997, Snow 1998.)
The Humean Thesis on Belief
. . .an opinion or belief is nothing but an idea, that is different from
a fiction, not in the nature or the order of its parts, but in the manner
of its being conceived. . . An idea assented to feels different from a
fictitious idea, that the fancy alone presents to us: And this different
feeling I endeavour to explain by calling it a superior force, or vivacity
The Humean Thesis on Belief
. . .an opinion or belief is nothing but an idea, that is different from
a fiction, not in the nature or the order of its parts, but in the manner
of its being conceived. . . An idea assented to feels different from a
fictitious idea, that the fancy alone presents to us: And this different
feeling I endeavour to explain by calling it a superior force, or vivacity,
or solidity, or firmness, or steadiness. [. . .] its true and proper name
is belief, which is a term that every one sufficiently understands in
common life. [. . .] It gives them [the ideas of the judgment] more force
and influence; makes them appear of greater importance; infixes
them in the mind; and renders them the governing principles of all
our actions. (Treatise, Section VII, Part III, Book I)
. . .the mind has a firmer hold, or more steady conception of what
it takes to be matter of fact, than of fictions. (Treatise, Appendix)
Tradition in Hume interpretation has it that beliefs are lively ideas.
In my interpretation, beliefs are steady dispositions. (Loeb 2002)
A disposition to vivacity is a disposition to experience vivacious
ideas, ideas that possess the degree of vivacity required for
occurrent belief. Some dispositions to vivacity are unstable in that
they have a tendency to change abruptly. . . Such dispositions, in
Hume’s terminology, lack fixity. Hume in effect stipulates that a
dispositional belief is an infixed disposition to vivacity. . . (Loeb 2010)
there must be a property that plays a twofold role. The presence
of the property must constitute a necessary condition for belief. In
addition, establishing that the beliefs produced by a psychological
mechanism have that property must constitute a sufficient condition
for establishing justification, other things being equal. My claim is that
stability is the property that plays this dual role, one within Hume’s
theory of belief, the other within Hume’s theory of justification.
(Loeb 2002)
In a nutshell:
Beliefs are stable dispositions to have ideas with high “degree of vivacity”
on which acting, reasoning, and asserting is then based.
This is plausible also independently of (Loeb on) Hume: our epistemic lives
depend on stability under perception, supposition, communication,. . ..
Now we are going to explicate this:
(Rational) Degree of vivacity ≈ subjective probability P (cf. Maher 1981)
Let us call then the following the Humean thesis on rational belief:
It is rational to believe a proposition just in case it is rational to have a
stably high degree of belief in it (“infixed disposition to vivacity”).
Now we are going to explicate this:
(Rational) Degree of vivacity ≈ subjective probability P (cf. Maher 1981)
Let us call then the following the Humean thesis on rational belief:
It is rational to believe a proposition just in case it is rational to have a
stably high degree of belief in it (“infixed disposition to vivacity”).
But what is a stably high degree of belief?
,→ Skyrms (1977, 1980) on resiliency with respect to conditionalization!
Now we are going to explicate this:
(Rational) Degree of vivacity ≈ subjective probability P (cf. Maher 1981)
Let us call then the following the Humean thesis on rational belief:
It is rational to believe a proposition just in case it is rational to have a
stably high degree of belief in it (“infixed disposition to vivacity”).
Let W be a finite non-empty set of possible worlds. A proposition is a
subset of W , and P is a probability measure over propositions.
Now we are going to explicate this:
(Rational) Degree of vivacity ≈ subjective probability P (cf. Maher 1981)
Let us call then the following the Humean thesis on rational belief:
It is rational to believe a proposition just in case it is rational to have a
stably high degree of belief in it (“infixed disposition to vivacity”).
Let W be a finite non-empty set of possible worlds. A proposition is a
subset of W , and P is a probability measure over propositions.
The Humean Thesis Explicated (First Approximation)
If Bel is a perfectly rational agent’s class of believed propositions at a time, and
if P is the same agent’s subjective probability measure at the same time, then
HTYr : Bel (X ) iff for all Y , if Y ∈ Y and P (Y ) > 0, then P (X |Y ) > r .
But stably high probability with respect to what class Y of propositions?
Smaller Y s yields braver Bels, larger Y s yields more cautious Bels:
The Humean Thesis Explicated
HTYr : Bel (X ) iff for all Y , if Y ∈ Y and P (Y ) > 0, then P (X |Y ) > r .
Y ∈ Y iff P (Y ) = 1:
The Lockean Thesis.
Reduces to: Bel (X ) iff P (X ) > r .
(In general, not stable enough.)
Y ∈ Y iff Bel (Y ) (and P (Y ) > 0):
A coherence theory of belief.
Y ∈ Y iff P (Y ) > s:
A modestly cautious proposal.
Y ∈ Y iff Poss(Y ) (and P (Y ) > 0):
Another cautious proposal.
| {z }
not Bel (¬Y )
Y ∈ Y iff P (Y ) > 0:
The probability 1 proposal.
Reduces to: Bel (X ) iff P (X ) = 1.
(In general, too stable.)
Smaller Y s yields braver Bels, larger Y s yields more cautious Bels:
The Humean Thesis Explicated
HTYr : Bel (X ) iff for all Y , if Y ∈ Y and P (Y ) > 0, then P (X |Y ) > r .
Y ∈ Y iff P (Y ) = 1:
The Lockean Thesis.
Reduces to: Bel (X ) iff P (X ) > r .
Y ∈ Y iff Bel (Y ) (and P (Y ) > 0):
A coherence theory of belief.
Y ∈ Y iff P (Y ) > s:
A modestly cautious proposal.
Y ∈ Y iff Poss(Y ) (and P (Y ) > 0):
Another cautious proposal.
| {z }
not Bel (¬Y )
Smaller Y s yields braver Bels, larger Y s yields more cautious Bels:
The Humean Thesis Explicated
HTYr : Bel (X ) iff for all Y , if Y ∈ Y and P (Y ) > 0, then P (X |Y ) > r .
Y ∈ Y iff P (Y ) = 1:
The Lockean Thesis.
Reduces to: Bel (X ) iff P (X ) > r .
Y ∈ Y iff Bel (Y ) (and P (Y ) > 0):
A coherence theory of belief.
Y ∈ Y iff P (Y ) > s:
A modestly cautious proposal.
Y ∈ Y iff Poss(Y ) (and P (Y ) > 0):
Another cautious proposal.
| {z }
not Bel (¬Y )
Theorem
r
The Humean thesis HTPoss
entails the Humean thesis for all other conditions
above (assuming not Bel (∅), and with appropriate thresholds).
The Humean Thesis Explicated
If Bel is a perfectly rational agent’s class of believed propositions at a time, and
if P is the same agent’s subjective probability measure at the same time, then
r
) Bel (X ) iff for all Y , if Poss(Y ) and P (Y ) > 0, then P (X |Y ) > r
(HTPoss
where Poss(Y ) iff not Bel (¬Y )
(and
1
2
≤ r < 1).
The Humean Thesis Explicated
If Bel is a perfectly rational agent’s class of believed propositions at a time, and
if P is the same agent’s subjective probability measure at the same time, then
r
) Bel (X ) iff for all Y , if Poss(Y ) and P (Y ) > 0, then P (X |Y ) > r
(HTPoss
where Poss(Y ) iff not Bel (¬Y )
(and
1
2
≤ r < 1).
This will be our official explication of belief!
‘Poss(Y )’ matches the probabilistic condition ‘P (Y ) > 0’.
The resulting constraint on Bel is not as strong as it might seem:
‘Poss’ expresses serious possibility (Levi 1980), not logical possibility.
It is not a reduction of Bel to P.
The explication is fruitful, as we are going to see.
Consequence 1: The Logic of Belief
The logic of belief follows from the Humean thesis (and probability theory).
Theorem
r
, and
If P is a probability measure, if Bel satisfies the Humean thesis HTPoss
if not Bel (∅), then the following principles of doxastic logic hold:
Bel (W ).
If Bel (X ) and X ⊆ Y , then Bel (Y ).
If Bel (X ) then Poss(X ).
If Bel (X ) and Bel (Y ), then Bel (X ∩ Y ).
r
So HTPoss
entails that there is a least believed proposition BW that is
non-empty and which generates the agent’s belief system Bel.
BW!
X!
Bel(X)!
Y!
Poss(Y)!
Moral: Once belief is postulated to be “sufficiently” stable,
the logical closure of belief becomes derivable.
)
An example from Bayesian networks: J. Pearl (1988), D. Barber (2012)
Krombholz (2013)
The)sprinkler)was)left)on)
It)rained)
R)
J)
J=1$ R$
1)
1)
0.2) 0)
Jack’s)lawn)is)wet)
S)
R=1$
0.2)
T=1$
1)
1)
T)
0.9)
Tracey’s)lawn)is)wet) 0)
R$
1)
1)
0)
0)
S=1$
0.1)
S$
1)
0)
1)
0)
“belief arises only from causation. . .” (Treatise, Section IX, Part III, Book I)
Choose a Humean threshold of r = 12 :
6. w8 : T = 0, J = 1, R = 0, S = 1
(“Ranks”)
5. w7 : T = 0, J = 0, R = 0, S = 1
4. w5 : T = 1, J = 1, R = 1, S = 1;
w6 : T = 1, J = 1, R = 0, S = 1
3. w4 : T = 1, J = 0, R = 0, S = 1
2. w2 : T = 1, J = 1, R = 1, S = 0;
w3 : T = 0, J = 1, R = 0, S = 0
1. w1 : T = 0, J = 0, R = 0, S = 0
r
The candidates for BW that satisfy the Humean thesis HTPoss
(given P):
{w1 , w2 , w3 , w4 , w5 , w6 , w7 , w8 }
{w1 , w2 , w3 , w4 , w5 , w6 , w7 }
{w1 , w2 , w3 , w4 , w5 , w6 }
{w1 , w2 , w3 , w4 }
{w1 , w2 , w3 }
{ w1 }
(“Spheres”)
Choose a Humean threshold of r = 12 :
6. w8 : T = 0, J = 1, R = 0, S = 1
(“Ranks”)
5. w7 : T = 0, J = 0, R = 0, S = 1
4. w5 : T = 1, J = 1, R = 1, S = 1;
w6 : T = 1, J = 1, R = 0, S = 1
3. w4 : T = 1, J = 0, R = 0, S = 1
2. w2 : T = 1, J = 1, R = 1, S = 0;
w3 : T = 0, J = 1, R = 0, S = 0
1. w1 : T = 0, J = 0, R = 0, S = 0
E.g., if BW = {w1 , w2 , w3 }, this means:
Bel (S = 0)
Bel (T = 1 ↔ R = 1)
Bel (¬(J = 0 ∧ R = 1))
Bel (S = 0 ∧ (T = 1 ↔ R = 1) ∧ ¬(J = 0 ∧ R = 1))
Poss(R = 1)
P (S = 0 ∧ (T = 1 ↔ R = 1) ∧ ¬(J = 0 ∧ R = 1) | R = 1) >
1
2
Consequence 2: The Lockean Thesis
r
As I said before, the Humean thesis HTPoss
also implies an instance of the
Lockean thesis (Foley 1993):
The Lockean thesis: Bel (X ) iff P (X ) ≥ s > 12 .
r
imply both the closure of belief under
But now you might think: how can HTPoss
conjunction and the Lockean thesis? (cf. Kyburg 1961 on the Lottery Paradox)
Consequence 2: The Lockean Thesis
r
As I said before, the Humean thesis HTPoss
also implies an instance of the
Lockean thesis (Foley 1993):
The Lockean thesis: Bel (X ) iff P (X ) ≥ s > 12 .
r
imply both the closure of belief under
But now you might think: how can HTPoss
conjunction and the Lockean thesis? (cf. Kyburg 1961 on the Lottery Paradox)
The answer is that the Lockean threshold in question depends on P and Bel:
Theorem
r
If P is a probability measure, if Bel satisfies the Humean thesis HTPoss
, and
if not Bel (∅), then an instance of the Lockean thesis holds:
Bel (X ) iff P (X ) ≥ P (BW ) (> r ≥ 12 ).
Choose a Humean threshold of r = 12 :
6. w8 : T = 0, J = 1, R = 0, S = 1
(“Ranks”)
5. w7 : T = 0, J = 0, R = 0, S = 1
4. w5 : T = 1, J = 1, R = 1, S = 1;
w6 : T = 1, J = 1, R = 0, S = 1
3. w4 : T = 1, J = 0, R = 0, S = 1
2. w2 : T = 1, J = 1, R = 1, S = 0;
w3 : T = 0, J = 1, R = 0, S = 0
1. w1 : T = 0, J = 0, R = 0, S = 0
r
The candidates for BW that satisfy the Humean thesis HTPoss
(given P):
{w1 , w2 , w3 , w4 , w5 , w6 , w7 , w8 } (≥ 1.0)
{w1 , w2 , w3 , w4 , w5 , w6 , w7 } (≥ 0.9984)
{w1 , w2 , w3 , w4 , w5 , w6 } (≥ 0.992)
{w1 , w2 , w3 , w4 } (≥ 0.9576)
{w1 , w2 , w3 } (≥ 0.9)
{w1 } (≥ 0.576)
(“Spheres”)
Moral: If the Humean thesis holds, then
belief corresponds to “high” degree of belief, but the threshold for “high”
depends on contextual factors including the agent’s degrees of belief.
In Lecture 2 we will see how this will take care of the famous Lottery paradox.
(The Preface Paradox will be discussed in Lecture 3.)
But we will also find that the theory comes with a price:
a strong context-sensitivity of rational belief.
“When I reflect on the natural fallibility of my judgment, I have less confidence
in my opinions, than when I only consider the objects concerning which I
reason” (Treatise, Section I, Part IV, Book I)
Consequence 3: Decision Theory
Here is a simple theory of decision-making based on categorical beliefs:
O is a set of outcomes.
u : O → R is an “all-or-nothing” utility function that takes precisely two
values umax > umin .
Actions A are all the functions from W to O (very tolerant!).
Use(A) = {w ∈ W | u (A(w )) = umax } (= {w ∈ W | A serves u in w }).
An action A is (practically) permissible given Bel and u iff Bel (Use(A)).
Theorem
r
, and if
If P is a probability measure, if Bel satisfies the Humean thesis HTPoss
not Bel (∅), then:
If O is a set of outcomes, u : O → R is a utility function that takes precisely two
values umax > umin , actions A are all the functions from W to O, and
Use(A) = {w ∈ W | u (A(w )) = umax } (= {w ∈ W | A serves u in w }), then
for all actions A, B: if Bel (Use(A)) and ¬Bel (Use(B ))) then
EP (u (A)) > EP (u (B )),
for all actions A: if
EP (u (A)) is maximal,
then Bel (Use(A)), and for all actions B with Bel (Use(B )) it holds that
EP (u (A)) − EP (u (B )) < (1 − r ) (umax − umin ).
Choose a Humean threshold of r = 12 :
2. w2 : T = 1, J = 1, R = 1, S = 0;
w3 : T = 0, J = 1, R = 0, S = 0
1. w1 : T = 0, J = 0, R = 0, S = 0
Goal: Tracey’s lawn should be wet (and wasting water is not an issue);
one should not engage with the sprinkler if it is on already.
A1 : Turning Tracey’s sprinkler on (or in any case attempting to).
A2 : If Tracey’s sprinkler is off, turning it on; else leaving it on.
Use(A1 ) = {w1 , w2 , w3 }. Use(A2 ) = W .
E.g., if BW = {w1 , w2 , w3 }, this means:
Bel (Use(A1 )), Bel (Use(A2 )).
With umax = 1, umin = 0: EP (u (A2 )) − EP (u (A1 )) = 1 − 0.9 = 0.1.
Moral: If the Humean thesis holds, then
rational qualitative decisions are probabilistically reliable.
“[belief] renders them [the ideas] the governing principles of all our actions”.
Conclusions
I have suggested a joint theory of rational belief and subjective probability:
The Humean Thesis on Belief: Rational belief corresponds to resiliently
high probability.
The theory has attractive consequences: belief is stable, closed under
logic, corresponds to high probability, supports reliable decisions,. . .
One can also prove converses: each combination of doxastic logic with
another of the consequences entails the Humean thesis.
,→ Lectures 2 and 3!
The theory has lots of applications, such as to qualitative Bayes net
descriptions (but also to theory choice, belief revision, assertion,
a new Ramsey test for conditionals, learning conditionals,
subjunctive supposition, acceptance, knowledge, vagueness,. . .).
,→ Lectures 2 and 3!
Postscript for the probabilistic sceptic:
In some situations it may be useful to determine Bel from P in line with the
theory: to put things qualitatively and simply; to make sense of 0-1-talk.
In some situations it may be useful to determine P from Bel in line with the
theory: when the probabilities are initially unavailable but would be useful.
In some situations it may be useful to do neither: each of P and Bel have
a life of their own (e.g., they can be updated on the same X separately).
In some situations it may be useful to do a little bit of both, and to check
P vs Bel for consistency (“coherence”), always in line with the theory.
In any case:
There is hope for a joint formal epistemology of belief & degrees of belief.
References:
The present paper will be for the Proceedings of the Aristotelian Society.
If you want to read an early draft, please check out
https://lmu-munich.academia.edu/HannesLeitgeb