1. Introduction
The problem of irrelevant conjunction (a.k.a., the tacking problem) was originally raised as a problem for naive hypothetico-deductive (H–D) accounts of confirmation.Footnote 1 According to the (naive) H–D account of confirmation, E confirms H relative to background K (roughly) if H & K (classically) deductively entails E (i.e., if H & K ⊨ E). Therefore, owing to the monotonicity of ⊨, we have the following fact about (naive) H–D-confirmation:
The problem with (1) is supposed to be that conjoining X's that are utterly irrelevant to H and E seems (intuitively) to undermine the confirmation E provides for the resulting (conjunctive) hypothesis H & X. For instance, intuitively, the return of Halley's comet in 1758 (E) confirmed Newton's theory (H) of universal gravitation (relative to the background evidence (K) available at the time). But, according to the H–D account of confirmation, this implies that the return of Halley's comet also confirms the conjunction of H and (say) Coulomb's Law (or any other proposition(s) one would like to conjoin to H). And, no matter how many irrelevancies are conjoined to H, E will continue to confirm the conjunction, according to the H–D account of confirmation.
Because probabilistic correlation is not monotonic, Bayesian (incremental) confirmation does not have the property expressed in (1). That is, according to Bayesianism, it does not follow from E's confirming H that E must also confirm H & X, for any X. For contemporary Bayesians, “ E confirms H relative to K” just means that for an appropriate Bayesian probability function Pr (for an agent), Pr(H| E & K) > Pr(H| K). And, in lots of cases, evidence E may confirm a hypothesis H in this probabilistic way without confirming some conjunction H & X—i.e., without also making Pr(H & X| E & K) > Pr(H & X| K). So, Bayesianism is immune from the original problem of irrelevant conjunction. However, Bayesian confirmation does still 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), provided that E is less than certain on K—i.e., provided that Pr(E| K) < 1.Footnote 2
That is, when H & K ⊨ E we have, for any X, Pr(E| H & X & K) = 1; thus Pr(H & X| E & K) = Pr(E| H & X & K) · Pr(H & X| K)/Pr(E| K) > Pr(H & X| K), provided that Pr(E| K) < 1. This is the traditional way of framing the problem of irrelevant conjunction.
2. Existing Bayesian Resolutions of the Traditional Problem
Several Bayesians have tried to resolve the traditional problem of irrelevant conjunction by proving various quantitative results about the degree to which E confirms H versus H & X (relative to K) in the case of deductive evidence. One obvious Bayesian move is to point out that for any X, Pr(H| E & K) ≥ Pr(H & X| E & K), and that given H & K ⊨ E, equality holds in the special case where Pr(X| H & K) = 1. So, in the deductive case H is always made at least as probable as H & X by E (relative to K), and in most cases is made more probable.
Although this may appear to solve the problem of irrelevant conjunction, the solution turns on the trivial fact that conjunctions are never more probable than their conjuncts. And it turns on reading the “confirms more” relation as the “makes more probable than” relation. But this is not a good interpretation of the “confirms more” relation. For, on this reading, when we compare the confirmation of hypotheses by evidence in more general cases it will often happen that “ E confirms H 2 more strongly than E confirms H 1 (relative to K)” (because Pr(H 2| E & K) > Pr(H 1| E & K)), while E actually lowers the probability of H 2 (Pr(H 2| E & K) < Pr(H 2| K)) and raises the probability of H 1 (Pr(H 1| E & K) > Pr(H 1| K)). So, the brute comparison of the relative sizes of posterior probabilities is intuitively an unappealing way to cash out the “confirms more” relation (see Popper Reference Popper1954 for an early and forceful argument for this claim).
How then are we to cash out “confirms more”? Clearly, we want to employ some measure of how much the posterior probability of H is raised by E over its prior probability. I will call such measures relevance measures. Several relevance measures of confirmation have occurred to Bayesians (see Kyburg Reference Kyburg, Machan and Lucey1983, Fitelson Reference Fitelson1999, and Festa Reference Festa, Galavotti and Pagnini1999 for surveys). The most common are the ratio measure, the difference measure, and the likelihood-ratio measure. The ratio measure would measure the amount by which E confirms H (relative to K) as the ratio r(H, E| K) = Pr(H| E & K)/Pr(H| K). This measure would have it that E confirms H 2 more strongly than H 1 (relative to K) just when the ratio r(H 2, E| K) is larger than the ratio r(H 1, E| K). The difference measure gauges the degree to which E confirms H (relative to K) in terms of the difference d(H, E| K) = Pr(H| E & K) − Pr(H| K). On this measure, E confirms H 2 more strongly than H 1 (relative to K) just when the difference d(H 2, E| K) is larger than the difference d(H 1, E| K). Finally, according to the likelihood-ratio measure, the degree to which E confirms H (relative to K) is given by the likelihood ratio l(H, E| K) = Pr(E| H & K)/Pr(E|
$\overline{H}$& K).Footnote 3 And, according to l, E confirms H 2 more strongly than E confirms H 1 just in case the ratio l(H 2, E| K) is larger than the ratio l(H 1, E| K). There is much controversy in the literature as to which of these three measures (or some other measure altogetherFootnote 4) is the “one true (relevance) measure of (incremental) confirmation”. No clear consensus has been reached amongst Bayesians (see Fitelson Reference Fitelson2001 for an extended discussion of this controversy and some attempts to resolve it). Until such a consensus is reached, the best policy is to look for highly robust solutions to problems involving these measures—solutions that hold (simultaneously) for as many of the competing relevance measures as possible.
With regard to the problem of irrelevant conjunction, Earman's analysis employs the difference measure d. Earman (Reference Earman1992, 63–65) proves the following:
In other words, Earman argues that, while E does (from a Bayesian point of view) continue to confirm H & X in the case of deductive evidence, the degree to which E confirms H & X will be less than the degree to which E confirms H (relative to background evidence K). And, as more and more irrelevant conjuncts are added, the degree to which E confirms the conjunction will tend to decrease. Earman's Bayesian approach to this problem is sensitive to choice of measure of confirmation—sensitive to precisely which function r, d, l, or some other function, is employed to represent the amount by which E confirms H (relative to K). That is, in Fitelson Reference Fitelson1999 and Fitelson Reference Fitelson2001, it is shown that (2) is not true if the difference measure d is replaced by the ratio r of the posterior probability of H (on E) to the prior probability of H (relative to K).Footnote 5 But, because Earman's (2) holds for all other Bayesian relevance measures that have been proposed, his argument can be inoculated against the measure sensitivity problem simply by providing compelling reasons to reject the ratio measure of degree of confirmation (see Fitelson Reference Fitelson2001 and Eells and Fitelson Reference Eells and Fitelson2002 for several such reasons). However, it seems to me that Earman's analysis of irrelevant conjunction has other, more telling shortcomings.
A closer look at (2) reveals that the irrelevance of X is absent from the Bayesian analysis and resolution of the problem of irrelevant conjunction. What (2) says is that, when H & K ⊨ E, as conjuncts X (simpliciter) are added to H, the degree to which E confirms H & X will tend to decrease. As far as (2) is concerned, a conjunct X could be (intuitively) relevant to H and E, but this would not prevent the conjunction H & X from being less strongly confirmed than H by E. This is unfortunate, for two reasons. First, it was supposed to be the irrelevant X's that made (1) and (1′) seem so unattractive. It's not so obvious that either (1) or (1′) is incorrect in the case of highly positively relevant X's. Moreover, Bayesian confirmation theory is founded on a perfectly precise and intuitive kind of relevance (viz., correlation), which is not mentioned anywhere in the “Bayesian” statements or resolutions of the problem of irrelevant conjunction. So, Bayesians who endorse Earman's resolution (grounded in (2)) have apparently both (i) lost track of which X's were supposed to make (1) and (1′) seem so unintuitive; and, in the process, (ii) forsaken the very notion of relevance that undergirds their own theory of confirmation.
Rosenkrantz (Reference Rosenkrantz1994) does seem somewhat sensitive to these points. Rosenkrantz motivates his (alternative) Bayesian resolution of the problem of irrelevant conjunction, as follows.Footnote 6
On H–D accounts, H is confirmed by a verified prediction, E, but E is equally a prediction of H & X, where the ‘tacked on’ X may be a quite extraneous hypothesis. … There are those who think that this sin of ‘irrelevant conjunction’ vitiates Bayesian confirmation theory as well. … I hope you will agree that the two extreme positions on this issue are equally unpalatable, (i) that a consequence E of H confirms H not at all, and (ii) that E confirms H & X just as strongly as it confirms H alone. … In general, intuition expects intermediate degrees of confirmation that depend on the degree of compatibility of H with X. Measuring degree of confirmation by [ d(H, E| K) = dfPr(H| E & K) − Pr(H| K)] … yields [ d(H & X, E| K) = Pr(X| H & K) · d(H, E| K), in the case of deductive evidence]. (Rosenkrantz Reference Rosenkrantz1994, 470–471)
In other words, Rosenkrantz suggests that we should measure the degree to which E confirms H (relative to K) by taking the difference (d) between the posterior probability of H (on E) and the prior probability of H (relative to K). Then, Rosenkrantz bases his alternative Bayesian resolution of the problem of irrelevant conjunction on the following theorem (which is really just a way of precisely filling out Earman's (2)):
Rosenkrantz deserves credit here for trying to bring the irrelevance back into the Bayesian resolution of this problem. However, his account has two serious flaws. First, it is shown in Fitelson Reference Fitelson1999 and Fitelson Reference Fitelson2001 that Rosenkrantz's account is even more sensitive to choice of relevance measure of confirmation than Earman's (2)-based account. That is, if we replace d with either r or l or any of several other currently proposed Bayesian relevance measures of the degree to which evidence confirms hypotheses, Rosenkrantz's analysis doesn't hold up.Footnote 7 So, Rosenkrantz's proposal doesn't have the kind of robustness that solutions should have under the present circumstances, where we have no compelling reasons to accept d as opposed to these other Bayesian relevance measures (e.g., l) of the degree to which bits of evidence confirm hypotheses.
In addition, Rosenkrantz's account makes use of a strange—and decidedly non-Bayesian—notion of “relevance.” Rosenkrantz seems to be suggesting that a conjunct X should be considered “irrelevant” to H (relative to background K) if Pr(X| H & K) < 1.Footnote 8 This suggestion is inadequate for two reasons. First, since when do Bayesians think that the degree to which X is relevant to H can be measured using only the conditional probability Pr(X| H & K)? Secondly, the inequality Pr(X| H & K) < 1 can, at best, only tell us when X is “irrelevant” to H—it can say nothing about whether X is “irrelevant” to E, or to various logical combinations of H and E. It seems to me that the cases in which (1) and (1′) are least intuitive are cases in which X is (intuitively) irrelevant to both H and E, and to all logical combinations of H and E. Indeed, it seems to me that an adequate Bayesian solution to the problem of irrelevant conjunction should be directly extendable to non-deductive cases, and should draw on a probabilistic notion of the relevance of X. The deductive case should turn out to be just a special case of a more general rule governing confirmationally irrelevant conjunctions. Thus, I suggest that we need a new understanding of the problem of irrelevant conjunction, as seen from a Bayesian perspective.
3. A New Bayesian Perspective and Resolution
I suggest that we go about this in an entirely different way. Let's start by saying what it means (in a Bayesian framework) to say that X is confirmationally irrelevant to H, E, and all logical combinations of H and E. Then, once we have this precisely defined, let's see if (and under what auxiliary assumptions) we can show that such irrelevant conjuncts lead to decreased confirmational power. The first of these tasks is already done. Bayesians already have a perfectly precise definition of confirmational irrelevance: probabilistic independence. So, let's say that A is confirmationally irrelevant to B (relative to K) just when A and B are probabilistically independent (given K)—i.e., when Pr(A & B| K) = Pr(A| K) · Pr(B| K). Therefore, we already know what it means (in a Bayesian confirmation-theoretic framework) to say that X is confirmationally irrelevant to H, E, and all logical combinations H and E. The next step is to note that Bayesian confirmation theory does fall prey to its own general, qualitative problem of irrelevant conjunction. That is, we have the following theorem (hereafter, “confirms” is used in the Bayesian, relevance sense—see the Appendix for proofs of theorems):
Theorem 1. If E confirms H, and X is confirmationally irrelevant to H, E, and H & E (relative to K), then E also confirms H & X (relative to K).
I claim that this is the “problem of irrelevant conjunction” that Bayesian confirmation theorists should be worrying about. The good news is that we can prove the following quantitative theorem, which seems to be the relevant result for Bayesian confirmation theory and the problem of irrelevant conjunction:
Theorem 2. If E confirms H, and X is confirmationally irrelevant to H, E, and H & E (relative to K), then (provided that Pr(X| K) < 1Footnote 9):
where 
may be either the difference measure d or the likelihood-ratio measure l of degree of confirmation (but, !
may not be the ratio measure r, since in cases of irrelevant conjunction we will have r(H, E| K) = r(H & X, E| K)).
Our Bayesian analysis and resolution of the problem of irrelevant conjunction has the following advantages over its existing rivals:
Our analysis makes use of the irrelevance of X. Moreover, our notion of confirmational irrelevance is not some peculiar one (like Rosenkrantz's), but just the standard Bayesian notion of probabilistic independence.
Our resolution is not restricted to the (not very inductively interesting) special case of deductive evidence; it explains why irrelevant conjuncts are confirmationally disadvantageous in all contexts (deductive or otherwise).
Our resolution is as robust (i.e., measure- insensitive) as any other existing resolution (e.g., Earman's), and more robust than any other existing account that tries to be sensitive to the “irrelevance” of the conjunct X (e.g., Rosenkrantz's). That is, our account is based on a theorem which holds for all Bayesian measures of confirmation currently being used in the philosophical literature, except for the ratio measure r. Footnote 10
Appendix: Proofs of Theorems
A Proof of Theorem 1
Theorem 1. If E confirms H, and X is confirmationally irrelevant to H, E, and H & E (relative to K), then E also confirms H & X (relative to K).
Proof. The proof of this theorem is easy.Footnote 11 We assume without loss of generality that K is tautologous, and we reason as follows:
Therefore, in the case of irrelevant conjunct X and confirmatory evidence E, Pr(H & X| E) > Pr(H & X), which completes the proof.
B Proof of Theorem 2
Theorem 2. If E confirms H, and X is confirmationally irrelevant to H, E, and H & E (relative to K), then (provided that Pr(X| K) < 1):
where 
may be either the difference measure d or the likelihood-ratio measure l of degree of confirmation (but, 
may notbe the ratio measure r, since in cases of irrelevant conjunction we will have r(H, E| K) = r(H & X, E| K)).
Proof. For the 
= d case of the theorem (indeed, for all three cases), we assume without loss of generality that K is tautologous, and we reason as follows:
For the 
= r case of the theorem, we will prove that r(H, E) = r(H & X, E), when X is an irrelevant conjunct (hence, that r violates the theorem).
We are now ready to prove the likelihood-ratio (l) case of the theorem (via reductio ad absurdum, and using the d and r results as lemmas).
This completes the proof of Theorem 2, as well as the Appendix.Footnote 12
