Grim Odds or Simple Sums?

Don't Fear the (Unreliable) Reaper

In a blog post entitled “Unreliable Grim Reapers,” Alexander Pruss presents a probabilistic twist on the classic Grim Reaper paradox. The argument imagines Fred, alive at 10:00 AM, facing an infinite sequence of Grim Reapers, each swinging a scythe at a precise time tₙ = 10:00 + 60/n minutes with a probability 1/nᵖ of killing him if he’s still alive. Pruss argues that if p ≤ 1, then the scenario is impossible because the probability of avoiding paradox (via Fred surviving or dying consistently) is zero, yet consistency requires it to be one: a contradiction! However, if p > 1, then the scenario seems possible, leading Pruss to find this dependence on p counterintuitive, especially when considering an emerging response to the original Grim Reaper paradox.

From the outset, I’m skeptical that this result is as counterintuitive as Pruss suggests. The outcome seems to follow straightforwardly from the mathematics of infinite series. To clarify this, I’ll first formally present Pruss’s argument, critically analyze it, and conclude with a potential objection to my analysis.

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The Unreliable Grim Reapers

The scenario unfolds as follows: Fred is alive at 10:00 AM. An infinite sequence of Grim Reapers, indexed by natural numbers (n = 1, 2, 3, …), stand ready. The n‑th Reaper’s alarm triggers at tₙ = 10:00 + 60⁄n minutes. So, Reaper 1 acts at 11:00 AM, Reaper 2 at 10:30 AM, Reaper 3 at 10:20 AM, and so on, with times converging toward 10:00 AM as n increases. When a Reaper’s alarm rings, it checks Fred’s status. If he’s alive, it swings its scythe, succeeding in killing him with probability 1⁄nᵖ where p > 0 is a fixed parameter. If the swing fails, Fred lives on. If Fred is already dead, the Reaper does nothing. Each Reaper’s success is independent of the others, and no other mechanism can kill Fred.

This setup modifies the original Grim Reaper paradox, where each Reaper kills with certainty, leading to a contradiction: Fred must die (since each Reaper ensures it), but no Reaper can be the killer (as an earlier Reaper would have acted first). The probabilistic version seems to sidestep this by allowing Fred to survive if all Reapers fail or to die if one succeeds before others. Pruss argues that paradox is avoided only if there’s a point where all subsequent Reapers fail to kill Fred—either because he’s dead or their swings miss. He formalizes this as the event A, the “Paradox Avoidance” event, which is the union of events Eₙ, where Eₙ means all Reapers m > n fail to kill Fred (and E₀ means all Reapers fail).

The catch lies in the probabilities, in calculating the probability of A. The probability that Reaper n fails, given it swings, is 1 − 1⁄nᵖ. For all Reapers m > n to fail, the probability is the infinite product P(Eₙ) = ∏ₘ₌ₙ₊₁^∞ (1 − 1⁄mᵖ).

Pruss notes that this product’s behavior hinges on p:

1. If p ≤ 1: To avoid the original paradox, the scenario requires Event A to occur—there must be some point after which all Reapers fail to kill Fred, ensuring either survival or a single death. This is necessary for consistency, so P(A) should be 1. However, the mathematics tells a different story. The series ∑ₘ₌₁^∞ 1/mᵖ diverges for p ≤ 1 (e.g., for p = 1, it’s the harmonic series, which grows without bound). A standard result in probability theory states that if ∑ aₘ diverges for 0 < aₘ < 1, then ∏ (1 − aₘ) = 0. Here, aₘ = 1/mᵖ, so the product ∏ₘ₌ₙ₊₁^∞ (1 − 1/mᵖ) = 0, meaning P(Eₙ) = 0 for all n. Since A is the countable union of these events (A = E₀ ∪ E₁ ∪ E₂ ∪ …), and each Eₙ has probability 0, the total probability is P(A) = 0. This creates a contradiction: the scenario demands P(A) = 1 for consistency, but the calculation yields P(A) = 0. Thus, for p ≤ 1, the scenario is impossible, as it requires an event with zero probability to occur with certainty.

2. If p > 1: The series ∑ 1⁄mᵖ converges (e.g., for p = 2, it’s π²⁄6), so ∏ (1 − 1⁄mᵖ) > 0. Thus, P(Eₙ) > 0, and P(A) is positive, avoiding the contradiction. The scenario seems possible.

Pruss finds it counterintuitive that the scenario’s possibility depends on whether p is above or below 1, suggesting a deeper issue with infinite causal sequences or the “inconsistent pair” resolution of the original paradox, otherwise known as the “Unsatisfiable Pair Diagnosis” (UPD). Briefly, the UPD offers a way to resolve the original Grim Reaper paradox without invoking heavy metaphysical commitments like causal finitism, which Pruss leans toward. According to the UPD, the paradox is logically impossible because it hinges on two conditions that can’t both hold:

• that there is an infinite sequence of Reapers with no first member, and

• each Reaper acts only if no earlier Reaper does.

These form an unsatisfiable pair. Proponents of the UPD argue this logical contradiction alone explains the paradox’s impossibility, no need for banning infinite causes.

Analyzing the Argument

The contradiction for p ≤ 1 arises because the infinite product ∏ (1 − 1⁄mᵖ) collapses to 0. This isn’t mysterious, it’s a direct consequence of the p-series test. For p ≤ 1, the terms 1⁄mᵖ don’t decay fast enough, so their cumulative effect drives the product to zero. For p > 1, the terms shrink rapidly, allowing the product to stabilize above zero. This is textbook analysis. Consider Fred’s survival probability (E₀, all Reapers fail):

• P(E₀) = ∏ₘ₌₁^∞ (1 − 1⁄mᵖ).

For p ≤ 1, this is 0; for p > 1, it’s positive. Similarly, the probability that Fred dies at tₙ (Reaper n succeeds, all m > n fail) is:

• P(Eₙ) = (1⁄nᵖ) · ∏ₘ₌ₙ₊₁^∞ (1 − 1⁄mᵖ).

Again, this is 0 for p ≤ 1, positive for p > 1. The total probability of consistent outcomes (P(A) = P(E₀) + ∑ₙ₌₁^∞ P(Eₙ) is 0 for p ≤ 1, implying inconsistent outcomes (e.g., infinitely many successes) have probability 1, a logical impossibility since consistency is required.

But is this dependence on p counterintuitive? Infinite products converging to positive values when their corresponding series converge is standard. Think of a simpler analogy: a coin with probability 1⁄n² of landing heads on the n-th flip. The probability of all tails is ∏ (1 − 1⁄n²) > 0, because ∑ 1⁄n² < ∞. If the probability were 1⁄n, the product would be 0, as ∑ 1⁄n diverges. The cutoff at p = 1 is where convergence flips, a natural boundary in analysis.

Pruss’s surprise seems to stem from the metaphysical implications: a scenario’s possibility hinges on a mathematical threshold. But this feels less like a paradox and more like a feature of infinity. Infinite sequences often produce sharp transitions, just consider Zeno’s paradoxes or supertasks. The assumption that infinite causal chains should behave intuitively may be the real culprit.

An Objection?

One objection might defend Pruss’s intuition, arguing that the dependence on p exposes a flaw in the “inconsistent pair” response. If the original paradox is impossible due to logical inconsistency, why does its probabilistic cousin’s possibility depend on a convergence criterion? Recall that the "inconsistent pair" response is one way to resolve the original Grim Reaper paradox. It states that the original setup is impossible because its description logically entails two contradictory propositions: (1) some Reaper must kill Fred, and (2) for any given Reaper n, it's impossible that Reaper n killed Fred.

Pruss imports this style of reasoning into the Unreliable case for p ≤ 1. He argues that the contradiction P(A) = 1 and P(A) = 0 functions similarly to the logical contradiction in the original. It makes the scenario impossible because it generates an inconsistency. He proposes using this probabilistic contradiction as the basis for declaring the Unreliable Reaper scenario impossible when p ≤ 1. The "counterintuitive" aspect, for Pruss, is that the very possibility of the Unreliable Reaper scenario seems to depend critically on the value of p. If the original paradox was deemed impossible because of its inherent logical structure, one might expect variations to be possible or impossible based on their logical structure too. The Unreliable Reaper scenario doesn't seem logically contradictory at first glance: Fred could survive, or one Reaper could kill him.

This suggests the original paradox’s impossibility might involve more than logical structure. Perhaps infinity itself imposes metaphysical constraints, supporting causal finitism.