Grim Reaper paradox

In philosophy, the Grim Reaper paradox is a paradox involving a supertask of an infinite sequence of grim reapers, each tasked with killing a person if no reaper has already killed them. The paradox raises questions about the possibility of continuous time and the infinite past (temporal finitism).

The paradox is inspired by J. A. Benardete's paradoxes from the 1964 book Infinity: An Essay in Metaphysics. In fact, various formulations of paradoxes involving beginningless sets, whose members perform a function only if no previous member performs it, are all labelled Benardete Paradoxes.

The paradox

The paradox supposes there is an infinite sequence of Reapers, each assigned a time to kill a particular person. Each Reaper will only kill this person if no earlier Reaper has already killed them.

It is 12pm, the first Reaper is set to kill the person at 1pm. The second Reaper is set to kill them at 12:30pm, the third at 12:15pm, and so on.

As a consequence of these propositions, the person will be certainly be killed by a Reaper before 1pm, however, no individual Reaper can kill them, as there is always an earlier Reaper who would do so first. Therefore, it is impossible that the person survive, but also impossible that any Reaper kills them.

Resolutions and Implications

Discrete time

One solution to the paradox is supposing that time must be discrete rather than continuous. If so, an infinite number of Reapers cannot all have a separate time in which they will kill you, as there are only finitely many "moments" in each period of time. A possible issue with this solution is that the Reaper paradox can take different forms which do not rely upon continuous time. One such example appears in Benardete's book, in which a god throws up a wall if a man travels 1/2 mile, another god throws up a wall after 1/4 mile, another at 1/8 mile, ad infinitum. Discrete time would do nothing to prevent this paradox.

Causal finitism

Another solution is the idea of Causal finitism, which asserts that there cannot be an infinite regress of causes. In other words, every causal chain must have a starting point. Thus, there cannot be an infinite number of Reapers whose actions depend on all previous Reapers. All Benardete paradoxes share this feature of an infinite causal chain, and so are all impossible.

Causal finitism could plausibly imply the discreteness of time, temporal finitism, infinitely large spatial regions, and continuously dense spatial regions, all of which are heavy metaphysical commitments.

The Unsatisfiable Pair Diagnosis

A third potential solution to the Grim Reaper paradox has been suggested, known as the Unsatisfiable Pair Diagnosis (UPD). The UPD asserts that Benardete paradoxes (including the Grim Reaper paradox) are simply logically impossible, and no metaphysical thesis needs to be adopted. In The Form of the Benardete Dichotomy Nickolas Shackel observes that all Benardete Paradoxes involve two conditions:

1. The linearly ordered set S has no first member
2. For all x in S, E at x iff E nowhere before x

Shackel shows these statements to be formally inconsistent, they logically cannot both be true. The paradox assumes that some set of items could satisfy both statements, but no set can.

Relevance to theism

According to Pruss, the Grim Reaper paradox provides grounds for thinking that the past is finite, i.e. that there must be a first period of time. This would support the Kalam cosmological argument, backing up the premise that the universe began to exist.

In 2018, Pruss provided a more thorough cosmological argument using causal finitism to motivate a necessary uncaused cause. The argument is as follows:

1. Nothing has an infinite causal history.
2. There are no causal loops.
3. Something has a cause.
4. Therefore, there is an uncaused cause.

Pruss then adds the following Causal Principle: 5. Every contingent item has a cause. From this the conclusion can be drawn that there is an uncaused cause which exists necessarily. Pruss states that it is still a major task to argue from a necessary first cause to theism.

Whilst The Kalam argument opposes sequences that go infinitely backwards in time, this argument denies all causally backwards-infinite sequences.