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.

My colleague and friend Dan Bonevac has discovered a new interpretation the Third Way that resolves the problems that have puzzled readers from medieval times. The argument seems to involve two highly problematic claims:
 

1. If it is possible for a thing not to exist, then there must be some point of time at which the thing does not exist in reality.
2. If for each thing in some class C there is some time at which it does not exist, then there is a single time at which nothing in C exists.

 
Dan proposes that we interpret the temporal adverbs in the argument (quandoque, aliquando, modo) as modal rather than temporal modifiers. Such an interpretation is quite natural in many (if not all)  languages, including Latin and English. Dan notes that St. Thomas never uses the word 'time' ('tempus') or any other explicitly temporal term. In fact, if we look at the parallel argument in the Summa Contra Gentiles (I.13, paragraph 33), we see a complete absence there of temporality. Under this interpretation, the two problematic claims become:
 

1. If it is possible for a thing not to exist, then there is some possible situation in which it does not exist.
2. If for each thing in class C, there is some possible situation in which it does not exist, then there is a single possible situation in which nothing in C exists.

 
Now principle 1 is simply a tautology of modal logic. Principle 2 is still a substantive principle, but it is a quite plausible one, as we shall see.
 
Here is the Third Way under this modal interpretation:
 
Here’s the Third Way under this interpretation:
 

1. There are contingent (non-necessary) beings (because we see things coming into and going out of actual existence).
2. That which is contingent is non-existent in some possible world. (Modal logic)
3. Suppose for contradiction that all actual beings are contingent (non-necessary).
4. Given 2 and 3, there is a possible world in which nothing exists.
5. If it were possible for nothing to exist, then nothing would exist in the actual world.
6. Since it is impossible for something that doesn’t exist to exist as a possibility except by the causal agency of something actually existing. (Nihil ex nihilo)
7. But, obviously, something exists in the actual world (namely, the contingent things mentioned in premise 1).
8. Therefore, by contradiction (of the hypothesis in line 3), some actual being is a necessary being. (From 3-7)

Now everything is crystal clear, except for premises 4 and 5. Dan shows how we can construct, using resources available to Thomas Aquinas, plausible arguments for each. He calls premise 4 the “Annihilation Lemma” and 5 “the Dead End Lemma”.
 
Proof of premise 4: the Annihilation Lemma.
 

1. Suppose that everything is contingent in the actual world, w0.
2. Necessarily, all causal chains are finite (no regresses or cycles).
3. So, there are uncaused contingent things in w0. Let the xx’s be the plurality of those things.
4. Subtraction rinciple: if a world x contains one or more uncaused contingent things, then there is a world y in which those things do not exist, and the set of uncaused things in y is a subset of the set of uncaused things in x (i.e., no additional things exist uncaused in y).
5. So, there is a world w1 in which none of the xx’s exist, and such that the set of uncaused things in w1 is a subset of those in w0.
6. Since the xx’s are all the uncaused things in w0, this means that the set of uncaused things in w1 is empty.
7. However, if all causal chains are necessarily finite, then a world with no uncaused things must be an empty world, a world in which nothing exists.
8. So, w1 is an empty world.
9. So, it is possible for nothing to exist.

 
Proof of Premise 5: The Dead End Lemma
 

1. Suppose that w1 is an empty world.
2. Something can possibly exist relative to world w only if there is something in w that can cause it to exist. (A causal principle: Nihil ex Nihilo)
3. So, nothing can possibly exist relative to world w1.
4. The actual world is necessarily possible. (Axiom B of standard modal logic)
5. So, the actual world is possible relative to world w1. (4, definition of necessity)
6. So, nothing exists in the actual world. (From 3, 5)

 
Why think the Subtraction Principle is true? Suppose that there is an uncaused thing x which, if deleted from the world, necessitated the introduction of a new uncaused thing y in its place. In that case, it seems that the existence of y in the new world would be caused by the absence of x (together with the other conditions that, jointly with the non-existence of x, necessitated the existence of y). This is doubly problematic. First, and most importantly, because we seem to have a contradiction: the existence of y would be both caused and uncaused. And, second, because it doesn’t seem that the existence of anything could be wholly caused (or explained) by the non-existence of something else.
 
This version of the argument requires two causal principles: (i) necessarily, every causal chain is finite, and (ii) necessarily, it is impossible for something to exist unless (a) it actually exists, or (b) it could be caused to exist by something that actually exists. The second principle (Nihil ex Nihilo) is pretty strong. It would imply (given S5 modal logic) that every contingent thing in the actual world has a cause in the actual world. Here’s the proof. Suppose for contradiction that x is contingent and uncaused in w0 (the actual world). Consider any possible world w1 in which x does not exist. The existence of x is possible but not actual in w1 (by axiom B), so by Nihil ex Nihilo there must be some y that exists in w1 and is capable of causing x to exist. This plausibly entails that, in any world w in which x does exist, x is caused to exist by some y that also exists in w. Hence, since x exists in the actual world w0, x must be caused to exist in this world, contrary to our original assumption.
 
An interesting question: could we do without the first causal principle (namely, no infinite regresses or cycles)? Here’s a possible way of doing so. Suppose that there are infinite series or cycles of contingent things. We could plausibly strengthen our subtraction lemma, so that it allows for the simultaneous subtraction of all uncaused contingent things and all infinite contingent series and cycles, without requiring the addition of any new uncaused things or any new infinite series. If so, we could run the original argument without the first principle

 All of Aquinas's Five Ways depend, in one way or another, on ruling out the possibility of infinite causal regresses. In the version of the First Way (the argument from motion) in the Summa Contra Gentiles (I.13), Aquinas follows Aristotle in offering two separate arguments against the causal regress. In the first three ways in the Summa Theologiae, he offers just one of the two Aristotelian arguments: an argument that depends on what I call the No-Intermediate-Real-Cause thesis. This is a thesis that states that if x is a cause of y, and y is a cause of z, then y is not really a cause in the strict sense but only secundum quid (only in a manner of speaking). An intermediate link in a causal chain is not in any sense the source of the reality of the ultimate effect--it is merely a conduit through which the first cause acts. Therefore, an infinite regress is impossible, because (as Aristotle and Aquinas note) every link in the regress would be only an intermediate cause. Hence, such a regress cannot contain any real causation.
 
This is a plausible argument, despite the fact that it is often dismissed as based on a fallacy of equivocation. The standard objection (going back at least to Cajetan, I believe) is that Aquinas equivocates on the phrase "removing the first cause." If we have a finite chain and we hypothetically remove the first cause from the series, it is obvious that none of the intermediate causes can act. Aquinas asserts that if we consider any infinite regress, we have a situation from which we have (in a sense) "removed the first cause". But, as critics point out, in this case there never was a first cause to be "removed", and so the cases are incomparable. However, Aquinas real point is simply to claim that intermediate causes are never causes in their own right but are always wholly parasitic on the first cause. Given that assumption, infinite causal regresses are indeed impossible.
 
The other strategy for dealing with infinite regresses was invented by Avicenna, followed by Scotus, Leibniz, and many subsequent thinkers (including me in 1997). This is the Aggregation Strategy. The Aggregation Strategy concedes, for the sake of argument, that infinite regresses are possible. However, the Strategy insists, if a regress consists entirely of contingent (or finite) things, then we can aggregate the whole series into a single entity and insist on a cause for it. Start now with a single finite thing, and consider all of the finite causes of that thing (whether immediate or remote). Either this series terminates in an uncaused thing, or else it constitutes an infinite regress. In the latter case, we can demand a cause for the whole series. This cause must be infinite, since any finite cause of the series would be a cause of the original entity and so would already be included in the series itself. A member of the series cannot cause the whole series. An infinite thing cannot be caused. And so we reach an uncaused first cause.
 
Aquinas was aware of this strategy, through his close reading of Avicenna. Why didn't he adopt it? I think he was worried that not all infinite series can be aggregated into a single entity. If an infinite series consists of entities of the same species (or a finite number of species), then it has the characteristic that Aquinas labels being accidentally infinite. An accidentally infinite series can be aggregated and must be caused as a whole. This is why Aquinas can concede that accidentally infinite series might exist without losing the force of his first cause argument. However, if an infinite series consists of entities of an infinite number of species, with the species climbing progressively higher and higher in the Great Chain of Being, the Aggregation Strategy could not be convincingly applied. Aquinas would call such a series essentially infinite, and he must (if the Second or Third Way is to work) deny the metaphysical possibility of such a series. This is why he appeals to Aristotle's argument and what I call the No-Intermediate-Real-Cause thesis, which should now be applied only to series that rise "vertically" through the ontological order of species.