Joyce’s Argument for Probabilism

Patrick Maher (p-maher@uiuc.edu)
Department of Philosophy, University of Illinois at Urbana-Champaign

Abstract. James Joyce’s ‘Nonpragmatic Vindication of Probabilism’ gives a new
argument for the conclusion that a person’s credences ought to satisfy the laws of
probability. The premises of Joyce’s argument include six axioms about what counts
as an adequate measure of the distance of a credence function from the truth. This
paper shows that (a) Joyce’s argument for one of these axioms is invalid, (b) his
argument for another axiom has a false premise, (c) neither axiom is plausible, and
(d) without these implausible axioms Joyce’s vindication of probabilism fails.

1. Introduction

James Joyce (1998) offered a new argument for the conclusion that a
person credences ought to satisfy the laws of probability. The argument
assumes that it makes sense to talk about the distance that a person’s
credence function is from the truth in any possible world. Joyce also
states six axioms that, he says, any adequate measure of this distance
must satisfy. Joyce’s “Main Theorem” is that these axioms imply that
if a person’s credence function b does not satisfy the laws of probability
then there exists another credence function b∗ which does satisfy the
laws of probability and which is closer to the truth in every possible
world. Joyce assumes that an epistemically rational person strives to
havecredences that areas close to the truthas possibleandso concludes
that an epistemically rational person’s credences will satisfy the laws
of probability.
Joyce’s Main Theorem assumes that the set of propositions is count-

able. However, the case in which the set of propositions is countably
infinite raises mathematical issues that I judge not worthpursuing here,
so in this article I will consider only the special case in which the set
of propositions is finite. In that special case, at least, Joyce’s Main
Theorem is correct.1 But there remains the question of whether his
axioms are true.
In Section 3 I show that Joyce’s argument for one of his axioms is

invalid. In Section 4 I show that his argument for another axiom rests
1 There are a few small slips in Joyce’s proof but they can be corrected. For

example, Joyce’s proposition II (p. 598) does not follow from the axiom Structure
that he cites but it does follow from Normality. Also, his Lemma-3 (p. 599) is false
if c = m, as is possible, but it is easy to show that the Main Theorem does hold if
c = m.

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on a false premise. In Section 5 I argue that both these axioms are
implausible. In Section 6 I show that without the implausible axioms
Joyce’s vindication of probabilism fails.

2. Concepts and notation

I will use the following concepts and notation. Except for ‘n’, ‘X → Y’,
and ‘I’, this notation is Joyce’s.

Ω: A finite algebra of propositions. (Joyce allows Ω to be countable
but I am here considering only the special case in which it is finite.)

n: The number of elements in Ω.

�: The set of real numbers.

X → Y: The set of all functions from X to Y.

B: The set Ω → �. Joyce refers to elements of B as “credence func-
tions.”

V : The set of functions in Ω → {0, 1} that correspond to consistent
truth value assignments to the elements of Ω (with 1 representing
truth and 0 falsity). Elements of V can be thought of as possible
worlds. Note that V ⊂ B.

I: Theset of all I ∈ B×V → � whichare suchthat, for any b ∈ B and
ω ∈ V , I(b, ω) is a rationallypermissible measure of the inaccuracy
of b in the possible world ω.

In the preceding definitions I said B and V are subsets of Ω → �.
Here I was following what Joyce seems to be saying in the following
passage (pp. 590f.):

B is the family of all credence functions defined on a countable
Boolean algebra of propositions Ω and V is the subset of B con-
taining all consistent truth-value assignments to members of Ω.

However, there are other places (p. 583 for example) where Joyce says
or implies that B and V are subsets of �n.
Let X1, . . . , Xn be some enumeration of the elements of Ω. Then we

can identify any b ∈ Ω → � with the point

(b(X1), . . ., b(Xn)) ∈ �n.

Although Joyce does not mention it, I take it that he is assuming such
an identification of Ω → � with �n. For simplicity, and to make my

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treatment correspond to Joyce’s, I will likewise assume this identifica-
tion in what follows.

3. Weak Convexity

The axiom that Joyce (p. 596) calls “Weak Convexity” may be stated
as follows:

Weak Convexity: If b, b∗ ∈ B, ω ∈ V , I ∈ I, I(b, ω) = I(b∗, ω),
and m = 1

2
b+1

2
b∗ then I(b, ω) ≥ I(m, ω)with identity only if b = b∗.

Here is Joyce’s argument for Weak Convexity quoted in full (pp. 596f.,
with two typographic errors corrected):

To see why Weak Convexity is a reasonable constraint on gra-
dational inaccuracy notice that in moving from b to m an agent
would alter each degree of belief b(X) by adding an increment of
k(X) = 1

2
[b∗(X)−b(X)].She wouldadd the same increment of k(X)

to each m(X) in moving from m to b∗. To put it in geometric terms,
the “vector” k that she must add to b to get m is the same as the
vector she must add to m to get b∗. Furthermore, since b∗ = b +2k
the change in belief involved in going from b to b∗ has the same
direction but a doubly greater magnitude than the change involved
in going from b to m. This means that the former change is more
extreme than the latter in the sense that, for every proposition X,
both changes alter the agent’s degree of belief for X in the same
direction, either by moving it closer to one or closer to zero, but
the b to b∗ change will always move b(X) twice as far as the b to
m change moves it. Weak Convexity is motivated by the intuition
that extremism in the pursuit of accuracy is no virtue. It says that
if a certain change in a person’s degrees of belief does not improve
accuracy then a more radical change in the samedirectionand of the
same magnitude should not improve accuracy either. Indeed, this
is just what the principle says. If it did not hold, one could have
absurdities like this: “I raised my confidence levels in X and Y and
my beliefs became less accurate overall, so I raised my confidence
levels in X and Y again, by exactly the same amounts, and the
initial accuracy was restored.”

Joyce is here claiming that Weak Convexity follows from a premise that
could be stated formally thus:

Premise 1: If b, k ∈ B, ω ∈ V , I ∈ I, and I(b + k, ω) ≥ I(b, ω)
then I(b + 2k, ω) ≥ I(b, ω).

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Define ρ ∈ B × Ω → � by the condition

ρ(b, ω) =
∑
X∈Ω

|b(X)− ω(X)|.

A proof of the following theorem is given in Section 8.1.
Theorem 1: Theassumptionthat ρ ∈ I is consistentwithPremise1
but not with Weak Convexity.

This theorem shows that Premise 1 does not entail Weak Convexity
and so Joyce’s argument for Weak Convexity is invalid.

4. Symmetry

The axiom that Joyce (p. 596) calls “Symmetry” may be stated as
follows:

Symmetry: If I ∈ I and I(b, ω) = I(b∗, ω) then, for any λ ∈ [0, 1],

I(λb + (1 − λ)b∗, ω) = I((1 − λ)b + λb∗, ω).

Joyce offers a “rationale” or argument for Symmetry. As he presents it,
this argument depends on Weak Convexity and we have just seen that
Joyce has failed to justify that. However, Joyce’s essential point here
seems not to depend on Weak Convexity, so I shall try to extract this
essential point from his presentation.
Speaking of a particular kind of violation of Symmetry, Joyce writes

(p. 597):
Given the initial symmetry of the situation this would amount to
an unmotivated bias in favor of one set of beliefs or the other.

I think he means that, in violating Symmetry, there would be a “bias”
in favor of either b or b∗ and this bias would be “unmotivated” because
of the symmetry involved in the fact that I(b, ω) = I(b∗, ω). A little
later on the same page Joyce writes:

[Symmetry] says that when b and b∗ are equally accurate there can
be no grounds, based on considerations of accuracy alone, for pre-
ferring a “compromise” that favors b to a symmetrical compromise
that favors b∗.

Although Joyce presents this as what Symmetry “says”, in fact Sym-
metry makes no reference to “grounds” at all. So perhaps Joyce’s
statement here is best interpreted as a restatement of his earlier claim,
that any distinction between the “symmetrical compromises” would
involve an “unmotivated bias”.

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Supposing that Joyce is here giving an argument for Symmetry, I
will now try to formulate more clearly what this argument is. I am
trying to find premises that fit what Joyce actually says as closely
as possible, are as plausible as possible, and entail Symmetry. The
following is the reconstruction that seems to me to best meet these
conflicting desiderata.
I will take the passages from Joyce that I quoted in the paragraph

before last to be asserting:
Premise 2: If a person judges b and b∗ to be equally accurate in
ω, then that person has no grounds for judging λb +(1 − λ)b∗ and
(1 − λ)b + λb∗ to have different accuracies in ω.

Premise 2 departs somewhat from Joyce’s own wording because he
talks, not of a person judging b and b∗ to be equally accurate, but of b
and b∗ being equally accurate (see my last quotation). However, Joyce
(p. 590) allows that there can be different measures of accuracy, so he
cannot assume that there is a fact of the matter as to whether b and b∗

are equally accurate in ω. Hence my formulation of Premise 2 in terms
of a person’s judgments.
In addition to Premise 2 I take Joyce to be tacitly assuming:
Premise 3: If a person has no grounds for judging that c and c∗

have different accuracies in ω then it is not rationally permissible
for the person to judge that c and c∗ have different accuracies in ω.

It follows from Premises 2 and 3 that if a person makes rationally
permissible judgments, and the person judges that b and b∗ are equally
accurate in ω, then the person does not judge that λb + (1 − λ)b∗ and
(1 − λ)b + λb∗ have different accuracies in ω. This, together with the
definition of I, entails Symmetry. So Symmetry does followvalidly from
Premises 2 and 3. But I will now argue that Premise 2 is false.
Consider the following example. (Here each b ∈ B will be regarded

as a point in �n with its ith coordinate denoted bi.)

ω = (1, 0, ω3, . . . , ωn)
b = (−1, 0, ω3, . . ., ωn)

b∗ = (2, 1, ω3, . . . , ωn)
λ = 2/3
c = λb +(1 − λ)b∗ = (0, 1/3, ω3, . . . , ωn)

c∗ = (1 − λ)b + λb∗ = (1, 2/3, ω3, . . . , ωn).

Figure 1 shows the situation in the plane of the first two coordinates.
Since

ρ(b, ω) =
∑
X∈Ω

|b(X)− ω(X)| =
n∑

i=1

|bi − ωi|

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�

b
(−1, 0)

�

ω
(1, 0)

� b
∗

(2, 1)

�

c
(0, 1/3)

�

c∗
(1, 2/3)

✏✏
✏✏

✏✏
✏✏

✏✏
✏✏

✏✏
✏✏

✏✏
✏✏

✏✏

Figure 1. Counterexample to Premise 2

we have
ρ(b, ω) = ρ(b∗, ω) = 2

but
ρ(c, ω) = 4/3 > 2/3 = ρ(c∗, ω).

So if I take ρ(b, ω) to measure the inaccuracy of b in ω then in this
example I will judge that b and b∗ are equally accurate in ω but c is
less accurate than c∗.
According to Premise 2, I can have no grounds for taking c to be less

accurate than c∗ in ω. But in fact there are perfectly sensible grounds
for it, as follows. Moving from b to c improves accuracy with respect
to X1 by 1 and moving from b∗ to c∗ does the same. On the other
hand, moving from b to c reduces accuracy with respect to X2 by 1/3,
while moving from b∗ to c∗ increases accuracy with respect to X2 by
the same amount.
Hence Premise 2 is false. A person who takes b and b∗ to be equally

accurate in ω can havegrounds for taking λb+(1−λ)b∗ and (1−λ)b+λb∗
to have different accuracies in ω.

5. These axioms not plausible

We have seen that Joyce’s arguments for Weak Convexity and Symme-
try are both unsound. It also seems to me that neither of these axioms
is self evident. I will now give a reason for thinking that in fact these
axioms are false.
It is natural to measure the inaccuracy of b with respect to the

proposition X in possible world ω by |b(X)− ω(X)|. It is also natural
to take the total inaccuracy of b to be the sum of its inaccuracies with
respect to each proposition. But these two assumptions together imply
that the overall inaccuracy of b in ω is ρ(b, ω). Hence, it is natural to

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use ρ(b, ω) as a measure of the inaccuracy of credence function b in the
possible world ω.
Furthermore, there are other contexts in which measures analogous

to ρ seem the most natural ones to use. For example, imagine that a
student takes three examinations, with the maximum possible score for
each examination being 100. Suppose the student scores 92 on the first
examination, 98 on the second, and 97 on the third. I now ask: How
far short has the student fallen from the goal of a perfect score on all
three examinations? I think the most natural answer to this question
is 8 +2 + 3 = 13 points. But that answer assumes a measure with the
same form as ρ.
These considerations make it plausible that ρ ∈ I. But if ρ ∈ I then,

by Theorem 1, it follows that Weak Convexity is false. The example
used in Section 4 shows that it also follows that Symmetry is false. I
conclude that, in the absence of any cogent argument for them, Weak
Convexity and Symmetry are both implausible.

6. Other axioms not sufficient

I have argued against two of Joyce’s axioms but have not criticized his
other four axioms. In this section I will consider what can be inferred
from these other axioms alone.
Joyce’s first four axioms may be stated as follows. (I have altered

Joyce’s formulations of these axioms to make them clearer and more
precise.)

Structure: If I ∈ I and ω ∈ V then

a. I(b, ω) is a non-negative continuous function of b.
b. For all X ∈ Ω and ε > 0 there exists δ > 0 such that, for all

b ∈ B, if |b(X)| > δ then I(b, ω) > ε.

Extensionality: I is a subset of B × V → �.
Dominance: If I ∈ I and b(X) = b∗(X) for every X ∈ Ω other
than Y then

I(b, ω) > I(b∗, ω) iff |b(Y ) − ω(Y )| > |b∗(Y ) − ω(Y )|.

Normality: If I ∈ I and |b(X)− ω(X)| = |b∗(X) − ω∗(X)| for all
X ∈ Ω then I(b, ω) = I(b∗, ω∗).

It is easy to prove:
Theorem 2: The assumption that ρ ∈ I is consistent with Struc-
ture, Extensionality, Dominance, and Normality.

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Hence these four axioms cannot be used to argue against the assump-
tion that ρ ∈ I.
As I mentioned in the Introduction, Joyce’s strategy for justifying

probabilism is to show that if a credence function b does not satisfy the
axioms of probability then there exists another credence function b∗

that does satisfy the axioms of probability and is closer to the truth in
every possible world. On the other hand, we have the following theorem
(proved in Section 8.2.)

Theorem 3: If Ω contains at least one contingent proposition then
there exists b ∈ B which does not satisfy the axioms of probability
and is such that, for all b∗ ∈ B that do satisfy the axioms of
probability, there exists ω ∈ V for which ρ(b, ω) ≤ ρ(b∗, ω).

So if ρ ∈ I then Joyce’s vindicationof probabilism fails. But Theorem 2
says that the assumption that ρ ∈ I is consistent with Joyce’s first four
axioms. Hence, without the implausible axioms of Weak Convexity and
Symmetry, Joyce’s vindication of probabilism fails.

7. The Norm of Gradational Accuracy

Besides the six axioms, Joyce’s argument for probabilism assumes that
an epistemically rational person strives to have credences that are as
close to the truth as possible. Joyce (p. 579) calls this “The Norm of
Gradational Accuracy (NGA)”. I do not think that NGA is obviously
true. Our concern with truth might more plausibly be understood as a
concern to accept true propositions and not false ones, rather than to
have credences that are close to the truth according to some favored
measure of distance. Credences help determine what propositions it is
rational to accept and this gives a connection between credences and
truth. (See Maher 1993 ch. 6 for an elaboration of this approach.) I see
no need to attach an additional epistemic value to having credences
close to the truth; an advantage of not doing so is that we then do not
need to make unjustifiable decisions about how to measure this dis-
tance. So NGA seems to me more dubious than the laws of probability
that Joyce is attempting to justify.
However, the main point of this paper has been to show that, even if

we grant Joyce NGA, his vindication of probabilism is still unsuccessful
because two of his axioms are unjustified and implausible.

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8. Proofs

8.1. Proof of Theorem 1

Thinking of b, k, and ω as points in �n, let their ith coordinates be
denoted bi, ki, and ωi respectively. Then we have:

ρ(b + k, ω) =
n∑

i=1

|bi + ki − ωi|

=
n∑

i=1

| 1
2
(bi − ωi)+ 12(bi +2ki − ωi)|

≤
n∑

i=1

(
1
2
|bi − ωi| + 12 |bi +2ki − ωi|

)

= 1
2
ρ(b, ω)+ 1

2
ρ(b +2k, ω).

Hence

ρ(b +2k, ω) ≥ 2ρ(b + k, ω) − ρ(b, ω)
= ρ(b + k, ω)+ [ρ(b + k, ω) − ρ(b, ω)]
≥ ρ(b, ω), if ρ(b + k, ω) ≥ ρ(b, ω).

Thus the assumption that ρ ∈ I is consistent with Premise 1.
Now consider the case in which, for some ω,

b = (ω1 + 1, ω2, ω3, . . . , ωn)
b∗ = (ω1, ω2 +1, ω3, . . . , ωn).

Then ρ(b, ω) = 1 = ρ(b∗, ω). Also, m = (ω1 +
1
2
, ω2 +

1
2
, ω3, . . . , ωn)

and so ρ(m, ω) = 1 = ρ(b, ω). Since b �= b∗, this shows that it is not
consistent with Weak Convexity to have ρ ∈ I.

8.2. Proof of Theorem 3

For each X ∈ Ω let nX denote the number of ω ∈ V for which ω(X) = 1.
Define b ∈ B by the condition that, for all X ∈ Ω,

b(X) =
{
1 if nX > n/2
0 if nX ≤ n/2.

}

For each ω ∈ V let Xω be the conjunction of all Y ∈ Ω for which
ω(Y ) = 1. Then ω(Xω) = 1 and ω′(Xω) = 0 if ω′ �= ω; hence nXω = 1
for all ω ∈ V . Since there is at least one contingent proposition in
Ω, n ≥ 2 and so nXω ≤ n/2. Hence b(Xω) = 0 for all ω ∈ V . But

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{Xω : ω ∈ V } is a partition of Ω, so b does not satisfy the laws of
probability.
For any X ∈ Ω,

∑
ω∈V

|b(X)− ω(X)| = nX|b(X)− 1| +(n − nX)|b(X)|

= nX(1 − b(X))+(n − nX)b(X),
since 0 ≤ b(X) ≤ 1

= nX +(n − 2nX)b(X)

=
{

n − nX if nX > n/2
nX if nX ≤ n/2.

}
(1)

Let b∗ be any element of B that satisfies the laws of probability. Then
we likewise have∑

ω∈V
|b∗(X) − ω(X)| = nX +(n − 2nX)b∗(X).

Since 0 ≤ b∗(X) ≤ 1, it follows that
∑
ω∈V

|b∗(X) − ω(X)| ≥
{

n − nX if nX > n/2
nX if nX ≤ n/2.

}
(2)

Comparing (1) and (2) we see that, regardless of the value of nX,
∑
ω∈V

|b(X)− ω(X)| ≤
∑
ω∈V

|b∗(X) − ω(X)|.

Since this holds for all X ∈ Ω, we have
∑
X∈Ω

∑
ω∈V

|b(X)− ω(X)| ≤
∑
X∈Ω

∑
ω∈V

|b∗(X) − ω(X)|.

Reversing the order of the summations and applying the definition of
ρ then gives: ∑

ω∈V
ρ(b, ω) ≤

∑
ω∈V

ρ(b∗, ω).

Hence ρ(b∗, ω) ≥ ρ(b, ω) for at least one ω ∈ V .

References

Joyce, J. M.: 1998, ‘A Nonpragmatic Vindication of Probabilism’. Philosophy of
Science 65, 575–603.

Maher, P.: 1993, Betting on Theories. Cambridge: Cambridge University Press.

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