"Whatever is Moved is Moved by Another"

In this post I am going to try to defend Aquinas's First Way, specifically against the attacks brought against it by my friend Alex. Alex has written a fine explanation and critique of Aquinas's first and most famous Way, the argument from motion. The paper can be found here. Unlike many attacks on Aquinas's argument, Alex's reading of Aquinas is sympathetic and charitable, and thus at the same time his criticisms are incisive and well-taken. Anyone who wants to fully understand my post should read Alex's paper first; nevertheless, I will summarize some of his most important results.

To be specific, I'm going to defend Aquinas's premise that whatever is moved is moved by another, which we shall call (MOV). I do not claim that Aquinas ever made the defense I am making. In fact, I think the argument I give is in some respects new. But when all is said and done what I am concerned with is whether Aquinas's premise is defensible and true.

First of all, Alex points out that the term 'motion' in scholastic philosophy really means change. And to say that an object is changing with respect to some feature P is to say that it is going from being potentially P to actually P (more on this terminology here). I will take this for granted in everything I say about change. Now, in summary, Alex's main objection to Aquinas's defense of (MOV) is that either it is (a) valid but palpably unsound or (b) all its premises are true yet it is invalid, i.e. does not prove the premise (cf. his paper for details). However, Alex does think that Aquinas can defend the following more modest premise, which David Oderberg attributes to Aquinas:

(ACT) If something changes from being potentially F to being actually F then there must be some actual being that initiates this change.

The problem is that the more modest and highly defensible premise (ACT) is not equivalent to (MOV), leaving (MOV) undefended and the First Way ultimately uncompelling.

Before I present my argument in favor of Aquinas's (MOV), we need some definitions. First, the definition of an external object, (EXT):

(EXT) x is an object external to y just in case x is not y and x is not a part of y. [def.]

Let's also define what I will call the notion of change per se. (This is my own terminology.) Intuitively, something changes something else per se if it is the most immediate and fundamental efficacious cause of the change, [or the only sufficient cause such that you can't get any 'closer' to the change]. For instance, my hand pushes a stick which pushes a rock; the idea is that the stick, specifically its tip, is what changes the location of the ball per se. Here is a somewhat more formal definition of changing per se, which we will call (CPS):

(CPS) x is a cause per se of a change in something y with respect to feature P by action E just in case (i) x changes y with respect to feature P by action E, (ii) if at the same time as action E there is an action F of some parts of x, and these parts also change y with respect to feature P by action F, then the action F taken alone is not sufficient for changing y with respect to P, and (iii) x's action E taken alone is causally sufficient for changing y with respect to P [def.]

This definition can be made more precise, but the concept should be somewhat clear. The idea behind what I've called change per se is that whatever changes something else per se is the thing that changes y in the most immediate sense and a sense more proper than other things. So, for instance, take the following objects: Me, my arm, my arm's atoms, and a stick. When my arm changes the location of the stick, I can be said to change the location of the stick; however, I cannot be said to change its location per se, since, arguably, if somehow my arm persisted in its motion without the rest of my body (maybe by a miracle it was detached and could float, pushing things around), it would still be sufficient for the stick's changing with respect to its location (contra (ii)). On the other hand, arguably, my arm, or at least some part of it, changes the stick's location per se by its motion, since clearly it can be said to be changing the stick's location, thus satisfying (i). It arguably satisfies (iii) for the reasons stated, and it arguably satisfies (ii) since intuitively if you removed most of the arm but left a chunk of it or a few of its atoms, and they did the same thing as they did when my whole arm's motion occurred, then they would not be able to bring about the stick's change of location.

Now maybe you will disagree with my example and say that given my definition the arm does not change the stick's location per se. But the example is simply to illustrate what I'm trying to get at. If you deny the example is an example of change per se then you should understand what I mean. Also, I would not be surprised if my definition requires chisholming; nevertheless, I think it is on the right track, and helps get my point across. What is most important is just that we have some intuitive understanding of what I mean by something's changing something else per se.

Now we need the following premises. I will translate them into predicate logic, and from my translations it should be clear which formulas correspond to which English phrases.

(1) For all x, if x is changed with respect to P by something y then there is some actual thing z which changes x with respect to P.

Translation 1: ∀x[∃yCxy→∃z(Az∧Cxz)]

(2) For all x and y, if y is actual and x is changed per se with respect to P by y, then y is either an object external to x or y is a proper part of x.

Translation 2: ∀x∀y[(Ay∧Dxy)→(Eyx∨Pyx)]

(3) For all x, if x is changed with respect to P by something actual y, then there is a z which is actual and changing x per se with respect to P.

Translation 3: ∀x[∃y(Ay∧Cxy)→∃z(Az∧Dxz)]

(4) For all x and y, if y is changing x per se with respect to P, then y is changing x with respect to P

Translation 4: ∀x∀y(Dxy→Cxy)

(5) For all x and y, if x is external to y, then x is not identical to y.

Translation 5: ∀x∀y(Exy→x≠y)

(6) For all x and y, if x is a proper part of y, then x is not identical to y.

Translation 6: ∀x∀y(Pxy→x≠y)

Let's examine whether these premises are plausible or not. 4, 5 and 6 can easily be shown to follow from the definitions of 'change per se', 'external object', and 'proper part' respectively, so I will not talk about them any more. 1 is basically just a more precise statement of (ACT), so I won't say too much in its defense, but the premise is eminently plausible: Upon a small amount of reflection it is simply obvious that what is merely potential cannot have any power to bring about something actual. Merely potential chemical reactions do not bring about any actual chemical reactions. So the only thing which can bring about something is something which actually exists, and doesn't merely potentially exist. 

The crucial premises then are 2 and 3. 3 is quite plausible on the face of it. For surely if something is changed at all, then there is something which changes it in the most immediate sense i.e. changes it per se. There must be some most immediate explanation or cause of a change right? If there isn't, then the change can never come about. This seems intuitive enough.

(The intuition is this: There seems to be some sort of infinity problem here, though the problem isn't with an infinite regress but rather with what we can call an infinite "progress" of causes. If there is no immediate cause, there has to always be another cause that's "closer" to the effect, but never one that actually "gets" to the effect. If it isn't clear what I mean I can elaborate.)

What about 2? The idea behind 2 is that some things can truly be said to bring about changes in themselves in some sense, but they can't be said to bring about per se changes in themselves; properly speaking, it is the parts which are bringing about the change in the whole. For instance, dogs can move themselves only because their legs do. So, the only thing which can bring about a per se change in something is either something external to it or else one of its parts.

Suppose to the contrary that the cause x of the per se change in y with respect to P is not one of y's parts and is not something external to y. Then since clearly whatever is not a proper part of y and is not external to y is identical to y, it follows  x = y. So y brings about a per se change in y. Now either (a) some of the parts of y bring about the change in y or (b) none do (either way, definitely no parts bring it about per se, as per our assumption).

Assume (a). If none do, then y's parts remain completely the same, yet there is a change in y. But surely y taken alone is not sufficient for explaining the change in y, and thus y does not cause a per se change in itself! After all, how could y change itself with no external influence and no action of any of its parts at all? It would have to be a spontaneous causa sui! So on the supposition that the parts do not act in any way so as to bring about the change in y, it follows y cannot be a per se cause of a change in itself. Since we assumed however that y does cause a per se change in itself, it follows we must reject this supposition and conclude that some of the parts do in fact bring about a change in y. In other words, we must reject (a) and assume (b).

Assume (b). Suppose on the other hand that some of the parts do help cause the change in y. By the definition of per se change, the action of these proper parts of y is not sufficient for explaining the change in y; but nevertheless the action of y taken apart from any external cause is. This seems to make little sense; y still appears to be acting as a causa-sui, since it is still causing a change in itself at least in part independently of the action of its parts. Since this is impossible--nothing can be a self-cause except by the action of its parts--we must conclude that the parts do not help cause the change in y. Thus (b) is false.

Since both (a) and (b) are false, and either (a) or (b) must be true given our assumption that y causes a per se change in itself, we must reject our assumption that y caused a per se change in itself. But if that is the case, then given that there is no external cause of y then x (the cause of the change in y) must be a proper part of y, as we set out to prove.

So much for premises 1 and 2 then. Now, given that all the above premises 1-6 are true we can prove:

(7) For all x, if x is changed with respect to P by some y, then x is changed by some z non-identical to itself.

Translation 7: ∀x[∃yCxy→∃z(Dxz∧x≠z)]

I won't explain the proof here; instead, for anyone who doubts me, I have attached a formal proof below. From 7 and 4 of course it can be shown quite easily that whatever is changed with respect to P is changed with respect to P by some non-identical z: That is to say, whatever is changed is changed by another. Hence, given my 1-6, Aquinas's premise is secure.

Proof of 7:

[Note: If you can't see the proof, right click and either open in a new tab or else save the image and zoom in with some image viewer. I did the proof rather quickly so it is not the most elegant and could be done in fewer steps, but it gets the job done.]

The Universe is Contingent (And Therefore Needs an Explanation)

One common fallacy is the fallacy of composition, where one argues from the fact that each part of a thing has a certain feature to the conclusion that the whole thing has that feature. For instance, one could argue that every brick of the house is cube-shaped, therefore the house is cube-shaped. Or one could argue that each part of one's brain is unconscious, therefore the whole brain is unconscious. These inferences are fallacious.

However, I think it is worth noting that not all inferences from properties of parts to properties of the whole are invalid. If each part of a wall is made entirely of stone, then the whole wall is made entirely of stone. Similarly, if each part of the ball is entirely red, then the whole ball is entirely red. And so on.

Contingency seems to be like this, at least in this case. So here's an argument that the universe must be contingent:

Two Inadequate Arguments for a Finite Past

In this post I will consider two arguments which have at times been brought up in connection with the Kalam Cosmological Argument (KCA), which I will call the "subtraction argument" and the "argument from traversing an infinite," the former of which I have heard from Dr. William Lane Craig. The KCA goes as follows:

1. Whatever begins to exist has a cause.
2. The universe began to exist.
3. So the universe has a cause.

The arguments in question are designed to defend the second premise, which is presumably implied by the past's being finite. I should note that I think the second premise is true and there is strong evidence in favor of its truth. Alexander Pruss has given an excellent argument here, to which I have heard no compelling reply. I also think there is very strong scientific indication of the premise's truth, which Craig has adequately demonstrated. I just don't think these two arguments demonstrate its truth.

The "argument from traversing an infinite" goes something like this:

1*. If the past were infinite, one would have to cross an infinite temporal distance to get to the present moment.
2*. If one had to cross an infinite temporal distance to get to the present moment, then one could not get to the present moment.
3*. So if the past were infinite, then one could not get to the present moment.
4*. But we are at the present moment.
5*. So the past is not infinite.

The argument requires some unpacking. First of all, to say the past is finite is to say there was a beginning of time, and to say the past is infinite is to say there was no beginning of time. Second, 'temporal distance' means the length of time between one moment and another. There is a perfectly good way to define finite temporal distance. If we take our measure of time as a second, we can assign the current time the number 0, the time one second ago -1, the time two seconds ago -2, and so forth. To find the temporal distance from one time t1 to another t2, we take the number assigned to t1 and the number assigned to t2, and take the absolute value of the difference between the two. For instance, take the time 1000 seconds ago. To find the temporal distance from that time to the present time you take the absolute value of -1000 minus 0, which is of course 1000 seconds. Pretty simple.

However, problems begin to arise when we start to talk about an "infinite temporal distance." This phrase is ambiguous, and depending on which interpretation of this phrase we take it will either cause problems for premise 1* or for premise 2*. First, the phrase could mean something analogous to the way finite temporal distance has been defined above. However, infinity is not a real number, so you simply cannot define an infinite temporal distance the same way as above. There is no number "-infinity" from which you can subtract, say, -5. So if this is what is meant, then premise 1* appears to be false, since no real sense can be given to an infinite distance in this way.

On the other hand, crossing an infinite temporal distance could just mean that the set of all the numbers assigned to the seconds is infinitely large. This makes perfectly good sense of the phrase, but then in that case it is not clear why premise 2* is true. As Thomas Aquinas points out, there being an "infinite temporal distance" in this sense is perfectly consistent with all the temporal distances from the past to the present being finite, where "temporal distance" is defined as it was earlier:

"Passage is always understood as being from term to term. Whatever bygone day we choose, from it to the present day there is a finite number of days which can be passed through. The objection, however, is founded on the idea that, given two extremes, there is an infinite number of mean terms." [ST Ia q.46 a.2]

So for instance, the distance from the present to one second ago is 1 second, the distance from the present to two seconds ago is 2 seconds, etc. and so on forever and ever back into time. Hence, no matter how far you go back in time, the distance in the way I've defined above from any given past moment to the present will be finite, and thus you will only have to cross a finite number of seconds to get to the present moment. But of course any finite number of seconds can at least in principle be crossed; hence, premise 2* is false.

So much for the "traversing an infinite" argument then. The "subtraction argument" goes something like this:

1'. If the past were infinite, then an actual infinity would be possible.
2'. If an actual infinity were possible, then one could perform subtraction on infinities.
3'. But if one can perform subtraction on infinities, then one will get contradictory results.
4'. So if the past were infinite, then one would get contradictory results.
5'. So the past is not infinite.

This seems to be one of the arguments William Lane Craig gave in his debate against Peter Millican. Let me first point out an ambiguity in the phrase "actual infinity," after which I'll assess the argument under each interpretation. Here are two possible meanings of the phrase "actual infinity":

(ACT1) An actual infinity exists just in case for some time, at that time there exist distinct concrete objects such that the size of the set containing all and only them is infinite.

(ACT2) An actual infinity exists just in case there is some set containing only distinct concrete objects whose size is infinite.

Some elaboration is in order. First off, both definitions presume when speaking of actual infinities that we are dealing with concrete objects. While my understanding is that Craig does not believe there are infinitely many numbers (he is a nominalist), presumably his argument doesn't presuppose this view; Craig only wants to rule out the possibility of infinitely many concrete objects. Now as for the definitions themselves, the difference between (ACT1) and (ACT2) is that in (ACT1) you only have an actual infinity when all the concrete objects exist at the same time. In (ACT2) you could have finitely many objects at t2, finitely many at t1, and so forth, yet if you take a set containing concrete objects from different times, and the times go back to infinity, you will still have an actual infinity. So both of these definitions make fine sense. However, the assessment of the argument will depend on which interpretation we take.

Let's deal with the first definition, (ACT1). Given our definition of actual infinity in (ACT1), premise 1' does not appear to be true, or at least not obviously true. It is consistent with holding that the past is infinite that at each time there are only finitely many concrete objects. And if you believe only objects in the present moment exist, then 1' is definitely not true. The fact that there were objects at each time in the eternal past in no way implies an infinite collection of simultaneously-existing objects.

My main concern is with 2' and 3' though. Take 2', since it is also ambiguous to a certain extent. The problem is it is not immediately clear what is meant by "perform subtraction on infinities." Craig acknowledges that the ordinary operation of subtraction is not defined for "infinity". If he did mean this, i.e. the ordinary operation of subtraction, then clearly 3' would be true but 2' would have no support at all. So it cannot mean the ordinary operation of subtraction.

Now, Craig gives us an example to support 2' and 3', and this may help us understand what he means. Suppose we have an infinite number of coins. Then we can take away all the coins except three of them. And in this sense we can be said to perform subtraction on infinity, i.e. taking away some number of things from an infinitely large collection. This definition of performing subtraction on infinities makes sense.  But then why is 3' true? Craig says, considering our infinite number of coins, that you can take away infinitely many coins and be left with 3 coins, and thus infinity minus infinity = 3; but you can also take away infinitely many coins and be left with 2 coins, and thus infinity minus infinity = 2; hence, 2 = 3, which is our contradiction.

The problem with this argument is that it runs on an equivocation: We agreed that we are not using "performing subtraction" or "minus" in the normal sense of the arithmetical operation, since this just makes no sense. So "infinity minus infinity = 3" must simply be shorthand for saying "taking away infinitely many objects from an infinite collection leaves us with 3", and similarly with "infinity minus infinity = 2." But then if "2 = 3" means that 2 is identical to 3, then it certainly does not follow that 2 = 3; all that follows is that you can take away an infinite number of things and be left with 3, and also take away an infinite number of things and be left with 2, and this is certainly not a contradiction! It only looks like a contradiction when we are illicitly inferring "2 = 3", as if the phrase "infinity minus infinity = 2" were using "minus" and "=" in the same way as "5 minus 3 = 2." It would be like if I had infinitely many pennies and dimes, and I said, "infinity minus infinity = a penny, infinity minus infinity = a dime, so a penny = a dime." Clearly I am making an illicit inference here, and for the same reason Craig's argument makes an illicit inference as well.

Now, this whole time I have been working under the assumption that throughout the argument "actual infinity" is meant in the sense of (ACT1). But under interpretation (ACT2) the situation is even worse, since it is not clear 2' is true. It seems that in order to "subtract" infinitely many coins in the sense defined above, all of them must exist at the same time. But if "actual infinity" is taken in the sense of (ACT2), then it is not required that all of the infinite number of coins exist at the same time, and thus 2' has no support. And of course, with the exception of premise 1', all the same criticisms I have just given apply equally well under (ACT2). So, interpreted charitably, the argument seems to be a failure, with the primary problem being in premise 3'.

I should note one more time that, in spite of all my criticisms of these two arguments, I think there are good reasons for thinking the KCA is sound. I just don't think these are among them.

All Perfections are Possibly Instantiated

My logic teacher and I were reflecting on the ontological argument and we came up with this. Forgive the cheap operators. A perfection is a property which it is better to have than to lack. Let A be any perfection (say, being perfect) and let B be any non-perfection (like being imperfect) and let 'PA' mean 'A is a perfection' and 'PB' mean 'B is a perfection'. Then let PP be the principle that perfections only imply perfections:

(PP) For any property J and any property K, [PJ & [](x)(Jx --> Kx)] --> PK

Since A is a perfection and B is a non perfection:

[Prem] PA & ~PB

Now, assume for reductio that ~<>(Ex)(Ax). This is equivalent to []~(Ex)(Ax) which is equivalent to [](x)~(Ax). Since necessarily A is not exemplified by anything, then trivially [](x)(Ax --> Bx). So by our premise, [PA & [](x)(Ax --> Bx)] But by (PP) [PA & [](x)(Ax --> Bx)] --> PB. It follows that PB. But by our premise, ~PB. This is a contradiction. So we must reject our assumption. So ~~<>(Ex)(Ax). So <>(Ex)(Ax).

Remarks on "Divine Hiddenness"

[Originally posted at the Rational Gang. Please direct any comments there.]

A lot of atheists say that if God were to perform a miracle in front of their eyes, then they would believe. So then, if God is all good, and "wills all men to be saved," then why doesn't he provide such obvious evidence for his existence? Since an all-loving God is expected to reveal himself to all, and there are atheists that have reasonable nonbelief, this is evidence that God does not exist.

1. Coercion and Moral Responsibility:
A first thing to note is that God wants to preserve our moral responsibility and freedom. Now, almost all accept that it is possible to coerce someone into doing something. This can be done, most obviously, by physical force, as when kidnappers pick up a hostage and put him into a van against his will. Clearly in this context the person is not responsible for this event. Another, more interesting case, is where a mother is forced by a cruel killer to choose between her two sons being shot, or else they will both be tortured to death. The mother is not morally responsible, or at least very responsible, if she ultimately chooses one child in order to save at least one; the ultimate responsibility goes to the wicked sadist who is forcing the "choice" upon her. The point here though is that the mother is not literally coerced by force; rather, she is coerced by reasons. She knows that if she doesn't pick one, they will both be lost. In cases of coercion, one loses moral responsibility for one's actions.

But then it's straightforward applying this to the case of God. God wants us to have the freedom to choose him. He doesn't want to coerce us and ultimately remove our responsibility. But if one can be coerced by reasons, and incontrovertible proof would coerce some people into belief, then it's easy to see why God would not provide incontrovertible proof to absolutely everyone in order that they might all have absolute certainty: It removes responsibility on our part.

2. Divine Obviousness:
A reply inspired by the Thomistic tradition is to say that, as a matter of fact, God isn't hidden at all. Rather, God, as the ultimate source of all being, completely and totally permeates throughout nature. This is how God is omni-present, not in the sense that the universe is God (a la pantheism), but rather that the very existence of things is just one step away from God himself. However, being creatures of habit, and being familiar with the world, we take the awful mystery of existence for granted, and fail to see the divinity which sustains it all. But this is no failure on God's part. It's a failure of us to appreciate God's creation.

3. They're Wrong:
While atheists say they know that if God were to provide them a clear sign such as a miracle then they would convert, that might, for all we (or they) know, not be the case. In fact, this is what we find from the Christian perspective in John 11, where the Pharisees, even upon hearing and accepting that Christ performed miracles, refused to acknowledge him: "So the chief priests and the Pharisees gathered the council, and said, 'What are we to do? For this man performs many signs. If we let him go on thus, every one will believe in him, and the Romans will come and destroy both our holy place and our nation.'" So for these people, even the greatest of miracles would not convince them. How would this apply in the case of others?

According to one theory of the relationship between God's foreknowledge and our freedom, Molinism, God knows infallibly what we would freely do in any given circumstance. Supposing this view is coherent, we can apply it in the case of atheists: God does not give incontrovertible evidence to people who he knows that, in any circumstance where they are given incontrovertible evidence, would not convert. This means that atheists cannot say, "Well, if only God had given me more evidence, then I would have converted." Not necessarily.

If this is too strong, we can even weaken the principle: God does not give reasonable evidence to people who he knows that, in any circumstance where they are given reasonable evidence, would not convert. This seems more plausible anyway. For one, it doesn't seem necessary for God to give everyone incontrovertible evidence. If we are given reasonable evidence and fail to see it, then that is our fault. Moreover, this widens the group of people we are dealing with because, though some may accept God given incontrovertible evidence, fewer will accept God given only reasonable and sufficient evidence, which may explain a large number of unbelievers.

What this ultimately shows is that, even if some atheists are not actually given incontrovertible, or even reasonable, evidence, this doesn't make it the case that God is doing something evil, the reason being that they would not accept the evidence in other feasible circumstances anyhow. It would be for God to cast pearls to swine.

4. Rationality and Reasonableness:
This leads to a further question: are atheists, in fact, given reasonable evidence? Their whole argument may be begging the question here. After all, theists believe that there is reasonable evidence for the existence of God. But then if God has provided everyone with reasonable evidence, the argument fails, because this is precisely what is in question, i.e. whether belief in God is reasonable.

The atheist finds himself in a bit of a pickle. In proposing that there is not enough evidence for Christian belief, he must give us a criterion of reasonable belief that is strong enough such that it supports the reasonableness of atheism, but is not so strong that it begs the question against the theist. To put it another way, there is a tension in the claim that "the reasonableness of atheism entails the unreasonableness of theism," because it is possible that, given the atheist's criterion of reasonableness, the theist can turn right around and say, "Well, it is reasonable to believe that God might not reveal himself to everyone with absolutely convincing evidence."

5. Five points.
To sum up, I think there are five reasons we can give as to why God's existence is not incompatible with "divine hiddenness."
(a) Reasons can coerce. So if one were given incontrovertible reasons to believe in God, then one would be coerced into belief in God. But since coercion is incompatible with significant moral responsibility, and God wants to leave us morally responsible, God does not want to coerce us. Therefore, God does not give incontrovertible evidence.
(b) God is not in fact hidden; rather, God is obvious. However, humans fail to recognize the mystery of their own existence and the creation before them, and fail to see the divine in nature.
(c) For all we know, God does not give those who have a reasonable non-belief in God incontrovertible evidence, because he knows that in all circumstances, they will not believe in God.
(e) For all we know, God does not give those who have a reasonable non-belief in God reasonable evidence, because he knows that in all reasonable circumstances, they will not believe in God.
(e) The atheist begs the question against the theist in two ways: first, by saying that his nonbelief in God is reasonable, since the theist believes there are reasonable grounds for belief, and second, by saying that his reasonable nonbelief implies God's nonexistence, since the the theist can say that, by the atheist's own criteria, it is reasonable to believe that God has reasons to stay hidden.

All in all, this provides a strong cumulative case for believing that the argument from divine hiddenness doesn't work, and that theism is left unharmed.

What Does It Take to Be An Atheist?

Consider some valid theistic arguments and what an atheist must do to deny their soundness.

Argument A - A Necessary Concrete Entity:
(1) Possibly a necessary concrete being exists implies a necessary concrete being exists.
(2) Possibly, a necessary concrete being exists.
(3) Therefore, a necessary concrete being exists.

Assume further that it can be shown that the necessary being is God.

What must the atheist do to deny this argument? (1) is just axiom S5. Therefore they must deny (2). This implies that (a) possibly nothing exists or (b) necessarily, there exists at least one contingent entity. This also implies that all concrete entities are contingent.

Atheist Theorem 1 (AT1): It is not possible that a necessary concrete being exists.
Atheist Theorem 2 (AT2): Either (a) possibly nothing exists or (b) necessarily there exists at least one contingent entity.
Atheist Theorem 3 (AT3): For any concrete entity x, x is contingent.

Disproving AT1, AT2 or AT3 is sufficient for refuting atheism.

Argument B - The Scotistic Cosmological Argument:
(1') Whatever is possible is contingent or necessary.
(2') A first cause is possible.
(3') Therefore, a first cause is contingent or necessary.
(4') Any contingent substance is possibly actualized by another substance.
(5') A first cause is not possibly actualized by another substance.
(6') Therefore a first cause is not contingent.
(7') Therefore, a first a cause is necessary.

What must the atheist do to deny this argument? The atheist can only really deny either (2) or (4).

Atheist Theorem 4 (AT4): Either (a) a first cause is impossible or (b) some contingent substance is not possibly actualized by another.

Hence, disproving this disjunction AT4 is sufficient for refuting atheism. Saying that (b) is false also leads to some interesting consequences about infinite regresses.

Argument C – A Modal Cosmological Argument:
(1*) Every contingent entity possibly has an external cause.
(2*) If the sum total of contingent concrete entities C has an external cause, that cause is necessary.
(3*) C is a contingent concrete entity.
(4*) Possibly, C has an external cause.
(5*) Therefore, possibly there is a necessary cause of C.
(6*) Therefore, there is a necessary cause of C.

What must the atheist deny? All premises seem quite strong. He must then deny (1*).

Atheist Theorem 5 (AT5): For some concrete contingent entity x it is impossible that x is caused.

Disproving AT5 is sufficient for disproving atheism.

Thus far we have Atheist Theorems 1-5:

AT1: It is not possible that a necessary concrete being exists.
AT2: Either (a) possibly nothing exists or (b) necessarily at least one contingent entity exists.
AT3: For any concrete entity x, x is contingent.
AT4: Either (a) a first cause is impossible or (b) some contingent substance is not possibly actualized by another.
AT5: For some concrete contingent entity x it is impossible that x is caused.

Disproving one of these theorems is sufficient for disproving atheism. Showing a contradiction between these beliefs and other atheist beliefs is sufficient for showing atheism as such to be incoherent. To me all of these propositions seem quite implausible in their own right.