Christophe de Ray's Conceptual Cosmological Argument

Introduction

This argument is due entirely to Christophe de Ray, and is presented in his excellent paper "Existence exists, and it is God" (2022). This post is effectively a summary of the argument as I understand it, and is written largely as a way to help me organize my own thoughts on the matter. I highly encourage all who read this to check out de Ray's original paper.

Summary of the Argument

The argument may be summarized as follows:

1. All particular things exist in virtue of their having or sharing in something, which we call Existence.
2. Existence could not enable particular things to exist unless it itself exists.
3. Therefore, Existence exists.
4. If Existence exists, then it is a supremely excellent being.
5. Therefore, Existence is a supremely excellent being.

(1) is motivated by the typical arguments for existence as a first-order property (e.g. the inability of the Frege-Russell view to properly analyze singular existence statements). (2) seems self-evident, while (3) follows logically.

Motivating (4): "If Existence exists, then it is a supremely excellent being"

(4) is the most substantive premise, and obviously stands most in need of defense. The idea here is that we need to give some account of how Existence itself is able to exist. After all, if all particular things receive their being from Existence, then how can Existence itself exist? Surely it cannot bestow being on itself, for this would require that Existence be ontologically prior to itself (which is surely incoherent). Hence, it seems that we must say that Existence itself exists in a different way from how other things exist: while all other things exist in virtue of their standing in a certain "sharing" relation to Existence, Existence itself is a paradigm existent, the existence of which consists in its self-identity. De Ray proposes the following account of existence:

Paradigm Theory (PT): x exists if and only if either x is identical to the paradigm existent, namely Existence, or x bears the ‘sharing’ relation to Existence.

In order to make sense of this account, de Ray proposes an analogy. There are two ways in which something might satisfy the predicate "is a The Starry Night painting": one way is to be a copy or imitation of the original van Gogh painting, while another way is simply to be the original van Gogh painting. The copies satisfy the predicate in virtue of their standing in the "being a copy" relation to the original painting; however, the original van Gogh satisfies the predicate simply of virtue of its self-identity. Similarly, while all other things exist in virtue of their standing in the "sharing" relation to Existence, Existence itself exists in virtue of its self-identity.

So, Existence is a paradigm existent. But what kind of thing would a paradigm existent be? Plausibly, to be a paradigm F is to be a "perfectly good or true instance of F. [...] Obviously, no particular F could be better or more truly F than the paradigm F, since if it [was], then it would be the paradigm. For example, it is impossible for any painting to be a better or truer The Starry Night painting than van Gogh’s original" (de Ray 2022, 6). "

This analysis implies that there can be better or worse ways of existing, with the paradigm existent being a perfect instance of existence. "This accords with our intuition that while both humans and oysters are real, the existence of the former is superior to that of the latter, in virtue of some of the attributes that humans have and that oysters lack, such as self-awareness, creativity, the ability to form relationships, etc. [...] But if the paradigm existent is a perfect instance of existence, and thus exists in a perfect way, then it follows that Existence must possess all the excellence that it is possible for a being to possess. This must surely include consciousness, intelligence, rationality, and indeed maximal power, knowledge and moral goodness, to name only some. Whatever the exact nature of such a being, it is surely appropriate to call it a supremely excellent being" (ibid., 6-7). Hence, (4) is true, while (5) follows logically from (1)-(4).

A Potential Objection (and its Resolution)

Are there any objections to all of this? Here is one: before an object can instantiate a property, it must first exist. But on this account, objects exist in virtue of their standing in a certain relation, i.e. that of imitating the paradigm existent. "It could therefore be objected that our account is incoherent, because imitation can never be prior to the existence of the imitator" (ibid., 12).

De Ray argues that this difficulty is best resolved by considering more carefully what it means for a particular thing to imitate the paradigm existent. Consider again the van Gogh case: what it is for a copy of The Starry Night to stand in the imitation relation to the original painting is simply for it to be modelled on the original, i.e. for it to be created in a certain way by a creative agent. It does not have to exist before it can be a copy of the The Starry Night; rather, it just is a copy of The Starry Night for as long as it exists, in virtue of its having been made in a certain way.

Returning to the case of existence, if we think of the imitation relation as a kind of creation, "the objection at hand vanishes, because it is clear that the existence of a created thing is not ontologically prior to its being created. It would be strange indeed to argue that ordinary existents are created by the paradigm because they exist, and far more plausible to say that they exist in virtue of being created" (ibid., 12).

It is also useful to note that while in the van Gogh case the creation is done by a "third party" (i.e. the painter), who is distinct from the paradigm case (i.e. van Gogh's painting) and the created imitations, in the case of existence there can be no third-party, since everything other than the paradigm existent is an ordinary existent. Any creative agent other than the paradigm existent itself "would thus need to create [itself] to resemble the paradigm existent, which is absurd. Thus, it is the paradigm itself that would need to do the creating, modelling ordinary existents after itself" (ibid., 12). We thus have the result that Existence is a supremely perfect being, which is responsible for creating all other things. "And this," as Aquinas put it, "everyone understands to be God."

Dispositionalism and Contingent Existence

[Note: Most of this post consists of fairly obvious observations about the consequences of modal dispositionalism; I just wanted to have this train of thought in writing somewhere.]

Critics of the cosmological argument will sometimes claim that while no contingent thing exists in all possible worlds, it might still be the case that all possible worlds contain at least one contingent thing. William Rowe provides the following analogy:

We know that although no horse in a given horse race necessarily will be the winner, it is, nevertheless, necessary that some horse in the race will be the winner. (1975, 164)

It turns out that dispositionalists cannot plausibly appeal to this possibility as a way of avoiding a necessary being. Here's why: either causal history is infinite, or it is not. If it is not, then the dispositionalist has very good reason to affirm the existence of a (set of) necessary being(s); see e.g. Vance (2014), Vetter (2015), and Kimpton-Nye (2021). So the dispositionalist who wishes to avoid a necessary being should assume that causal history is infinite. But, as it turns out, this strategy will not work either.

Let S be the plurality of all actually-existing contingent concrete things. (If nothing is necessary, then S will include all actually-existing concrete things.) Suppose that the cosmological arguer asks for an explanation of why the beings in S exist, and the critic gives the aforementioned reply that it is necessary for some contingent things to exist. But note that on a naturalistic dispositionalism, "every possible world contains some natural thing that actually exists... necessarily, every world contains some stretch of our actual natural past" (Leftow 2017, 326). This means that the critic's reply commits them to claiming that it is necessary for at least some of the beings in S to exist.

Is this reply at all plausible? I think not; after all, every being in S is contingent, and so fails to exist in some possible world. But it seems obvious that if x could fail to exist, and if y could fail to exist, then both x and y could jointly fail to exist; otherwise, we would have to suppose that x's nonexistence somehow forces y to exist, which seems implausible. As Vetter puts it, "the possibility, for each contingent object, that it does not exist, together with what we might call a principle of independence—that the non-existence of contingent objects can never force other contingent object into existence—yields the global possibility that none of the actual contingent objects exist" (2015, 275).

A further problem is that while the critic might claim that it is necessary for some of the beings in S to exist, there can be no particular beings in S which exist necessarily. This should be obvious merely from the fact that, as stipulated, S is the plurality of all actual contingent things; however, the point becomes more interesting when put in terms of branching causal histories. Recall that, as Leftow points out, the theory under consideration entails that "every world contains some stretch of our actual natural past" (2017, 326), which, given an infinite past-eternal universe, "would be an infinite stretch" (ibid., 326). However, this would not be the same stretch of our actual past; rather, different possible worlds would share different stretches of the actual past. To make the problem clearer: if causal history is infinite, then for any causal node n, there is an earlier node n-1. But this entails that there is no single node n which is part of the shared history of all possible worlds. Hence, the aforementioned critic of the cosmological argument must claim that it is a necessary truth that all possible worlds share a stretch of causal history with the actual world, but not any particular stretch. This does not seem like a very good explanation of the entire causal series. 

It is also worth noting that the naturalist dispositionalist is committed to claiming that "there could not have been other natural laws... [nor] could there have been a different total amount of mass-energy" (Leftow 2017, 325). Their view will also threaten modal collapse, since "If [this] theory is true and determinism is true, the actual world is the only possible world: there are no chancy causes, so there are no branches off the tree of actual history" (ibid., 326). Thus, on this view "we should take modal Spinozism precisely as seriously as we take determinism" (ibid., 326).

The upshot is that the dispositionalist must either admit that there is a (set of) necessary being(s), or else claim that it is a necessary truth that some actually-existing contingent things exist. Given that this latter claim seems extremely unlikely (if not flat-out nonsensical), entailing as it does many implausible consequences, it appears that the dispositionalist has strong reason to accept the existence of a (set of) necessary being(s).

A Convenient Way to Explain Metaphysical Modality

Consider the following question: could the laws of nature have been different?

According to physical modality, the answer is trivial: no, of course the laws of nature could not have been different. After all, physical modality just is the range of what is the modal space governed by the laws of nature.

According to logical modality, the answer is similarly trivial: yes, of course the laws of nature could have been different. After all, there is nothing logically incoherent about the idea that, say, the speed of light could have been slightly faster or slower than it in-fact is. 

Metaphysical modality, then, is that form of modality which renders non-trivial the question "could the laws of nature have been different?"

Dore on Rowe on the Cosmological Argument

I recently came across an old paper by Clement Dore, entitled "Rowe on the Cosmological Argument" (1983). In it, Dore objects attempts to uphold Hume's objection to the PSR, by critiquing Rowe's defense of the principle. He writes:

[In] order to support the cosmological argument, PSR must be interpreted this way: "For any infinite set, S, each of whose members can be explained by another member, the question 'Why are there any members of S?.' can (only) be truly answered by referring to a cause which is not itself a member of S." But how much credibility has PSR, thus interpreted? Consider the question, "Why are there any things which are at least as large as a quark?" Rowe's claim that PSR (as presently interpreted) may be plausible appears to commit him to the thesis that the following may be plausible: "If there is a finite number of members of the set of things which are at least as large as a quark (call it S'), then S' is like my philosophy of religion class, in that we do not need to infer an irreducibly distinct cause of its having members. However, if S' has an infinite number of members, then it is subject to a causal explanation which does not reduce to the individual explanations of the individual members, namely, the following one: The individual members of S' exist because there is something smaller than a quark which is their ultimate cause." 

But of course, Dore argues, it is absurd to suppose that there must be something smaller than a quark which is the ultimate cause of the set S'. And since PSR (allegedly) commits one to such a claim, it seems that PSR must be rejected.

I found this line of argument noteworthy, if only because of how absurdly weak it seems. Firstly, because the cosmological argument doesn't require one to assume any infinite set of objects; the argument from contingency (for example) works identically well whether the totality of all contingent things is finite or infinite.

But there's another issue with Dore's line of reasoning. For assuming that S' has infinitely many members, it seems that we may still avoid his claim of absurdity with relative ease. Example: suppose that the universe had a beginning in time, with an initial state and physical laws that were/are metaphysically necessary (this is the view taken by, among others, Graham Oppy). In that case, S' would be explained by the initial state of the universe, in tandem with the laws of nature that necessarily follow from it. This claim hardly seems absurd, and it is hard to believe that it should lead one to abandon PSR.

Alternatively, suppose that the universe has existed for eternity, and that its existence is metaphysically necessary. In that case, we have a number of options. For instance, S' might be metaphysically necessary (e.g. if the universe is necessary and past eternal, and S' includes everything in the universe, it follows that S' is metaphysically necessary). Alternatively, S' might be explained by the laws of nature in tandem with the most fundamental physical entities, and these things might be metaphysically necessary. Or else, if nothing in the physical universe is necessary, then S' would need to be explained by something outside of the universe, in which case the cosmological argument is sound. None of these options seem so absurd as the merit the rejection of PSR.

[Note: Dore might consider the last result (i.e. the cosmological argument being sound) to be absurd; however, this would hardly constitute a genuine reply to that argument. For then Dore would be arguing as follows: "The PSR as required by the cosmological argument is absurd, because if it were true, it would entail the success of the cosmological argument." This might be sufficient grounds for rejecting PSR, but it will certainly not convince anyone who regards the cosmological argument as sound.]

The Contents of a Foundational Mind (Or: A Weird Objection I Found on the Internet)

I recently came across somebody who argued that the foundation of reality could not be a mind, because if it were, then this mind would have no referent beyond its own existence (no "intrinsic content," as they put it). This would (supposedly) prevent the foundational mind from proceeding beyond its initial state. 

I must say, this strikes me as obviously wrong. After all, if God exists, and is omniscient, then He presumably has perfect modal knowledge; that is to say, even before the moment of creation, God knew all possible worlds that He could actualize. How might God know/have known this? Well, one potential answer is that modality is grounded in divine power. This would allow God to have perfect modal knowledge simply by knowing Himself, and specifically, His own power to actualize various states of affairs.

Indeed, classical theism has traditionally held that God's knowledge is rooted in His knowledge of Himself. For instance, here's what Aquinas has to say on the matter:

Inasmuch as He knows His own essence perfectly, He knows it according to every mode in which it can be known. Now it can be known not only as it is in itself, but as it can be participated in by creatures according to some degree of likeness. But every creature has its own proper species, according to which it participates in some degree in likeness to the divine essence. So far, therefore, as God knows His essence as capable of such imitation by any creature, He knows it as the particular type and idea of that creature; and in like manner as regards other creatures. So it is clear that God understands many particular types of things and these are many ideas. (ST I.15.2)

And of course, one does not need to be a Thomist to accept my initial point (i.e. that God could have perfect modal knowledge simply by knowing His own power).

Richard Carrier's Argument from Specified Complexity

Noted pseudo-philosopher Richard Carrier has an argument which, he thinks, demonstrates that "we should think the supernatural is impossible." Specifically, he wants to claim that "God's mind appears to be logically impossible." His line of reasoning goes (more-or-less) as follows:

1. (Assume for contradiction) God's mind is simple and immaterial.
2. God's mind must be able to store and process information.
3. Information storage and processing require specified complexity.
4. Such complexity requires some material on which information can be stored and circuitry stamped.
5. God's mind is complex and material (from 2-4).
6. Contradiction (from 1 and 5). 

Unfortunately, Carrier seems to have gotten himself tripped up on the nature of God's knowledge. For he writes:

A god has to know things (store information), and think things (process information). But there is a difference between correct and incorrect information, present and absent information; between conflation and distinction. A god has to be able to distinguish one person’s face, from another; and correctly connect each face, with other information about the corresponding person, like that Joe’s face goes with Joe’s job in sales and wife of eleven years, and Mark’s face goes with Mark’s service in the military and husband of eleven years. That information could be connected up differently—wires crossed, and Joe’s face gets incorrectly linked to Mark’s husband, producing the false information that Joe has a husband, and so on. And notice how many different ways connections can be crossed up: the more information, the more different connections are possible. And most of them (in fact, all but one of them; out of effectively infinitely many) will be false.

However, as the physicist Aron Wall pointed out in response to a near-identical objection (ironically put forward by a couple of conservative Mormons), "This is only an issue if you assume that God's knowledge is, like ours, representational, that is, that it proceeds by means of making something like an image or duplicate of the object known, in some other physical system (in our case, the brain)." But of course, classical theism does not make such an assumption; rather, it has always held that God knows all things via direct awareness, simply by knowing Himself as creator and sustainer of all things (hint: this is part of what CTs mean when we say that God's knowledge, creative act, etc. are really all one thing). Thomas Aquinas provides a useful summary of this concept, when he writes:

Now, it is not repugnant to the simplicity of the divine mind that it understand many things; though it would be repugnant to its simplicity were His understanding to be formed by a plurality of images. [...] Inasmuch as He knows His own essence perfectly, He knows it according to every mode in which it can be known. Now it can be known not only as it is in itself, but as it can be participated in by creatures according to some degree of likeness.  But every creature has its own proper species, according to which it participates in some degree in likeness to the divine essence. So far, therefore, as God knows His essence as capable of such imitation by any creature, He knows it as the particular type and idea of that creature; and in like manner as regards other creatures. So it is clear that God understands many particular types of things and these are many ideas. (ST I.15.2)

The idea that God's knowledge does not consist in stored-up beliefs or mental images may also be found outside of the Thomistic tradition. For example, William Alston (1986), a critic of divine simplicity, was nevertheless a major proponent of the idea that God has no beliefs whatsoever, but rather knows all things via direct awareness. To quote:

A creature in our condition needs inner representations in order to be able to think about absent states of affairs, since the facts are rarely if ever directly present to our consciousness. But since God enjoys the highest form of knowledge He is never in that position, and so He has no need for inner representations that He can ‘carry around with him’ for use when the facts are absent. The facts are never absent from His awareness; thus it would be fatuous to attribute to Him any such mental map. When we have arrived at our destination we can fold the map away.

In a recent paper defending Alston's view, Saeedimehr (2021) discusses two non-propositional accounts of God's knowledge, each of which would seemingly allow us to avoid Carrier's argument. To quote:

Since God is absolutely simple, He (His essence) is identical with his knowledge. Therefore, God’s knowledge is totally simple and hence it is beyond any kind of complexity, including the complexity due to having a propositional structure.

Since Alston finds the principle of Divine simplicity quite problematic, he seeks another basis for the non-propositional position. He argues that as human beings, our propositional knowledge stems from two of our limitations: first we ‘cannot grasp any concrete whole in its full concreteness’, and second ‘we need to isolate separate propositions in order to relate them logically’ and then ‘extend our knowledge inferentially’. But since God is obviously beyond these limitations, His knowledge is not required to be propositional.

It would seem, then, that a non-propositional account of God's knowledge allows us to reject premise (2) of Carrier's argument; namely, that God's mind stores and processes information. God does not store information at all (or to quote Alston, "carry [it] around with Him"); rather, at any one moment He has an infallible intuitive grasp of all facts. Similarly, God does not need to "process information" (or to quote Alston, "isolate separate propositions in order to relate them logically"). We need to do that, because our cognition is fallible and limited; God, on the other hand, is far beyond such things.

One hopes that Carrier himself never gets wind of this post, lest I be subjected to his, uh... scholarly rigor.

Do Theistic Arguments Support the Simulation Hypothesis?

Richard Brown believes that "the traditional arguments for God’s existence provide better support for the simulation hypothesis than they do for the traditional God of Theism." He basis this view on the following argument:

1. The traditional a posteriori arguments (cosmological, teleological, etc.) point, if one accepts their conclusions, to a creator, but not to what kind of creator.
2. The traditional theistic God (all-powerful, all-knowing, and morally perfect) is one candidate for being the Creator.
3. The Simulators are another candidate.
4. The traditional arguments do not distinguish between (2) and (3).
5. The problem of evil (evidential) suggests that (2) is not the creator.
6. Therefore, the traditional arguments for God’s existence provide better support for the simulation hypothesis than they do for the traditional God of Theism.

I think the theist can challenge almost every premise here. Let's go through them in order

While premise (1) might apply to some arguments for God's existence (such as the fine-tuning argument), it seems obvious that it does not apply to many others, including some of the most popular in the current literature. For example, Rasmussen's contingency argument tells us that the creator must be a metaphysically necessary being, while his various stage-two arguments (if they succeed) tell us that it must be eternal, unlimited, unsurpassably powerful, and possessing a supreme nature. Similarly, the Pruss-Koons argument from basic natural facts seeks to establish the existence of a "supernatural being," that is to say, "a simple being that is unbounded and wholly infinite in intrinsic measure."

Premise (2) is uncontroversial (especially amongst theists), and so it seems like we can skip over that one.

Premise (3) seems to fail for the same reasons as premise (1); namely, the most popular theistic arguments today (such as the contingency argument) would seem to rule out a simulator. After all, a simulator would presumably be a natural being, rather than a supernatural one. We also have no reason to assume that they would be necessary, rather than contingent. In addition, stage-two contingency arguments (such as those made by Rasmussen) are specifically designed to solve the so-called "gap problem," by showing that the necessary being in question must have some or all of the traditional divine attributes.

Premise (4) fails for the same reasons as premises (1) and (3).

Premise (5) only works if one agrees that the problem of evil really does provide evidence against theism. However, even if one does agree with this, it seems like this premise still fails to support the simulation hypothesis. This is because, if my criticisms of premises (1) and (3) succeed, the traditional theistic arguments rule out the idea of a natural contingent being as the cause of our universe. 

The conclusion (6) stands and falls with the rest of the argument, so if (as I've argued) the rest of the argument fails, then so does (6).

I think Brown has also neglected an important point, namely that theistic arguments must be approached in tandem, rather than one-by-one. For example, assume that the argument from contingency is correct, and the existence of a necessary being has been established. Then when we consider the fine-tuning argument, we must take into account the fact that a necessary being exists. It would be deeply strange for there to be a necessary being which explains contingent reality, and also a fine-tuner who fixed the laws of nature. Therefore, any evidence that we have for design should be interpreted as evidence for design by the necessary being. But surely the traditional simulation hypothesis does not regard the simulating beings (whoever or whatever they are) as necessary. So Brown's argument would seem to fall flat yet again.

There is a potential way that one could escape this last argument: namely, one could hold that our world really is a simulation, but that reality as a whole was created by God. This view is not unheard of in the literature; David Kyle Johnson (2012) has argued that some theists might be committed to this view, while theists Barry Dainton (2020) and Dustin Crummett (2021) have both approached it as a potential theodicy. That being said, I don't personally endorse this view (nor am I one of those whom Johnson argues are committed to it), and I'm not sure how important it is for my above argument (since it is ultimately still a theistic view).

A New Paper on Physical Necessity

Alexander Roberts has published an interesting new paper in Mind, which argues for the claim that "those who believe in physical necessity should not harbour any scepticism towards merely metaphysical possibilities." How he reaches this result is quite complex, and I'll leave it to the full paper to provide an explanation; however, "the result requires those who believe in physical necessity to recognize merely metaphysical possibilities which are forbidden by our physical laws." The key line of argument in the paper revolves around the notion of lawless worlds (i.e. worlds in which no laws of nature, whether ours or any others, obtain). Roberts argues that such worlds are accessible from all other possible worlds by a finite chain of physical possibility.

Causal Explanations Need Not Increase the Probability of Events

I recently came across this argument, which purports to refute the argument from contingency:

1. Suppose that some necessary proposition q causally explains a contingent proposition p.
2. A proposition q only causally explains a proposition p if q is true.
3. So, q is a necessary truth. (From 1 and 2)
4. The probability of a necessary truth is 1.
5. So, P(q) is equal to 1. (From 3 and 4)
6. Then, P(p|q) is equal to P(p). (From 2 and 5)
7. But, if q causally explains p, then P(p|q) > P(p).
8.  So, q does not causally explain p.
9. Then, q both does and does not causally explain p, which is absurd. (From 1 and 8)
10. Therefore, no necessary proposition causally explains a contingent proposition.

The essential premise here is (7), which the author summarizes thusly: "though causal explanations may neither entail nor make probable what they explain, it is at least true of causal explanations that their being true increases the probability of that which they explain." 

However, on closer examination, we see that this is flatly false. For it is well-known in philosophy of science that an explanation can reduce the probability of an event that it explains (thanks to Alexander Pruss for the following example). Suppose that Todd is a skilled assassin, who has been hired to kill Carl. When Todd fires his gun, there is a 99% chance that his target will die. As he is lining up his shot, Bob, a less effective assassin, also shows up. When Bob fires his gun, there is a 60% chance that his target will die. Bob, who compensates for his lack of precision with speed, gets his shot off first, and Todd flees the scene, never firing his gun. As it happens, Bob's shot kills Carl (though if he had missed, due to Todd's flight, Carl would have lived).

Here we have a case of a clear causal explanation, which nevertheless reduces the probability of the event that it explains. Bob's gunshot provides a causal explanation of Carl's death, despite the fact that it reduced Carl's chance of dying from 99% down to 60%. In other words, P(Carl's death | Bob's shot) is 60%, while P(Carl's death) without Bob's shot is 99%. So, by premise (7), we should say that Bob did not cause Carl's death, which is absurd.

At the end of the aforementioned "refutation," the author attempts to rebut these sorts of examples. He claims that, while the less effective assassin may reduce the target's chance of death relative to the more effective assassin, they nevertheless increase the probability simpliciter. I would argue that his objections fail; by premise (7), the probability of p occurring without q must be lower than it is with q in order for q to count as a causal explanation. Therefore, if the probability of p would be higher in the absence of q, premise (7) would tell us that q cannot be a causal explanation of p. This would require us to embrace the absurd conclusion that Bob did not cause Carl's death. The author of this supposed "refutation" should either revise premise (7), or acknowledge that his argument is a failure.

Just for the sake of argument, there are other examples we can give to illustrate this general issue. Suppose that there were an infinite series of snipers lined up to kill Carl, all firing one after another, and all with a non-zero chance of hitting their target. Here Carl's chance of death is 100%, and no individual sniper can affect this probability in any way (since even if he were to pack up his rifle and go home, there would be an infinite number of substitutes). But of course, one of the snipers must inevitably be the one to fire the fatal shot. From this, we can generate a problem for our interlocutor.

Suppose snipers one through thirty miss their target, and sniper thirty-one makes the kill. If he had missed, there would have been an infinite number of snipers after him, one of whom would have hit the mark; that is, the probability of Carl's death is 100%, whether sniper thirty-one is there or not. Therefore, P(Carl's death | Sniper Thirty-One) = P(Carl's death). But then, by premise (7) we should deny that sniper thirty-one caused Carl's death, which is absurd.

The conclusion, therefore, is that causal explanations need not increase the probability of the events that they explain. Thus, the supposed refutation of the argument from contingency fails.