By delaying this issue until Question 11 in the First Part of Summa Theologiae (and chapter 42 in the SCG, I), Thomas is indicating that it is easier to prove that there is at least one God than it is to prove that there is no more than one. It is also important to realize that the ‘one’ that we appeal to in proving God’s unity is not the number 1, but the concept of oneness that is convertible with being. God is supremely one precisely because He is supremely a being. We don’t count gods in the same way that we can count apples or doors. As we shall see, the nature of God excludes the very possibility of there being two or more gods. At the same time, since the oneness involved is not that of the numerical one, Thomas is leaving open the door for God’s being, in some sense, a plurality or multitude. This will help in working through both the problem of God’s ideas  and the Trinity.
 
In Summa Theologiae I, Q11, article 1, Thomas explains that the focal meaning of ‘one’ in ‘there is one God’ is that of being undivided. God is supremely indivisible, and this follows from His simplicity. Thomas argues that if oneness were not convertible with being, an infinite regress would result. If oneness is not equivalent to being, then it would have to be something that is added to being. But then we can ask what makes this addition one addition, and an infinite regress follows.
 
But what about multitudes, like crowds of people? If they exist, they must also be one. And, indeed, a crowd is one crowd. But how can something be both one and many? Isn’t that inconsistent? Thomas anticipates here an answer given much later by the German logician Gottlob Frege. A multitude is one in one way, and many in another. It is, for example, one crowd but many people. Thomas elaborates this point in article 2, again distinguishing between one as the principle of number and one as convertible with being. We can ask for the number of a crowd, and this question must appeal to some way of dividing the crowd—into families, or individuals, or human cells.  At the same time, a crowd must be, like anything that exists, in some more basic sense one thing. Some more recent metaphysicians, following the work of American logician George Boolos, have suggested that multitudes can exist without being one thing at all. Very large proper classes, like the class of all sets, for example, seems to be a real multitude that is not in any sense one thing. If this is right, it could create some difficulty for some of Thomas’s proofs (in article 3) for God’s oneness. Or, it might simply point to the fact that the oneness of God is consistent with His comprising a kind of multiplicity, so long as these things are not parts or attributes of God.
 
In objection 4, Thomas addresses the problem of the definition of one as undivided. Being undivided is a negative notion, signifying the absence of division. But being is perfectly positive. So, how can being be convertible with oneness? Thomas replies that division is prior to one only in the order of our understanding. Ontologically, being undivided is prior to being divided. It’s just that we are first aware of composite things before we are aware of their simple parts.
 
In article 3, Thomas offers three arguments for the oneness of God. The first argument appeals to God’s simplicity. God is made to be God by His divine nature, and that divine nature also makes Him exist as a particular being. For there to be two gods, there would have to be two divine natures, each of the same species. But for two natures to exist with the same species, there would have to be something responsible for making each distinct from the other. So, for example, two men can be two by virtue of being combined with two packets of prime matter. Two packets of prime matter have no actual nature of their own, and so they can be fundamentally or primitively distinct. The divine nature is an actual nature (it is maximally actual), and so two divine natures cannot be fundamentally distinct. Since God is identical to His own nature, there cannot be two instances of the divine nature, just as there cannot be two instances of a single angelic species.
 
In the second argument, Thomas appeals to God’s infinity. (This is a new argument, not present in the SCG.) In fact, he appeals to the infinity of God’s perfection, by which he means that nothing can be superior to God in perfection. Suppose that there were two such maximally perfect beings. In this argument, Thomas concedes (for the sake of argument) that there could be two distinct species of god. If there were two such species, something would have to differentiate them. One would have to have something that the other did not have. But this means that one would have to have some form of perfection that was lacking in the other. But God has all perfections. So, in fact, this argument actually appeals to God’s perfection, rather than His infinity.
 
Third, Thomas appeals to the apparent unity of the world. This is one of the relatively few cases in which Thomas appeals to some form of the Fifth Way—pointing to God as the cause of the world’s systematic harmony, the fact that the active and passive powers of the world’s created substances fit together in order to make a stable, scientifically intelligible universe. Thomas gives a more detailed version of this argument in Summa Contra Gentiles 1.42, paragraph 7.
 
In Summa Contra Gentiles 1.42, paragraph 5, Thomas also appeals to some details of Aristotle’s natural philosophy, especially the assumption that the movement of the heavenly spheres are regular and continuous. It’s not clear to me whether any of this can be salvaged, given the falsity of Aristotle’s astronomy.
 
In paragraph 8, Thomas argues that if there were two gods, at least one would have to be composite. But no composite being could be necessary through itself (as established in the Third Way). It’s not obvious here why one of the gods would have to be composite, but the two arguments that Thomas gives in the Summa Theologiae support this premise: either the two gods would belong to the same species, in which case each would have to have a part that individuates it from the other, or they would belong to two species, in which case each would have to add some differentia to their common genus.
 
Paragraphs 9-11 contain another interesting argument that appeals to God’s necessity of being. This argument involves a complex dilemma: if there were two necessary beings (each necessary per se), then either (1) the two differ by something required for the completion of the necessity of being, or (2) not.
 

• Assuming (2). (Paragraph 9) If not, then they are different by something accidental. This accident must have a cause (since all accidental additions must be caused), which is either (a) the essence of the necessary thing, or (b) some external thing. We can rule out (a), since both necessary beings would have the same accident, if it is a proper accident determined by their common essence. If (b), then the existence of two necessary beings depends on the action of this third thing. But then the existence of each thing would also depend on the third thing, and so neither could be necessary in itself.
•  
• Assuming (1). (Paragraph 10) That which differentiates the two necessary beings would be either (a) something included in the common nature of necessary being, or (b) something that differentiates the two natures into two species. Case (a) is impossible because in that case, the factor will be present in both necessary beings. In case (b), the common nature must be made actual by each of the two differentiae, in which case, each necessary being would derive its existence from something else. Hence, the two things cannot be necessary in themselves, nor could each be simple and identical to its own act of existence. This complex argument seems to be mirrored in the first two arguments in the Summa Theologiae. The SCG helps us to see that these are really two parts of a single argument, based on a dilemma (either one species of god or two).

 
Paragraph 12 involves a very simple argument, based on the thesis that God is identical to His nature and to His act of existence. If there were two gods, then each god would have the same divine nature. But this divine nature would then have to be identical to two distinct acts of existence. But nothing can be identical to two distinct things.
 
Paragraphs 13 and 14 contain another complex argument.
 

1. If x is a per se necessary being, then x’s necessary being belongs to x insofar as it is x. That is, the necessity of x is grounded by x’s being x.
2. Suppose there were two per se necessary beings, x and y. Then x’s being x would also consist in x’s not being y, since if x and y are distinct, x’s being x depends on its not being y.
3. Consequently, x’s necessity would belong to it insofar as x is distinct from y.
4. This means that x’s necessity depends on the existence of y.
5. Consequently, x’s necessity would not be per se. Contradiction
6. So, there cannot be two per se necessary beings.

 
Thomas defends premise 1 in paragraph 14. If x’s necessary being depends only on x, then x’s necessary being must belong to x insofar as it is x.
 
In paragraph 18, Thomas appeals to the fact that God is supreme being. Since being is convertible with oneness, God must be supremely one, and so undivided. This applies equally well to the divine nature.
 
Thomas also appeals in paragraph 20 to the superiority of monarchy as the form of government. Since God is the perfect governor of the universe, God must be one.

Anthony Kenny was right (in his The Five Ways) to connect the Fourth Way with the claim (first stated by St. Thomas in the early De Ente et Essentia) that God is identical to His own act of existence. I think that he’s also right in thinking that St. Thomas did not change his theory of esse (act of existence) but did change his understanding of essence or quiddity. In De Ente, Aquinas did not distinguish between understanding the meaning of a word (like ‘phoenix’) and grasping the essence of the kind of thing that the word names. In fact, if there are no phoenixes, it is impossible for us to grasp the essence of a phoenix. This does undercut St. Thomas’s first argument for the real distinction between essence and existence in De Ente, but, in my opinion, this is no great loss. The real case for the distinction lies in the fact that there can be only one thing whose essence is its existence. Hence, the real distinction is easy to establish for everything but God.
 
Kenny approaches the problem as you would expect a mid-20th-century analytic philosopher to do so: from a grammatical-linguistic analysis of the verb ‘to be’ (and ‘est’ in Latin). Not surprisingly, he concludes that St. Thomas’s account of God is “unintelligible”. Aquinas’s strategy is to argue that we are forced to the “unintelligible” conclusion by the facts of causation and the natural world. The exceptional nature of God is a feature, not a bug. It is not surprising if the grammar of ordinary language finds it difficult to accommodate the conclusion.
 
I thought it was surprising that Kenny doesn’t mention Exodus 3 in this context, in which God tells Moses that His name is “I am that I am,” which (as St. Thomas notes) is a striking anticipation of Thomas’s theory. The phrase “that I am” clearly refers to God’s act of existence, and the phrase “I am…” in this context clearly asks for a phrase delineating God’s nature or essence.
 
The second thing that Kenny fails to take into account is that Aquinas’s theory of ‘esse’ and ‘actus essendi’ (acts of existence) is a substantive metaphysical proposal, not merely an analysis of ordinary language and thought. Aquinas is offering an interesting and attractive theory about actuality, something that philosophers have wrestled with from antiquity to the present time. How are my actual daughters different from all of the possible but not actual daughters that I could have had? There have been relatively few accounts of this fact in the history of philosophy:
 

• Actualism. There is no such thing as a non-actual daughter. Talk about such things must be paraphrased as talk about our ideas or suppositions.
• Mysterious quality. There is a special, ineffable quality that all and only actual things have.
• Kantianism. To be actual is to be connected in appropriate ways to sensations (as opposed to acts of imagination or supposition).
• Leibnizian optimism. To be actual is to be a denizen of the best possible world.
• Thomism. To be actual is to contain an act of existence (actus essendi). These acts of existence are all exactly the same, with one exception—all but God are cases of participated existence, while God is the unique case of unparticipated existence.
• Centrality. To be actual is to belong to the central portion of reality, upon which all merely potential realities depend (by being related to the powers or dispositions of actual things).

 
It’s easy to dispose of options 2, 3, and 4. We have no acquaintance with any actual-ish quality, and even if we did, it would be easy to conceive of non-actual things with that quality (disposing of 2). Kant’s attempted definition ignores the fact that it is only actual sensations that are relevant to the actuality of a physical thing, rendering his definition of ‘actual’ viciously circular (disposing of 3). Being part of the best possible world intuitively has nothing to do with being actual. If God chooses to make the best possible world actual, He must do something. Being best isn’t sufficient to make it actual on its own (disposing of 4).
 
I don’t think 6 is really a competitor with 5. Necessary beings (like God) are essentially “central” in this way, so 6 would provide some basis for identifying God as actual. However, many actual things are contingent. This means that although they are in fact metaphysically central, they could have been peripheral. We still need an explanation of what makes one contingent thing metaphysically central and another peripheral.
 
So, that leaves only 1 and 5. Aristotelians will reject 1 (actualism) on the ground that it denies the metaphysical significance of the actuality/potentiality distinction. If everything non-actual is completely unreal, then we face the Parmenidean problem of explaining how substantial change (generation and corruption of substances in nature) is possible. Even more importantly, we cannot treat active powers or passive potencies to change as aspects of reality. If something has the potential to become hot, for example, this consists in the thing’s having a real relation to a merely potential accident of heat. If there are no merely potential entities, then we would have to embrace some form of Platonic realism, understanding the potentiality for heat as a relation to the universal idea of Heat Itself.
 
Kenny puts a great deal of emphasis on Aquinas’s principle that everything receives its existence through its form. Thus, for Socrates to exist, existence must come to Socrates through his form of humanity. Consequently, Socrates cannot exist without being a living human being. Thus, for Socrates, to be is to be human, a living human being. However, Aquinas’s theory is that this is true only of creatures. God does not receive His existence from anything. Consequently, it does not have to come to Him through any limiting form. His existence is simple, unqualified, and unlimited. For God to be is simply for God to be, full stop.
 
As Kenny notes, Aquinas insists that God’s existence is not the greatest-common-factor kind of existence that is common to every actual thing (SCG I.26). That kind of generic existence is shared by both God and creatures—in God it is unlimited, in creatures it is limited by essence. God’s existence is the sort that is incompatible with any kind of limitation or restriction.

​As in Maimonides’ case, Aquinas adds a second stage to the argument. He wants to establish not only the existence of a necessary being, but of a necessary being that has existence “in and of itself”, that doesn’t derive its necessary existence from something else.  Here again Aquinas has recourse to his no-infinite-regress assumption: the chain of causation explaining why derivatively necessary beings are necessary must terminate in a thing that is non-derivatively necessary, and this being will be God (a being whose essence is its existence).
 
Here’s a way of thinking about this second stage. Let’s suppose for contradiction that there is an infinite regress of necessary beings, each of which derives its necessity from its predecessor. So, N1 is caused to be necessary by N2, N2 is caused to be necessary by N3, and so on. And let’s assume that all necessary beings belong to such a regress: nothing is necessary in and of itself (unconditionally). Now, a world in which none of N1, N2, N3, etc. exist is an impossible world, since each of these beings exists necessarily and so exists in every possible world. So, the scenario in which none of the N’s exist is an “impossible world”, if you’ll allow me to talk of it that way. Let’s call this impossible world w!.
 
Let’s assume that if a scenario S is impossible, and this scenario S can be derived from some possible world w simply by deleting entities that exist in w, then there must be some ground or explanation of S's impossibility. Let's stipulate that the impossible world w! comes from the actual world (which is possible) by deleting all of the conditionally necessary beings in the actual world. Then the impossibility of w! must be explained in one of two ways: it fails to include something that is unconditionally necessary, or it violates some constraint of conditional necessity, i.e., it contains A but not B, even though A would (if it existed) necessitate B’s existence (which it could do by necessitating B’s necessary existence). But w! is not impossible in either of these ways. There is (by hypothesis) no unconditionally necessary being, so it isn’t impossible for that reason. And it satisfies all of the conditional constraints by never including any of the N’s. Its non-inclusion of Ni is permissible, because it also fails to include N(i+1), and Ni is necessary only conditional on N(i+1)’s existence. So, w! is possible, after all, which means that none of the N’s is necessary.
 
Therefore, it is impossible for anything to be necessary unless something is necessary unconditionally. And to be necessary unconditionally is to be necessary in and of oneself.

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

The Third Way is clearly building on the Second Way. It adds an important component to the conclusion: we reach the conclusion that the uncaused cause of things must be a necessary being, in contrast to the contingent beings that are familiar to us. This immediately raises the question of what Aquinas means by 'necessary'. There are a range of possible interpretations:
 

1. Logically necessary. Its non-existence “entails a contradiction.” Or, perhaps, its non-existence is not ideally conceivable. Or, perhaps, its existence is an “analytic” truth (true by virtue of the meanings of the terms involved).
2. Metaphysically necessary. The narrowest form of real necessity. Not relative to a system of logic, and not epistemically or semantically defined.
3. Naturally necessary. Necessary given the metaphysically contingent order of nature within creation. Its existence is necessitated by the existence of anything or any kind of thing found in the actual creation.
4. Actually existing at every time.
5. Naturally incorruptible.
   

         a. Actually exists at some time and cannot naturally be corrupted.
         b. Actually exists at all times and cannot naturally be corrupted.
 
We can set aside meaning 1, since the argument has nothing to do with logical deducibility, analyticity, or conceivability.
 
Anthony Kenny argues (in his book, The Five Ways) that the Third Way must intend ‘necessary being’ in sense 5a or 5b. This is based on Aquinas’s assertion in the Summa Contra Gentiles (Book II, chapter 30) that the heavenly spheres are “necessary beings,” despite being created by God. However, this evidence does not rule out 3 or 4 either, although it does rule out 2.
 
​The next point of complexity concerns the distinction between things that are necessary per se (in and through themselves) and those whose necessity is caused by another. Are each of the four relevant categories of necessity (2, 3, 5a, and 5b) divided into two sub-categories by this distinction, or can some forms of necessity exist only per se or only when caused by another?
 
Now, suppose that the distinction cuts across all four meanings. Suppose, further, that Aquinas intended ‘necessary’ to correspond to meaning 3, 5a, or 5b. If so, the only result that Aquinas could reach would be that God is an incorruptible being that exists at all times or with merely natural (created) necessity. God is certainly incorruptible, but it wouldn’t make sense to attribute to Him existence in and through time, or merely natural necessity.
 
So, we must assume either that Aquinas intended meaning 2 throughout, or that he intended ‘necessary’ to include things that are necessary in senses 2, 3, 5a, or 5b, but he believed that only things with necessity in sense 2 (metaphysical necessity) could be necessary per se. Given his affirmation of created, metaphysically contingent beings (like the heavenly spheres) as ‘necessary’ in SCG II.30, the second assumption seems most likely. So, we can assume that necessary beings include all beings that are either metaphysically or naturally necessary, and that only metaphysically necessary beings are necessary per se. This leaves open the question of whether there could be metaphysically necessary beings (meaning 2) whose necessity is caused by something else. The most plausible candidates would be the divine ideas, although Aquinas never says that the necessary being of the divine ideas is caused (Summa Th. I q15, a1 and a2). He suggests instead that each divine idea is in some sense “identical” to the divine essence, despite the existence of many ideas.
 
The safest interpretation might be that only God is metaphysically necessary and only God is necessary per se (so those two categories coincide). Consequently, all of the merely naturally necessary beings have their necessity caused by God.
 
We still have to consider meanings 5a and 5b, which were favored by Kenny. I would rule out 5a immediately, since Aquinas never, as far as I know, labels the human soul as ‘necessary’, even though it is certainly incorruptible. It is true that Aquinas states (in Summa Theologica I q75 a6 ad 2) that the human soul has no ‘potentiality for non-existence’ but it isn’t clear that that is the same as necessary being.
 
The heavenly spheres are necessary beings in both sense 3 and 5b. So, the acid test will be to see which meaning makes most sense of the premises of the first part of Thomas’s argument. I will turn to this problem next week.