Imagine, if you will, a guy named Fred.

Fred is immortal. The only way he can die is if he is decapitated by a Grim Reaper’s scythe. Nothing else will kill him.

Now, imagine it is 10 o’clock. Fred is alive at exactly 10 o’clock.

Earlier in the day, Old Grim Reaper (we’ll call him GR1) gets the call. He is tasked with decapitating Fred by 11 o’clock that night.

GR1 is a little lazy, however. He decides, “I’m going to take a nap. I’ll set my alarm for 10:30. If Fred is alive at 10:30, I’ll decapitate him. If he’s already dead, then my job is done anyway, and I can go back to bed.” He turns in.

Meanwhile, word gets out. Fred is to be dead by 11.

A younger, up-and-coming Grim Reaper (we’ll call him GR2) sees GR1 setting his alarm and he decides, “Boy, if I beat Old Grim Reaper to Fred, I’ll get promoted! I’ll set my alarm to 10:15. If Fred is alive at 10:15, I’ll decapitate him. If he’s dead, well, my job’s done!” He lays down.

Another Grim Reaper, GR3, has the same idea; but, he sets his alarm for halfway between 10 and 1:15… 10:07.5.

GR4 sets his alarm for 10:03.75.

GR5 sets his alarm for 10:01.875. … and so on.

In fact, an infinite number of Grim Reapers set their alarms between 10 and 11, each one setting their alarm half the time between the later GR and 10, all with the intent of decapitating Fred when they wake up if he’s still alive.

Surely, with an infinite number of Grim Reapers set to decapitate Fred if he’s still alive when they wake up, Fred is toast.

Right?

Well, if GR1 woke up and Fred was still alive at 10:30, he would surely decapitate him. But hold on. If Fred was alive at 10:30, that means he must have been alive at 10:15. Which also means that he was alive at 10:07.5, and 10:03.75, and so on. Each GR would have the opportunity to decapitate Fred.

So is Fred decapitated by 11 o’clock?

Yes. Definitely.

But… which of the infinite Grim Reapers decapitated him?

Well…

None of them!

It turns out, with an actual infinity of Grim Reapers, none of them will have decapitated Fred when their alarms went off because there would have always been another Grim Reaper before them who would have done the job. Yet by 11, he’s decapitated anyway.

Strange, right?

A New Look at the Kalaam Cosmological Argument

This particular paradox with the Grim Reapers is my new favorite! I guess it appeals to my quirky sense of humor. This is just one paradox explored by Dr. Alexander Pruss in his recorded lecture given here:

Dr. Alexander Pruss is a Professor of Philosophy at Baylor University in Texas. He takes an interesting and unique look at the Kalaam Cosmological Argument for God’s existence by presenting it with a focus on the uncaused cause.

In his paper entitled “Causal Finitism and the Kalaam Argument,” he presents the following iteration of the Kalaam as follows:

(1) There is a cause.
(2) There is no circle of causes.
(3) There is no infinite regress of causes.
(4) If (1)–(3), there is an uncaused cause.
(5) So, there is an uncaused cause.
(6) If there is an uncaused cause, God exists.
(7) So, God exists.

Causal Finitism and the Kalaam Argument (Pruss), page 1

Pruss points out that in philosophy, the only premises that are actually controversial are (3) and (6). He briefly points out that premise (6) has been intimately examined by other philosophers (see, for instance, William Lane Craig’s Reasonable Faith Podcast from December 2019 entitled “Why the Kalam Cosmological Argument?“); the very properties of an “uncaused cause” must be the classically defined God (ie, not necessarily the Christian God). He cites the work of others and then moves on, since Premise (6) is not the main focus of the paper.

Premise (3) is the main focus.

The Problem With Actual Infinities

Pruss goes on to demonstrate, both in the video above and his paper, a number of reasons why infinite causal regresses are impossible, thereby supporting Premise (3) of the Kalaam Argument, by exploring three paradoxes. The Grim Reaper example from earlier shows a paradox related to an actual infinity:

(i) Fred is dead,

(ii) Fred can only be dead by the hand of a reaper, but

(iii) no reaper raised a hand to harm him.

page 8

He offers two more illustrations of the absurdity of a causally linked infinite regress.

This image illustrates the kind of halving of time used in the Grim Reaper and Thomson’s Lamp illustrations.
  1. Grim Reapers
  2. Thomson’s Lamp
    • Consider an indestructible lamp with an indestructible toggle switch, toggled on and off between 10 am and 11 am exactly.
    • At 10 am the Lamp is off. At 10:30 you push the toggle and it’s on. Then at 10:45, you toggle it off again. Then at 10:52.5, it’s on. At 10:56.25, it’s off. And so on, ad infinitum.
    • The question is, at 11 am, is the lamp toggled on or off?
    • If the lamp is toggled an actual infinite number of times, there is no way to determine the outcome, since there’s always another toggle after the one “right before” 11 o’clock hits.
  3. Die Guessing
    • You are kidnapped and forced to play a game once a year where someone throws a “fair” six-sided die. Before the die is thrown, you must guess “SIX” or “NOT SIX”. If you’re right, nothing happens. If you’re wrong, you’re tortured.
    • The odds for each side of the fair die is 1/6. Therefore, if you call “NOT SIX,” you have a 5/6 chance of being right and avoiding torture.
    • Let’s say 20 years go by and a six has yet to be rolled. You might start thinking, “Is this the year I call ‘SIX’ instead?” Don’t! This is called the Generalized Gambler’s Fallacy. Basically, no matter how many finite times you roll the die, each individual roll has the same odds of getting “NOT SIX” as all the previous rolls: 5/6. The only prudent strategy is to choose “NOT SIX” every time, year after year.
    • However, if you were immortal and so were your captives, and you were forced to play this game an infinite number of times, the two options’ odds actually change! Calling “SIX” actually becomes just as good of a strategy to avoid torture as calling “NOT SIX!”
    • This violates the Generalized Gambler’s Fallacy for each individual roll, thereby producing a paradox.

These paradoxes illustrate the impossibility of actual infinites… but how does this relate to the uncaused cause of Kalaam??

Causal Finitism

Dr. Pruss’ uses these paradoxes as evidence for what he calls causal finitism. Causal finitism is simply the claim that “an infinite number of things cannot be causally prior to one thing.” (p. 3) This is not to suggest that infinities themselves are impossible (as that would destroy mathematics as we know it) but rather causal chains of events specifically cannot continue backward in time forever.

His argument is phrased in two ways. The first is inductive, appealing to the best explanation:

(8) Scenarios P1, P2 and P3 are impossible because they are paradoxical.

(9) A single version of causal finitism gives an elegant unified explanation why P1, P2 and P3 are impossible.

(10) There is no good competitor.

(11) So, probably, causal finitism is true.

page 12

…and the other is a deductive version, where Pi represents any of these paradoxes:

(12) If causal finitism is not true, scenario Pi is possible.

(13) Scenario Pi is not possible.

(14) So, causal finitism is true.

page 15

The next section is fairly technical.

The logic behind these is straightforward, so philosophers must find other solutions to, or ways out of, these paradoxes to counter Pruss’ principle. To avoid causal finitism as the best explanation as to why these paradoxes are impossible, one might instead hold to (a) finitism, (b) alternate theories of causation like retrocausality, or (c) the theory that “infinite intensive magnitudes are impossible.” (p. 13)

The first two alternatives are fairly intuitive. First, finitism, which says there are absolutely no infinities of any kind, period, seems unlikely considering how useful infinities are to mathematics and physics; adopting this principal would be detrimental to our current understanding of the universe (ie, how do we talk about black holes if we can’t discuss infinities?). Second, it is unlikely that retrocausality, which have future causes influencing past events, are true. An interesting and cool theory in quantum mechanics to be sure, but I am personally not convinced.

The impossibility of “infinite intensive magnitudes” is a third alternative explanation given by Dr. Michael Huemer in his book “Approaching Infinity” (which I have not read, so take all of this with a grain of salt; I just present Pruss’ perspective as I understand him).

According to Pruss, Huemer differentiates between intensive magnitudes and extensive magnitudes. Extensive magnitudes are derived from the sum of their parts (ie, the mass of an object is the sum of the masses of the object’s components); Huemer says these infinites are possible. For example, if there was an infinite number of stars in the universe, the total mass of the universe would be infinite. Every other kind of magnitude would be intensive, which, Huemer claims, are not possible.

Huemer’s approach attacks the physical limitations of each paradox. For instance, in Thompson’s Lamp scenario, since the switch has to travel a distance between being toggled on and off an infinite number of times, the average speed of the toggling back and forth would end up being ∞ km per second, which makes it impossible. This explains why the Lamp paradox is impossible without needing to appeal to causal finitism.

However, as indicated in Dr. Pruss’ assessment, Huemer’s approach has issues. For instance, are these categories truly meaningful in the real world (ie, do the laws of nature even care about “average velocity” like it does “velocity”)? Also, Huemer relies heavily on the laws of nature of this universe without accounting for possible worlds to postulate a universal philosophical principle. Causal finitism does not have such limitations and has more explanatory power.

If Causal Finitism is Right…

If we accept that causal finitism is an accurate description of the nature of reality, if there truly cannot be an infinite regress of causal events, then we are left with an incredibly robust iteration of the Kalaam Cosmological Argument for the existence of God. The only thing left to do is hash out why and how this God caused our universe… and perhaps what He wants… from us. 🙂

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Check out Dr. Pruss’ blog. I just discovered his work and am excited to dig into it!

http://alexanderpruss.blogspot.com/