From local to nonlocal CPs

A local CP is a principle that every localized contingent item of a certain sort has a cause. Thus, a local CP about contingent substances holds that every substance has a cause. A cosmological argument making use of a local CP needs to rule out infinite regresses. On the other hand, a nonlocal CP lacks the restriction that the items be localized in the way substances and events are, and this allows one to get out of infinite regresses. Using a CP instead of the PSR has the advantage that avoiding the van Inwagen problem and its relatives is easier.

I shall argue that the intuitions that typically make local CPs plausible apply just as well to nonlocal CPs. The locality restrictions are objectionably ad hoc, and if we should accept a local CP, we should accept a nonlocal CP. I will then give an argument for a powerful CP.

Consider first a restriction of CPs to (localized) substances as opposed to substance-like aggregates such as heaps of sand or the mereological sum of all the contingent substances now existing. The basic intuition behind such CPs is that bricks and other objects cannot come into existence without cause. But suppose that we learned from the correct metaphysics that bricks are not actually substances but are heaplike (as is indeed what Aristotle's metaphysics says about bricks). That would not affect our commitment to the impossibility of bricks coming into existence ex nihilo.

Now, maybe, we could argue that a CP restricted to substances would suffice to show that nonsubstantial items such as bricks that are made up of a finite number of substances (maybe elementary particles are substances even if bricks are not) have causes. For we could just apply the CP separately to each of the component substances, and while some of them could be causes of others, it could not be true of all the component substances that they are caused by other component substances, since that would require either a causal loop or an infinity of component substances. Thus, the restricted CP is sufficient to do justice to our intuition that bricks do not causelessly pop into existence even if bricks are heaps.

But this argument only works if bricks are made up of a finite number of substances. However, suppose we found out that a brick was, in fact, made up of an infinite number of particles. It does not look right now as if physics is heading in the direction of positing an infinite number of particles in ordinary material objects, but unless there is some logical problem with actual infinities - which problem would then be grist for the kalam arguer's mill, so an atheist will probably not want to embrace that option - the possibility is not absurd. Finding this out would not, I think, shake our conviction that a brick cannot pop into existence. Should it not be, if anything, harder for more particles to pop into existence? Nor would we be impressed by being told that the brick made of infinitely many particles popped into existence by the following pattern. At time t0 + 1 second, particle number 1 was caused to exist by particle number 2; at time t0 + 1/2 second, particle number 2 was caused to exist by particle number 3; at time t0 + 1/3 second, particle number 3 was caused to exist by particle number 4, and so on, with none of the particles existing at time t0. That would still count as an objectionable causeless popping into existence of the brick. But if bricks are not substances, this possibility cannot be ruled out by a CP restricted to substances. However, our intuitions call for this to be ruled out.

Perhaps we can restrict the CP to entities that consist of less than the sum total of all contingent beings. But that will gain the opponent of global CPs nothing. For instance, let

50 be any simple particle that has a contingent cause (there are in fact many such) and let

51 be the aggregate of all other contingent beings now in existence. Let C1 be a cause of S1, by the restricted CP. As the contingent cause of S0 is outside of S0 (since a particle cannot be caused by itself), this cause must be a part of S1, and hence is caused by C1. By transitivity, C1 will also be the cause of S0. If there are no simple particles, the argument is slightly more elaborate and is left as an exercise to the reader (hint: just let S0 be a cat, and note that the cat surely has a cause that is outside of it).

And, certainly, it will not do to restrict the CP based on size, absurdly as if objects that are less than 10 m in diameter needed causes, but larger objects like the universe did not. Here, it is worth recalling Taylor's example of the universe being like a walnut (Taylor 1974, chap. 10). If we accept that then it should have a cause, we should also accept it when it is much larger.

A different kind of restriction is diachronic in nature. Perhaps, the CP can only be applied to entities that exist all at one time and cannot be applied to causal chains of entities, or at least to infinite such chains. However, once again, such a CP will fail to rule out a brick's doing what intuitively should count as popping into existence causelessly. Suppose we saw a brick pop into existence in midair. We would be deeply puzzled. To allay our puzzlement, a scientist tells us that study of the phenomenon reveals that what actually happened was this. The brick popped into existence at t0 + 1 millisecond. There was no cause at t0. However, at t0 + 1/2 milliseconds, the particles of the brick were caused by a set of earlier particles making up a brick, which then immediately annihilated themselves (or perhaps underwent substantial change into the new ones). At t0 + 1/3 milliseconds, the earlier particles were caused by a yet earlier set. And so on, ad infinitum. The whole infinite sequence took 1 millisecond to complete, but nonetheless each synchronic collection of particles had an earlier cause.13 Surely, this would still be as objectionable as a brick popping into existence causelessly. That there was an infinite sequence of bricks, or of sets of particles, in that millisecond does not seem to affect the idea that this cannot happen.

Thus, to rule out the popping of bricks into existence ex nihilo, we need a CP not restricted in a way that rules out infinite chains. Similar considerations rule out CPs concerning events that do not generalize to infinite chains of events. A fire could start for no cause via an infinite chain of events, each temporal part of the fire being caused by an earlier temporal part of the fire, and so on, with the whole infinite chain only taking a second, and the temporally extended conflagration having no cause. That is absurd, and CPs for events should rule it out.

But perhaps there is a difference between infinite chains that take an infinite amount of time and infinite chains, like the one in the previous examples, that take a finite amount of time. Maybe we can restrict CPs to chains of causes that take a finite amount of time?

I do not think this is plausible because an interval from minus infinity to a finite number is order-isomorphic to a finite interval. For instance, the function f(t) = —1/(t - 1) maps the half-infinite interval14 (—«>, 0] onto the half-open finite interval (0, 1], while preserving order relations so that t1 < t2 if and only if f(t1) < f(t2) for t1 and t2 in 0].

But perhaps there is something relevantly metaphysically disanalogous about infinite amounts of time as opposed to finite ones. One difference would be if infinite amounts of time were impossible. But if so, then the kalam argument again shows up, and in any case, if an infinite amount of time is impossible, then a restriction of the CP to chains that take a finite amount of time is no restriction at all.

Another potential difference is that one might argue that a finite temporal interval either is preceded by a time or at least could be preceded by a time (if time has a beginning at the start of the interval), but a temporal interval infinite at its lower end could not be preceded by a time. This would be in support of a restriction of the CP to causal chains that are not temporally infinite in the direction of the past.

Consider first the following version of this disanalogy: the interval 0] is not preceded by an earlier time, while a finite interval (0, 1] is. But suppose that we are dealing with a causal chain spread over a half-open finite interval that is not preceded by an earlier time, since time starts with this half-open finite interval. In that case, we need a cause for the chain as a whole just as much as in the case where the half-open finite interval is

13. Examples like this go back at least to Lukasiewicz (1961), who tried to use them to show how free will could be reconciled with determinism (see also Shapiro 2001).

14. Here I use the standard notation where (a, b] = {x : a < x < b}.

preceded by an earlier time. Suppose that we have a chain of causes spread over (0, 1], tending to being temporally positioned at 0 in the backwards limit, and suppose that there is no "time 0." Surely, the nonexistence of a time prior to the interval makes it, if anything, "harder" for a chain of causes to arise without an external cause. After all, if even time does not exist prior to the chain of causes, then the chain is even more a case of coming into existence ex nihilo, since there is even less there. The absence of an earlier time does nothing to make it easier for things to arise causelessly.

Maybe, though, the idea is that we should require causes where causes can be reasonably demanded. But, the argument continues, a cause can only be reasonably demanded if there is a prior time, since causes must be temporally prior to their effects. However, the latter thesis is dubious. Kant's example of a metal ball continually causing a depression in a soft material shows that simultaneous causation is conceivable. And apart from full or partial reductions of the notion of causation to something like Humean regularity and temporal precedence, I do not think there is much reason to suppose that the cause of a temporal effect must even be in time.

Now consider the second version: even if there is no "time 0," there could be one. The finite interval (0, 1] could be preceded by a time, while the interval (-«>, 0] could not. But it is quite unclear why this alleged modal difference is at all relevant to the existence of a cause for the chain. The absence of the possibility of an earlier time does not seem relevant, unless perhaps one thinks that causation requires temporal priority, a thesis one should, I think, reject.

There is also another, more controversial, objection to such a restricted CP. An infinite interval 0] can be embedded in a larger temporal sequence [-«>, 0] obtained by appending a point, which we may call and which stands in the relation < x to every point x of (-«>, 0]. It may well be that such a sequence could be the temporal sequence of some possible world. And, if so, then the interval 0] could be preceded by an earlier time, and the disanalogy disappears.

It appears, thus, that local CPs are restricted in an ad hoc way. If we have strong intuitions in favor of local CPs, then we likewise should accept unrestricted CPs that can be applied to infinite chains of global aggregates of entities.

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