Note: this post is out of date. An updated reply to Koons' Grim Reapers argument can be found here.
A brief reply to Koons' Kalam Argument: the Grim Reapers
1. Introduction
The question of whether there are infinitely many past times is central to the Kalam Cosmological Argument.
In a 2011 paper[1], [2], Robert Koons uses a 'Grim Reapers' argument to support the hypothesis that the past is finite, and moreover, the hypothesis that time is not dense, in the sense that a finite temporal interval can only be divided in finitely many subintervals.
In this article, I will argue that Koons' Grim Reapers argument fails to support either hypothesis.
2. Intrinsicality
Koons' gives the following two definitions of 'intrinsic' in two versions of his paper; one is in terms of propositions, the other one in terms of properties.
Koons: [1]
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ﾒ of world Wﾒ whose contents exactly duplicate the contents of R in W also has P throughout Rﾒ.
Koons: [2]
A proposition P is intrinsic to spatiotemporal region R if P consists of a finite conjunction of atomic propositions ascribing simple powers and dispositions to things located entirely within R at times wholly within R.
In any case, applicability of the principles of binary patchwork and infinitary patchwork require that the powers and dispositions of each of the reapers be intrinsic. I will use the first definition above, but the second wouldn't affect the objections I will raise.
3. Binary Patchwork and Infinitary Patchwork
Koons' proposes two principles, binary patchwork and infinitary patchwork.
Koons: [1]
Binary 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.
Koons: [1]
P2. Infinitary Patchwork (PInf). 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 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 Ti exactly resembles the part of W' within region f'(f(Ti)).
4. Grim Reapers
Grim Reaper#n (henceforth, GRn) is an entity that checks whether there is a Fred particle at some designated position at a distance d/2j from a plane P, for some j >n.
If there is one such particle, then GRn does nothing.
If there is no such particle, GRn places one particle at a specific location, at a distance d/2n from plane P.
Also, GRn exists at a region Rn in Wn, and there is a world W and a function from the series {Rn} into the spatiotemporal regions of W such that no two values of f overlap.
Each region Rn has a temporally closed boundary in the direction of the future, and an open one in the direction of the past; also, the regions are temporally adjoining, and there is an infinite series towards the past: for instance, if Rn is extended over an interval (tn+1,tn], Rn+1 is extended in (tn+2,tn+1]. [3]
According to Koons', patching those worlds would result in a contradiction.
However, the following scenario is apparently not contradictory.
Let's say that, at world W', the following obtains:
For every n natural number n, in the interval (tn+1,tn], there is a reaper GRn and a Fred particle Fn at the specified location, at a distance d/2n of plane P.
GRn and Fn do not exist at any other time in W'.
There appears to be no contradiction in the previous scenario, so it seems that patching them is possible after all.
5. Grim Signalers
Koons' considers an objection such as the one raised above, and says that what matters is that a signal of some kind has to persist, and so the Grim Reapers (or Grim Signalers) would have an intrinsic power to send a signal to a following reaper, and to receive a signal from a previous ones.
Also, sending and receiving those signals would be intrinsic powers of the reapers or signalers, with the corresponding intrinsic dispositions.
Moreover, he avoids the objection that powers and dispositions may fail by saying that whether powers and dispositions are used successfully is intrinsic to the situation in which they're used.
However, adjoining non-overlapping intervals aren't good enough. The power to send the signal to the next signaler is akin to the power to make the particle last into the next interval. That seems to defeat the argument, because the intervals would have to overlap.
6. Conclusion
Given the previous considerations, we can tell that the fact that Koons obtains a contradiction from the scenario he constructs is not related to the issue of whether time is dense or whether there are infinitely many past non-overlapping temporal intervals, and generally not related to infinities, since a similar contradiction can be obtained in the binary case.
So, in particular, Koons' 'Grim Reapers' (or Grim Signalers) scenarios fail to establish any conclusions about whether time is dense, or whether there are infinitely many past non-overlapping temporal intervals.
Notes and references
[1] Source: http://www.robkoons.net/media/83c9b25c56d629ffffff810fffffd524.pdf
[2] Source: http://www.lastseminary.com/cosmological-argument/The%20Grim%20Reaper%27s%20Revenge%20-%20A%20New%20Kalam%20Argument.pdf
[3] That the regions have a temporally closed boundary towards the future and an open one towards the past, plus the conditions that the regions are adjoining are mentioned by Koons' in his reply to the 'vanishing particle' objection.
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