I have a developed a suite of arguments, based on the Grim Reaper paradox of Jose Benardete (Infinity, 1968) for two related conclusions: the impossibility of infinite causal regresses and the necessary finitude of the past. (See my related posts: The anti-regress argument and the unsatisfiable pair diagnosis.
Here are my published papers and articles:
• “A New Kalām Argument: Revenge of the Grim Reaper,” Noûs 48 (2014):256-267.
• “Does the Universe Have a Cause?” in Contemporary Debates in Philosophy of Religion, 2nd edition, Raymond VanArragon and Michael Peterson (eds.), Wiley-Blackwell, 2020, pp. 3-13, 22-3.
• “Did the Universe have a Cause?” in The Big Question: The Debate over God’s Existence, Andrew Drinkard, ed., Eerdmans 2025
Here is my simplified version of the Benardete paradox, adapted to show the impossibility of an infinite past/regress:
I will tell here a slightly simpler version of the story. Let’s suppose that there is an infinite regress of Grim Reapers, each of which is assigned a particular time on which to perform its duty. Grim Reaper 1 is assigned noon, January 1st, 1 B.C., Reaper 2 the same day in 2 B. C, 3 in 3 B. C., and so on throughout an infinite past. Each Grim Reaper n + 1 passes a death warrant on to its successor (Reaper n) at the end of its assigned period (Jan. 1st n B. C.). Each Grim Reaper n follows the following script: (1) If the warrant already contains a numeral larger than n, then Grim Reaper n leaves the warrant unchanged and passes it on, in turn, to Reaper n – 1; (2) otherwise, Grim Reaper n signs the warrant by writing on it the numeral n.
This story obviously involves some kind of impossibility, since we can derive from it a contradiction. We can prove both that the warrant will and will not contain a numeral on Jan. 1st, 1 A. D. First, it must have a numeral on it on Jan. 1st 1 A. D. The warrant either did or did not have a numeral on it on the morning of Jan. 1st 1 B. C. If it did, then Grim Reaper 1 would have passed it on unchanged, and it would still have that numeral on it on Jan. 1st 1 A.D. If it did not have a numeral in 1 B. C., then Grim Reaper 1 would have written the numeral ‘1’ on it. So, in either case, the warrant will contain some numeral in 1 A. D.
However, there is no number n such that the corresponding numeral could be on the warrant at that time. Suppose for contradiction that the warrant in 1 A. D. contains the numeral n. Then Grim Reaper n must have received a blank warrant from Reaper n + 1 and then written the numeral n on the warrant. Now, either the warrant already had a numeral m on it at the beginning of n + 1 B. C., or not. If it did, then Reaper n + 1 would have passed this warrant unchanged to Reaper n, and Reaper n would not have written his own number on it. If the warrant did not contain a numeral the beginning of n + 1 B. C., then Reaper n + 1 would have written his numeral on it, and, again, Reaper n would not have written his numeral. This argument applies to any natural number, so the warrant cannot have a number on it in 1 A. D. Contradiction.
I made use of this contradiction as part of the following argument:
Main Argument
1. If an infinite past (or regress) were possible, the Grim Reaper scenario would be possible.
2. The Grim Reaper scenario is not possible.
Therefore, neither an infinite past nor an infinite causal regress is possible.
I argued for premise 1 by relying some version of David Lewis’s Patchwork Principle. The Patchwork Principle has the following form:
(PP) If there is a possible world w1 containing an event of type E, and there is a possible world w2 containing a spatiotemporal or causal structure with room for k non-overlapping copies of events of type E (where k is any cardinal number, including transfinite ones), then there is a possible world w3 containing k non-overlapping copies of events of type E.
It is clear that each individual Grim Reaper story (event of type E) is possible. Premise 1 simply applies the Patchwork principle to this type.
In the last few years, these papers have attracted a number of replies, mostly from Joe Schmid and his collaborators. These responses can be grouped thematically
1. Appeals to Shackel’s Unsatisfiable Pair Diagnosis
Schmid, Joseph C. and Alex Malpass (2025), Bernadete Paradoxes, Causal Finitism, and the Unsatisfiable Pair Diagnosis, Mind 134(534):397-421, 2025
2. Appeals to Branching Actualism
Joseph Schmid and Alex Malpass (2023), Branching Actualism and Cosmological Arguments, Philosophical Studies180(7):1951. https://doi.org/10.1007/s11098-023-01958-y
3. Other Challenges to the Patchwork Principle
Schmid, J. C. (2024). Benardete paradoxes, patchwork principles, and the infinite past. Synthese, 203(2), Article 51. https://doi.org/10.1007/s11229-023-04481-z
Daniel Linford and Alex Malpass (2025) “The Grim Reaper Argument is an Argument for No One” Synthese 206:254. https://link.springer.com/article/10.1007/s11229-025-05361-4
4. Parody Arguments concerning the Simultaneous and Time-Reversed Causation
Schmid, Joseph C. (2024). The End is Near: Grim Reapers and Endless Futures. Mind 133 (532). https://academic.oup.com/mind/article/133/532/1057/7492631
Dana, Troy and Schmid, J.C.(2026), “Grim Reaper Paradoxes and Patchwork Principles: Severing the Case for Finitism,” The Journal of Philosophy 123(3):137-163, 2026
I will take up these objections in this post and in the succeeding three. Let me turn now to the Unsatisfiable Pair Diagnosis. I already addressed this objection in a post in 2025, but I would like to revise and extend my remarks here.
The Schmid and Malpass 2025 article relies on a 2005 article by Nicholas Schackel, “The Form of the Benardete Dichotomy,” British Journal for the Philosophy of Science 56(2):397-417.
In that article, Schackel restates the assumed structure of the Grim Reaper series by Condition ANB (all iff nowhere before):
Condition ANB: for all x in S, E at x iff E nowhere before x
Schackel claims that he can derive a contradiction from two assumption: that S satisfies Condition ANB and that S is ‘unbegun’ (has no first member). Hence, Shackel proposes what he calls the Unsatisfiable Pair Diagnosis of the paradox. Contradiction results from assuming two inconsistent assumptions about the series of Reapers.
As I have pointed out, the fact that the beginningless Reaper series is logically impossible is a feature rather than a bug in my argument. In fact, the unsatisfiability of the conditions is exactly premise 2 of my main argument.
Shackel’s proof of the contradiction in his 2005 is stated very loosely and informally. Let’s see if I can work it out. A variable contained by two parentheses (like (x)) will represent the universal quantifier. A variable preceded by an E is an existential quantifier. The ‘if and only if’ biconditional is represented by <->, the ‘if-then’ conditional by ->, negation by ~, and the relation of temporal or causal priority by <.
0. Let S be linearly ordered by < (i.e., transitive and asymmetric)
1. NB: (x)Ey y < x
2. ANB: (x)(Fx <-> (y)(y < x -> ~Fy))
3. Fa (for contradiction)
4. Ey y < a From 1
5. b < a From 4
6. (y)(y < a -> ~Fy) From 2, 3
7. b < a -> ~Fb From 6
8. ~Fb From 5, 7
9. ~(y)(y < b -> ~Fy) From 2, 8
10. Ey(y < b & Fy) From 9
11. d < b & Fd From 10
12. d < a by transitivity of < From 5, 11, 0
13. ~Fd Contradiction From 6, 12
14. (x)~Fx Reductio of 3 by 11, 13
15. ~Fe From 14
16. ~(y)(y < e -> ~Fy) From 2, 15
17. Ey( y < e & Fy) From 16
18. f < e & Ff From 17
19. ~Ff From 14
Contradiction (18, 19)
This formalization bring out the fact that there are really three conditions needed for a contradiction: NB, ANB, and the transitivity of <. The linearity or transitivity of the temporal and causal orders is a metaphysical rather than a logical or mathematical truth. In the absence of premise 0, NB and ANB are logically consistent.
Here is a proof that transitivity is needed. Let S be the set of negative integers, and define ’n < m’ as n = m – 1. Then we can have a model of NB and ANB with alternating F and ~F integers.
So, Shackel’s diagnosis is really the Unsatisfiable Trio Diagnosis. And that is important, since it shows that we cannot avoid the conclusion of the main argument simply by adding a logical or mathematical consistency proviso to the Patchwork Principle, as Schmid and Malpass proposed.
Instead, they would have to add a metaphysical possibility proviso:
(PP-M&S) If there is a possible world w1 containing an event of type E, and there is a possible world w2 containing a spatiotemporal or causal structure with room for k non-overlapping copies of events of type E (where k is any cardinal number, including transfinite ones), then there is a possible world containing k non-overlapping copies of events of type E, provided there is a metaphysically possible world containing k non-overlapping copies of type E.
This proviso obviously renders the PP completely vacuous.
