Philosophy of Science, 71 (October 2004) pp. 505–514. 0031-8248/2004/7104-0005$10.00
Copyright 2004 by the Philosophy of Science Association. All rights reserved.

505

Discussion: Re-solving Irrelevant
Conjunction with Probabilistic

Independence*

James Hawthorne and Branden Fitelson†

Naive deductivist accounts of confirmation have the undesirable consequence that if
E confirms H, then E also confirms the conjunction H7X, for any X—even if X is
completely irrelevant to E and H. Bayesian accounts of confirmation may appear to
have the same problem. In a recent article in this journal Fitelson (2002) argued that
existing Bayesian attempts to resolve of this problem are inadequate in several im-
portant respects. Fitelson then proposes a new-and-improved Bayesian account that
overcomes the problem of irrelevant conjunction, and does so in a more general setting
than past attempts. We will show how to simplify and improve upon Fitelson’s solution.

1. Introduction. We will begin by recalling the problem of irrelevant con-
junction. Then we describe Fitelson’s (2002) solution. And finally we show
how to improve on it.

The problem of irrelevant conjunction was originally raised as a prob-
lem for hypothetico-deductive (H-D) accounts of confirmation. On H-D
accounts, E confirms H relative to background K when H7K deductively
entails E (i.e., if H7K ). This leads to the following result:X E

(1) If E H-D-confirms H relative to K, then E H-D-confirms H7X
relative to K, for any X.

The problem with (1) is that when E confirms H, any other hypothesis
X compatible with H gets a free confirmational ride. Merely tack X onto
H, and E confirms them together, regardless of the fact that X may be
utterly irrelevant to E in the presence of H.

*Received October 2003; revised January 2004.

†To contact the author write to James Hawthorne, Department of Philosophy, Uni-
versity of Oklahoma, 605 Dale Hall Tower, Norman, OK 73019; e-mail: haw-
thorne@ou.edu; or Branden Fitelson, University of California-Berkeley, 314 Moses
Hall #2390, Berkeley, CA 94720-2390; e-mail: branden@fitelson.org.

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506 HAWTHORNE AND FITELSON

Bayesian confirmation may seem to have an advantage over the H-D
account regarding irrelevant conjunctions. For, the Bayesian account does
not generally have the property expressed in (1). On a Bayesian account,
“E confirms H relative to K” just means that “ ”Pr (HFE7K ) 1 Pr (HFK )
for an appropriate Bayesian probability function Pr. And in lots of cases
evidence E may confirm H in this probabilistic way without confirming
some conjunction H7X—i.e. without also making Pr (H7XFE7K ) 1

. So, Bayesianism is generally immune to the original problemPr (H7XFK )
of irrelevant conjunction.

However, Bayesian confirmation still seems to suffer from this problem
in the case of deductive evidence. That is, Bayesian confirmation and H-
D confirmation both satisfy the following special case of (1) whenever E
is less than certain on K—i.e. when :Pr (EFK ) ! 1

(1′) If , then E confirms H7X relative to K for any X con-H7K X E
sistent with H7K.

That is, when we have, for any X, . So,H7K X E Pr (EFH7X7K ) p 1
provided X is logically consistent with H7K and is probabilistically con-
sistent with it as well (i.e. , we havePr (H7X7K ) 1 0)

Pr (H7XFE7K ) p Pr (EFH7X7K ) Pr (H7XFK )/ Pr (EFK )

1 Pr (H7XFK ).

So, Bayesians must concede that the problem persists for evidence that
is deductively entailed by H7K.

Bayesians do have some wiggle room, however. They can concede (1′),
but argue that in the context of deductive evidence, H simpliciter will
always be better confirmed than H7X. One way to attempt to legitimate
this claim is to point out that it follows from the axioms of probability
that , and that when (andPr (HFE7K ) ≥ Pr (H7XFE7K ) H7K X E

), equality only holds in the special case wherePr (H7X7K ) 1 0
(sincePr (XFH7K ) p 1

Pr (H7XFE7K ) p Pr (XFH7E7K ) Pr (HFE7K )

p Pr (XFH7K ) Pr (HFE7K )).

So, in the deductive case H is always made at least as probable as
by E (relative to K), and in the interesting cases (whereH 7 X

, it’s made more probable.Pr (XFH7K ) ( 1)
This solution turns on reading the “E confirms H1 more than H2”

relation as “H1 is more probable given E than is H2 given E.” But that
is not generally a good way to understand “confirms more.” For, given
this reading, in more general cases it will often happen that “E confirms
H2 more than H1 relative to K” (because ),Pr (H FE7K ) 1 Pr (H FE7K )2 1

T h i s  c o n t e n t  d o w n l o a d e d  f r o m  1 9 2 . 1 2 . 8 8 . 1 4 0  o n  T h u ,  1 3  F e b  2 0 1 4  0 0 : 4 7 : 2 5  A M
A l l  u s e  s u b j e c t  t o  J S T O R  T e r m s  a n d  C o n d i t i o n s



RE-SOLVING IRRELEVANT CONJUNCTION 507

while E actually lowers the probability of H2 ( )Pr (H FE7K ) ! Pr (H FK )2 2
and raises the probability of H1 ( ). So, the brutePr (H FE7K ) 1 Pr (H FK )1 1
comparison of the relative sizes of posterior probabilities is an intuitively
unappealing way to cash out the “confirms more” relation. Thus, the
proposed solution to irrelevant conjunction depends on a flawed account
of “confirms more.”

So, is there a better way to cash out “confirms more” that gives the
desired solution to irrelevant conjunction? Several accounts of “E confirms
H given K” have been proposed in the literature that provide measures
of how much E incrementally increases the probability of H, given K—
increases it over the probability of H prior to the evidence. The most
common are the ratio measure, the difference measure, and the likelihood
ratio measure:

ratio measure:

Pr (HFE7K ) Pr (EFH7K )
r(H, EFK ) p p ;

Pr (HFK ) Pr (EFK )

difference measure:

d(H, EFK ) p Pr (HFE7K ) � Pr (HFK );

likelihood-ratio measure:

Pr (EFH7K )
l(H, EFK ) p .

Pr (EF∼H7K )
It should be clear how these measures might be applied to the problem
of irrelevant conjunction. The ratio measure would have it that E confirms
H2 better than H1 (relative to K) just when the ratio is largerr(H , EFK )2
than the ratio . The difference measure says that E confirmsr(H , EFK )1
H2 better than H1 (relative to K) just when the difference isd(H , EFK )2
larger than the difference . And likelihood-ratio measure saysd(H , EFK )1
that E confirms H2 better than H1 (relative to K) just when isl(H , EFK )2
larger than . Thus, a Bayesian solution to the problem of ir-l(H , EFK )1
relevant conjunction might be obtained if it can be shown that

is larger than where c is some appropriate measurec(H, EFK ) c(H7X, EFK )
of incremental confirmation such as r, d, or l.

It is now well known that when each of these measures is applied to
the same issue or problem in Bayesian treatments of confirmation, they
may provide divergent results. Indeed, Fitelson (1999, 2001) has exten-
sively analyzed this issue, and finds strong grounds to prefer l over other
measures. Be that as it may, what we want to know is whether the irrel-
evant conjunction problem has a Bayesian solution based on any, or all
of these measures. The Fitelson paper under discussion provides an answer

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508 HAWTHORNE AND FITELSON

to this query. Indeed, Fitelson’s answer goes beyond the call. For, Fitelson
holds, a proper Bayesian solution to the problem of irrelevant conjunction
should be directly extendable to the case of non-deductive evidence as
well. And Fitelson’s analysis applies to non-deductive cases.

2. Fitelson’s Original Solution. Fitelson suggests that a proper Bayesian
analysis of the problem of irrelevant conjunction should start by saying
what, in a Bayesian framework, it means for X to be irrelevant to the
confirmation of H by E, given K. He suggests that this should mean that
X is irrelevant to H, E, and logical combinations of H and E, where
irrelevance is to be understood in the usual Bayesian sense of probabilistic
independence. Formally, the proposal is this:

Definition. Confirmational Irrelevance. A is confirmationally irrelevant
to B relative to K just when A and B are probabilistically independent,
given K i.e., when .Pr (A7BFK ) p Pr (AFK ) 7 Pr (BFK )

Fitelson then proves two theorems that bear on the problem of irrel-
evant conjunction for both deductive and non-deductive evidence. The
first theorem shows that the apparent problem persists even in non-de-
ductive cases—that if E confirms H, but X is confirmationally irrelevant
to H, E, and H7E, then on each of the measures r, d, and l, E also confirms
(i.e. incrementally raises the probability of) H7X.

Theorem 1. If E confirms H, and X is confirmationally irrelevant to
H, E, and H7E (relative to background K), then E also confirms
H7X (relative to K).

Fitelson’s second theorem then comes to the rescue of the Bayesian
account by showing that if E confirms H, but X is confirmationally ir-
relevant to H, E, and H7E, then on measures d and l (but not on r) E
incrementally raises the probability of H more than it does H7X.

Theorem 2. If E confirms H, and X is confirmationally irrelevant to
H, E, and H7E (relative to background K) and , thenPr (XFK ) ( 1

, where c may be either the difference mea-c(H, EFK ) 1 c(H7X, EFK )
sure d or the likelihood-ratio measure l of degree of confirmation
(but, c may not be the ratio measure r, since in cases of irrelevant
conjunction we will have ).r(H, EFK ) p r(H7X, EFK )

This result seems quite satisfactory, but for one thing. The antecedents
of these theorems appear to be a little too strong, and perhaps less in-
tuitively compelling than we might like.

3. An Improvement on the Solution. In the deductive case the intuition
about the irrelevance of X flows from the idea that a hypothesis is tested

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RE-SOLVING IRRELEVANT CONJUNCTION 509

by what it says about the evidence. Adding X to H (given K) says nothing
more about E than H (given K) already says. So, intuitively, E should
confirm H7X (given K) no more than it confirms H alone (given K).
Indeed, intuitively E should perhaps confirm H7X less than H alone (given
K). A natural Bayesian extension of this idea about X adding nothing to
what H says about E (given K) is this:

Natural Conjunctive Irrelevance Criterion. X is an irrelevant conjunct
to H given K with respect to evidence E just in case

i.e. just when E is independent of XPr (EFH7X7K ) p Pr (EFH7K )
given H together with K.

This criterion involves an intuition about probabilistic independence
that is central to Fitelson’s previous theorems, but draws on much more
intuitively direct version of the idea that X is an irrelevant conjunct. This
describes one sense in which the antecedents in Fitelson’s theorems seem
overly strong.

The antecedents of the theorems are too strong in a more literal sense
as well. For it turns out that:

i. If “X is confirmationally irrelevant to H, E, and H7E” in the sense
employed in the theorems, then “X is an irrelevant conjunct to H
given K with respect to evidence E” in the sense of the Natural
Conjunctive Irrelevance Criterion just defined. And the Natural
Conjunctive Irrelevance Criterion is strictly weaker than supposing
X to be confirmationally irrelevant to H, E, and H7E. Furthermore:

ii. The Natural Conjunctive Irrelevance Criterion suffices in place of
Fitelson’s original irrelevance conditions, to establish of the con-
sequents of the theorems.

To see that claim (i) holds, notice that Fitelson’s original independence
conditions imply that

Pr (EFH7X7K ) 7 Pr (HFK ) 7 Pr (XFK )

p Pr (EFH7X7K ) 7 Pr (H7XFK )

p Pr (H7X7EFK )

p Pr (H7EFK ) 7 Pr (XFK )

p Pr (EFH7K ) 7 Pr (HFK ) 7 Pr (XFK );

so . But Fitelson’s original condition is notPr (EFH7X7K ) p Pr (EFH7K )
equivalent to the revised condition, since we can easily have cases where

while (i.e.Pr (EFH7X7K ) p Pr (EFH7K ) Pr (XFH7K ) ( Pr (XFK )
). Consider, for example, cases where XPr (H7XFK ) ( Pr (XFK ) Pr (HFK )

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510 HAWTHORNE AND FITELSON

is a kind of evidence statement E* (e.g. “the second toss will be heads”)
and hypothesis H is relevant to E* (e.g. H says “the coin is fair” and

), and where E (e.g. “the first toss is heads”) isPr (E*FH7K ) ( P(E*FK )
independent of E* given H7K.

To verify claim (ii), first observe that the following revised version of
Theorem 1 holds:

Revised Theorem 1. If E confirms H relative to K and
, then E also confirms H7X relative toPr (EFH7X7K ) p Pr (EFH7K )

K, where the notion of confirmation is given by either the ratio mea-
sure r or the difference measure d or the likelihood-ratio measure l.

This is a simplified version of the Bayesian “bad news” theorem. In the
case of the ratio measure r it says that if Pr (EFH7X7K ) p Pr (EFH7K )
and (i.e. H is confirmed by E given K on measure r), thenr(H, EFK ) 1 1

(i.e. is confirmed by E given K on measure r).r(H7X, EFK ) 1 1 H 7 X
For the difference measure d the revised theorem says that if

and (i.e. H is confirmed byPr (EFH7X7K ) p Pr (EFH7K ) d(H, EFK ) 1 0
E given K on measure d), then (i.e. H7X is confirmedd(H7X, EFK ) 1 0
by E given K on measure d). In the case of the likelihood-ratio measure
l the revised theorem says that if andPr (EFH7X7K ) p Pr (EFH7K )

(i.e. H is confirmed by E given K on measure l), thenl(H, EFK ) 1 1
(i.e. H7X is confirmed by E given K on measure l).l(H7X, EFK ) 1 1

In spite of the bad news from Revised Theorem 1, the corresponding
revised version of Theorem 2 shows that even on the weaker, more intuitive
notion of irrelevance, two of the three Bayesian measures of incremental
confirmation say that hypotheses are always better confirmed than their
conjunctions with irrelevant alternatives.

Revised Theorem 2. If E confirms H relative to K and
and , thenPr (EFX7H7K ) p Pr (EFH7K ) Pr (XFH7K ) ( 1

, where c may be either the difference mea-c(H, EFK ) 1 c(H7X, EFK )
sure d or the likelihood-ratio measure l of degree of incremental
confirmation. But, c may not be the ratio measure r, since for r, when
E confirms H relative to K and , wePr (EFX7H7K ) p Pr (EFH7K )
have that .r(H, EFK ) p r(H7X, EFK )

This revision of Fitelson’s original Bayesian resolution of the irrelevant
conjunction problem significantly strengthens that result. Like Fitelson’s
earlier version, this resolution is not restricted to the special case of de-
ductive evidence. However, this version has the advantage of employing
a notion of an irrelevant conjunct that is precisely analogous to that in
the original deductive evidence case. It subsumes the deductive irrelevant
conjunction issue as a special case in a natural way. And, like Fitelson’s

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RE-SOLVING IRRELEVANT CONJUNCTION 511

original approach, this solution is more robust, more measure-insensitive,
than other suggested resolutions of the problem.

Appendix: Proofs of Theorems

Proof of Revised Theorem 1

Suppose that in each case, 1, 2, and 3,Pr (EFX7H7K ) p Pr (EFH7K )
below.

1. For ratio measure r: If (i.e. H is confirmed by E givenr(H, EFK ) 1 1
K on measure r), then

Pr (HFE7K ) Pr (EFH7K )
1 ! r(H, EFK ) p p

Pr (HFK ) Pr (EFK )

Pr (EFH7X7K )
p p r(H7X, EFK )

Pr (EFK )

(i.e. H7X is confirmed by E given K on measure r).
2. For the difference measure d: Suppose (i.e. H is con-d(H, EFK ) 1 0

firmed by E given K on measure d). Then 0 ! d(H, EFK ) p
, soPr (HFE7K ) � Pr (HFK )

Pr (HFE7K ) Pr (EFH7K )
1 ! p

Pr (HFK ) Pr (EFK )

Pr (EFH7X7K ) Pr (H7XFE7K )
p p ,

Pr (EFK ) Pr (HFK )

so (i.e. H7X is confirmed by E given K0 ! Pr (H7XFE7K ) � Pr (H7XFK )
on measure d).

3. For the likelihood-ratio measure l: Suppose (i.e. H isl(H, EFK ) 1 1
confirmed by E given K on measure l). Then

Pr (EFH7K )
1 ! l(H, EFK ) p ;

Pr (EF∼H7K )
so . This impliesPr (EF∼H7K ) ! Pr (EFH7K ) 1 ! Pr (EFH7K )/Pr (EFK )
[because

Pr (EFK ) p

Pr (EFH7K ) 7 (1 � Pr (∼HFK )) � Pr (EF∼H7K 7 Pr (∼HFK ),

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512 HAWTHORNE AND FITELSON

so

Pr (EFK ) � Pr (EFH7K ) p

(� Pr (EFH7K ) � Pr (EF∼H7K )) 7 Pr (∼HFK ) ! 0].

Then,

Pr (EFH7X7K ) Pr (H7XFE7K )
1 ! p .

Pr (EFK ) Pr (H7XFK )

So we have , which impliesPr (H7XFK ) ! Pr (H7XFE7K )

Pr (∼(H7X )FK ) 1 Pr (∼(H7X )FE7K ).

These two inequalities together yield

Pr (H7XFK ) Pr (H7XFE7K )
!

Pr (∼(H7X )FK ) Pr (∼(H7X )FE7K )

Pr (EFH7X7K ) Pr (H7XFK )
p 7 .[ ] [ ]Pr (EF∼(H7X )7K ) Pr (∼(H7X )FK )

Thus (i.e. H7X is confirmed by E1 ! Pr (EFH7X7K )/ Pr (EF∼(H7X )7K )
given K on measure l).

Proof of Revised Theorem 2

Suppose that and in eachPr (EFX7H7K ) p Pr (EFH7K ) Pr (XFH7K ) ( 1
case, 1, 2, and 3, below.

1. For the case:c p r

Pr (EFH7K ) Pr (HFE7K )
r(H, EFK ) p p

Pr (EFK ) Pr (HFK )

Pr (EFH7X7K )
p p r(H7X, EFK ).

Pr (EFK )

2. For the case: Suppose E confirms H relative to K. Thenc p d

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RE-SOLVING IRRELEVANT CONJUNCTION 513

—i.e. . Thend(H, EFK ) 1 0 Pr (HFE7K ) � Pr (HFK ) 1 0

Pr (EFH7X7K ) Pr (H7XFK )
Pr (H7XFE7K ) p

Pr (EFK )

Pr (EFH7K )
p 7 Pr (H7XFK )[ ]Pr (EFK )

Pr (HFE7K )
p 7 Pr (H7XFK ).[ ]Pr (HFK )

So,

d(H7X, EFK ) p Pr (H7XFE7K ) � Pr (H7XFK )

Pr (HFE7K )
p Pr (H7XFK ) 7 � 1[[ ] ]Pr (HFK )

Pr (H7XFK )
p 7 d(H, EFK )[ ]Pr (HFK )
p Pr (XFH7K ) 7 d(H, EFK ) ! d(H, EFK )

(unless , in which case they’re equal).Pr (XFH7K ) p 1
3. For the case: Suppose E confirms H relative to K. Thenc p l l(H,

—i.e. . Notice thatEFK ) 1 1 Pr (EFH7K )/ Pr (EF∼H7K ) 1 1
Pr (EF∼X7H7K ) p Pr (EFH7K )

(since and ). Also noticePr (XFH7K ) ( 1 Pr (EFX7H7K ) p Pr (EFH7K )
that (otherwise , soPr (∼HF∼(X7H )7K ) ! 1 0 p Pr (HF∼(X7H )7K )

0 p Pr (H 7 ∼(X7H )FK ) p Pr (H 7 ∼XFK )
p Pr (∼XFH7K ) 7 Pr (HFK ),

so , which contradicts our assump-Pr (XFH7K ) p 1 � Pr (∼XFH7K ) p 1
tion that ).Pr (XFH7K ) ! 1

Now

Pr (EF∼(X7H )7K )
p Pr (EFH 7 ∼(X7H )7K ) 7 Pr (HF∼(X7H )7K )

� Pr (EF∼H 7 ∼(X7H )7K ) 7 Pr (∼HF∼(X7H )7K )
p Pr (EFH7K ) 7 Pr (HF∼(X7H )7K )

� Pr (EF∼H7K ) 7 Pr (∼HF∼(X7H )7K ).

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514 HAWTHORNE AND FITELSON

So,

Pr (EF∼(X 7 H ) 7 K )
p Pr (HF∼(X 7 H ) 7 K ) � Pr (∼HF∼(X 7 H ) 7 K )

Pr (EF(X 7 H ) 7 K )

Pr (EF∼H 7 K ) Pr (EF∼H 7 K )
7 1 .[ ]Pr (EFH 7 K ) Pr (EFH 7 K )

REFERENCES

Fitelson, Branden (1999), “The Plurality of Bayesian Measures of Confirmation and the
Problem of Measure Sensitivity”, Philosophy of Science 66 (Proceedings): S362–S378.

——— (2001), Studies in Bayesian Confirmation Theory. Ph.D. Dissertation. Madison, WI:
University of Wisconsin—Madison.

——— (2002), “Putting the Irrelevance Back Into the Problem of Irrelevant Conjunction”,
Philosophy of Science 69 (4): 611–622.

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