2. NOR-confirmation

The new agglomeration condition I would like to present is the following:

(3) $$P(\neg {H_1} \wedge \neg {H_2}|E) \gt P\left( {\neg {H_1} \wedge \neg {H_2}} \right).$$

Less formally, it states that the evidence in question $E$ confirms that neither hypothesis ${H_1}$ nor ${H_2}$ is true, or, equivalently, that both target hypotheses are false. Due to the obvious relationship to Peirce’s (Reference Peirce, Hartshorne and Weiss1933) NOR-connective ${H_1} \downarrow {H_2}$ , I will call condition (3) NOR-confirmation.

Intuitively, NOR-confirmation (3) is at odds with both the agglomeration antecedent (1) and the consequent (2). After all, the two latter conditions state that the evidence in question $E$ confirms that both target hypotheses ${H_1}$ and ${H_2}$ are true, namely individually and conjunctively. It might therefore come as a surprise that NOR-confirmation (3) guarantees that agglomeration is valid for Bayesian confirmation. That is, whenever (3) holds, (1) entails (2). For a proof, first observe that (3) is equivalent toFootnote ²

$$P({H_1} \vee {H_2}|E) \lt P\left( {{H_1} \vee {H_2}} \right),$$

which, by general additivity, expands to

$$P({H_1}|E) + P({H_2}|E) - P({H_1} \wedge {H_2}|E) \lt P\left( {{H_1}} \right) + P\left( {{H_2}} \right) - P\left( {{H_1} \wedge {H_2}} \right).$$

By simple algebra, this is equivalent to

$$P({H_1}|E) - P\left( {{H_1}} \right) + P({H_2}|E) - P\left( {{H_2}} \right) \lt P({H_1} \wedge {H_2}|E) - P\left( {{H_1} \wedge {H_2}} \right),$$

and, by condition (1), the following equivalent of (2) follows:

$$0 \lt P({H_1} \wedge {H_2}|E) - P\left( {{H_1} \wedge {H_2}} \right).$$

To see that NOR-confirmation (3) is a non-trivial condition for agglomeration, i.e., (3) is consistent with (1), consider an urn containing ten balls with three binary attributes, distributed as shown in Table 1. Let the evidence $E$ be that a randomly drawn ball is blue, ${H_1}$ that it is small, and ${H_2}$ that it is clean. Then, NOR-confirmation (3) is satisfied, i.e., the evidence confirms that the drawn ball is not small and not clean:

$$P(\neg {H_1} \wedge \neg {H_2}|E) = 4/10 \gt P\left( {\neg {H_1} \wedge \neg {H_2}} \right) = 3/10.$$

The agglomeration antecedent (1) is satisfied, i.e., the evidence confirms that the drawn ball is small and clean individually,

$$\forall i \in 1,2:P({H_i}|E) = 6/10 \gt \left( {{H_i}} \right) = 5/10,$$

and hence the agglomeration consequent (2) is satisfied, i.e., the evidence confirms that the drawn ball is small and clean conjunctively:

$$P({H_1} \wedge {H_2}|E) = 6/10 \gt P\left( {{H_1} \wedge {H_2}} \right) = 3/10.$$

This shows that NOR-confirmation (3) is a non-trivial condition for agglomeration.

3. Carnap and Salmon

The observation that NOR-confirmation (3) and the agglomeration antecedent (1) are consistent is not entirely new: this fact was already noted implicitly by Carnap (Reference Carnap1950) and Salmon (Reference Salmon and Achinstein1983). The two authors discussed examples in which a piece of evidence $E$ confirms two hypotheses ${H_1}$ and ${H_2}$ individually while disconfirming their disjunction ${H_1} \vee {H_2}$ , the latter being equivalent to NOR-confirmation (3). What is new, however, is that this makes Carnap’s and Salmon’s examples rather peculiar instances of agglomeration. To see this more clearly, consider Salmon’s example:

Salmon found the fact that such situations can arise “shocking and counterintuitive” (Salmon Reference Salmon and Achinstein1983, section 3). But he overlooked that being an instance of agglomeration, it follows that the evidence also confirms the hypothesis that Jones has viral and bacterial pneumonia. Just imagine the following dialogue:

I suspect that most readers will find the researcher’s utterances confusing and unhelpful. Perhaps, some will even question the validity of her inference, arguing that the lab results should disconfirm the hypothesis that Jones has viral and bacterial pneumonia. But the researcher’s inference is valid and everything she says is consistent.Footnote ⁴

4. The rarity objection

One might try to relativize the phenomenon above by arguing that it is probably very rare. My response to this objection is twofold: I admit that the phenomenon is not very prevalent, but this does not make it less unsettling. More precisely, the conjunctive prevalence of cases where NOR-confirmation (3) and (1) are jointly satisfied is around 2.5%. And the conditional prevalence of cases where NOR-confirmation (3) is satisfied if (1) is satisfied is around 10%. This can be shown using Monte Carlo integration based on 10 million regular probability functions over an algebra generated by three variables (Metropolis and Ulam Reference Metropolis and Ulam1949). The left-hand graph in figure 1 shows how the prevalence stabilizes with increasing number of probability functions.

Figure 1. Prevalence of the NOR-effect (left) and the Simpson-effect (right).

To put these values into context, compare them with Simpson’s (Reference Simpson1951) paradox, a different but similarly puzzling probabilistic phenomenon where a piece of evidence $E$ confirms a hypothesis $H$ conditional on some assumption $X$ and conditional on $\neg X$ , but $E$ fails to confirm $H$ unconditionally (Sprenger and Weinberger Reference Sprenger, Weinberger and Edward2021). The conjunctive prevalence of such cases is only around 0.83%, and their conditional prevalence is around 3.33%, as shown on the right-hand side of figure 1. But the low prevalence of Simpson’s paradox has not kept researchers from finding the phenomenon unsettling. So, even if cases where NOR-confirmation (3) and the agglomeration antecedent (1) are jointly satisfied are rare, they are prevalent enough to care about.

5. Previous agglomeration conditions

NOR-confirmation (3) is not the only agglomeration condition for Bayesian confirmation. As Reichenbach (Reference Reichenbach1956) showed in his analysis of common-cause structures, agglomeration is also valid if the evidence screens off both hypotheses from each other:

(4) $$P({H_1}|E \wedge {H_2}) = P({H_1}|E){\rm{\;\;\;\;and\;\;\;\;}}P({H_1}|\neg E \wedge {H_2}) = P({H_1}|\neg E).$$

And, as Falk (Reference Falk1986) pointed out in his discussion of Cohen’s (Reference Cohen1977) corroboration theorem, agglomeration remains valid even if screening-off is relaxed as follows:Footnote ⁵

(5) $$P({H_1}|E \wedge {H_2}) \geqslant P({H_1}|E){\rm{\;\;\;\;and\;\;\;\;}}P({H_1}|\neg E \wedge {H_2}) \le P({H_1}|\neg E).$$

Salmon (Reference Salmon and Achinstein1983) uncovered another interesting condition. While agglomeration can also fail for independent hypotheses, it cannot if, additionally, the two target hypotheses are independent conditional on the evidence:

(6) $$P({H_1}|{H_2} \wedge E) = P({H_1}|E){\rm{\;\;\;\;and\;\;\;\;}}P({H_1}|{H_2}) = P\left( {{H_1}} \right).$$

Finally, there are two more recent conditions from the literature on the problem of irrelevant conjunction (Schurz Reference Schurz2022). The first is part of Fitelson’s (Reference Fitelson2002) confirmational irrelevance condition,

(7) $$P({H_2}|{H_1} \wedge E) = P\left( {{H_2}} \right){\rm{\;\;\;\;and\;\;\;\;}}P({H_2}|{H_1}) = P\left( {{H_2}} \right),$$

and the second is Hawthorne and Fitelson’s (Reference Hawthorne and Fitelson2004) conditional irrelevance condition which states that the evidence is irrelevant for one hypothesis conditional on the other:

(8) $$P({H_2}|{H_1} \wedge E) = P({H_2}|{H_1}).$$

Now, interestingly, NOR-confirmation (3) is logically independent of each of the aforementioned conditions (4)–(8). That is, NOR-confirmation (3) is consistent with each of them but neither entails nor is entailed by any of them. A proof of this statement is provided in Appendix A.

Notice, however, that most of these logical independence relationships break down once the agglomeration antecedent (1) is satisfied. More precisely, if (1) holds, then NOR-confirmation (3) is inconsistent with screening-off (4), full independence (6), confirmational irrelevance (7), and conditional irrelevance (8). A proof of this is provided in Appendix B. With these remarks, I close my discussion of NOR-confirmation (3).