The Third Way: A New Interpretation

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