W. L. Craig and J. P. Moreland (eds.), Blackwell Companion to Natural Theology, Oxford : Blackwell, 2009

Copyright © 2009 Blackwell Publishing Ltd.

Leibnizian Cosmological Arguments

Alexander R. Pruss

Baylor University

A cosmological argument takes some cosmic feature of the universe—such as the existence of contingent things or the fact of motion—that calls out for explanation, and argues that this feature is to be explained in terms of the activity of a first cause, which first cause is God. A typical cosmological argument faces four different problems. If these problems are solved, the argument is successful.

The first problem is that although some feature, such as the existence of contingent things, calls for explanation, it can be disputed whether an explanation exists. I shall call this the Glendower Problem in honor of this exchange from Shakespeare’s Henry IV, Part 1, Act III:

Glendower: I can call spirits from the vasty deep.

Hotspur: Why, so can I, or so can any man;

But will they come when you do call for them? (Shakespeare 2000, p. 59)

A typical solution to the Glendower Problem involves a causal or explanatory principle, such as the claim that all things have causes or that all contingent facts possibly have explanations, together with an argument that the principle applies to the cosmic feature in question and implies the existence of an explanation of it.

The second issue that must be faced in defending a cosmological argument is the Regress Problem, the problem of how to deal with an infinite regress of causes or expalanations. Hume stated that if we had an infinite regress of explanations, E 1 explained by E 2 , E 3 by E 4 , and so on, then everything in the regress would be explained, even if there were no ultimate explanation positing some first cause.

The third difficulty is the Taxicab Problem, coming from Schopenhauer’s quip that in the cosmological argument, the Principle of Sufficient Reason (PSR) is like a taxicab that, once used, is sent away. The difficulty here is to answer what happens when the explanatory principle that was used to solve the Glendower Problem gets applied to the first cause. A popular formulation is: “If God is the cause of the universe, what is the cause of God?” Typical solutions argue that the case of the first cause is different in some way that is not merely ad hoc from the cases to which the explanatory principle was applied.

The final difficulty for cosmological arguments is the Gap Problem. Granted, there is a first cause, but does anything of religious interest follow? There is a gap between the statements that there is a first cause and that there is a God. Aquinas in his Five Ways proves the existence of an unmoved mover, and then says: “et hoc omnes intelligent Deum” (“and all understand this to be God”). Some critics have taken this to be his way of papering over the difficulty of moving from a first cause to God; however, that reading is mistaken in light of the fact that succeeding sections of the Summa Theologiae give careful and elaborate arguments that the first cause is wholly actual, unchanging, simple, one, immaterial, perfect, good, and intelligent. Rather, Aquinas is simply marking the fact that the theist will recognize the unmoved mover to be God. Aquinas recognizes that an argument that the first cause has at least some of the attributes of the God of Western monotheism is needed and offers such an argument.

The solutions to the Glendower and Regress Problems tend to go hand in hand, and probably the best way to classify cosmological arguments is by how they address them. There are then three basic kinds of cosmological arguments: Kalaam, Thomistic and Leibnizian. The Kalaam and Thomistic arguments posit an intuitively plausible Causal Principle (CP) that says that every item of some sort, e.g., event, contingent being, instance of coming-into-existence, or movement, has a cause. The arguments then split depending on how they handle the Regress Problem. The Kalaam argument proceeds by arguing, on a priori or a posteriori grounds, that the past is finite, and hence in fact no infinite regress occurred. The Thomistic argument, exemplified by Aquinas’ first three ways, does not rule out the possibility of an infinite past, but uses a variety of methods to argue against the hypothesis that there is an infinite regress of causes with no first cause. The most distinctive of these methods is an attempt to show that there is an intrinsic distinction between intermediate and non-intermediate causes, where an intermediate cause of E is an item C that is itself caused by something else to cause E, and that this distinction is such that intermediate causes are, of necessity, dependent for their causal activity on non-intermediate causes, which then end the regress.

Leibnizian arguments, on the other hand, invoke a very general explanatory principle, such as the PSR, which is then applied to the cosmos or to some vast cosmic state of affairs, or else a non-local CP that can be applied to an infinite chain or the universe as a whole. In the PSR-based versions, the Regress Problem is typically handled by showing that an infinite chain of causes with no first cause fails to explain why the whole chain is there. The main challenge for Leibnizian arguments here is to argue for an explanatory principle or CP that is (a) plausible, (b) applicable to the cosmic state of affairs in question, and (c) not so strong as to lead to implausible conclusions such as the denial of contingency or of free will. In this chapter, I shall defend several Leibnizian arguments.

The basic Leibnizian argument has the following steps:

(1) Every contingent fact has an explanation.

(2) There is a contingent fact that includes all other contingent facts.

(3) Therefore, there is an explanation of this fact.

(4) This explanation must involve a necessary being.

(5) This necessary being is God.

We shall see, however, that the first step, the assumption of the Principle of Sufficient Reason, can be modified in various ways, with the resulting argument maintaining the distinctive feature of Leibnizian arguments that the relevant explanatory or causal principle is to be applied to a global state or proposition.

2. The Principle of Sufficient Reason

2.1. The scope of the PSR

For simplicity, I shall stipulatively use the term “fact” for a true proposition. The PSR states that every fact, or every contingent fact, has an explanation, and this is the standard tool in Leibnizian arguments for handling the Glendower and Regress Problems.

Some authors restrict the PSR to contingent facts. The advantage of a restriction to contingent facts is that we do not know very much about how the explanation of necessary truths works, and hence may not be in a position to justify the PSR for necessary truths. To explain the Pythagorean Theorem, presumably I should prove it from the axioms. But which proof counts as explanatory? Which axioms are the right ones to start from? Is there a fact of the matter here?

On the other hand, maybe the case of necessary facts is not a real worry. For it might be that any necessary truth p can be explained by citing its necessity: p holds because p necessarily holds. This leads into a regress, since that p necessarily holds will also be a necessary truth by axiom S4 of modal logic, but perhaps this regress is somehow to be distinguished from vicious ones.

Alternately the defender of an unrestricted PSR can say that while we do not yet know how the explanation of necessary truths works, we do know some cases of it. For instance, it might be that the proposition that 1=1 is self-explanatory, namely explained by the very same proposition 1=1, while the proposition that necessarily 1=1 is explained by the proposition that 1=1 together with the fact that mathematical truths are necessary truths. The necessary truth that all dogs are mammals, assuming this is indeed metaphysically necessary, is explained by the genetic similarity between dogs and the first mammals, together with some necessary truths about how biological classification works. The necessary truth that making false promises is wrong might be explained by the fact that promising falsely treats the promisee as a mere means. In other words, while we have no general account of the explanation of necessary truths, we do have many examples. And, anyway, the requirement that we have a general account of explanation would also be a problem for a PSR restricted to contingent propositions, since it is not clear that we yet have a general account of explanation of contingent propositions, though we have many clear examples.

2.2. Why should we believe the PSR?

2.2.1. Self-evidence

Many of those who accept the PSR do so unreflectively, because they take the PSR to be self-evident. I do not think that there is any good argument against the propriety of doing so. We are perfectly within our epistemic rights to accept the Law of Excluded Middle (LEM)—the claim that for all p, p or not-p, because of its self-evidence, absent any further argument for it. However, it will be of no use to opponents of the PSR or of the LEM to be told that the claim they deny is self-evident to us. Presumably, the claim is not self-evident to them, and we can all agree that there are many things that people have claimed to be self-evident which in fact are false, so the fact that the claim is said by us to be self-evident does not provide these opponents with much reason to accept it. There may be a presumption that what people take to be self-evident is in fact more likely true than not, but this presumption is often easily defeated.

One might think that philosophical disagreement about the PSR shows that the PSR is not self-evident, or at least that those of us who take it as self-evident should not see this as providing any reason to believe it to be true. Otherwise, how could competent philosophers like David Hume or Graham Oppy fail to see it as self-evident? Or, worse, how is it that some of these philosophers take as self-evident claims incompatible with the PSR?

If we think we should accept the LEM because of its self-evidence despite some brilliant intuitionist mathematicians’ denials of it, we will be unimpressed by this argument. And it is not clear on what grounds we could accept the LEM other than self-evidence. Is there some inductive argument like: “For many propositions p, we have concluded that the LEM holds. Hence, the LEM holds for all propositions p”? I doubt it. The problem is that an inductive argument of the form “Many Fs are Gs, thus all Fs are Gs” is epistemically close to worthless by itself. Many dogs are spotted, thus all dogs are spotted? We would do slightly better if we could show that most Fs are Gs, though even that would be very weak (“Most humans are female, thus all humans are female”). But how would we check that the LEM holds for most propositions? To check that the LEM holds for a proposition is, presumably, to determine that this proposition is true or to determine that this proposition is false, since in either case the truth of the LEM follows for the proposition. But most propositions are such that we cannot determine whether they are true or false.

In any case, the argument from philosophical disagreement is weak. It might be that our judgment as to what is or is not self-evident is fallible, and Hume and Oppy have simply judged wrongly. Or it might be that it is possible to be talked out of seeing something as self-evident, just as it is possible to be (rightly or wrongly) talked out of all sorts of commonsensical beliefs. Finally, it could be that the PSR’s opponents have failed to grasp one or more of the concepts in it, due to their substantive philosophical positions. Thus, Hume’s equating constant conjunction with causation suggests that he does not have the same concept of causation that I do—that he is talking of something different—and the fact that he thinks causation thus understood yields explanations, as well as his belief that infinite regresses can be explanatory, show that his concept of explanation is different from mine. Differences in views of modality are also relevant. As a result, it is far from clear to me that Hume has even grasped the PSR in the sense which I assign to it. And if not, then his failure to see it as self-evident is irrelevant.

I can give a similar story about Hume’s seeing as self-evident propositions that are incompatible with the PSR, such as that no being’s existence is necessary. Hume’s concept of the necessity of p is that a contradiction can be proved from the denial of p. If LEM is true, this is equivalent to equating necessity with provability. But defenders of Leibnizian cosmological argument typically use a notion of broadly logical necessity when they claim God is a necessary being, and broadly logical necessity is weaker than provability.

At this point it may seem as if the defense of the self-evidence of the PSR destroys all possibility of philosophical communication. If philosophers all mean different things by the same terms, how can they even disagree with one another? Two points can be made here. The first is that in many cases when philosophers use a word like “cause”, they both mean by it what ordinary language does and they have an account of what the word says which account they think is faithful to the ordinary meaning. And if this is true, then when one philosopher says “A causes B” and the other says “A does not cause B”, there is a genuine disagreement between them even if their analyses of causation are different, since the first philosopher holds that A causes B in the ordinary English sense of “causes” (which she rightly or wrongly thinks is identical with her analysis of the term) and the second denies this. Second, disagreement is possible because even though philosophers may use the term “causes” differently, they will tend to agree on some entailments, such as that if A causes B, then both A and B occurred and B’s occurrence can be explained at least in part in terms of A’s occurrence. So differences in meaning do not undercut philosophical communication, but they seriously damage the argument against self-evidence.

Self-evidence might well give those of us to whom the PSR is self-evident a good reason to believe it. But if we want to convince others, we need arguments.

2.2.2. The epistemological argument

This argument is based on ideas of Robert Koons (1997), though I am simplifying it. Start with the observation that once we admit that some contingent states of affairs have no explanations, a completely new sceptical scenario becomes possible: No demon is deceiving you, but your perceptual states are occurring for no reason at all, with no prior causes.

Moreover, objective probabilities are tied to laws of nature or objective tendencies, and so if an objective probability attaches to some contingent fact, then that situation can be given an explanation in terms of laws of nature or objective tendencies. Hence, if the PSR is false of some contingent fact, no objective probability attaches to the fact.

Thus we cannot even say that violations of the PSR are improbable if the PSR is false. Consequently, someone who does not affirm the PSR cannot say that the sceptical scenario is objectively improbable. It may be taken to follow from this that if the PSR were false or maybe even not known a priori, we wouldn’t know any empirical truths. But we do know empirical truths. Hence, the PSR is true, and maybe even known a priori.

2.2.3. Evolution

One of my graduate students suggested in discussion that if one rejects the PSR, our knowledge of evolution may be undercut. We can use this insight to generate an ad hominem argument for the PSR. Most atheists and agnostics (and many theists as well, but it is to atheists and agnostics that the argument is addressed) believe that there is a complete naturalistic evolutionary explanation of the development of the human species from a single celled organism. I claim that they are not justified in believing this if they do not accept the PSR.

For consider what the argument for thinking that there is such an explanation could be. We might first try an inductive argument. Some features of some organisms can be given naturalistic evolutionary explanations. Therefore, all features of all organisms can be given naturalistic evolutionary explanations. But this argument is as bad as inductive arguments come. The error in the argument is that we are reasoning from a biased sample, namely those features for which we already have found an explanation. Such features are only a small portion of the features of organisms in nature—as always in science, what we do not know far exceeds what we know.

Once we admit the selection bias, the argument becomes: “All the features of organisms for which we know the explanation can be explained through naturalistic evolutionary means, and so all the features of organisms can be explained through naturalistic evolutionary means.” There are at least two things wrong with this argument. The first is that it might just be that naturalistic explanations are easier to find than non-naturalistic ones, and hence it is no surprise that we first found those explanations that are naturalistic. But even if one could get around this objection, it would not obviate the need for the PSR. For the argument at most gives us reason to accept the claim that those features that have explanations have naturalistic evolutionary explanations. The inductive data is that all the explanations of biological features that we have found are naturalistic and evolutionary. The only conclusion that can be drawn without the PSR is that all the explanations of biological features that there are are naturalistic and evolutionary, not that all biological features have naturalistic evolutionary explanations.

A different approach would be to suppose that natural occurrences have naturalistic explanations, and evolution is the only naturalistic form of explanation of biological features that we know of, so that it is likely that the development of the human race has a naturalistic evolutionary explanation. But what plausibility is there in the claim that natural occurrences have naturalistic explanations if one does not accept the PSR for contingent propositions? After all, if it is possible for contingent propositions to simply fail to have an explanation, what reason do we have for confidence that at least those contingent propositions that report natural occurrences have explanations? If “natural occurrence” is taken as entailing the existence of a naturalistic explanation, the argument for an evolutionary explanation of the development of the human race becomes question-begging in its assumption that the development was a natural occurrence. But if “natural occurrence” is taken more weakly as a physical event or process, whether or not it has a natural explanation, then the naturalness of the occurrence does not give us reason to think that the occurrence has an explanation, much less a naturalistic one, absent the PSR. If we had the PSR in play, we could at least try to use a principle, perhaps defeasible, that the cause is ontologically like the effect, so that if the effect is natural, the cause is likely such as well. (It is interesting that this principle itself could be useful to theist with respect to the Gap Problem—see the perfection axiom in Section 5.4.)

Consider a final way to justify the evolutionary claim. We have good inductive reason to think that everything physical obeys the laws of physics. But everything that is governed by the laws of physics has a naturalistic explanation. Hence, the development of the human race has a naturalistic explanation, and an evolutionary one is the best candidate we have.

The claim that everything that obeys the laws of physics has a naturalistic explanation, however, has not been justified. The claim was more plausible back when we thought that everything could be explained in a Newtonian manner, but even then the claim could be falsified. Consider John Norton’s ball-on-dome example (Norton 2003). We have a rigid dome, on the exact top of which there sits a perfectly round ball, and the dome is in a constant downward gravitational field of acceleration g. The dome is rotationally symmetric, and its height as a function of the distance r from its central axis is h=(2/3g)r3/2. It turns out to be consistent with Newtonian physics that the ball should either remain still at the top of the dome or start to roll down in any direction whatsoever, in the absence of any external forces. One might wonder how this squares with Newton’s second law—how there could be an acceleration without an external force. It turns out, however, that because of the shape of the dome, in the first instant of the ball’s movement, its acceleration would be zero, and after that it would have an acceleration given by the gravitational force. The physics would fail to explain the ball’s standing still at the top of the dome or the ball’s moving in one direction or another; it would fail to explain this either deterministically or stochastically. Thus, even Newtonian physics is not sufficient to yield the claim that everything that obeys the laws of physics can be explained in terms of the laws of physics.

And I doubt we do any better with non-Newtonian physics. After all, we do not actually right now know what the correct physics is going to be, and in particular we do not know whether the correct physics will make true the claim that everything that obeys the laws of physics can be explained in terms of the laws of physics. Besides, surely it would be an implausible claim that justification for the claim that the human race developed through evolutionary means depends on speculation about what the final physics will be like.

I do not have an argument that there is no other way of arguing for the evolutionary claim absent the PSR. But, intuitively, if one weren’t confident of something very much like the PSR, it would be hard to be justifiedly confident that no biological features of the human species arose for no reason at all—say, that an ape walked into a swamp, and out walked a human, with no explanation of why.

2.2.4. Inference to best explanation

Suppose we have a phenomenon and several plausible explanations. We then reasonably assume that the best of these explanations is probably the right one, at least if it is significantly better than the runner-up. How we measure the goodness of an explanation is, of course, controverted: prior probability, simplicity, explanatory power, etc. are all candidates. Or, if we’ve ruled out all explanations but one, we take the remaining one to be true (White, 1979)—this is what the maxim that “when you have eliminated the impossible, whatever remains, however improbable, must be the truth” comes down to in Sherlock Holmes’ actual practice (Doyle 1890, p. 93, original italics).

But suppose we admit, contrary to the PSR, the possibility that the phenomenon has no explanation at all. What reason do we have to suppose that the best or the only explanation is likely to be true? To argue for that explanation, we compared it to its competitors. But the hypothesis the phenomenon has no explanation at all was not one of these competitors. Indeed, we do not know how to compare this hypothesis to the competitors. The hypothesis that there is no explanation is in one sense simpler than any explanatory explanation. On the other hand it altogether lacks explanatory power. Still, it is unfair to rule it out just because it lacks explanatory power unless one believes in the PSR.

Perhaps the no-explanation hypothesis can be ruled out not because it is impossible, as the defender of the PSR will say, but because it is simply less probable than its competitors. But does it make any sense to assign a probability to the hypothesis that a brick comes to exist ex nihilo in mid-air in front of us for no reason at all, assuming this is possible? We certainly cannot assign a probability grounded in the laws of nature to a brick’s coming into existence ex nihilo, in the way in which we can to the electron’s moving upwards in the Stern-Gerlach experiment, since the brick’s entry into existence would arguably not be governed by the laws if it happens “for no reason at all.”

But maybe we can argue that such an arising ex nihilo is impossible, since it is contrary to the laws. However, the laws of nature only specify what happens in the absence of external influence. They do not, thus, exclude the possibility of a brick coming into existence by the power of a non-physical being, say God. But if the PSR does not hold, intuitively any laws that do not preclude the possibility of a brick coming into existence by the power of a non-physical being should not exclude the possibility of the brick coming into existence ex nihilo. The possibility of a non-physical being’s producing such a brick shows that there is no innate contradiction between the brick’s coming into existence and there being such-and-such laws of nature. And it would be odd indeed if the laws of nature entailed that any bricks that come into existence should have causes of some sort or other, whether natural or not. Furthermore, if my argument is taken seriously, then we may not have good reason to believe in the laws of nature in the first place—for the phenomena which we tried to explain in terms of them just might be lacking in explanation.

Suppose, however, that we grant that the laws of nature exist and entail that physical events have causes, natural or not, but continue to balk at the full PSR because we are not sure whether non-physical facts have to have explanations. Then at least on probabilistic grounds we cannot exclude the following explanatory hypothesis, available for any phenomenon F: there came into existence, ex nihilo and for no reason at all, a non-physical being whose only basic non-formal property was the disposition to cause F as soon as the being is in existence, a property that the being has essentially, and this being came into existence for precisely the amount of time needed for the activation of this disposition. Why did Jones fall asleep? Because a non-physical being came into existence for no reason at all, a being characterized by an essential dispositio dormitiva and by nothing else. No nomic probabilities can be assigned to the hypothesis of such a non-physical being’s coming into existence. (It might be that there is some argument available that only God can create ex nihilo, and so such a being cannot create a brick ex nihilo. Fine, but at least it should be able to create it out of air.)

One might try to assign non-nomic probabilities to the no-explanation and non-natural-being-ex-nihilo hypotheses in terms of a principle of indifference. But then, the no-explanation hypothesis would be on par with each explanatory explanation. And there would be an infinitude of explanatory hypotheses in terms of non-natural beings that came into existence ex nihilo, for we could suppose that in addition to the disposition to cause F they do have some other essential property (say, being happy, or being beautiful), and differ in respect of it. Why would we take a “normal” scientific explanation over one of these, then?

It is tempting here to say: “Well, we don’t know anything either way about the likelihoods of these weird hypotheses that contradict the PSR. So we should just dismiss them all.” As practical advice for doing our best in finding predictions, this may be fine. But if we are to hope for scientific knowledge, that surely will not do. A complete inability to estimate the likelihood of an alternate hypothesis is surely a serious problem.

It is easy not to take these odd hypotheses seriously. And that may well be because we do in fact have a deep commitment to the PSR and maybe even to a defeasible principle that causes have a resemblance to their effects. If I am right, the PSR is essential to the practice of science, even outside of evolutionary biology.

2.2.5. Why aren’t there widespread violations of the PSR all around?

If the PSR were false, we would expect a profusion of events that would not appear to fit into any kind of nomic causal order. After all, for each way that things could go in accordance with the laws of nature, there is an uncountable infinity of ways—of arbitrary cardinality—that things could, for no reason at all, go contrary to the laws of nature. For instance, if we deny the PSR, then for no reason at all, a cloud of photons, À 9314 in number, could suddenly appear ex nihilo just near the moon, heading for San Francisco. (Because the cardinality is so high, some of the photons would have to share the same quantum state; but photons are bosons, so they should be able to do that.) And the number of ways such things could happen seems to have no limit if the PSR fails. Or perhaps À 9314 non-natural beings could come into existence, each of which could then produce one photon.

Our empirical observations suggest that the probability of such events is very low. On the other hand, if we get our probabilities a priori from some sort of principle of indifference, supposing all arrangements to be equally likely, the messy PSR-violating arrangements would seem much more probable. How to explain the fact that bricks and photon clouds do not show up in the air for no discernible reason? I suggest that the best explanation is that the PSR holds, and that whatever beings there may be (e.g., God) who are capable of causing bricks and photon clouds to show up in the air for no discernible reason are in fact disposed not to do so. We need both parts for the explanation: without the PSR, the possibility of this happening for no reason at all would be impossible to rule out, and without the claim that existing beings are unlikely to cause it, the PSR would be insufficient (this suggests that if the cosmological argument can establish the existence of a first cause, there is reason to think the first cause has a predilection for order, a fact relevant to the Gap Problem).

It may seem that I am caught in a vicious circularity here. I have produced a phenomenon—the lack of weird apparently causeless events—and have suggested that its explanation needs to involve the PSR. But am I not invoking the PSR in supposing that there is an explanation here? No. I am only invoking inference to best, or only, explanation, an ampliative principle that we should all accept. Nor am I applying this principle to some strange fact, like the conjunction of all contingent states of affairs. I am applying the principle to the homely fact that bricks and photon clouds do not show up in the air ex nihilo. And the best explanation of this fact is that they, simply, cannot do that, absent some cause, and there does not in fact exist a cause likely to produce such effects.

One might think that some physical law, say a conservation law, would do the explanatory work here, a principle other than the PSR. But the logical possibility of miracles shows that it should be possible for a supernatural being to cause photon clouds to show up ex nihilo, and if the PSR is false, such supernatural beings could be coming into existence all the time, and causing the weird effects. Our best explanation for why this is not happening is that there is nothing in existence that would be likely to cause such supernatural beings to come into existence, and by the PSR they can’t come into existence uncaused.

2.2.6.1. Alethic modality

Alethic modality is a deeply puzzling phenomenon. Whence the difference between a golden mountain and a square circle? Why is it necessary that 2+2=4, but merely contingent that horses exist? I could become a biologist, but I couldn’t ever be a number or a point in space. What makes that so?

The question here is as to the truth ground of these kinds of facts. I am not asking the explanatory question of why these facts obtain. That in at least some cases is easy to find. A square circle is contradictory, for instance, and had evolution gone somewhat differently, the niche occupied by horses would have been occupied by medium sized and fast reptiles. But what features of reality make these alethic modal facts hold?

Five main kinds of non-revisionist theories have been offered here: narrowly logical, Lewisian, Platonic, Aristotelian-essentialist, and Aristotelian-causal. The first three will be seen to be unsatisfactory and only the Aristotelian theories will remain. Of these the, Aristotelian-essentialist account will have some serious problems with it—and, moreover, seems to require theism, so the agnostic or atheist cannot embrace it as an alternative the Aristotelian-causal one. The remaining theory, the Aristotelian-causal one, turns out to entail a PSR sufficiently strong to run a cosmological argument, given some plausible auxiliary assumptions. Hence we should accept the PSR, unless we have a better account of alethic modality.

I shall now argue for the unsatisfactoriness of the first four theories. I have no argument that there is no better story possible than the Aristotelian-causal one. But until one is found, we should accept this account, and hence the PSR.

In a number of other early modern thinkers, we have the following “narrowly logical” account of modality, probably best developed in Leibniz. A proposition p is necessary if and only if a contradiction can be proved from its negation. Assuming classical logic, as these thinkers did, it follows that necessity is equivalent to provability. And a proposition is possible if and only if no contradiction can be proved from it.

There are counterexamples to this account.

First, we learn from Gödel that for any axiomatization within our reach (any set of axioms we can generate recursively), there will be truths of arithmetic that we can’t prove from the axiomatization. On the narrowly logical account, thus, there are contingent truths of arithmetic. This seems absurd. (For one, what kind of truthmakers would they have?)

Second, necessarily, all horses are mammals. But this is an empirical discovery. We cannot prove it by narrowly logical means. A posteriori necessities like this provide a large family of counterexamples.

Third, it is impossible for anything to cause itself. (If, like Descartes, you disagree, choose another example—maybe, the claim that it is necessarily possible for something to cause itself.) But how would we go about proving this? We might start with some partial analysis of causation. Perhaps a cause has to temporally precede the effect (a dubious thesis in my opinion, but what I say will apply to any story we could fill in here). And nothing can temporally precede itself. But how could we prove that a cause has to temporally precede the effect, and how do we prove that nothing can temporally precede itself?

In two ways, I suppose. First, we might derive these claims from some definitions, say of causation or temporal priority. But, leaving aside the somewhat implausible suggestion that “causation” and “temporal priority” can both be defined, how do we prove that this definition is in fact the right way to define the terms? To show that a definition is correct is beyond the powers of logic narrowly conceived, unless the definitions are stipulative, in which case the proof is trivial. But a stipulative route is unsatisfactory for two reasons. First, the claim that nothing can cause itself is not just a claim involving a stipulative concept of “cause”. Second, even if I have a stipulative definition, I need the principle that if D is stipulatively defined as E (where E is some linguistic expression), then necessarily anything that satisfies D satisfies E. But what grounds the latter necessity? If I say that I can prove it from the definition of “stipulated”, then I go around in a circle—for either the definition of “stipulative” is non-stipulative, in which case it seems we need to go beyond logic narrowly conceived to prove the definition of “stipulative” correct, or else we have a stipulative definition of “stipulative”, and to prove that anything that satisfies D must satisfy E whenever E is the stipulative definition of D, I need to know that necessarily whatever is stipulative has the properties in terms of which the word has been defined.

So the stipulative route to proving that nothing can cause itself won’t work. The only other route is that among our axioms there are substantive axioms about the nature of causation or that there are substantive rules of inference in our logic. Without such axioms or rules of inference, we get nowhere when dealing with a non-stipulative concept. But now note that any axiom gets to be necessary for free on the narrowly logical account. So what would it be that would make it be the case that among our axioms is the claim that, say, causes temporally precede their effects, or whatever other truth it would be from which we were going to prove that nothing can cause itself, while the equally true claim that there are horses is not among the axioms? The intuitive answer is that the claim about causation is more plausibly a necessary truth, while the claim about horses is plainly contingent, but that would be viciously circular. Similarly, if there are substantive rules of inference in our logic, say ones that allow us to infer from x causes y and y causes z that x is not identical with y, the question of what makes these but not other substantive rules of inference (say, the rule that one can derive there are horses from every statement) appropriate is equally problematic as the question of what gets to count as an axiom.

And so the narrowly logical account is of little help—a part of what makes a proposition an axiom seems to be that it is necessary, and a part of what makes a proposition be a rule of inference is that embodies a necessary implication. Moreover, the necessity here is the same sort of necessity we were trying to explicate, so there is really very little gain. Alethic modality remains ungrounded.

Our last example has shown the general problem with narrowly logical accounts of modality: the grounding burden simply shifts to the question of the choice of the axioms and/or rules of inference and that question we cannot answer with the resources of the view in question.

An early modern answer one might try is this: We take as axioms all and only the claims that are clear and distinct. An anachronistic objection is that this does not solve the Gödelian problem. A counterexample-based answer is that the claim that I exist seems to be as clear and distinct as anything can be, and yet is contingent. Moreover, plausibly, there are necessary truths that are far beyond our ken and cannot be derived from clear and distinct truths within our ken. (If we assume the existence of God, this is very plausible: there surely are many such facts about him.) And, besides, we no longer have much of a handle on the notion of clear and distinct claims, and to use them to ground necessity would be to confuse facts about our doxastic faculties with metaphysics.

The narrowly logical view is distinctly unsatisfactory. Let us thus continue our brief survey.

2.2.6.3. Lewisian account of modality

The Lewisian account, also known as Extreme Modal Realism (EMR), says that a proposition is possible if and only if it holds in some possible world, and necessary if and only if it holds in all possible worlds. This is only going to be of help if we have an independent account of possible worlds, and indeed EMR supplies one. A possible world is a maximal spatiotemporally interconnected aggregate of things. (We can also stipulate that abstract entities count as existing in every world.) We live in one of these worlds, the actual world, and there are infinitely many others. Every way that things could have been is a way that things are in some world. We then make a distinction between existence and actuality. Something exists provided it exists in some world or other. Something is actual provided it exists in the actual world.

EMR has a number of problematic consequences. For instance, if EMR holds, consequentialistic moral reasoning breaks down completely, because no matter what I do, the overall consequences in reality are the same, since reality always already contains all possible worlds. Lewis thinks that we can restrict our concern to those who exist in our world, and only count what happens to them as relevant. But this neglects the importance of overall consequences. Even deontologists need consequentialistic moral reasoning. If I am to give money to one of two charities, and everything is otherwise morally on par, I should choose the one giving to which will produce better consequences.

Lewis, however, thinks that what matters ethically is not just the consequences, but that I have produced them (Lewis 1986, p. 127). I cannot affect what happens in other worlds, but I can be the cause of goods in our world. Of course, this makes no difference in the space of all possible worlds—in infinitely many of them, people very much like me are causes of goods and in infinitely many of them, people very much like me are not causes of goods, and the distribution of worlds is not affected by my action. But my relationship to the goods is affected.

However, this unacceptably reduces the moral weight of consequences. Suppose that either you or I can operate on a patient. The operation is perfectly safe, but I am better than you at this particular operation, and so the patient will recover somewhat faster after the surgery if I do it. I thus have good reason, when we are deciding which of us will perform the operation, to volunteer to do it. And if I do perform the operation, then I additionally gain the agent-centered good of my being the cause of the patient’s improvement. However, the latter consideration is surely of very little moral weight. After all, the same kind of consideration would also give you reason to do the surgery, but this consideration should be trumped by the good of the patient. Even if my skill at this operation is only slightly better than yours, so that the patient will likely recover slightly better, all other things being equal this fact should trump your reason to be the cause of the patient’s improvement. Thus, the agent-centered reason of wanting to be the cause of good is in a case like this of very low weight—the consequences are the main consideration.

This isn’t so in every case. When there is a close relationship between me and someone else, then it may matter very much that I be the one to benefit that person. However, when there is no particularly morally important relationship—and merely being spatiotemporally connected is very low on the scale of moral importance—it should not matter or at least matter much.

On Lewis’s view, however, my reason to help strangers is only the agent-centered reason to be the cause of goods, because the consequences are always the same. But since the agent-centered reason to be the cause of goods has extremely low weight, it follows that EMR radically lowers the weight of reasons to help strangers. If we accept a more traditional assessment of the weight of these reasons, we shall have to reject EMR.

Instead of cataloguing further problems entailed by EMR, I shall give what I take to be one of the deepest criticisms, which I believe is due to van Inwagen. The criticism is simply that the existence of infinitely many maximally spatiotemporally interconnected aggregates has nothing to do with modality. If we found out that reality contains infinitely many maximally spatiotemporally interconnected aggregates, we would simply have learned that the actual world is richer than we thought—that it contains all of these island universes—rather than learning something about the space of possibilities.

Here is a variant on the objection. Suppose that there exist infinitely many maximally spatiotemporally interconnected aggregates, and some of them contain golden mountains but none contains unicorns. It would follow that golden mountains are possible, simply because what is actual is also possible, but surely it would not follow from this fact that unicorns are impossible. And if there were only one spatiotemporally interconnected aggregate, namely ours, it would not follow that modal fatalism is true—that every actual truth is necessary. Yet on Lewis’s view, if no unicorns were found in any island universe, it would follow that unicorns are impossible, and if there were only one island universe, it would follow that every actual truth is necessary, since things couldn’t be otherwise than they are then.

Now Lewis, of course, thought there was more than one universe, and indeed that there was a universe that contained unicorns. He believed this because he accepted a recombination principle that said that one can cut up the ingredients of one world and rearrange them in any geometrically available way, and the resulting rearrangement would be exemplified in some world or other. However, while he accepted the recombination principle, the recombination principle is not, on his view, a part of what makes alethic modal claims true. What makes alethic modal claims true just are the facts about universes, and we have seen that that is not correct.

We should thus reject EMR and keep on searching for a good account of modality.

2.2.6.4. Platonic account of modality

The most promising contemporary realist alternative to Lewis’s account of possible worlds are the abstract worlds accounts promoted by Robert M. Adams (1974) and Alvin Plantinga (1974). On their accounts, worlds turn out to be abstract Platonic entities, exactly one of which is instantiated by the universe, where “the universe” is defined to be the aggregate of all existing or occurring concrete entities, and this is the world that is absolutely actual. I will focus primarily on the Adams permutation of this account.

We thus start off by introducing propositions as theoretical abstract entities that are the bearers of truth-values and are needed to explain what it is that sentences express, what the objects of beliefs and propositional attitudes are and what paraphrases preserve, somewhat as electrons are needed to explain various physical phenomena. Some propositions, namely the true ones, are related to things and events in the universe, with the relation being one of the propositions being made true by or representing these things and events in the universe. If things in the universe were otherwise than they are, then different propositions would stand in these relations to things in the universe—if there were unicorns, then the proposition that there are unicorns would stand in the relation of being made true by to some things, namely the unicorns in the universe.

Note that the theoretical reason for believing in these Platonic propositions is largely independent of issues of modality. Adams then constructs a possible world as a maximal consistent collection of propositions. (An argument is needed that such collections exist, but let that pass.) Exactly one world is then absolutely actual: it is the one all of whose propositions are true. A proposition can be said to be true at a world providing it is one of the propositions that are members of the collection of propositions that the world is identical with. Note that because the worlds are Platonic entities, I had to distinguish between the concrete universe, which we physically inhabit, and the actual world which is the collection of all true propositions.

One might object to the Platonic approaches on the grounds that they all involve queer entities. Not only are we required to believe in Platonic beings, but, as Lewis notes, we are to believe that there is a magical relation of representation holding between Platonic beings such as propositions and the concrete entities that make them true, with it being contingent which propositions enter into those relations since it is contingent which propositions are true. What is it, then, that picks out one relation in the Platonic heaven rather than another as the relation of representation?

The proponents of these Platonic worlds can argue, however, that they have no need to answer this question. The relation of representation is one of the primitive terms in their theory, and it is not a primitive chosen ad hoc to explain possible worlds, but a primitive needed for other explanatory purposes, such as for making sense of our practices of claiming, believing and paraphrasing. Nonetheless, if we had some way of pointing out this relation within the Platonic universe of all relations, we would be happier as theorists.

These Platonic theories are expressly non-reductive as accounts of possibility, unlike Lewis’s theory. For Adams, a possible world is a maximal consistent collection of propositions, which is just the same as saying it is a maximal compossible collection of propositions. On this theory, there is a primitive abstract property of possibility or consistency that applies to individual propositions and to collections of them. One could also take necessity to be the primitive concept, but this would not change anything substantially.

That the Platonic accounts are non-reductive is only a problem if a reductive account of possibility is available. However, the most plausible account claiming to be reductive is Lewis’s, which is too paradoxical to accept. But while a complete reduction is probably impossible, it could be desirable to give at least a partial reduction, on which the whole realm of alethic possibility would be seen to have its root in some more comprehensible subclass. An example of an otherwise implausible theory that would provide such a reduction would be an account on which a proposition is possible if and only if Alvin Plantinga could conceive its being true: all of modality would then be reduced to Alvin Plantinga’s considerable powers of imagination. Claims about Plantinga’s powers are still modal claims, but of a more comprehensible sort than claims about the possibilities of unicorns and zombies. However, these Platonic accounts do not succeed in performing this more limited reduction either.

Adams’ theory is an actualist one. His possible worlds are built up out of things that are actual. These abstracta actually exist—indeed, necessarily so—and an actualist theory is one that grounds possibility in actually existent realities. On the other hand, Lewis’s other worlds are not actual entities by Lewis’s indexical criterion, as they are not the world in which my tokening of the word “actual” in this sentence occurred. If we think of possible worlds as possibilities for our universe, then there is a sense in which Adams and Plantinga have grounded possibilities in actuality, thereby answering to the Aristotelian maxim that actuality is prior to possibility.

However, in a deeper way, the Platonic approach is not faithful to what the Aristotelian maxim affirms. When Aristotelians say that a possibility is grounded in an actuality, they mean that actuality includes some powers, capacities or dispositions capable of producing that possibility, which of course once produced would no longer be a mere possibility. This is clearest in the paradigm case where the actuality is temporally prior to the possibility. Aristotle’s favorite illustration is how the actuality of one man makes possible the existence of a future man through the first man’s capability for begetting a descendant. If we find attractive the idea that possibilities should be grounded in actuality in the stronger Aristotelian sense, then the Platonic approach will be unsatisfactory, because Platonic entities, in virtue of their abstractness, are categorially barred from entering into causal relations, and hence cannot make possibilities possible by being capable of producing them.

Moreover, an Aristotelian can argue that in fact there are capabilities and dispositions sufficient to ground the truth of at least some possibility claims. That I could have been a biologist is very plausibly made true by my capacities and dispositions and those of various persons and things in my environment. These capacities and dispositions are concrete real-worldly things, albeit ones having modal force. Hence, in fact, we do not need a Platonic realm to make at least some possibility claims true. Indeed, the facts about the Platonic realm—about propositions’ having or not having some primitive property—are interlopers here. Just as the statement that I could have been a biologist was not made true by what my Lewisian counterparts in other worlds do, so too it is not made true by abstract properties of Platonic abstracta. The common intuition behind both cases is that it is something in me and my concrete environment that makes the statement true.

This, however, creates a major problem for the Platonic approach. On the Platonic approach, what makes it possible that I have been a biologist is that the abstract proposition (an entity in the Platonic heaven) that I have been a biologist has the abstract property of possibility. But we have just seen that there are concrete capacities and dispositions in the universe that are by themselves sufficient to make it possible that I have been a biologist. We thus have two different ways of characterizing possibility: one is via concrete this-worldly Aristotelian properties of concreta which really do exist—the Platonist should not deny this—and the other is via abstract Platonic primitive properties of abstracta. Moreover, anything that is possible on the Aristotelian grounds will be physically possible, and hence also logically possible, and thus possible on Platonist grounds (though prima facie perhaps not conversely). But now we can ask: Why is this so? Why is there this apparent coincidence that anything made possible by this-worldly powers and capacities and dispositions happens to correspond to a proposition in the Platonic realm that has a certain abstract property? The Platonist is unable to explain this coincidence between powers in our universe and abstract facts about the Platonic realm given the lack of causal interaction between the two realms.

2.2.6.5. Aristotelian-essentialist account of modality

Aristotle’s own account of modality seems to have been based on the idea that a sentence is necessarily true if and only if it holds always. Then, a sentence is possibly true if it holds at some time. I shall not consider this account further. It is not clear that in characterizing “necessarily” in this way one is really talking of the same thing we are when we say that necessarily there are no square circles. We certainly mean more by saying that there can be no square circle than that just that there have never been, nor are, nor ever will be any square circles. Granted, if we adopt some kind of principle of variety, on which given infinite time every possibility is realized, we might get out of this Aristotelian story an account that is extensionally acceptable. However, that account would still face many of the same problems Lewis’s account faces—indeed, it would be just like Lewis’s account, but with time-slices replacing universes. In particular, the objection that we’re not talking about modality at all would be to the point. If it should turn out that the past, present and future of our world contain no golden mountains, that would say nothing about whether golden mountains are possible.

But while Aristotle’s own account of modality was flawed, two somewhat different accounts have been derived from ingredients of Aristotelian ontology. One of these grounds modality in the essences of things and takes necessity to be the primitive notion. The other account grounds modality in causal powers and takes possibility to be more primitive. I shall begin by discussing the account based on essences (cf. O’Connor 2008).

Things that exist have essences. These essences, on this account, constrain what properties these things can have. Thus, a horse cannot be immaterial, and a dog cannot become a cat. A proposition is impossible provided that it affirms something contrary to the essences of things.

There are several objections to this rough sketch of a view. First, maybe it is plausible that the essence of a horse encodes that a horse must occupy space. But what makes it necessary that horses must occupy space or be green? Do we really want to suppose that for every property P, the essence of a horse contains in itself the specification that a horse occupies space or has P? An affirmative answer appears implausible. Why should the essence of a horse include the specification that horses occupy space or are cats?

This objection is not just an incredulous stare. Horses could surely exist without any cats in existence. But the essence of a horse in some way presupposes catness. It follows that it makes sense to talk of catness—the essence of cats—apart from cats, since horses could exist apart from cats, and hence the essence of a horse could exist apart from cats. The Aristotelian, however, cannot tolerate this, unless the Aristotelian is a theistic Aristotelian who accepts that all essences have some kind of an existence in the mind of God. Thus, unless one accepts theism, the theory seems to be unsatisfactory.

But maybe I was too fast. Perhaps it is not that the essence of a horse contains all the necessary truths about horses, but that all the necessary truths about horses can be derived from the essence of a horse as combined with all other essences there are. That every horse occupies space or is a cat can be derived from the essence of a horse and the essence of a cat.

But “derived” surely means “logically derived”. And so it turns out that the Aristotelian-essentialist needs elements of the narrowly logical view. Once again, the same question comes up: What grounds the choice of axioms or rules of inference? However, the Aristotelian is better off here than the proponent of just the narrowly logical view, because the truths contained in the essences of things provide a rich set of non-arbitrary axioms.

Aristotelian-essentialists might then be able just to specify, say, some plausible version of logic (e.g., some second order quantified modal logic), and claim that our thought and language presupposes the truth of this logic. They could then say one of two things about the status of this logic. First, they could say that the basic rules of this logic are grounded in some or all essences. For instance, maybe every essence encodes the rules of logic, or maybe one could make the theistic move of saying that the essence of God encodes these rules. In this way, the rules of logic would be on par with other truths within the essences of things, such as the truth that horses occupy space that is encoded within the essence of a horse. This construal of the rules of logic would be to make the rules of inference effectively into facts or propositions written into essences, such as:

(6) For all p and q, if it is the case that if p then q, and if it is the case that p, then it is the case that q.

But the rules of logic cannot be construed in this way without losing what is essential to them, namely their applicability. If modus ponens is just the fact (6) or maybe the necessary truth of (6), then how do you apply modus ponens? You have p, you have if p then q, and then you know that the antecedent of the big conditional in (6) is satisfied. But how do you know that the consequent of the big conditional in (6) holds, namely that it is the case that q? You know it by modus ponens. But modus ponens is just the truth (6), so you need to once more go back to (6) to apply it to the case where you have (6) and the antecedent of (6). In other words, you need modus ponens to apply modus ponens if modus ponens is just a truth like (6), and a vicious regress ensues. Applicability requires that the truths of logic be more than just statements.

A better solution for advocates of the Aristotelian-essentialist account of modality would be to say that logic narrowly construed is something deeper than the necessities they are grounded in essences. One could, for instance, take the Tractarian line that narrowly logical impossibilities cannot even be thought.

But we have not exhausted all the objections to the Aristotelian-essentialist view. Consider truths that hold of all things no matter what essence they might have. No entity has a shape that is both a square and a circle (at the same time and in the same respect), and no entity is cause of itself. What makes these be necessary truths? Granted, it may be encoded in the essence of every actually existing thing that nothing having that essence is a square circle, or is causa sui, or exists in a world where some (actually true) Gödelian unprovable arithmetical claim fails to hold, but a seemingly stronger claim is true: there could not be anything, whether with one of these essences or with some other essence, that is a square circle or that is causa sui or that exists in a world where some particular (actually true) Gödelian unprovable arithmetical claim fails to hold. Maybe the square circle case can be handled through a narrowly logical move as above, but it may not be plausible that this can be done with the causa sui case, though perhaps there is some Tractarian line that one can take that self-causation cannot even be thought. But in any case the Tractarian line does not seem to help much with the Gödelian worry.

Moreover, consider the question of what essences can possibly exist (in mind or reality). The story we have so far is that something is possible provided its existing is not contradictory to the truths encoded in those essences that exist. This, however, seems to let in so many essences that a certain amount of skepticism is engendered. For instance, it seems that there will be a possible world w which is just like this one, with this exception. The essence of human beings does not exist at w, but instead there are entities that physically behave just like human beings, except that instead of being a single natural kind, they are divided up into two natural kinds defined by different essences: there are those who have an even number of hairs on their bodies and there are those who have an odd number of hairs on their bodies. As soon as one of these beings gains or loses a hair, it perishes, and a new being comes to exist, physically and psychologically just like it, apart from that hair. Otherwise, everything is as it is in our world. After all the existence of such being and such essences does not seem to contradict the truths encoded in any of the essences that exist, such as the essence of the live oak or the photon. But once we allow that w is possible, do we have good reason to suppose that it isn’t our world, that in our world there aren’t different essences for people with even numbers of hairs and for people with odd numbers of hairs?

The problem, thus, is with what constrains what essences there could be. One answer, inspired by the static character of Aristotle’s universe, would be that all the essences that can exist in fact do exist, or at least existed, exist or will exist. However, a crucial difficulty remains as to what “can” could mean here. What constrains which essences can exist?

Some of these problems can be solved by going a theistic route. Perhaps there is a God whose essence encodes necessary truths not just about himself but about others, such as that there can be no square circles, and that certain weird essences can’t exist.

In fact I think one can argue that only a necessarily exemplified essence can solve the difficulties here. For, on the present account it seems very likely that an essence cannot in any way constrain what happens in any worlds in which that essence is not exemplified. An essence E can exclude some worlds containing an exemplification of it from including something incompatible with E, but it does not have anything to say about what things are like in worlds where there is no exemplification of E.

Suppose now that none of the essences that are exemplified in our world is necessarily exemplified. We should then be able to describe a world full of really, really weird things—beings with essences that make their kinds be defined by the number of hairs, self-caused beings, and the like—as long as we do not transgress narrowly logical norms and as long as we take care to include none of the beings of our world. And such a world will be possible, since the essences that exist in our world will be irrelevant to what goes on in that world, since our world’s essences will be unexemplified there. Likewise, a completely empty world would be possible then, a world with no essences exemplified. In that world it will be true that everything that is narrowly logically possible is metaphysically possible, since there will be no constraining essences at all. In particular, in that world it will be possible that Gödelian claims of arithmetic that are true at our world are false. And of course it would then be the case that S5 is false, but the Aristotelian-essentialist may not mind that consequence.

If we think that the space of all possible worlds is not such a slum as to include all such worlds, we have to think that at least one of the beings that exist in our world is such that its essence is necessarily exemplified, and that the essences of the necessary beings place constraints on what sorts of essences there can be, what sorts of arithmetical truths there can be, and so on.

There are now two difficulties. First, what does it mean that the essence of some being is necessarily exemplified? If an essence E cannot constrain what happens in worlds where it doesn’t exist, it is unclear how E could prevent the actuality of worlds that do not contain an exemplification of E. Second, just how does an essence place such global constraints on worlds and on what essences are exemplified in them?

The first difficulty forces us, I think, to modify the account. Let N be one of the necessarily exemplified essences. Even if some necessities are grounded in essences, the necessity of N’s being exemplified cannot be grounded in an essence, at least not in the sense in which essences exclude their being exemplified together with something incompatible, since by doing so the essences do not exclude their not being exemplified. So there is some other kind of necessity that the exemplification of N has. This is in general not going to be narrowly logical necessity, since unprovable arithmetical truths will follow from the exemplification of all the necessarily exemplified essences.

The account now becomes rather less attractive. It posits three kinds of modality as together yielding metaphysical alethic modality: the necessity of the exemplification of certain essences, the necessities encoded in essences, and narrowly logical necessity. Moreover, our best story as to what a necessarily exemplified essence that constrains reality outside of itself is like is that it is the essence of God, so this is not an escape an atheist is likely to want to take. And we have no story yet about what necessary exemplification is grounded in.

There is a way of making something similar to the above story work. If we posit that all contingently exemplified essences must originate from something, then we might get the idea of an essence that does not itself originate from anywhere, an essence which is necessarily exemplified, so that the contingently exemplified essences get their reality from at least one necessarily exemplified essence or from the exemplifier of such an essence (in the case of God, if divine simplicity holds, the two options will come to the same thing). It will also be plausible that just as the essences originate from something, so do their exemplifications—on an Aristotelian view, essences aren’t completely independent of their exemplifications. All of this focuses the attention, however, on causation, and leads us to the last account of modality, the causal one.

Another thing that leads us away from the Aristotelian-essentialist account of modality is the intuition that I used against the Platonic view. One can give a simple account of why I could be a biologist in terms of my abilities and the powers of various entities in my environment. On the Platonic side, I wondered why there is this coincidence between what happens in the Platonic realm and earthly powers and capacities. Now one can wonder why there is a coincidence between powers and essences. Why is it that I can’t do anything that contradicts the essence of any entity in existence? Perhaps this question is somewhat less pressing than on the Platonic side. After all, maybe my powers are grounded in my essence. But it is still not clear why something couldn’t have the power to act contrary to its essence.

The critiques of the Platonic and Aristotelian-essentialist accounts point the way toward an account where causation is central. Here is a sketch of an account that does this. Say that a non-actual state of affairs S is merely possible provided that something—an event or substance or collection of events or substances, say—exists (in the tenseless sense: existed, exists presently, exists eternally or will exist) with a causal power of bringing about S, or with a causal power of bringing about something with a causal power of bringing about S, or with a causal power of bringing about something with a causal power of bringing about something with a causal power of bringing about S, or more generally provided that something exists capable of originating a chain of exercises of causal power capable of leading to S. We then say that a state of affairs is possible if it is either actual or merely possible, and that it is necessary when its non-occurrence is impossible. A proposition, then, is possible provided it describes a possible state of affairs, and necessary if it describes a necessary state of affairs.

This account has the advantage of reducing metaphysical possibility to causal possibility. One might think this is not much of a gain—we’re still stuck with some primitive modality. Yes, but the primitive modality we are left with is a modality that we have a better handle on, and a better epistemological story about. We ourselves exercise causal powers all day long, and run up against the causal powers of other entities. Our scientific observation of the world gives us information as to what is and what is not within the powers of things. For instance, we know that unicorns are possible, because we know that it would be within the powers of natural selection and variation processes to have produced unicorns.

Moreover, we are probably going to need causal powers, or something like them, in our metaphysics even if we have an independent story about metaphysical alethic modality. It does, after all, seem to be a feature of the world that entities can produce effects. So by reducing metaphysical to causal modality, we seem to make a real gain in elegance and simplicity.

Furthermore, this account lets us handle a spectrum of modalities in a uniform framework by restricting the entities in the causal chains that define mere possibility and the causal relations between them. For instance, a non-actual state of affairs is physically causally merely possible provided that it can be produced by a causal chain consisting purely of physical entities, and starting with something physical. A state of affairs is temporally merely possible provided that it is not actual but can be produced by a chain of exercises of causal power starting with something in the present or future.

But what is of most relevance to this chapter is that, given some plausible assumptions, the Aristotelian-causal account, perhaps surprisingly, entails a version of the PSR: every contingent state of affairs has a causal explanation, i.e., an explanation based on facts about contingent exercises of causal powers, perhaps combined with some necessary truths.

For the argument, I need a prima facie weaker version of the Brouwer axiom. The Brouwer axiom in general states that if p holds, then it is a necessary truth that p is possible. The weaker version of it that I need is:

(7) If p holds contingently, then it is possible for p to be both possible and false.

This follows from the full Brouwer axiom, since if p holds contingently, then p possible, and so it is necessarily possible, but since it is contingent it is possibly false, so possibly it is both false and possible. And Brouwer, in turn, follows from S5.

Suppose for a reductio that a contingent state of affairs E has no causal explanation. Let E* be the state of affairs of E’s obtaining without causal explanation. Then E* is a contingent state of affairs. By the weaker version of the Brouwer axiom, it is possible that E* does not obtain but is nonetheless possible. Let us suppose a possible world w where that happens. Here, the use of possible worlds is inessential, but it helps make the argument clear. In w, E* does not obtain but is possible. Thus, there is a cause C in w which could initiate a chain of exercises of causal powers capable of leading to E*’s obtaining. But that is absurd, since in doing so the chain would give a causal explanation of E as well as leading to E’s not having a causal explanation!

One might deny Brouwer, as well as Brouwer’s weaker cousin (7), and hold on to the Aristotelian-causal account in the absence of the PSR. But the Brouwer axiom is intuitively plausible: however else things might gone than they did, it would still be true that they could have gone as they actually did.

Without the Brouwer axiom, we can give an alternate argument for the PSR based on the following highly plausible material conditional:

(8) If the PSR is true in all possible worlds with the possible exception of the actual world, then the PSR is in fact true in all possible worlds.

It would be incredibly bad luck for us to inhabit the one world where the PSR is false, if there were one. Moreover, if (8) is false, the following absurdity ensues: the PSR is false, but had I skipped breakfast this morning, it would have been true (since it is true in all possible worlds in which I skip breakfast this morning, as it is true in all possible worlds but the non-actual one, and in the actual one I had breakfast). And even someone who is willing to embrace this absurdity should still accept the cosmological argument, since the cosmological argument could be run in the world where I skip breakfast this morning, and it would be an even greater absurdity to suppose that God does not in fact exist but would have existed had I skipped breakfast this morning. And this would in fact be a contradiction, not just an absurdity, if God is a necessary being.

To show the PSR to be true given (8), for a reductio suppose that there is a possible world w, distinct from the actual world, but in which the PSR does not hold. Let E be a state of affairs in w that has no causal explanation. If E does not obtain in the actual world, let F=E. Otherwise, let F be the conjunction of E with some other state of affairs obtaining in w that does not obtain in the actual world—there must be such, since w is not the actual world, and hence different states of affairs obtain in w than in the actual world. In either case, F is a state of affairs in w that has no causal explanation. Let F* be the state of affairs of F’s obtaining with no causal explanation. Then F* is a possible state of affairs, but is not actual since F does not obtain in the actual world. But then there is something that can initiate a chain of causes leading to F*, which, as in the Brouwer-based argument, is absurd, since the chain of causes will lead to F’s obtaining, as well as to F’s not having a causal explanation.

Thus, the Aristotelian-causal account of modality leads to the PSR, while the main alternatives to this account of modality are unsatisfactory and/or require something like theism anyway. This gives us a powerful reason to accept the PSR.

2.2.7. Philosophical argumentation

It is morally acceptable to redirect a speeding trolley from a track on which there are five people onto a track with only one person. On the other hand, it is not right to shoot one innocent person to save five. What is the morally relevant difference between the two cases? If we denied the PSR, then we could simply say: “Who cares? Both of these moral facts are just brute facts, with no explanation.” Why, indeed, suppose that there should be some explanation of the difference in moral evaluation if we accept the denial of the PSR, and hence accept that there can be facts with no explanation at all?

Almost all moral theorists accept the supervenience of the moral on the non-moral. But without the PSR, would we really have reason to accept that? We could simply suppose brute contingent facts. In this world, torture is wrong. In that world, exactly alike in every other respect, torture is a duty. Why? No reason, just contingent brute fact.

The denial of the PSR, thus, would bring much philosophical argumentation to a standstill.

An interesting thing about this argument is that it yields a PSR not just for contingent truths but also for necessary ones.

2.2.8. Justification via the sense of deity

If God exists, then the PSR for contingent propositions is true. Why? Because God’s activity ultimately explains everything. This is going to be clearest on views on which God’s activity alone explains everything, and that is going to be most plausible on Calvinist-type views, but also seems correct on any theological account that has a strong view of divine concurrence with creaturely activity. Moreover, the inference from God’s being the creator and sustainer of everything to the claim that divine activity provides the explanation of everything contingent, or at least of everything contingent that is otherwise unexplained (this variant might be needed to handle creaturely free will), is a highly plausible one. Thus, someone who has good reason to accept theism has good reason to accept the PSR.

Now one might think that this is a useless justification for the PSR if we’re going to use the PSR to run a cosmological argument, since then the cosmological argument will be viciously circular: the conclusion will justify the PSR, whereas the PSR is a premise in the argument.

However, recently Daniel Johnson (forthcoming) has come up with a very clever account showing that a cosmological argument based on the PSR could still be epistemically useful even if the PSR is accepted because of the existence of God (he also applies the view to the possibility premise in the ontological argument). Suppose that, as Calvin and Plantinga think, there is a sensus divinitatis (SD) which non-inferentially induces in people the knowledge that God exists—at least absent defeaters—and tells them something about God’s power and nature.

Suppose that Smith knows by means of the SD that God exists. From this, Smith concludes that the PSR is true—this conclusion may not involve explicit reasoning, and it is one well within the abilities of the average believer. Smith then knows that the PSR is true. Next, Smith sinfully and without epistemic justification suppresses the SD in himself, and suppresses the belief that God exists. If Calvin’s reading of Romans 1 is correct, this kind of thing does indeed happen, and it is why non-theists are responsible for their lack of theism. However, the story continues, the suppression is not complete. For instance, Smith’s worshipful attitude towards God turns into an idolatrous attitude towards some part of creation. It may very well happen, likewise, that Smith does not in fact suppress her belief in the PSR, though she forgets that she had accepted the PSR in the first place because she believed in God. Indeed, this situation may be common for all we know.

Johnson then claims that Smith remains justified in believing the PSR, just as we remain justified in believing the Pythagorean theorem even after we have forgotten from whom we have learned it and how it is proved. Thus, Smith continues to know the PSR. The cosmological argument then lets Smith argue to the existence of God from the PSR, and so Smith then can justifiably conclude that God exists. Of course, unless Smith has some additional source of justification for believing the PSR, Smith has no more justification for believing that God exists than she did when she learned about God from her SD. So the argument has not provided additional evidence, but it has restored the knowledge that she had lost.

We have a circularity, then, but not one that vitiates the epistemic usefulness of the argument. Irrational suppression of a part of one’s network of belief can be incomplete, leaving in place sufficient beliefs allowing the reconstruction of the suppressed belief. A similar thing happens not uncommonly with memory. Suppose I am trying to remember my hotel room number of 314. I note to myself that my hotel room number is the first three digits of p. Later I will forget the hotel room number, but remember that it is identical to the first three digits of p, from which I will be able to conclude that the number is 314. My reason for believing the number to be identical to the first three digits of p was that the number is 314, but then after I will lose, through a non-rational process of forgetting, the knowledge that the number was 314, I will be able to recover the knowledge by using a logical consequence of that very piece of knowledge. In doing so, I do not end up with any more justification for my belief about the room number than I had started out with, but still if I started out with knowledge, I end up with knowledge again.

This means that an argument where a premise was justified in terms of the conclusion can be useful in counteracting the effects of non-rational or irrational loss of knowledge. This means that the cosmological argument could be useful even if none of the arguments for the PSR given above worked, and even if the PSR were not self-evident. For some people may know that the PSR is true because they once knew that God exists. They lost the knowledge that God exists, but retained its shadow, the entailed belief that the PSR is true.

2.3. Objections to the PSR

2.3.1. Modal imagination argument

One can, arguably, imagine that a brick pops into existence uncaused. Therefore, one might conclude that it is possible that a brick pops into existence uncaused, and hence that the PSR is not a necessary truth. This is a popular Humean argument against the PSR.

The defender of the PSR can, of course, simply insist that the inference from imaginability to possibility is defeasible. After all, someone might imagine that a certain straightedge and compass construction trisects an angle, and if the inference from imaginability to possibility were indefeasible, it would follow that the construction possibly trisects an angle. But a mathematical construction possibly (in the metaphysical sense) trisects an angle if and only if it actually does so, and in fact we know that angles cannot be trisected with straightedge and compass. So the inference had better be defeasible. The defender of the PSR can then claim that the arguments for the PSR are so strong that the argument from imaginability of PSR-failure, being defeasible, does little to shake our confidence in the PSR.

However, there is a better solution for the defender of the PSR, and this is to question the claim that the opponent has actually imagined a brick popping into existence uncaused. It is one thing to imagine something without simultaneously imagining its cause, and another to imagine something along with the absence of a cause. In fact, the task of imagining absences as such is a difficult one. If I tell an ordinary person to imagine a completely empty room, the subject is likely to imagine an ordinary room, with walls but no furniture. But has the subject really imagined an empty room? Likely not. Most likely the imagined room is conceptualized in a way that implies that it has air in it. For instance, we could ask our subject what it would be like to sit in that empty room for eight hours, and our subject is unlikely to respond: “You’d be dead, since the room has nothing in it, and hence no oxygen either.”

Could one with more directed effort imagine a room without any air in it? I am not at all sure of that. While we have the concept of vacuum as the absence of anything, it is not at all clear that we can imagine vacuum. Our language may itself be a giveaway of what we imagine when we imagine, as we say, a room “filled” with vacuum—perhaps we are not really imagining an empty room, but one filled with some colorless, frictionless, zero-pressure substance. Moreover, most likely we are imagining the room as embedded in a universe like ours. But a room in a universe like ours will be pervaded with quantum vacuum as well as with electromagnetic and other fields, and perhaps even with spatial or spatiotemporal points. Whether these “items” count as things or not is controversial, of course, but at least it is far from clear that we’ve really imagined a truly empty room.

It is true that philosophers sometimes claim that they can imagine a world that, say, consists only of two iron balls (Black 1952). But a claim to imagine that is surely open to question. First of all, the typical sighted person’s imagination is visual. The balls are, almost surely, imagined visible. But if so, then it is an implicit part of what one is imagining that there are photons bouncing off the balls. Furthermore, unless one takes care to specify—and I do not know how one exactly one specifies this in the imagination—that the balls obey laws very different from those of our world, there will constantly be occasional atoms coming off the edges of the balls, and hence there will be a highly diffuse gas around the balls. Suppose all of this physics is taken care of by our careful imaginer. Still, have we really imagined a world containing only two balls? What about the proper parts of the billiard balls—doesn’t the world contain those? What about properties such as roundness, or at least tropes such as this ball’s roundness? And aren’t there, perhaps, spatial or other relations between the balls? We see that unless one is a most determined nominalist, the content of the imagined world is going to be rather richer than we initially said. There are details implicit in the imagined situation which we have omitted.

There may, however, be a way we can imagine an absence. We can probably imagine absences of particular kinds of things in a particular area of space-time. Certainly I can imagine a room free of talking donkeys, or even of donkeys in general. Moreover, I can probably imagine a room with no particles or electromagnetic fields in it. But that is not the same as imagining a truly empty room. A truly empty room does not have any other kinds of fields in it at least if fields are things; there are no points of space or spacetime in it; and it certainly has no ghosts, angels or demons in it. But no list of kinds of things that we imagine as absent from the room will assure us of the literal and complete emptiness of the room. For there may always be a different kind of being, one utterly beyond the powers of our imagination, whose absence from the room we have failed to imagine. Nor will it do to imagine “unimaginables” as missing, since “unimaginables” are not a genuine kind of thing, but surely a mix of very different kinds of possibilia—it seems highly plausible that there are many kinds of possible things beyond our wildest imagination.

Similarly, we can imagine a brick coming into existence in the absence of a brickmaker, a brick not resulting from the baking of clay, a brick not made by an angel, demon or ghost. But that is not the same thing as imagining a brick that comes into existence completely causelessly. To imagine that, we would need to imagine every possible kind of cause—including the unimaginable ones—as absent. That seems to be a feat beyond our abilities. We can, of course, say the words “This is causeless” both with our lips and with our minds while imagining the brick, but the claim that whenever one can imagine an F and say of it, with lips or minds, that it is a G, then possibly there is an F that is a G, would not only be highly defeasible, but would surely be a non-starter. I can imagine a circle and say the words “This is a square” while imagining it.

Moreover, in general, when we imagine a situation, we imagine not a whole possible world, but a part of one, and our imagination is neutral on whether there are further support structures. I imagine three billiard balls on a billiard table. Probably it’s part of my imagining that there is gravity. Something, then, has to hold the table up, but what it is is not a part of the imagined situation. But I am not imagining a table miraculously suspended in a gravitational field—I am simply not imagining what the outside of the situation has to be like to support the part I care about.

Maybe with a lot of work one can imagine a situation involving a brick and involving enough imagined detail that one can with confidence say that the situation is one not only one where the ordinary causes of bricks are not present near the brick, but where nowhere in the universe are there any causes of the brick and where there are no non-physical causes of the brick either. But now we see that the situation imagined took rather more effort, and the above examples of how there may be more to an imagined situation than one initially thought should severely reduce one’s confidence that one has been successful at the task of imagining a causeless brick. And even if one has been successful at it, the inference to the possibility of a causeless brick is still defeasible.

I want to end this discussion by comparing the imaginability argument for a causeless brick to the imaginability argument against Platonism. One might claim that it is possible to imagine a brick that does not stand in an instantiation relation to any other entities. If one can, then defeasibly it follows that possibly a brick does not stand in an instantiation relation to any other entities. But that, of course, contradicts Platonism which holds that, necessarily, all bricks instantiate brickness. While I am not a Platonist, this argument against Platonism strikes me as weak. The Platonist can answer as I did above: Have we really imagined a brick that doesn’t stand in an instantiation relation to another entity, or have we merely imagined a brick without imagining its standing in an instantiation relation?

But there is also a further answer the Platonist can make. The Platonist can say: “For all you know, by imagining it as a brick you have implicitly imagined a situation where it is related to brickness, though your description of the contents of what you imagined contradicts this.” Compare this to the point one should make against someone who claims that to have imagined a cube without any space or spatial relations—surely by imagining it as a cube, you have implicitly imagined it as occupying space or as involving spatial relations (say, between the vertices).

Can the defender of the PSR make this point, too? Perhaps. The brick we allegedly imagine coming into existence ex nihilo is a contingent brick. But it might be that the nature of contingency involves being caused (cf. Section 2.2.6.6, above). Moreover, the brick has existence. But it seems implausible to claim that we have plumbed the depths of the nature of existence. It could, for instance, be that to be is either to be necessary or to be caused—that the esse, the existence, of a contingent being is its being caused (it may be that Thomas Aquinas thought this; I explore this kind of a view in Pruss 2006, Chapter 12). It could even be that the esse of a contingent being is its being caused by that particular set of causes by which it is caused—that would cohere neatly with and explain the essentiality of origins.

A variant of the argument from modal imagination is to say that one can without overt logical contradiction state the claim that a brick exists without a cause:

(9) $x(brick(x) & ~$y(causes(y,x)).

However, that is a bad argument. That one can state something without overt contradiction does not imply that there is no hidden contradiction. After all, compare (9) to:

(10) $x(sculpture(x) & ~$y(causes(y,x)).

This claim is impossible, since it is a necessary truth that sculptures have sculptors—that is what makes them be sculptures. In the case of (10) the contradiction lies pretty close to the surface. But how do we know that in (9) there is no contradiction somewhat further from the surface? Maybe there is even a hidden complexity in the concept represented by the existential quantifier.

2.3.2. Van Inwagen’s modal fatalism argument

2.3.2.1. The basic argument

Peter van Inwagen (1983, pp. 202–204) has formulated an influential and elegant reductio ad absurdum of the PSR. Let p be the conjunction of all contingent truths. If p has an explanation, say q, then q will itself be a contingent truth, and hence a conjunct of p. But then q will end up explaining itself, which is absurd. We can formulate this precisely as follows:

(11) No necessary proposition explains a contingent proposition. (Premise.)

(12) No contingent proposition explains itself. (Premise.)

(13) If a proposition explains a conjunction, it explains every conjunct. (Premise.)

(14) A proposition q only explains a proposition p if q is true. (Premise.)

(15) There is a Big Conjunctive Contingent Fact (BCCF) which is the conjunction of all true contingent propositions, perhaps with logical redundancies removed, and the BCCF is contingent. (Premise.)

(16) Suppose the PSR holds. (For reductio.)

(17) Then, the BCCF has an explanation, q. (By (15) and (16).)

(18) The proposition q is not necessary. (By (11) and (15) and as the conjunction of true contingent propositions is contingent.)

(19) Therefore, q is a contingent true proposition. (By (14) and (18).)

(20) Thus, q is a conjunct in the BCCF. (By (15) and (19).)

(21) Thus, q explains itself. (By (13), (15), (17) and (19).)

(22) But q does not explain itself. (By (12) and (19).)

(23) Thus, q does and does not explain itself, which is absurd. Hence, the PSR is false.

Versions of this argument has been defended by James Ross (1969, pp. 295–304), William Rowe (1975, 1984) and more recently Francken and Geirsson (1999).

The argument is plainly valid. Thus, the only question is whether the premises are true. Premise (14) is unimpeachable.

Premise (13) bears some discussion. In favor of it, one might note that the explanation of the conjunction might have more information in it than is needed to explain just one of the conjuncts, but if it has enough information to explain the conjunction it also have enough information to explain the conjuncts. We may, however, worry about Salmon’s remark that irrelevancies spoil explanations (Salmon 1990, p. 102). If we’re worried about this, however, we can replace “explains” with “provides material sufficient for an explanation” throughout the argument, and whatever was plausible before, will remain plausible. Alternately, we may say that if q explains a conjunction, then the only reason it might fail to explain a conjunct r is because q might contain irrelevant information. But surely when the conjunct r is equal to q itself, this worry won’t be real—how could q contain information irrelevant to itself? So even if (13) is questioned, (21) still very plausible follows from (15), (17) and (19).

This leaves the technical premise (15) about the existence of a BCCF, and two substantive claims, (11) and (12), about explanation. Leibnizian cosmological arguments based on the PSR need something like a BCCF, so questioning (15) is probably not a fruitful avenue for questioning for a defender of the Leibnizian cosmological argument (see further discussion in Section 4.1.1.3, below, as well as Pruss 2006, Section 6.1).

But we should not accept (11). We shall see that the main reason for believing (11) rests on a misunderstanding of how explanation works. Moreover, I shall argue that someone who accepts the logical possibility of libertarian free will should deny at least one of (11) and (12).

Premise (11), that no necessary proposition can explain a contingent one, needs some justification. The main reason to accept (11) is the idea that if a necessary proposition q explained a contingent proposition p, then there would be worlds where q is true but p is false, and so q cannot give the reason why p is true. This sketch of the argument can be formalized as follows:

(24) If it is possible for q to be true with p false, then q does not explain p. (Premise)

(25) If q is necessary and p is contingent, then it is possible for q to be true with p false. (A theorem in any plausible modal logic)

(26) Therefore, if q is necessary and p is contingent, then q does not explain p.

Instead of attacking (11) directly, I shall focus my attack on (24). Without (24), premise (11) in the modal fatalism argument does not appear to be justified. Now, granted, someone might one day find a powerful argument for (11) not dependent on (24), in which case more work will need to be done, but (24) seems to capture just about all the intuition behind (11).

By contraposition (24) is equivalent to:

(27) If q explains p, then q entails p.

Let me start with a quick ad hominem argument against (27). It seems a perfectly good explanation of why the dog did not bark that neither a stranger came by the dog nor did any other potential cause of the dog’s barking occur. But the explanans here only entails the explanandum if we suppose that it is a necessary truth that if the dog barked, its barking had a cause. But opponents of the PSR are unlikely to grant that this is a necessary truth, unless they have some principled to reason to argue that dogs’ barkings metaphysically require causes, but some other things don’t need any explanation, whether causal or not. But I doubt that there is a good way of drawing the line between barkings and other states of affairs.

Now, (27) does seem to hold in the case of many conceptual explanations. These explain a state of affairs by saying what the state of affairs is constituted by or consists in. For instance, in Metaphysics Z, Aristotle suggests explaining an eclipse of the sun by noting that an eclipse of the sun is identical with the earth’s entry into the moon’s shadow. Likewise, one might explain a knife’s being hot by noting that its being hot consists in, or maybe is constituted by, its molecules having high kinetic energy.

However, (27) is falsified by just about every modern scientific non-conceptual explanation that I know of. Scientific causal explanations in general simply do not give conditions that entail the explanandum. This is obvious in the case of statistical explanations, since in these the explanans gives laws of nature and states of affairs that do not entail the explanandum but either render the explanandum more probable than it would otherwise be, or at least are explanatorily relevant to the explanandum. Why d