II. THE REPETITION ARGUMENT

Dougherty gives an argument that Many–Few and Risk Tolerance are inconsistent which he calls the Repetition Argument. His argument is so-called because it examines the likely effects of repeating risky actions. This section will outline the Repetition Argument.

Dougherty begins by asking us to consider the following scenario:

Dougherty then reasons as follows. By Risk Tolerance, you are permitted to bring a victim Chancy Antidote. If one is unconvinced, the size of the risk that Chancy Antidote may do nothing can be altered until one is satisfied that Risk Tolerance entails that it is permissible to bring a victim Chancy Antidote. Consequently, Risk Tolerance entails (1):

Since you may bring Chancy Antidote to a victim in Poison once, Dougherty argues that you may bring Chancy Antidote to any number of people in a similar situation: ‘If you may bring the Chancy Antidote to [one victim], then you may also bring the Chancy Antidote to [that victim] and then bring it to [another victim]’.Footnote ⁷ Consequently, Dougherty argues that Risk Tolerance entails (2):

Given (2), by repeatedly administering Chancy Antidote, it follows that for every billion people to whom you bring Chancy Antidote, we would expect that you would fail to save someone's life while everyone else's lives are saved and their headaches cured. As a result, (2) has the problematic consequence of entailing (3):Footnote ⁸

The trouble is that Risk Tolerance entails (1) and Many–Few contradicts (3). Consequently, Dougherty concludes that these principles are inconsistent.

Dougherty's argument is deeply worrying. For if we reject Many–Few, then we must call into question our judgement that we may act to benefit the few even when the aggregate benefits to a larger group of individuals is greater. In turn, revising this judgement calls into question a powerful argument against consequentialist approaches to ethics. And if we reject Risk Tolerance, then we are forced to revise our judgements about everyday risky actions. In particular, we would have to avoid any beneficial action that would expose potential beneficiaries to a small risk of serious harm. Revising our actions in this way would be severely limiting.

Dougherty attempts to resolve this problem by searching for alternative principles that capture our intuitions about risk-imposition and aggregation. By his own lights, he is ‘unsuccessful in discovering any such principles’.Footnote ⁹ He then throws down the gauntlet for others.

III. A SCOPE AMBIGUITY IN PREMISE (2)

This section contends that Dougherty's problem need not be resolved by searching for alternative principles to Many–Few or Risk Tolerance. Rather, the solution is that the Repetition Argument trades on an ambiguity, namely, that (2) is ambiguous. Once (2) is disambiguated, the purported inconsistency between Many–Few and Risk Tolerance dissolves: the Repetition Argument provides no reason to think that Many–Few and Risk Tolerance are not both true.

Let us make this argument clearer by adopting the following formalization to illuminate the ambiguity of (2). Suppose that there is a finite number of people N = 1, 2, 3,. . ., n, each of whom is a victim in Poison. Let p ₁, p ₂,. . ., p _n be, respectively, the sentences ‘Person 1 is administered Chancy Antidote’, ‘Person 2 is administered Chancy Antidote’, and so on. Let Aϕ be the sentence ‘You bring it about that ϕ’ and let ◆ϕ be the sentence ‘It is permissible that ϕ’.Footnote ¹⁰ Then we may formulate (2) in two ways, depending on the scope of ‘may’. First, we can assign ‘may’ a narrow scope reading, such that you may bring it about that person 1 is administered Chancy Antidote, that you may bring it about that person 2 is administered Chancy Antidote, and so on:

The second reading assigns ‘may’ a wide scope reading, such that you may bring it about that persons 1 to n are administered Chancy Antidote all at once:

On the one hand, (2_WIDE) has the problematic consequence of entailing (3) and thus conflicting with Many–Few. For if you are permitted to give Chancy Antidote to the first person, to the second person, and so on, then, as Dougherty argues, you are permitted to cure the headaches of 999,999,999 people rather than saving someone's life. Given Many–Few, you are not permitted to cure the headaches of 999,999,999 people rather than saving someone's life. Consequently, if (2) is read as (2_WIDE), it generates a seriously problematic consequence. On the assumption that Many–Few is true, this is good reason for thinking that (2_WIDE) is false.

Furthermore, there is independent motivation for rejecting (2_WIDE) because it conflicts with the following highly plausible principle:

If you give everyone Chancy Antidote, then it is near certain that someone will die, even though it is not the case that there is someone for whom it is near certain that they will die. Exposure to near certain death runs a sufficiently high risk of serious harm. Consequently, given Constraint on Permissible Risk Imposition, it is impermissible to give everyone Chancy Antidote. So, the fact that (2_WIDE) conflicts with Constraint on Permissible Risk Imposition, as well as Many–Few, is good independent reason for thinking that (2_WIDE) is false.

On the other hand, (2_NARROW) does not have this problematic consequence, since (2_NARROW) does not entail (2_WIDE) and so does not entail (3). This is because deontic permissions do not necessarily agglomerate; that is, deontic permissions do not satisfy the following principle:

In other words, given the falsity of Agglomeration, if it is permissible that ϕ and it is permissible that ψ, it does not follow that it is permissible that ϕ and ψ. Such failures of Agglomeration occur in single-agent cases when two independently permissible actions, each of which affect only one and the same individual, are collectively impermissible. Consider the following case:Footnote ¹¹

In Insulin, it is permissible for B to give an insulin shot to A and it is permissible for C to give an insulin shot to A, since either action will save A from insulin shock. But it is impermissible for both B and C to give A each of their insulin, since this conjunctive action will cause A to become hyperglycaemic and die. Consequently, Insulin is a case where agglomeration fails.

Failures of Agglomeration also occur in multi-agent cases when two or more independently permissible actions, each of which affect different individuals, are collectively impermissible. Consider the following case:

In Lifeboat, for each individual, it is permissible to save that individual from drowning. But it is impermissible to save every individual, since doing so will cause the certain death of everyone including yourself, even though there is no individual for whom it is impermissible for you to save. Consequently, Lifeboat is a case where Agglomeration fails.

Given the general failure of Agglomeration, it should be no surprise that you may give Chancy Antidote to person 1, and you may give Chancy Antidote to person 2, and so on, even though it does not follow that you may give Chancy Antidote to person 1, and to person 2, and so on. Consequently, if (2) is read as (2_NARROW), it neither entails (2_WIDE) nor (3). That is, it does not follow from (2_NARROW) that you may cure the headaches of 999,999,999 people rather than save the life of someone else.

Risk Tolerance provides good reason for thinking that the narrow reading of (2) is true. We can also provide independent reason to think that (2_NARROW) is true. For the permissibility of exposing someone to a risk does not depend on whether you have previously exposed other people to this risk. This point is captured by the following highly plausible principle endorsed by Dougherty:Footnote ¹²

Given Only Effects Matter, the permissibility of giving Chancy Antidote to an individual in Poison depends upon the possible or actual effects of that action for that individual. Consider an arbitrary individual in Poison. Since only that individual is affected if we give her Chancy Antidote, and the risk of serious harm to her by giving her Chancy Antidote is sufficiently low, then it seems permissible to give her Chancy Antidote. Consequently, Only Effects Matter provides independent reason for thinking that the narrow reading is true.

We are now in a position to present the solution to Dougherty's problem. Dougherty argues that Risk Tolerance and Many–Few are inconsistent because Risk Tolerance entails (2), (2) entails (3) and (3) contradicts Many–Few. However, (2) is ambiguous between (2_NARROW) and (2_WIDE). Once (2) is disambiguated, we see that the Repetition Argument must implicitly assume the following instance of Agglomeration which takes (2_NARROW) as its antecedent and (2_WIDE) as its consequent:

However, proponents of Many–Few and Risk Tolerance need not accept (2_BRIDGE), since Risk Tolerance entails that its antecedent is true and Many–Few entails that its consequent is false, and a conditional with a true antecedent and a false consequent is itself false. Moreover, the rejection of (2_BRIDGE) is strongly supported by independent arguments for the truth of (2_NARROW), the falsity of (2_WIDE) and the invalidity of Agglomeration. Consequently, since the Repetition Argument relies on an appeal to a false instance of the invalid principle of Agglomeration, there is no argument from Risk Tolerance to something that contradicts Many–Few: you may bring Chancy Antidote to a victim in Poison once, but this does not mean that you may bring Chancy Antidote to person 1 and to person 2, and so on. Therefore, the Repetition Argument fails: Many–Few and Risk Tolerance are not inconsistent.

IV. WHY AGGLOMERATION FAILS AND THE LIMITS OF ACCEPTABLE AGGLOMERATION

The previous section argued that the Repetition Argument provides no reason for believing that either Many–Few or Risk Tolerance is false. Advocates of both principles may accept (2_NARROW), because it is entailed by Risk Tolerance, and reject (2_WIDE), because it contradicts Many–Few, without worrying about inconsistency because Agglomeration fails. However, it would be good to have a general, independent rationale for why Agglomeration fails, especially in the case of permissions to expose people to small risks. A satisfying answer to this question would explain why we should accept the narrow and not the wide reading of Dougherty's second premise.

We do not need anything other than the two principles Constraint on Permissible Risk Imposition and Only Effects Matter to explain the failure of Agglomeration in cases involving risk imposition.Footnote ¹³ To see this, let q ₁ and q ₂ be two propositions differing at most in the individual whom they are about. Let us assume that the risk of serious harm caused from bringing about q ₁ is sufficiently low, and the risk of serious harm caused from bringing about q ₂ is sufficiently low, while the risk of serious harm caused from bringing about both q ₁ and q ₂ is sufficiently high. This may be true even if the risks of the individual events are independent.Footnote ¹⁴ Given Only Effects Matter, it follows that:

Given the highly plausible principle that action is closed under conjunction (that is, if one brings it about that ϕ and one brings it about that ψ, then one brings it about that both ϕ and ψ):

But, by Constraint on Permissible Risk Imposition, it is impermissible to bring about q ₁ and q ₂, since the risk of serious harm caused by bringing about both q ₁ and q ₂ is sufficiently high:

By modus tollens, it is impermissible to bring about q ₁ and to bring about q ₂:

The Repetition Argument requires that something like the following substitution instance of Agglomeration is true:

However, it follows from (4) and (7) that (8) is false, since it is a conditional with a true antecedent and a false consequent. Since (4) and (7) are entailed by Only Effects Matter and Constraint on Permissible Risk Imposition, respectively, only these principles are required to show how agglomeration fails in the above case. By appealing to these principles, we can make similar remarks about the failure of Agglomeration in the Repetition Argument.

It would also be good to discuss the limits of acceptable agglomeration. More specifically, it seems that the above picture of risk-imposition is committed to the following seemingly problematic claim:Footnote ¹⁵

This claim is seemingly problematic because it implies the following two claims:

Suppose that an agent has already performed the set of actions (Ap ₁ &. . .& Ap_k ), and is now confronted with a scenario where they could perform Ap _k+1. If we accept the above claims, then the agent has not yet violated a requirement. The question is whether she would be violating a requirement by performing Ap _k+1. The answer is: yes and no. Yes; after all, the agent will be doing something she is not permitted to do, namely, perform the set of actions (Ap ₁ &. . .& Ap _k+1). But also no; for although that set of actions is impermissible, it does not follow that performing Ap _k+1 is impermissible nor that any other particular action is impermissible. In fact, the agent will be doing something that she is permitted to do, namely, performing the action Ap _k+1. This should not strike us as unusual. Recall Lifeboat; if I save all 10 individuals, it is clear that I have done something impermissible since we will all die, even though there is no individual for whom it is impermissible to save. Therefore, (*) is not an undesired feature of any theory of risk-imposition.

V. SOME OBJECTIONS

To summarize, this article has argued that once the wide-scope and narrow-scope readings of (2) have been distinguished, the Repetition Argument gives no reason to think that Many–Few and Risk Tolerance are inconsistent. However, the picture of risk that falls out may strike many as unnatural. This section aims to address some objections that might be raised to it.

One may object that my solution embodies a mistaken way of thinking about risk.Footnote ¹⁶ To see the purported mistake, suppose I am deciding whether to ride my bicycle without my helmet. I judge that there is a small risk of serious harm if I were not to wear my helmet. On the present view of risk, it is permissible for me to ride my bicycle without a helmet just this once, since the convenience of not wearing it is worth the risk. However, it also follows from the present view that I am not permitted to continually take these small risks, since it would become likely that one time I suffer serious harm and this is not worth the small inconvenience.

Dougherty objects that this view of risk does not appreciate the ‘importance of the independence of the risks in question’:Footnote ¹⁷

Dougherty argues that this thought can be captured by Only Effects Matter. The problem may then be stated as follows. My argument holds that (i) it is permissible to take a risk just this once and (ii) it is impermissible to repeatedly take risks. But Only Effects Matter shows that any account of risk that entails (i) and (ii) is mistaken. As a result, my solution opposes Only Effects Matter in spirit, if not in word. Once we appreciate that the correct way to think about risks is by assessing the permissibility of actions on the grounds of the effects of each action independently, we should reject my argument above.

Moreover, Dougherty claims that Only Effects Matter entails the following iterative premise, the antecedent of which is (1) and the consequent is (2):

But, if Only Effects Matter entails Iterative Premise, this provides another argument for the purported inconsistency between Many–Few and Risk Tolerance.

However, appealing to Only Effects Matter is not sufficient for rejecting my argument. Let us begin with the second objection first. Before we can agree with Dougherty that Only Effects Matter entails the iterative premise, we must disambiguate between a narrow-scope and a wide-scope reading of its consequent:Footnote ¹⁸

It should be clear that Only Effects Matter entails only (IP_NARROW), and not (IP_WIDE). Moreover, the consequent of (IP_NARROW) is my (2_NARROW), and I have argued that only (2_WIDE), and not (2_NARROW), leads to a contradiction with Many–Few. Consequently, one cannot appeal to Only Effects Matter in defence of Dougherty's argument.

Second, we can show that (i) it is permissible to take a risk just this once and (ii) it is impermissible to repeatedly take risks do not jointly conflict with Only Effects Matter.

It is easy to show that (i) is compatible with Only Effects Matter, for when we judge that it is permissible to bring the Chancy Antidote in Poison once, we are only considering the effects of that action on that person.

To show that (ii) is also compatible with Only Effects Matter, suppose that one brings the Chancy Antidote to any sequence of people in Poison:

Then, given the closure of action under conjunction, it follows from (11) that (12):

When we contemplate the permissibility of (12), it might be tempting to appeal to Only Effects Matter. However, such an appeal is illegitimate, since Only Effects Matter only governs the permissibility of risky actions that affect one person and the action in (12) affects more than one person. Consequently, we align our judgements about the permissibility of (12) with (ii), concluding that (12) is impermissible, without fear of Only Effects Matter legislating against this judgement.

Moreover, we can appeal to the following multi-agent version of Only Effects Matter to support the intuitive verdict that (ii):

Such a principle supports the intuitive verdict (ii). For if there is sufficiently large risk that an action may harm some person whomever that might be, then that action is impermissible. Appealing to Only Effects Matter 2, the multi-agent version of Only Effects Matter, then, provides further support for the compatibility of (i) and (ii). Consequently, Dougherty's objections to the present view of risk-imposition fall short.

Lastly, it is worth considering a version of Only Effects Matter that abstracts away from the permissibility of an action to violations of requirements more generally:Footnote ¹⁹

The picture of risk sketched here is not incompatible with Generalized Only Effects Matter. When assessing whether to perform a risky action that affects only one agent, we need only look at the possible or actual effects of that action for that person. However, when a risky action affects more than one agent, such as the conjunctive actions involved in Poison and Lifeboat, we must broaden our normative horizons and consider the possible or actual effects of that action for all agents involved. Sometimes a conjunctive action composed of a number of independent risky actions makes it near certain that some agent will suffer serious harm, even though it is not certain that any specific agent will suffer serious harm. In such cases, we are not permitted to perform the conjunctive action, even though we are permitted to perform each of its constituent actions.Footnote ²⁰