cause this being to come into being out of nothing. But why couldn't this cause be itself a
contingent being and it, in turn, be caused to begin to exist by an even earlier contingent
being, and so on ad infinitum? Thomas's answer as to why this regress of contingent
beings is impossible seems to commit an egregious quantificational blunder. For he says
that if there were to exist only contingent beings, then, since for each of them there is a
past time at which it doesn't exist, there is a past time at which each one of them does not
exist. And, if there ever were nothing, then, given the PSR, nothing would subsequently
exist, which contradicts the patent existential fact that there now exists at least one
contingent being. This argument seems to commit the same howler as is committed by
inferring from the fact that for every woman there is a man that there is a man who is for
every woman (talk about polygamy!). In logical terms, that fallacy is
(x)(y)xRy(y)(x)xRy. But it is hard to believe that a great philosopher committed so
obvious a blunder. With a little charity and imagination something interesting can be
made out of the Third Way, but we shall not attempt to do so here.^1
The Kalam cosmological argument of the medieval Islamic philosophers, which has been
defended in recent times by William Lane Craig (1979), also invokes the impossibility of
infinite regress but in a different way than Thomas did in his first two Ways. It selects as
its contingent existential fact that there now exists a universe—an aggregate comprised of
all contingent beings. It then argues that the universe must have begun to exist, for
otherwise there would be an actual infinite series of past events or time, which is
conceptually absurd. Because something cannot come out of nothing, there had to be a
cause for the universe coming into being at some time a finite number of years ago. And
this cause is identified with God, which again occasions the gap problem. Notice that the
version of the PSR that is appealed to is a restricted and thus less vulnerable version of
the PSR; for whereas the unrestricted version requires explanation for every thing that
exists or fails to exist, the restricted version requires an explanation only for a being's
coming into existence.
Just why is it impossible for there to be an actual infinity of past events or times? The
answer is not obvious. Thomas, for one, did not think it to be impossible. Two kinds of
arguments have been given. First, there are descendants of Zeno's arguments. It is not
possible actually to go through an infinite series of events, for before going through the
last event of the series, one would already have to have gone through an infinite series,
and before the second last event, one would already have to have traversed an infinite
series, and so on: the task could never have got started. But if there was an actual infinity
of past events, then our world has traversed an infinite set of events, which is impossible.
This argument depends on an anthropocentric notion of “going through” a set. The
universe does not go through a set of events in the sense of planning which to go through
first in order to get through the second, and so on.
The other kind of argument given by Kalam arguers is that the very concept of infinity is
incoherent. Imagine Hilbert's hotel, where there are infinitely many rooms, numbered 1,
2, 3, and so on, and where even if all rooms are occupied, space can always be found for
a new visitor by shifting the occupant of room 1 to room 2, moving room 2's occupant to
room 3, and so on. The slogan outside the hotel would say: “Always full, always room
for more,” and the Kalam arguer takes this to be incoherent. Or consider an infinite series
of events, again numbered 1, 2, 3, and so on. Then, the subseries consisting of the even-
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