Although full belief affords decision-makers convenience, less than full belief in scientific propositions may be inevitable due to uncertainty. Consider the novel coronavirus that has spread globally and affected over a hundred million people worldwide at the time of writing. Since medical researchers are still seeking an effective treatment for symptomatic patients, the proposition that treatment t effectively reduces symptoms (denoted as Reduce) remains uncertain, where t is in the domain of treatments currently under consideration.
In accommodating uncertain matters within an expert community, an expert’s (rational) degrees of belief or credences may be formally modeled by a probability function p on a (finite) Boolean algebra of propositions B, relative to a set of possible worlds W. But on a more liberal conception of subjective probability, an expert’s credences may instead be modeled by a non-empty set of probability functions P on the Boolean algebra. Credences under this more general mathematical representation are set-based rather than point-valued and can be imprecise.Footnote 3
On occasions where an expert’s credences are imprecise, the imprecision might be due to the evidence made available at that time. Consider the COVID-19 pandemic, for example. Many things remain unknown at the moment, including which course of treatment is most effective for reducing symptoms, whether the newly developed vaccines provide immunity against different strains of the virus, how long face coverings should continue to be worn after a sizeable portion of the global population has been vaccinated, etc. Gaps in the evidence might compel an expert to form set-based credences in relevant propositions, e.g., P(Reduce) = {0.5, 0.7} and P(~ Reduce) = {0.3, 0.5}, and rationally so, as their credal state is fixed by the evidence in hand, no more and no less. The rationality of imprecise credences on occasions where evidence is incomplete but not absent ought to convince us that probabilistic representations should not be dismissed in characterizing the doxastic commitments of decision-makers under the EP.
But even in admitting that experts occasionally form rational credences and the credences may either be precise or imprecise, credences often differ from individual to individual, causing difficulty for decision-makers in forming judgments of their own. Consider, for example, a group of virology experts that agree on a class of antivirals to experimentally test on positive COVID-19 patients, but the virologists disagree about the effectiveness of each treatment individually, where credences are low, high, and somewhere in between across the set of possible outcomes. How should decision-makers go about taking into account the diverging expert credences? Classical probabilists might suggest employing an aggregation strategy such as linear pooling, where an expert’s opinion, for all propositions X, is weighted based on the expert’s reliability and combined through weighted averaging (Dietrich & List, 2016). However, it is this kind of procedure that leads to the issue Peterson is concerned about.
Consider a pair of medical experts, A and B, that have credences 0.3 and 0.9 in the proposition Reduce, respectively, and suppose that the experts are equally competent and reliable. The equal-weighted average in credence between them is 0.6. But as a collective or consensus opinion presented to decision-makers, it fails to convey the degree to which the two experts’ opinions diverge. Furthermore, 0.6 credence implies that collectively, the medical experts are more confident in Reduce than ~ Reduce under the weighted averaging rule, suggesting that Reduce has more evidential support when that is not the case. One can see now why Peterson is inclined to reject probabilistic principles for reconciling conflicting probabilistic judgments. His concern is further reinforced by the fair chance that the weighted averaging rule is the first aggregation strategy that comes to mind for many, given its familiarity.
But these observations of a specific kind of aggregation procedure should not convince us that all hope is lost for a feasible credal interpretation of the doxastic commitments of decision-makers under the EP. For linear pooling, of course, is not the only way to aggregate beliefs. Some have recently illustrated the epistemic advantages of alternatively aggregating beliefs through imprecise probabilities (see Stewart & Quintana, 2018). Drawing on the imprecise probability approach to belief aggregation and a recent account given by Elkin and Wheeler (2018) concerning peer disagreement, I propose that a decision-maker’s doxastic state is constrained by the following principle in case the views of experts diverge on a given matter.
IP. For all permissible doxastic states D, a decision-maker should adopt D if and only if D just consists of the union of expert opinions, \(\bigcup_{i}\varvec{P}_{i}(X)\), for all n experts and propositions X in some Boolean algebra B.Footnote 4
What this principle implies in the earlier case, for example, is that decision-makers should adopt imprecise credences P(Reduce) = {0.3, 0.9} and P(~ Reduce) = {0.1, 0.7} in light of the disagreement between medical experts A and B.
The proposed IP principle has several advantages. First, IP sidesteps an inconsistency that Peterson’s account gives rise to from his endorsement of the right to left direction of (PER)—if at least some experts believe X, then it is permissible to believe X. It follows from the right to left direction of (PER) that if the most influential experts believe X, then a decision-maker is permitted to believe X, despite some non-influential experts believing ~ X. But the EP prohibits decision-makers from ignoring or discarding the views of experts having less influence or prominence within the community. Hence the inconsistency. By comparison, IP does not satisfy the right to left direction (see below). Consequently, IP is less permissive. But failing to satisfy the right to left direction of (PER) is fortunate, not fatal, since the inconsistency is avoided, unlike on Peterson’s overly permissive qualitative account.
Second, IP is consistent with the EP as defined. It follows from IP that every expert view is taken into account under the required doxastic state D, as every expert view, Pi, is a subset of D. So, no expert’s view is ignored or discarded for lacking influence or prominence within the community. If, however, some expert’s view is not a subset of D, then, trivially, D does not consist of the union of all expert opinions for all propositions X, thereby violating IP. We see then that the general idea underlying the EP of accounting for every expert view is implied by IP, thus making the doxastic principle consistent with the EP.
Third, IP is partially consistent with Peterson’s constraints. On the (OB) requirement, IP satisfies the right to left direction. If all experts hold the same view, i.e., P1 = … = Pn, then the required doxastic state D by IP is P such that P = Pi, for i = 1,…, n. IP, however, does not satisfy the left to right direction. Suppose that the required doxastic state D by IP is P, but 1 and 2 differ in their views such that P1 ≠ P2. It is obviously false then that P = Pi, for i = 1,…, n.
On (PER), since IP implies that D is permissible if and only if D just consists of the union of all expert opinions for all propositions X, it logically follows that the views held by some experts are contained in D. IP, however, does not satisfy the right to left direction. But again, this is fortunate since IP avoids the inconsistency resulting from (PER) and the EP. To see how the right to left direction is not met, suppose P1 is a subset of D, but P2 is not, for experts i = 1, 2. The antecedent of the right to left direction of (PER) is true, but the consequent is false given that D does not consist of the union of all expert opinions for all propositions X.
On (PRO), IP satisfies the right to left direction provided that if the intersection of \(\bigcup_{i}\varvec{P}_{i}\) and doxastic state D is empty, then D obviously does not consist of the union of all expert opinions for all propositions X and is impermissible. IP does not satisfy the left to right direction. Consider the above instance where P1 is a subset of D but P2 is not, for experts i = 1, 2. D is forbidden by IP, but the intersection of D and \(\bigcup_{i}\varvec{P}_{i}\) is not empty. Thus, we find that IP only satisfies weaker versions of Peterson’s constraints. But violating the original constraints is quite reasonable, especially considering the inconsistency generated by (PER) and the EP.
Fourth, IP is immune to Peterson’s criticism against permitting probabilistic rules in characterizing the doxastic commitments of decision-makers under the EP. IP implies that a decision-maker’s doxastic state is fixed in accordance with the information in hand, no more and no less. In fact, a hallmark of imprecise probabilities generally is that they match the character of the evidence (Sturgeon, 2010). So much for overstating the evidence.
Finally, IP is more transparent than Peterson’s qualitative account on evidential support by realizing lower and upper bounds on credence. Consider the case of medical experts A and B from before. Under the IP principle, Reduce is given a minimum credence of 30% and a maximum credence of 90%. Decision-makers are thus committed to unique ranges of credence when conflicting evidence yields different levels of support for and against propositions. That is, IP commits decision-makers to adopting ranges of credence bounded by the least and greatest amounts of credence supported by their evidence.
The qualitative account, on the other hand, can mask the evidential support for some propositions. Unfortunately for Peterson, this exposes his view of the EP to an opposite concern. In particular, Peterson’s qualitative account is prone to understating the evidence. To see this, suppose that A categorically believes Reduce, whereas B categorically believes ~ Reduce. Decision-makers are not permitted to believe both propositions by logic alone, but they are entitled to choosing a side, given Peterson’s endorsement of (PER). Whichever side is taken, though, leads to a neglect of evidence since one of the experts’ opinions that is supposed to be recognized is completely discounted. The (PER) constraint consequently results in understating the evidential support that exists for the proposition(s) decision-makers choose to disbelieve.
Fortunately for IP, it safeguards against both exaggerating and neglecting evidence by stopping short of saying too much and saying too little. If precautionary-based reasoning prohibits both overstating and understating the evidence, then the IP proposal should be preferred to Peterson’s qualitative account in representing the doxastic commitments of decision-makers under the EP.