1. Introduction
Think of a series of propositions 
such that E supports H1, H1 supports H2, … , and 
supports Hn. If the support relation is logical entailment, then, regardless of the length of the series, E supports Hn since logical entailment is transitive. We cannot say the same though if the support relation is confirmation in the incremental sense (hereafter simply “confirmation”), where, for any two propositions E and H, E confirms H just in case E increases the probability of H (i.e., 
). Even if each member (except the last) supports the next member in the series, E might not support Hn.Footnote 1 This is because confirmation, unlike logical entailment, is nontransitive. To illustrate the point, suppose a card is randomly drawn from a standard deck of cards, where E = the card drawn is a heart, H1 = the card drawn is a red jack, and H2 = the card drawn is a diamond. Proposition E confirms H1 since 
, and H1 confirms H2, as 
, but E does not confirm H2 because 
. It is somewhat troubling that confirmation is nontransitive since we often construct a chain of reasoning even when the support relation is only confirmation. In fact, such reasoning seems unproblematic in many cases. Take a case of testimony: E = Smith said that the card drawn is a red, where H1 = the card drawn is a red and H2 = the card drawn is a heart. Proposition E confirms H1, at least on certain ways of filling in the details; H1 confirms H2 since 
; and it seems clear that E confirms H2.
A second issue of interest, regarding a series 
where each member (except the last) confirms the next, concerns degree of confirmation. Suppose, unlike in the first card case above, E confirms each of the other members in the series. Intuitively, at least in many cases of this sort, confirmation dwindles in that the degree to which E confirms Hn is less than the degree to which it confirms 
, the degree to which it confirms 
is less than the degree to which it confirms 
, and so on.Footnote 2 Here is an example: E = Smith said that miracle M occurred, where H1 = miracle M occurred and H2 = God exists. Proposition E confirms H1, at least on certain ways of filling in the details; H1 in turn confirms H2; and, it seems, E confirms H2 (although the probability of H2 given E need not be high and, in fact, might be quite low). But, it seems, the degree to which E confirms H2 is less than the degree to which E confirms H1. The sense of dwindling confirmation becomes stronger as the chain gets longer, for example, if E = Jones said that Smith said that miracle M occurred, H1 = Smith said that miracle M occurred, H2 = miracle M occurred, and H3 = God exists. It turns out, however, that confirmation does not always dwindle. Returning to the card case, suppose E = the card drawn is a face card or the ace of hearts, H1 = the card drawn is a heart, and H2 = the card drawn is a king, a queen, or the ace of hearts. It is hard to deny that the degree to which E confirms H2 is greater than the degree to which E confirms H1, for E only slightly raises the probability of H1 (from 1/4 to 4/13) but substantially raises the probability of H2 (from 9/52 to 9/13). Another point (not unrelated to the first) in favor of this judgment is that 
, while 
; in other words, the probability of H2 is initially lower than that of H1, but it ends up higher given E, so that the increase in probability is greater for H2 than for H1.Footnote 3
It would be welcome, then, if there were conditions sufficient for transitivity in confirmation and conditions sufficient for dwindling confirmation. The former conditions could be used to verify in a given case, where E confirms H1, H1 confirms H2, … , and 
confirms Hn, that E confirms Hn. The latter could be used to verify that, even when E confirms Hn, the degree to which E confirms Hn is less than the degree to which E confirms 
.
It turns out, fortunately, that confirmation is transitive under the following condition:
Screening-Off Condition (SOC). 
and 
.Footnote 4
SOC says in effect that 
screens off E from Hk in that given the truth or falsity of 
, E has no impact on the probability of Hk.Footnote 5 In other words, E affects the probability of Hk only indirectly through its impact on the probability of 
. SOC is a condition sufficient for transitivity in confirmation in that if 
, 
, and SOC holds, then 
.
The main question we aim to answer is whether SOC is also a condition sufficient for dwindling confirmation. We argue in the affirmative. The article proceeds as follows. In section 2, we provide a list of the main confirmation measures in the literature (13 in total) and introduce an adequacy condition on confirmation measures, which is called the Weak Law of Likelihood (WLL) and which is satisfied by each of the measures in the list. Then, in section 3, we show that confirmation dwindles under SOC on any confirmation measure meeting WLL, hence on any adequate confirmation measure. This is the main result of the article.Footnote 6 We apply our result to testimony cases of the sort given above (where the subject uses information obtained from testimony for further inference) and relate our result to the Data-Processing Inequality in information theory. In section 4, we extend our result in two respects. Finally, in section 5, we conclude.
2. Preliminaries
It is a matter of controversy among Bayesian confirmation theorists how exactly to measure confirmation. The main confirmation measures in the literature are as follows:Footnote 7
Some of these measures are ordinally equivalent to each other: they impose the same ordering in degrees of confirmation on any two ordered pairs of propositions 
and 
.Footnote 8 For example, K, L, and L∗ are ordinally equivalent to each other. The measures taken as a group, however, are motley.
Consider now the following condition on any adequate confirmation measure X:
Weak Law of Likelihood (WLL). If (a) 
and (b) 
, then 
.
We argue in the next section that, under certain conditions, confirmation dwindles under SOC on all measures meeting WLL. First, though, some comments are in order regarding WLL.
Bayesian confirmation theorists, while disagreeing about how exactly to measure confirmation, all agree (as far as we are aware) that any adequate confirmation measure should satisfy WLL.Footnote 9 This is manifest in the fact that each of measures (1)–(13) meets WLL. This point is easiest to see with respect to measures (1)–(3) and (5)–(13). If 
, then
So, G meets WLL. Similarly, if 
, then
Thus, R too meets WLL. To see that the same is true of measures (1), (2), (5)–(9), and (11)–(13), consider the following equalities:
Take the first equality: 
is fully determined by G, R, and 
and is a monotonically increasing function of G and R. So, since 
is a constant in the inequalities in WLL, and since 
and 
increase, respectively, as 
decreases and 
increases, it follows that 
also increases as 
decreases and 
increases. But that is just what satisfaction of WLL demands. So C satisfies WLL. Similar remarks can be made with respect to measures (2), (5)–(9), and (11)–(13). Measure J (as far as we are aware) is not “reducible” to G, R, and 
. But it too meets WLL (see app. A for proof).
WLL is similar to the “weak likelihood principle,” which is endorsed by Joyce (Reference Joyce and Zalta2008) and which is equivalent to
WLL∗. If (a) 
and (b) 
, then 
.Footnote 10
In defense of WLL∗, Joyce writes:
(2.1e) [i.e., the weak likelihood principle] captures one core message of Bayes’ Theorem for theories of confirmation. Let’s say that H is uniformly better than H∗ as predictor of E’s truth-value when (a) H predicts E more strongly than H∗ does, and (b) ∼H predicts ∼E more strongly than ∼H∗ does. According to the weak likelihood principle, hypotheses that are uniformly better predictors of the data are better supported by the data. For example, the fact that little Johnny is a Christian is better evidence for thinking that his parents are Christian than for thinking that they are Hindu because (a) a far higher proportion of Christian parents than Hindu have Christian children, and (b) a far higher proportion of non-Christian parents than non-Hindu parents have non-Christian children.
In fact, the weak likelihood principle (2.1e) encapsulates a minimal form of Bayesianism to which all parties can agree. … Indeed, the weak likelihood principle must be an integral part of any account of evidential relevance that deserves the title “Bayesian.” To deny it is to misunderstand the central message of Bayes’ Theorem for questions of evidence: namely, that hypotheses are confirmed by the data they predict. (Reference Joyce and Zalta2008, sec. 3)
WLL∗ is slightly stronger than WLL, as one of the inequalities in its antecedent is weakened slightly from “less than” in b of WLL to “less than or equal to” in b of WLL∗. This fact, it turns out, is rather significant for our purposes. Unlike WLL, WLL∗ is not met by all of measures (1)–(13).Footnote 11 So, since not all Bayesian confirmation theorists will accept WLL∗, we want to remain neutral on whether confirmation measures ought to satisfy WLL∗. However, what Joyce says on behalf of WLL∗ carries over to WLL. If 
and 
, then H is “uniformly better than” H∗ at predicting E. The key insight behind WLL is that if H is uniformly better than H∗ at predicting E, then E provides more confirmation to H than to H∗.
One final comment on WLL is in order. While WLL is a rather undemanding adequacy condition on confirmation measures, it is not entirely toothless. Consider the measure
Measure D′ is similar in an obvious respect to D, which is a popular measure. But D′ does not meet WLL. So, it is significant that the main confirmation measures in the literature, motivated by various disparate considerations, all satisfy WLL.
The question of dwindling confirmation under SOC can now be put precisely. Let 
be a series of propositions. Consider the conditions:
A. H1 is confirmed by E, and Hk is confirmed by
for all ;B. E is screened off in the sense of SOC from Hk by
for all ;Footnote 12C. E neither entails nor is entailed by H1, and
neither entails nor is entailed by Hk for all .Footnote 13
for all 
;
for all 
;
neither entails nor is entailed by Hk for all 
.
The question of dwindling confirmation under SOC is this: Is it the case that if A, B, and C hold and X meets WLL, then 
for all 
?
3. Main Result
The task now is to answer the question of dwindling confirmation under SOC. We do this in section 3.1. We then turn to testimony cases in section 3.2 and to the Data-Processing Inequality in section 3.3.
3.1. Dwindling Confirmation under SOC
The answer to the question of dwindling confirmation under SOC is affirmative:
Theorem 1. If A, B, and C hold and X meets WLL, then 
for all 
.
(See app. B for proof.) The key in the proof of theorem 1 is the relations of likelihood that if A, B, and C hold, then (i) 
and (ii) 
for all 
. Theorem 1 follows immediately from i and ii by WLL.
Each of measures (1)–(13) meets WLL. So, theorem 1 entails that confirmation dwindles under SOC on each of measures (1)–(13). It is well known that certain claims about confirmation hold on some measures but not on others (Fitelson Reference Fitelson1999; Brossel Reference Brossel2013). There is no such “problem of measure sensitivity” with dwindling confirmation under SOC: confirmation dwindles under SOC on all of the main confirmation measures in the literature and, indeed, on any adequate confirmation measure extant or otherwise.
3.2. Testimony Cases
Theorem 1 has application in many cases involving testimony where 
said that H1, but the proposition of interest is not H1 itself but H2 that is related to it. In many such cases, E neither entails nor is entailed by H1, H1 neither entails nor is entailed by H2, but E confirms H1, which in turn confirms H2. Moreover, and crucially, SOC holds in many such cases: given the truth of H1 (which is the proposition to which S testified) or given the falsity of H1, that S said that H1 has no impact on the probability of H2 (Shogenji Reference Shogenji2003). The second testimony case from section 1, where E = Smith said that miracle M occurred, H1 = miracle M occurred, and H2 = God exists, is plausibly such a case. Proposition E increases the probability of H1 (at least on some ways of filling in the details of the case)—even if just barely. Proposition H1 in turn increases the probability of H2, E (we can suppose) neither entails nor is entailed by H1, and H1 neither entails nor is entailed by H2.Footnote 14 Further, it seems, SOC holds: E has an impact on the probability of H2 only indirectly through H1. To put this in terms of rational credence, if one already knows whether miracle M occurred, then learning further that Smith said that miracle M occurred has no impact on one’s credence in God’s existence. On this construal of the case, theorem 1 applies so that E confirms the existence of God less than it does the occurrence of miracle M.
Things are much the same in cases involving higher-order testimonies. For example, the first member of the series is E = S1 said that S2 said that H2, the second member is H1 = S2 said that H2, the third member is H2 itself, and the fourth member is some proposition H3 related to H2. In many such cases SOC still holds at each step, so that theorem 1 applies if other conditions are also met. Since confirmation keeps dwindling as the chain becomes longer, there is less confirmation, other things being equal, in cases involving more intermediary steps.
A caveat is called for here. Theorem 1 by itself has no implications concerning how much dwindling there is in a given case. Even if the degree to which E confirms Hk is less than the degree to which E confirms 
for all 
, it might be that the amount of dwindling at each step is slight and that the total amount of dwindling is not substantial. It is worth noting in this regard that measures (1)–(13) differ in terms of how much dwindling they allow in a given case. Consider R and Z, for instance, and a three-member series E, H1, H2 such that A, B, and C hold. There are distributions of probabilities on which each of 
and 
is very high, say, roughly equal to 100,000, but 
is very low, say, roughly equal to 1.01. Degree of confirmation can drop off precipitously on R in a single step of mediation under SOC. Things are quite different with Z. When SOC holds, 
equals the product of 
and 
.Footnote 15 So, if each of 
and 
is very high, then it cannot be the case that 
is very low.
3.3. Data-Processing Inequality
The Data-Processing Inequality concerns “mutual information.” Let X and Y be discrete variables. The mutual information 
between X and Y is the (weighted) average amount of information provided about (a proposition specifying) the state of the one variable by (a proposition specifying) the state of the other variable:
Here x and y are propositions specifying the states of X and Y, respectively. Mutual information is symmetric in that 
. Now let D, P, and E be discrete variables (D for “distal cause,” P for “proximate cause,” E for “effect”) and suppose SOC holds in that each state of P screens off each state of E from each state of D.Footnote 16 Then, by the Data-Processing Inequality, the mutual information between E and D is less than the mutual information between E and P: 
.Footnote 17 This result can be glossed: mutual information dwindles under SOC.
The Data-Processing Inequality has application to the issue of information loss in various settings. Consider, for example, an evolutionary process in which each member of a population has trait A or trait B (but not both).Footnote 18 Suppose the state of the population (the distribution of A and B in the population) at a given time screens off the state of the population at any earlier time. In other words, the state of the population at a given time has a direct impact only on the immediately succeeding state, so that if it has an impact on any subsequent states, it does so only indirectly through its impact on the immediately succeeding state. It follows from the Data-Processing Inequality that the current state of the population provides more information on average about the state of the population at more recent times than about the state of the population at less recent times.
This result is interesting and resembles dwindling confirmation under SOC, but there are important differences. First, mutual information (as defined above) is the expected amount of information—the amount of information about the state of one variable we should expect to receive upon learning the state of the other variable.Footnote 19 Suppose we learn E, where E is a specific proposition about the current state of the population in the evolutionary case above, and our interest shifts from the expected amount of information to the degree of confirmation provided by E. Suppose, more specifically, we want to know whether E confirms H1 better than H2, where H1 is a proposition about the state of the population at an earlier time t1 and H2 is a proposition about the state of the population at an even earlier time t2. The Data-Processing Inequality is of no help here since, as explained above, the Data-Processing Inequality concerns mutual information, and mutual information is the expected amount of information. Of course, information theory also allows us to determine how much information particular proposition x provides about particular proposition y in the form of 
. So, we can compare the amounts of information E provides about H1 and H2, respectively. But, notice that the measure used here is the log-ratio measure of confirmation R∗. So, when we translate the result into the language of confirmation theory, we are forced to use the specific measure R∗. Theorem 1 above reveals not only that E provides less information about H2 than about H1 under SOC but also that there is no need to use the specific measure R∗. If E confirms H1, H1 confirms H2, SOC holds, E neither entails nor is entailed by H1, and H1 neither entails nor is entailed by H2, then by theorem 1 the degree to which E confirms H2 is less than the degree to which E confirms H1 on any adequate confirmation measure. Theorem 1 thus serves as a welcome supplement to the Data-Processing Inequality.
4. Further Results
We now show that theorem 1 can be extended in two respects. The first respect (discussed in sec. 4.1) concerns a screening-off condition weaker than SOC. The second respect (discussed in sec. 4.2) concerns cases where condition C—the condition that E neither entails nor is entailed by H1, and 
neither entails nor is entailed by Hk for all 
—does not hold.
4.1. Dwindling Confirmation under PISOC
SOC is the standard screening-off condition. But there are others. A less restrictive screening-off condition is
Positive Impact Screening-Off Condition (PISOC). 
and 
.
PISOC says in effect that given the truth or falsity of Hk-1, E has no positive impact on the probability of Hk—either E has no impact on the probability of Hk or the impact is negative. PISOC is weaker than SOC, in that PISOC holds if SOC holds but not vice versa.
PISOC, unlike SOC, is not a condition sufficient for transitivity in confirmation.Footnote 20 But PISOC, like SOC, is a condition sufficient for dwindling confirmation. Consider the condition:
B∗ E is screened off in the sense of PISOC from Hk by
for all .
for all 
.Condition B∗ is simply B with SOC replaced by PISOC. PISOC is a condition sufficient for dwindling confirmation in that
Theorem 2. If A, B∗, and C hold and X meets WLL, then 
for all 
.
(See app. C for proof.) The crucial thing here is that A, B∗, and C—as with A, B, and C—together imply that (i) 
and (ii) 
for all 
.Footnote 21
Theorem 2 marks a further respect in which dwindling confirmation under SOC differs from the Data-Processing Inequality. The latter cannot be extended to the case where PISOC holds but SOC does not.Footnote 22
4.2. Nonincreasing Confirmation under SOC and PISOC
Recall that each of the main confirmation measures in the literature—measures (1)–(13)—meets WLL. The same is true of a slightly different condition:
WLL∗∗. If (a) 
and (b) 
, then 
.
First, if 
, then
and if 
, then
So G and R meet WLL∗∗. Next, given that G and R meet WLL∗∗, and given the list of equalities provided in section 2, it follows that C, D, K, L, L∗, M, N, R∗, S, and Z meet WLL∗∗. These measures are fully determined by G, R, and 
and are monotonically increasing functions of G, R, or both. As noted in section 2, J is not “reducible” to G, R, and 
, as far as we can tell, but it can be shown (in a way parallel to the proof given in app. A) that J also meets WLL∗∗.
A few remarks are in order concerning WLL∗∗ as it relates to WLL and WLL∗. First, all inequalities in WLL are strict (“>” or “<”), while all inequalities in WLL∗∗ are nonstrict (“≥” or “≤”). WLL states in essence that if H is uniformly better than H∗ at predicting E, then E confirms H more than it does H∗. Similarly, WLL∗∗ states in essence that if H is uniformly at least as good as H∗ at predicting E, then E confirms H at least as well as it does H∗. WLL∗∗ is every bit as intuitive as WLL. Any adequate confirmation measure should satisfy WLL∗∗. Second, whereas all inequalities in WLL∗∗ are nonstrict, this is not the case with WLL∗—the first and third are strict, while the second is nonstrict. WLL∗ states in essence that if H is in some way better than H∗ and in another way as good as H∗ at predicting E, then E confirms H better than it does H∗. Recall that not all of measures (1)–(13) meet WLL∗. This is because on some such measures the first condition in the antecedent of WLL∗ is irrelevant to the truth or falsity of the consequent so that WLL∗ is tantamount to the dubious claim that if H is at least as good as H∗ at predicting E, then E confirms H better than it does H∗. Things are different with WLL∗∗. The measures in question imply that the first condition in the antecedent of WLL∗∗ is irrelevant to the truth or falsity of the consequent, but they also imply that WLL∗∗ is tantamount to the plausible claim that if H is at least as good as H∗ at predicting E, then E confirms H at least as well as it does H∗.
Recall condition C that E neither entails nor is entailed by H1, and 
neither entails nor is entailed by Hk for all 
. It is essential to theorem 1 that C holds in that if A and B hold but C does not, then, even supposing that X meets WLL, it might be that 
for some 
. However, we now have
Theorem 3. If A and B hold and X meets WLL∗∗, then 
for all 
.
(See app. D for proof.) Theorem 3, unlike theorems 1 and 2, is not limited to cases where C holds. As a result, in some instances of theorem 3, 
, and thus confirmation does not dwindle. Theorem 3 can be glossed: confirmation in a chain of support never increases under SOC.
It is straightforward to show that, similarly, confirmation never increases under PISOC:
Theorem 4. If A and B∗ hold and X meets WLL∗∗, then 
for all 
.
(The proofs for theorems 2 and 3 can be readily adapted to establish theorem 4.)
5. Conclusion
To state the main result of the article in nonformal terms, as the chain of probabilistic support becomes longer, confirmation dwindles provided that the first proposition supports each of the subsequent propositions only indirectly through the chain. This is true on each of the main confirmation measures in the literature and, more generally, on any adequate confirmation measure. This result applies to many testimony cases, and it serves as a welcome supplement to the Data-Processing Inequality. One question for future research is whether expected confirmation dwindles under SOC—not just on the log-ratio measure of confirmation, as the Data-Processing Inequality implies, but also on any adequate confirmation measure. If so, we can make claims about expected confirmation without taking a stand on which of the many possible confirmation measures is to be preferred.
Appendix A: Proof of J’s Meeting WLL
Suppose the antecedent of WLL holds. Then, since G and R meet WLL, it follows that 
and 
. Given that 
, there are four cases to consider:
We begin with cases (A2) and (A3) and then turn to cases (A1) and (A4).
Rewrite 
and 
as follows:
Since 
and 
, it follows from (A2) and (A3), respectively, that
in cases Cases (A1) and (A4)—the two remaining cases—imply, respectively, that
In both case (A1) and case (A4), either (a) 
or (b) 
, from which it follows, respectively, that
a∗)
;b∗)
.
;
.
To show that 
in case (A1), it suffices to show that 
in the two subcases: (A1)a and (A1)b. Likewise, to show that 
in case (A4), it suffices to show that 
in the two subcases: (A4)a and (A4)b.
First subcase, (A1)a.—It follows immediately from (A1∗)a∗ that 
. Second subcase, (A1)b.—Let 
, where 
from b. Since 
, 
. Given this, and given that
we have
Meanwhile, it follows from 
and 
that 
is a strictly increasing function of x. But 
from (A1). So,
It follows from (A5) and (A6) that
Hence, 
. So,
It follows from (A8) and 
that
Equation (A9) implies
It then follows that 
as follows:
Hence, 
in case (A1).
Third subcase, (A4)a.—Let 
, where 
from a. Equation (A5) holds by the same reasoning as in the second subcase. Meanwhile, it follows from 
that 
is a strictly decreasing function of x. But 
from (A4). So (A6) holds. The argument then continues in parallel to the argument given in the second subcase. Fourth subcase, (A4)b.—It follows immediately from (A4∗)b∗ that 
. Thus, 
in case (A4). So, as 
in cases (A1)–(A4), it follows that J meets WLL. QED
Appendix B
Proof of Theorem 1: We establish theorem 1 in two steps. First, we establish a lemma:
Lemma. If A, B, and C hold, then (a) 
and (b) 
for all 
.
Second, we argue from the lemma to theorem 1.
Consider a three-member series E, H1, H2 and suppose the antecedent of the lemma holds. It is known (Shogenji Reference Shogenji2003) that if H1 screens off E from H2, as B implies, then
Equation (B1) implies
But, given C, we have
Equation (B2) and inequality (B4) together entail
Inequality (B5) implies
Next, from C, we have
Equation (B3) and inequality (B7) together imply
But, we have
It follows from (B8) and (B9) that
So, if the antecedent of the lemma holds for a three-member series E, H1, H2, then 
and 
.
Now we generalize this result to the n-member series 
. Suppose the antecedent of the lemma holds. SOC is a condition sufficient for transitivity in confirmation, so, given A and B, it follows that E confirms Hk for all 
. Next, given A–C, and given the result from the previous paragraph, it follows that 
and 
.
The lemma is but a short step away: 
and 
. So, as 
and 
, we have
It follows immediately from (B11) and (B12), respectively, that 
and 
. The lemma is thus established.
Suppose now the antecedent of theorem 1 holds. Then, the antecedent of the lemma also holds. So, it follows by the lemma that (a) 
and (b) 
. It then follows, as X meets WLL, that 
. QED
Appendix C
Proof of Theorem 2: Suppose the antecedent of theorem 2 holds. Then, by reasoning similar to that for (B1), we have
It follows from (C1) that
With (C2) and (C3), instead of (B2) and (B3), the proof continues in parallel to the proof given above for theorem 1 (details omitted so as to avoid repetition). First, it is shown that if A and B∗ hold for a three-member series E, H1, H2, then 
and 
. Next, it is shown that the point generalizes to the n-member series 
, and thus 
and 
. Finally, since X meets WLL, it follows from the antecedent of theorem 2 that 
. QED
Appendix D
Proof of Theorem 3: The proof for theorem 1 can be adapted to establish theorem 3. Suppose the antecedent of theorem 3 holds. It follows from A and B that (B1), (B2), and (B3) hold. Then, instead of (B4), which is a strict inequality, we have
Equation (B2) and inequality (D1) imply
Inequality (D2) entails
Further, instead of (B7), which is a strict inequality, we have
Equation (B3) and inequality (D4) together entail
But we have
It follows from (D5) and (D6) that
So if A and B hold for a three-member series E, H1, H2, then 
and 
. The argument now proceeds in parallel to the argument given above for theorem 1 (details omitted so as to avoid repetition). First, it is shown that the point just established about a three-member series can be generalized to the n-member series 
, and thus 
and 
. Then, from the antecedent of theorem 3, which says in part that X meets WLL∗∗, it follows that 
. QED
and 
if and only if (iff) E logically entails H. Second, confirmation in the incremental sense is equivalent to confirmation in the “relevance” sense: E incrementally confirms H just in case E is positively relevant to H. Put formally, 
just in case 
. Third, and finally, confirmation in the incremental sense is neither sufficient nor necessary for confirmation in the “absolute” sense, where E confirms H iff the probability of H given E is sufficiently high. See Carnap (
) concerns just the probabilities 
and not degrees of confirmation (i.e., incremental confirmation). See McGrew (
and 
.
and 
, 
iff 
. Note that it follows by contraposition that 
iff 
.
iff 
, (b) 
iff 
, and (c) 
iff 
. This condition, called Qualitative Discrimination in Crupi, Festa, and Buttasi (
, (b) 
, and (c) one of the two inequalities is strict, then 
and 
. Each of these likelihood conditions—WLL and WLL∗ on its two construals—is much weaker than the Law of Likelihood (LL): if 
, then 
. See also Fitelson (
and 
. It should be the case by WLL∗ that 
, but it is not since 
. The case of Z is similar.
, 
, 
, and 
.
to be less than 
, it also needs to be the case that 
is greater than 0. This condition should be understood throughout this subsection.
and 
. So, PISOC is not sufficient for transitivity in confirmation.
and 
. NISOC is like PISOC in that given the truth or falsity of Hk-1, E may still have some impact on the probability of Hk. But whereas PISOC precludes the impact from being positive, NISOC precludes it from being negative. NISOC, like SOC, is a condition sufficient for transitivity in confirmation (Roche 
has value 1, 
has value 2, and similarly for y1, y2, z1, z2. First, one can require that each member of the partition 
be screened off in the sense of PISOC by each member of the partition 
from each member of the partition 
. Second, one can require just that x1 be screened off in the sense of PISOC from z1 by y1. Both options are problematic. The first option fails because there is no case in which each member of 
is screened off in the sense of PISOC by each member of 
from each member of 
but each member of 
is not screened off in the sense of SOC by each member of 
from each member of 
. For, if, say, 
, then 
. So, 
and 
only if 
and 
, as required by SOC. The second option fails because there are counterexamples; more specifically, there are cases where x1 is screened off in the sense of PISOC from z1 by y1 and yet 
is less than 
.
