As Alexander Pruss has observed (Pruss 2009), the Grim Reaper paradox suggests not only that no finite time period can be divided into infinitely many sub-periods but also that it is impossible that there should exist infinitely many time periods, all of which are earlier than some event. It seems to provide grounds for thinking that time must be bounded at the beginning: that there must be a first period of time. If not, we could simply construct a new version of the Grim Placer paradox. As in the original version, we postulate the possibility of a Grim Placer, who creates a particle and places it at a designated spot, if and only if no particle is already located at a spot corresponding to any earlier Placer. In this version, Placer 1 is set to act at the first moment of 1 B.C., Placer 2 at the first moment of 2 B.C., and so on ad infinitum. Once again we can generate the contradiction: some particle must be placed within d meters of the plane, but there is no finite distance from the plane such that a particle could have been placed there.
 
Let us try to be more explicit about the premises needed to generate the paradox. First of all, we must assume that a single, isolated Grim Placer scenario is metaphysically possible:
 
P1. Possible Grim Placer (PGR). There are  numbers d and m such that for every positive integer n there is a possible world W and a region R such that R has a finite temporal duration d seconds, there is a Grim Placer wholly contained within R, and throughout R the Grim Placer has the power and disposition to create a “Fred” particle and place it at a designated position m/2^n  meters from the plane P if there is no unique particle located at m/2^i meters from P for some i >n (eliminating all other particles located within m meters of P, if there are more than one), and otherwise to maintain the unique Fred particle that is located at m/2^i meters from P in its initial position.
 
Secondly, we appeal to some version of David Lewis’s Patchwork Principles (Lewis 1983, 76-7). Much, if not most, of our knowledge of possibility is based on patchwork principles, since we have little direct access to alternative possibilities. Instead, we have to rely on our direct knowledge of the actual world, as well as the license to cut-and-paste or recombine various regions of the actual world into a new arrangement.
 
Binary Spatiotemporal Patchwork. If possible world W1 includes spatiotemporal region R1, possible world W2 includes region R2, and possible world W3 includes R3, and R1 and R2 can be mapped onto non-overlapping parts of R3 (R3.1 and R3.2) while preserving all the metrical and topological properties of the three regions, then there is a world W4 and region R4 such that R3 and R4 are isomorphic, the part of W4 within R4.1 exactly duplicates the part of W1 within R1, and the part of W4 within R4.2 exactly duplicates the part of W2 within R2.
 
Following Lewis, I will assume that ‘intrinsicality’ and ‘exact duplication’ are inter-definable:
 
Definition of Intrinsicality: a property P is intrinsic to a thing x within region R in world W if and only if x is P throughout R in W, and every counterpart of x in any region R’ of world W’ whose contents exactly duplicate the contents of R in W also has P throughout R’.
 
Binary Spatiotemporal Patchwork licenses recombining region R1 from world W1 with region R2 from world W2 in any way that respects the metrical and topological properties of the two regions, so long as there is enough “room” in spacetime as a whole to fit the two regions in non-overlapping locations (as witnessed by the two regions R3.1 and R3.2 in world W2). The Binary Patchwork principle can plausibly be generalized to the case of infinite recombinations:
 
P2. Infinite Spatiotemporal Patchwork (PInfSP). If S is a countable series of possible worlds, and T a series of regions within those worlds such that Ti is part of Wi (for each i), and f is a metric and topology structure-preserving function from T into the set of spatiotemporal regions of world W such that no two values of f overlap, then there is a possible world W* and an isomorphism f* from the spatiotemporal regions of W to the spatiotemporal regions of W* such that the part of each world Wi within the region Ri exactly resembles the part of W* within region f*(f(Ri)).
 
In order to apply the Patchwork principles to Benardete's story, we must assume that the relevant powers and dispositions are intrinsic to the things that have them when they have them. Otherwise, we cannot assume that the joint possibility of an infinite number of Grim Placer scenarios follows from the possibility of a single scenario, taken in isolation.
 
Intrinsicality of the Grim Placers’ Powers and Dispositions (PDIn). The powers and dispositions ascribed to each Grim Placer are properties intrinsic to that Placer in its corresponding region and world.
 
Our hypothesis for the reductio will be the possible existence of a world with an entity that has an infinite past:
 
HIP. Hypothesis of the Possibility of an Infinite Past. There exists a possible world W´ and a spatiotemporal region R´ in W´ such that R´ has infinitely many temporally extended parts such that these parts can be put into a sequence (ordered by the natural numbers) in which each successive part in the sequence is within the backward time cone of its predecessor, and each part is large enough to contain a Grim Placer.
 
1. Start with a possible Grim Placer in world W and region R, with finite duration d. (From PGP, the Possibility of Grim Placer)
2. Next, locate a world W' with a region R' containing a non-well-founded infinite series of non-overlapping temporal parts, each of duration d and each in the backward time cone of its predecessor. (Assumption of HPIF, for reductio)
3. Find a single possible world W* with region R* containing a non-well-founded infinite series of non-overlapping temporal parts (R1, R2, etc.), with each Ri containing a counterpart of the Grim Placer. (From 1, 2, and Infinite Spatiotemporal Patchwork)
4. Assume that, in world W*, there is after period R1 no particle located at any distance m/2^n from P, for any n > 0. (Assumption for second reductio)
5. Therefore, there is after period R2 no particle located at any distance m/2^n, for any n > 1. (From 4)
6. Grim Placer #1 in period R1 in world W* placed a Fred particle at distance m/2 from P. (From 5, and the Possibility of Grim Placer)
8. Contradiction (4 and 6). So, after R1 in W*, there is some particle located at some distance m/2^n from P, for some n > 0. 
9. Therefore, no particle is located any distance m/2^j from the plane P, for any j > n. (From 8, the Possibility of the Grim Placer)
10. Therefore, no particle is located any distance m/2^j from the plane P, for any j > n+1. (From 9)
11. Therefore, Grim Reaper n + 1 placed a particle at distance m/2^(n+1) from P. (From 10, and the Possibility of the Grim Placer).
12. Contradiction (9 and 11).
13. So, there is no possible world containing a non-well-founded infinite series of non-overlapping temporal parts, each of duration d0 and each in the backward time cone of its predecessor. (Negation of HPIF)
 
From the conclusion of this argument (step 12), we can deduce premise P3:
 
P3. Every non-eternal thing began to exist at some point in time (since the past of each non-eternal thing is finite in length).
 
If any temporal thing had an infinitely long past, then that past would include an infinite series of non-overlapping periods of length d seconds, all in the past light cone of the current state of the thing in question, in contradiction to step 12. Thus, to reach the conclusion of an eternal first cause, we need only add the assumption of causal finitism. In the next section, I will argue that the Grim Placer paradox can be generalized into an argument for causal finitism.

In Jose Benardete’s paradox, we are to suppose that there is an infinite number of Grim Reaper mechanisms, each of which is engineered to do two things: first, to check whether the victim, Fred, is still alive at the Grim Reaper’s appointed time, and, second, if he is still alive, to kill him instantaneously, and, if he is already dead at the appointed time, to do nothing. The last Grim Reaper (Reaper 1) performs this dual task at exactly one minute after noon. The next-to-last Reaper, Reaper 2, is appointed to perform the task at exactly one-half minute after noon. In general, each Reaper number n is assigned the moment 1/2^(n-1) minutes after noon. There is no first Reaper: for each Reaper n, there are infinitely many Reapers who are assigned moments of time earlier than Reaper n’s appointment.
 
It is certain that Fred does not survive the ordeal. In order to survive the whole ordeal, he must still be alive after one minute after twelve, but, we have stipulated that, if he survives until 12:01 p.m., then Reaper 1 will kill him. We can also prove that Fred will not survive until 12:01, since in order to do so, he must be alive at 30 seconds after 12, in which case Reaper 2 will have killed him.  In the same way, we can prove that Fred cannot survive until 1/2^(n-1) minutes after 12, for every n. Thus, no Grim Reaper can have the opportunity to kill Fred. Thus, it is impossible that Fred survive, and also impossible that any Reaper kill him! However, it seems also to be impossible for Fred to die with certainty and yet to do so without any cause.
 
The original Grim Reaper paradox requires some assumption about causality: that Fred cannot die unless someone or something kills him. I would like to eliminate that dependency. Consider the following variation: the Grim Placer. In place of asking whether a pre-existing victim Fred is dead or alive, we will focus instead on the question of whether or some Grim Placer has issued a death warrant. Let’s say that each Grim Placer #n can issue a death warrant by placing a particular kind of point-sized particle in a designated position, at exactly the distance of d/2^n meters from a plane P. Each Grim Placer #n checks to see if a particle is already at a distance of d/2^i  meters from plane P, for some i > n: that is, he checks to see if any earlier Placer has issued a “warrant”. If a particle has already been placed in one of the designated spots, then the Grim Placer #n does nothing, other than maintaining the status quo.
 
If there is no particle in an appropriate location, then the Grim Placer #n issues his warrant, placing a particle exactly d/2^n meters from P. We can now prove both that at 12:01 that some particle is located within d meters of the plane, and that no particle is located there. Suppose that there is no particle at any location d/2^i meters from plane P, for any i. This is impossible, since if there were no particle d/4 meters from P, then Grim Placer #1 would place a particle in the position d/2 meters from P.
 
Thus, there must at 12:01 pm be some particle in an appropriate position. Suppose that the particle is located at that time in position d/2^n meters from P, for some n. This means that every Grim Placer whose number is greater than n did nothing, contrary to our hypothesis. Thus, this option is also impossible.

​As Alexander Pruss has observed (Pruss 2009), the Grim Reaper paradox suggests not only that no finite time period can be divided into infinitely many sub-periods but also that it is impossible that there should exist infinitely many time periods, all of which are earlier than some event. It seems to provide grounds for thinking that time must be bounded at the beginning: that there must be a first period of time. If not, we could simply construct a new version of the Grim Placer paradox. As in the original version, we postulate the possibility of a Grim Placer, who creates a particle and places it at a designated spot, if and only if no particle is already located at a spot corresponding to any earlier Placer. In this version, Placer 1 is set to act at the first moment of 1 B.C., Placer 2 at the first moment of 2 B.C., and so on ad infinitum. Once again we can generate the contradiction: some particle must be placed within d meters of the plane, but there is no finite distance from the plane such that a particle could have been placed there.

Here again is my Pruss-inspired version of the Kalam argument, relying on causal finitism:

P1. Every event has a finite causal history (no causal loops or infinite regresses).
P2. For everything that begins to exist (at some point in time), the event of its beginning to exist must have a cause.
P3. Every non-eternal thing began to exist at some point in time (since the past of each non-eternal thing is finite in length).
P4. If the yy cause the xx to begin to exist at t, then the yy exist at t or at some time earlier than t or eternally. 
Therefore, every non-eternal thing is ultimately caused to exist by some eternal (godlike) thing.

There is, however, a problem with premise P3. Causal finitism alone does not seem to give us a finite past, not even a finite history for a given thing. 

​Suppose that we have a thing x that exists in time and suppose that causal finitism is true. This means that each event in the history of x must have a finite causal history. Is this enough to entail that x must have begun to exist at some point in the past? Couldn’t the history of x begin with an event or state that is infinitely extended in the past direction? Let’s call such an initial state a simple infinitely long past state or SILPS.
 
We can refute the possibility of a SILPS by posing a dilemma: either time itself has an intrinsic measure (in which sense time can pass in the absence of change) or it does not. If time does not have an intrinsic measure, and the initial state of x is a simple state, without discrete parts, then that state cannot have any temporal duration, much less an infinite duration (since there are, ex hypothesi, no changes concurrent with this state by which time could be extrinsically measured). Hence, we must suppose that time itself has an intrinsic measure.
 
However, this is also inconsistent with a SILPS, since if time has an intrinsic measure, then any extended period of time has discrete proper parts corresponding to the measurable proper parts of that period of time. If an event or state has a duration corresponding to that extended period, then it too must have temporal parts corresponding to the proper parts of the period of time. Thus, the state is not simple or “uneventful” after all. This is a strong argument, although it will not persuade those who think that extended simples (like extended Democritean atoms) are metaphysically possible.
 
Here is a version of the argument without the assumption of a finite past (P3):
 
P1. Every event has a finite causal history (no causal loops or infinite regresses).
P2. For everything that begins to exist, the event of its beginning to exist must have a cause.
P3.1. If something has existed for an infinite period of time, then it must have an infinite causal history (because a simple infinitely long past state is impossible).
Therefore, every non-eternal thing is ultimately caused to exist by some eternal (godlike) thing.
 
Since my argument for P3.1 is less than ironclad, I will argue in future posts both for causal finitism and for the finitude of the past of each temporal thing. This provides support for both arguments: the original argument (which depends on both causal finitism and the finite duration of the past) and the revised argument (which depends on causal finitism and the impossibility of SILPS).

The Kalam argument for God’s existence, which was pioneered by John Philoponus (490-570), developed by Islamic philosophers such as al-Kindi and al-Ghazali, and championed in recent years by William Lane Craig (Craig 1979) and by me (Koons 2014), is an attempt to prove that the universe must have had a cause, a role which God seems best suited to fit. The argument typically takes the following form:
 
1. Whatever begins to exist must have a cause.
2. The universe began to exist, because time itself is bounded in the past.
Therefore, the universe had a cause.
 
The first premise has a great deal of intuitive appeal, and there are severe epistemological costs to countenancing the idea of uncaused origins. For instance, the skeptical scenario popularized by Bertrand Russell—How do we know that the universe didn’t simply appear 5 minutes ago?—would be a live possibility in the absence of an a priori causal principle similar to premise 1. So, let’s focus on premise 2.

The typical Kalam strategy for defending premise 2 is to argue that time past is not eternal, that is, that there is some finite temporal bound to all past events. Now, it is not immediately obvious that a finite bound to the past entails that the “universe” began to exist. First, it is not obvious that there is such a thing as the universe: perhaps the plurality of things that exist at a time t do not compose a single whole at t. We might try to avoid this composition question by modifying premise 2 into 2.1:
 
2.1 There is a time t such that everything existing at t began to exist at t, and nothing existed at any time prior to t.
 
In order to get the desired conclusion, we would also have to modify premise 1 as follows:
 
1.1 If some things xx began to exist at time t, then there must be some thing y or things yy not among the xx such that y (or the yy) caused the xx to begin to exist at t. (I am using double letters as plural variables, following George Boolos’s plural quantification (Boolos 1984). One should read ‘yy’ as ‘the y’s (plural)’.)
 
We will also have to rule out the possibility that the things coming into existence at the first moment of time might have been caused by things existing at later times:
 
3. If the yy cause the xx to exist at t, then the yy exist at t or at some time earlier than t or eternally.

Form 1.1, 2.1, and 3, we can reach the conclusion that something that exists eternally caused the beginning-to-exist of all the things that existed at the first moment of time (if there is such a first moment).
 
There is, however, a further lacuna to fill: from the fact that the past is finite in extent or duration, it does not follow that there is a first moment of time. For example, it could be that no event occurs 14 billion or more years ago, but for every length of time L years less than 14 billion years, there are events that occurred exactly L years ago. That is, there might be a finite bound on the past, with past moments that approach arbitrarily close to that boundary, but no moment that reaches it, i.e., no absolutely first moment. (Think of the set of positive real numbers, which approach arbitrarily close to zero without actually including it.)

Instead of looking for proof of the finitude of the past, we should look instead for support of what Alexander Pruss (2016) has called causal finitism. If we can show that every event has a finite causal history (i.e., no causal loops and no causal infinite regresses), then we can infer that there are uncaused events. If we can further assume that everything that begins to exist at a time must have a cause and that every non-eternal or fully temporal thing must have begun to exist at some time (because the past is finite), then we can conclude that all uncaused things must be eternal in nature (i.e., existing “outside” or “beyond” time itself). At that point, we might be able to show that such an eternal cause of temporal events must be relevantly godlike.
 
Here is a version of this Pruss-inspired argument:
 
P1. Every event has a finite causal history (no causal loops or infinite regresses).
P2. For everything that begins to exist (at some point in time), the event of its beginning to exist must have a cause.
P3. Every non-eternal thing began to exist at some point in time (since the past of each non-eternal thing is finite in length).
P4. If the yy cause the xx to begin to exist at t, then the yy exist at t or at some time earlier than t or eternally. [Premise 3 above]
Therefore, every non-eternal thing is ultimately caused to exist by some eternal (godlike) thing.

This proof assumes (in premise 3) that, for anything that begins to exist, there is a first moment of its existence. That seems pretty reasonable. In addition, one could probably derive this from causal finitism. Suppose, for contradiction, that some x has a finite past but no first moment of existence. Then it seems that there must an infinite regress of periods of x's existence, each caused by its predecessor, in contradiction to the principle of causal finitism.

But suppose one doesn't buy either of these moves. Then there would have to be a single initial period P of x's existence, a period which lacks a first instant. In that case, premises P2 and P4 (suitably modified) would entail that there must be some cause of x's beginning to exist, a cause that is either timeless or active at a time t that is prior to and adjacent to period P. And so the proof will go through.

The proof is pretty simple. Suppose x is some non-eternal thing. By P2 it begins to exist, and by P3 its beginning to exist must have a cause. By P4, this cause must either exist eternally or at the same or earlier time than that of the beginning of x's existence. If the cause is an eternal being, we're done. If the cause is a non-eternal being, then it must have a beginning of its existence. Premise P1 rules out an infinite regress of temporal causes. So, there must be an eternal cause.