An impossible proof of God

Abstract

A new version of the ontological argument for the existence of God is outlined and examined. After giving a brief account of some traditional ontological arguments for the existence of God, where their defects are identified, it is explained how this new argument is built upon their foundations and surmounts their defects. In particular, this version uses the resources of impossible worlds to plug the common escape route from standard modal versions of the ontological argument. After outlining the nature of impossible worlds, and motivating the need for positing them, the new argument is delineated and its premises justified. It is taken for granted that the argument cannot be sound, since it would prove too much. However, its premises are all plausible, and their denial promises to have significant ramifications. Several intuitive lines of objections are then explored in order to illuminate their shortcomings. The puzzle that the argument poses is therefore not whether the argument is sound, for it clearly cannot be. Rather, it is to place pressure on its plausible premises, so some plausible account of how the argument fails can be identified, and that the devising of such an account promises to be insightful. In the process, we should gain an improved understanding of how such ontological arguments work.

To believe a thing is impossible is to make it so.

—French Proverb.

In what follows, I will outline what I believe to be a new, and somewhat more compelling, version of the ontological argument for the existence of a God. However, I do not think that the argument is sound. I am not a theist, but even if I were, I do not think that I would want my theistic conviction to rest upon any mere conceptual grounding. Such grounds for how things are, seem, to me at least, insuppressibly suspect. Theists and non-theists alike should be sceptical of positing the existence of things, any things, on such bases. Why, then, you might ask, have I taken the trouble of constructing and presenting an argument that, if sound, would prove the existence of a God on purely conceptual grounds? It is because the premises of the argument are so very plausible, and their denial appears to have significant, and perhaps unwelcome, ramifications. This at least is my interest in the forthcoming argument.

Moreover, I think this new version of the ontological argument makes significant advances upon previous versions, and thus has the potential to help further this long-continued, and seemingly inexhaustible, dialectic. It is not part of my present purpose to explain where the argument goes wrong, for it inevitably must err somewhere, or so I will contend. But I will instead attempt to draw out some uncomfortable consequences of various options that initially seem promising, whilst counselling against journeys down blind alleys, pointing instead towards potentially more fruitful avenues (“Mission IMPOSSIBLE” section). The present work, nonetheless, should be construed as a piece of cartography, rather than navigation; there will be no conclusive destination, but instead, the aim is to establish and detail a challenge for better minds to contend with. That is, I am merely mapping some terrain; journey as you please.

However, before my exposition of the argument (“An impossible ontological argument” section), I will briefly describe the context from which the argument emanated (“The ontological argument” section). This serves to aid comprehension by making clear how my argument builds and improves upon earlier, and more familiar, versions. I will then introduce and outline the impossible worlds account of impossibility upon which that argument crucially depends (“Impossible worlds” section). Only once this solid foundation has been set will I build my argument upon it. So, it is to those foundations that we first proceed.

The ontological argument

Ontological arguments for the existence of a God tend to share the common theme of arguing from the mere concept, to the actual existence, of a God. The first known version originated in the work of the eleventh century C.E. theologian, St. Anselm of Canterbury.¹ He argued that we have in our minds the concept of a being than which none greater can be conceived. And accordingly, asserted that this greatest conceivable being exists in our minds. Anselm then asks us to assume (for reductio) that the greatest conceivable being exists in the mind only. And claims that to exist merely in the mind alone is a limitation. For it is greater to exist also outside of the mind, to exist in reality, than to exist in the mind only. Indeed, we can conceive of a being that is greater in this respect. Ergo, it is a contradiction to suppose that the greatest conceivable being exists in the mind alone. Hence, via reductio ad absurdum, the greatest conceivable being must exist in reality as well as in the mind.

Of course, this argument is very crude, and among other faults, it founders on its fallacious inference from our mere possession of the concept God, to the existence of the thing, God, in our minds. When we conceive of things, we do not have the conceived of things themselves in our minds, but rather, we have a mere representation of those things in our minds. Thus, it is not the case that the thing—in this instance, God—exists in the mind when we conceive of it. All that exists in our mind when we conceive of a thing is the representation of that thing.

In recent times, the argument has been reinvigorated in light of developments in modal logic. Most notably, Hartshorne (1941b, 1965), Malcom (1960), Plantinga (1974, Chap. 10) developed modal versions of the ontological argument that do not rely on many of the dubious assumptions afflicting some traditional versions of the argument, such as that assumption just shown to be fallacious—that when we conceive of a thing, that thing conceived of must itself exist in the mind, rather than a mere representation of it.² Here is one way to formulate a modal version of the ontological argument:

• P1. The concept God is that of the greatest conceivable being.³

• P2. Conceivability entails possibility.

• P3. It is part of the concept the greatest conceivable being that it is satisfied by a being, x, only if x has the essential property of being a necessary existent.⁴

• P4. If it is possible that an individual, x, exists and has the property of being a necessary existent, then x actually exists.

• C1. A God actually exists

Call this argument ‘MODAL’. MODAL is clearly valid. However, despite its merits, it has been subject to much criticism. This is unsurprising, since many share my intuition that, ‘one cannot build bridges from the conceptual world to the real world.’ (Plantinga 1974, p. 196).⁵, ⁶ Some have disputed P2, and argued that despite the conceivability of a God, conceivability does not entail possibility, so that the conceivability of a God does not itself entail that something satisfies that concept. This might be because it seems we can separately conceive of things, x and y, such that x and y are essentially necessary existents, yet are mutually incompatible with each other. Or likewise, it seems we can separately conceive of the truth of incompatible propositions, p and q, such that, if true, they are necessarily so.⁷

Rather than surrendering the entailment from conceivability to possibility, several philosophers have taken this point to indicate that, only if something is conceivable in sufficient detail does its conceivability entail its possibility.⁸ And that these problematic cases are symptomatic of a lack of strictness in the level of detail to which they are conceived, such that they do not blight the entailment construed with the stricter level of conceivability.⁹

One common example from the literature concerns Goldbach’s conjecture, which states that all of the positive non-prime even integers are expressible as the sum of two primes. To date, there is no known proof to show whether or not the conjecture is true. Now, I can conceive of a mathematician waving a few sheets of paper with alien-looking scribbles on them, and claiming to have proved that conjecture. Accordingly, it might be said that I can conceive of Goldbach’s conjecture being true. But this surely cannot be the level of detail required for the conceivability of something—in this case, the truth of Goldbach’s conjecture—to entail its possibility. Since, I can equally conceive of that same mathematician claiming to have disproved Goldbach’s conjecture. And by the same lights, it might be said that I can conceive of Goldbach’s conjecture being false. Yet Goldbach’s conjecture cannot be both possibly true and possibly false, since, as a conjecture of number theory, it bears its truth-value necessarily.

However, it may also seem too strict to some to claim that for conceivability to entail possibility, the level of detail required in that which we conceive is such that we must conceive of the proof itself.¹⁰ If that were the case, the conceivability of Goldbach’s conjecture would be redundant to our assessment of its possibility, since the actual proof alone suffices to establish its possibility—because what is actually the case is possibly the case. It should be evident, then, that there are varying degrees of detail to which we can conceive of something. And that those who want to hold onto the principle, that conceivability entails possibility, could still dispute the soundness of MODAL by arguing, that the level of detail at which we can conceive of a God is not sufficient to entail the possibility of a God. Accordingly, this would provide a reason to be sceptical of the inference from P1 and P2 to:

• P5. It is possible that something satisfies our concept God.

There would seem to be some degree of equivocation on the notion of ‘conceivable’ between P1 and P2, where the notion of ‘conceivable’ used in P2 is stricter than that used in P1, on pain of preserving the truth of the two premises—or so the objection runs.

But for those who dislike P2, for whatever reason, the argument could be run instead by replacing P2 with the perhaps weaker premise, P5.¹¹ As just noted, P5 is in fact a consequence of P1 and P2, and to that extent is justified by those premises. By removing P2 from the argument, we would thus make the argument dialectically weaker, since we would need to find alternative grounds for believing P5. The absence of such grounds would promote agnosticism about the soundness of the ensuing argument (cf. Le Poidevin 1996, pp. 25–27). However, I do not want to dismiss the possibility that there could be some alternative justification for P5. The plausibility of the argument depends on the extent to which you find P5 more plausible than:

• P5*. It is possible that nothing satisfies our concept God.

Given that our concept God is such that whatever satisfies it would have the essential property of being a necessary existent (from P1 and P3), if it is possible that nothing satisfies our concept God (i.e. P5*), then it follows that nothing actually satisfies our concept God. Since, if a God were to actually exist, it would exist in every possible world, contrary to P5*.

Perhaps those who would reject the ensuing version of MODAL that supplants P2 with P5, would do so on the grounds that they are generally sceptical of the possibility of necessary existents. Such philosophers would either take issue with P3, claiming that our concept God is not such that it is satisfied only by entities that have the property of being a necessary existent. Or they would reject P5 by contending that our concept God is such that nothing could satisfy it.

Still others have questioned whether MODAL shows that there is something wrong with the axiom ‘◊ p → p’ of the modal logic system B.¹² Consider the following remarks by Kane:

The modal ontological argument is valid in some standard systems of alethic modal logic, e.g., S5 (which is assumed by advocates of the argument like Hartshorne and Plantinga), and in at least one weaker system [i.e., B]. But it is not valid in all. For example, it is not valid in T (Von Wright’s M) or any Lewis system weaker than S5, including S4. What should all this mean for our assessment of the argument? Is choosing a modal system in theological contexts as innocent as choosing hors d’oeuvres? Can it be so with a proof for the existence of God at stake? (1984, p. 336)

Those philosophers, who are tempted to undermine MODAL by rejecting the modal logic system B, must take issue with P4. But this manoeuvre is ill-advised as a response to MODAL, since MODAL can be modified so that it does not rely on modal system B. This is done by replacing P3 and P4 with:

• P3*. It is part of the concept the greatest conceivable being that it is satisfied by a being, x, only if x has the essential property of being an actually necessary existent.

• P4*. If it is possible that an individual, x, exists and has the property of being an actually necessary existent, then x actually exists.

Here we have made a subtle alteration to P3 and P4, where the important property is not that of being a necessary existent, but rather that of being an actually necessary existent. The term ‘actually’, here, and elsewhere, unless stated otherwise, should be read as expressing the rigid, rather than the non-rigid, notion of actually. That is, that ‘actually’ always takes us back to the perspective of our world, so that we consider which worlds are accessible from our world when considering what is actually possible or impossible.¹³ When we consider whether some possible entity satisfies the concept God, then we need to look to all the worlds accessible from our world, not those worlds accessible from the world of the entity in question. Given these two modifications to MODAL, the resulting argument would rely only on the extremely weak modal system T to support the conditional expressed in P4*.¹⁴ This is because we require the inference rule ‘ p→ p’ to get us from the actual necessity of a God, to its actual existence.

Furthermore, it might be that this meagre assumption could be done away with also if we modify P3* and P4* accordingly:

• P3**. It is part of the concept the greatest conceivable being that it is satisfied by a being, x, only if x has the essential property of being an actual existent.

• P4**. If it is possible that an individual, x, exists and has the property of being an actual existent, then x actually exists.

Here we have exchanged the essential property of being an actually necessary existent, to that of being an actual existent. Admittedly, this does appear prima facie to be an improvement to the argument; the resulting argument would be simpler. However, I worry that those following in the footsteps of Kant, Frege, et al., might have ingrained in them the belief that existence is not a (first-order) property, and may well find the property of being an actual existent equally suspect. It is unclear to me that the arguments offered for thinking existence is not a property are at all adequate, but I will assume there is some substance to them for the sake of this essay.

Finally, an objector might take issue with P1. That is, they might claim that our concept God is not in fact that of the greatest conceivable being. Perhaps this is because they cannot make sense of terms like ‘greatest’, ‘best’, ‘perfect’, ‘supreme’, and others of their kind, without their being combined with some qualifying determinate. This qualifying determinate would be required to provide the relative ordering of things with respect to it. These bare superlative terms, when lacking a qualifying determinate, appear to be semantically incomplete. So though I can understand what is meant by ‘the greatest trampolinist’, or ‘the best botanist’, or ‘the perfect sphere’, I cannot grasp what is meant by ‘the greatest’, or ‘the best’, or ‘the perfect’ simpliciter (i.e. without qualification). Perhaps the qualification is meant to be that of ‘being’. But if so, I see no reason to think that any one existent is better with respect to being than any other.¹⁵ So even this charitable interpretation of the expression, ‘greatest conceivable being’, appears to be incoherent. Or maybe an objector will contest P1 on the grounds that the concept God is such that we can conceive of a greater being. For example, it might be thought that a God’s power would be limited with respect to the ability to sin, and we can conceive of a being that, ceteris paribus to a being that satisfies our concept God, has in addition the ability to sin.

Again, both these strategies seem fruitless,¹⁶ since we can easily alter the argument to avoid the problematic phrase, ‘greatest conceivable being’, seemingly without loss to its persuasiveness. Instead of P1 and P3*, we could have:

• P1*. It is conceivable that a being satisfies the concept God.

• P3***. It is part of the concept God that it is satisfied by a being, x, only if x has the essential property of being an actually necessary existent.

Nevertheless, it remains probable that some objectors will dispute even P1*. They might do this by claiming that the concept God is incoherent. Perhaps this will be because the concept God is such that some of the properties of any being satisfying it would have to be incompatible. Or alternatively, that our concept God is relational, such that the defining characteristics are mostly or wholly negative. This would then allow us only negative conceivability—that the concept is coherent (cf. Chalmers 2002, Sect. 2). It might then be argued that the relevant notion of conceivability contained in P1*, that concords with the use of conceivability in the antecedent of P2, is that of positive conceivability—whereby conceiving here means roughly to ‘…imagine (in some sense) a specific configuration of objects and properties.’ (Ibid., p. 150). I make no judgements about the success of these responses. This merely serves as a brief introductory sketch to a version of the kind of ontological argument that I intend to develop in this paper.

Impossible worlds

An impossible world is a world at which an impossibility obtains; where a proposition that cannot be true, is true. If we think of possible worlds as maximal consistent sets of propositions—as I take it most, or at least many, philosophers do—then we can think of impossible worlds as maximal inconsistent sets of propositions.¹⁷ Whilst possible worlds are ways the world could have been, impossible worlds are ways the world could not have been.¹⁸ Both kinds of ways are plentiful. Just as there are many ways things could have been, it intuitively seems as though there are also many ways that things could not have been. In this respect some have thought it inadequate to lump impossibility into one broad category of that which is not possible—impossibility is more fine-grained than that. For example, not only could Socrates not have been a frog, he could not have been a rhombus either.¹⁹ Instead of just asserting that neither of these statements expresses a possible truth—that there is no possible scenario that represents them—if we accept an impossible worlds account, we can claim that the two statements describe distinct impossible scenarios, each of which is represented by a distinct impossible world.

Another reason to believe in impossible worlds is that they can be employed to great effect in offering an account of the semantics of counterpossible conditionals (cf. Nolan 1997; Yagisawa 2010, Sect. 8.3). On the standard Lewisian semantics, counterfactuals have the following truth-schema: ‘If it were the case that P, then it would be the case that Q’ is true at some world, w, if, and only if, there is some metaphysically possible world, w′, at which P and Q is true, and w′ is closer than any metaphysically possible world, w, where P and not-Q is true, or there is no metaphysically possible world where P is true (cf. Lewis 1973).²⁰ As per the second disjunct of the truth conditions for the Lewisian semantics, counterfactuals with impossible antecedents are vacuously true. This may or may not have some intuitive plausibility for counterpossible conditionals with logical or mathematical falsities in the antecedent. However, things are less satisfactory for other kinds of antecedent impossibilities. Consider the following counterpossible conditionals:

• (A) If Hobbes had squared the circle, then geometricians would have been amazed.

• (B) If Hobbes had squared the circle, then Socrates would have been an astronaut.

• (C) If the futon were red-all-over and green-all-over, then it would have been a spectacle to see.

• (D) If the futon were red-all-over and green-all-over, then the moon would have been hollow.

Whilst A intuitively seems true, it does not give the impression of being vacuously so. Its truth appears to be owed to something else. On the other hand, and to support this assessment of A, intuitively, B seems false. Likewise, while C intuitively seems true, it too does not give the impression of being vacuously so; whereas D gives the impression of being false. The Lewisian semantics cannot adequately accommodate these intuitions; it cannot accommodate the fact that we want to offer a different assessment of conditionals like A and C, compared to those like B and D. However, with the tool of impossible worlds at our disposal, the Lewisian semantics for counterfactuals can be modified to accommodate our intuitions. We can offer the following alternative truth-schema: ‘If it were the case that P, then it would be the case that Q′ is true at some world, w, if, and only if, there is some metaphysically possible or impossible world, w′, at which P and Q is true, and w′ is closer than any metaphysically possible or impossible world, w″, where P and not-Q is true.²¹ This account has the added plausibility of giving a less disjunctive account of the semantics for counterfactual conditionals. For these reasons, among others,²² impossible worlds are becoming increasingly popular.

Now, just as the advancement of modal logic provided the capacity to develop some sophisticated versions of the traditional ontological argument, I think the adoption of impossible worlds, whatever they may be, suggests a way of developing certain modal versions of the ontological argument.

An impossible ontological argument

Let us now proceed by delineating that new version of the ontological argument; a version that makes vital use of the impossible worlds described in the previous section. The proposed argument has the following form:

• P1*. It is conceivable that a being satisfies the concept God.

• P6. Conceivability entails either possibility or impossibility.²³

• From P1* and P6, via modus ponens:

• I-C1. It is either possible or impossible that a being satisfies the concept God.

• P7. It is part of the concept God that it is satisfied by a being, x, only if x has the constitutive property of being an actually necessary existent.

• A1. It is possible that a being satisfies the concept God.

• P4*. If it is possible that an individual, x, exists and has the property of being an actually necessary existent, then x actually exists.

• From A1, P7, and P4*, via modus ponens:

• I-C2. There actually exists a being that satisfies the concept God.

• A2. It is impossible that a being satisfies the concept God.

• P8. If it is impossible that a being satisfies the concept God, then there is an impossible world, w, such that a being in w satisfies the concept God.

• P9. If some individual, x, exists at an impossible world, and x has the property of being an actually necessary existent at that world, then x actually exists.

• From A2, P7, P8, and P9, via reductio ad absurdum:

• I-C3. It is not the case that it is impossible that a being satisfies the concept God.

• And from I-C1, I-C2, and I-C3, via disjunctive syllogism:

• C2. There actually exists a being that satisfies the concept God.

Call this argument ‘IMPOSSIBLE’. As should be clear from the above presentation, IMPOSSIBLE is valid. Nevertheless, I shall guide the reader through some of the inferences, as I examine and justify the premises. To begin with, let us grant P1* for the sake of argument. As we shall later see, nothing much rests upon it for the wider significance of the argument kind. The “conceivable” here is to be understood in a very weak sense of the term. Perhaps all that is needed for us to conceive of a God is that we grasp the concept God. That is, P1* merely requires an understanding of what it would take for something to satisfy the concept God. I take it that those who think our concept God is incoherent will find all arguments for theism unattractive. But if this is your response, independent reasons must be provided for the rebuttal. And given the slipperiness of the concept, this is no mean feat.²⁴

However, it is perhaps not entirely implausible that it is inconceivable that something satisfies the concept God. The ‘inconceivable’ here permits of further divergent readings. Firstly, it might be thought that something is inconceivable if it is inconceivable by us. This might be due to our limited cognitive capacities. Secondly, it might be thought that something is inconceivable because it is inconceivable in principle. Such things would be inconceivable even for an ideal conceiver. The worry that the existence of a being satisfying the concept God is inconceivable in the former sense poses no threat to the truth of P1* if we thought it were not inconceivable in the latter sense. Though our justification for P1* would rely on our actually being able to conceive of the existence of a being satisfying the concept God. Since if we could not conceive of such a being, due perhaps to our limited cognitive capacities rather than any manifest incoherencies in the concept, then we would not know whether or not such a being is inconceivable in principle. And thus, we would not know whether P1* is true.

If you have these concerns about P1*, fear not, as we can easily modify the argument by supplanting P1* and P6 for I-C1, and beginning the argument from there. My preference, however, remains with the argument that includes P1* and P6, for two reasons. Firstly, that way of presenting the argument lends more support to P7. Some philosophers may well attempt to reject P7 on the grounds that they do not think there is a property of being an actually necessary existent. Yet, all that need be under consideration for P7 in the preferred version of the argument is that that property is conceivable, in that weak sense of conceivable mentioned earlier. If that property is conceivable, as it must be if the concept God is conceivable—where it is constitutive of the concept God that a being satisfying that concept has this property, at least given the concept God that is currently under consideration—then it must be either possible or impossible that it has some instance. Once this much has been granted, there would seem to be no principled reason why this property cannot be built into the concept God, or why it would affect whether that concept is satisfied at a possible or impossible world.

Without the justification for I-C1 that comes from P1* and P6 combined, and assuming no alternative justification can be found, the doors would be open for a potential objector to deny P7, on the grounds that a being instantiating the property of being an actually necessary existent cannot constitute a part of our concept God, since the latter is, whilst the former is not, possibly or impossibly satisfied. And if this were the case, the theist would then be faced with the option of either rejecting P7, or rejecting I-C1 and admitting that our concept God has no instance. I assume that given this choice the theist would opt for the former option, unless they were to betray their theistic commitments. So the inclusion of P1* and P6 blocks one potential objection to the argument.

Secondly, an objector might similarly be tempted to reject I-C1 on the grounds that the concept God is neither satisfied at a possible nor an impossible world. The conceivability of a being satisfying the concept God (P1*), and the entailment from conceivability to possibility or impossibility (P6), allows the theist a foot-in-the-door to force the dilemma proposed in I-C1. It should be clear then that P1* and P6 provide good justificatory grounds for believing I-C1. Nonetheless, this does not close off provisions from alternative justificatory grounds for believing I-C1. Other forms of justification could only make the argument stronger. But to merely state I-C1 without appeal to P1* and P6, or any other justification for that matter, would result in a dialectically weaker position for a defender of IMPOSSIBLE.

And whilst we are drawing our attention towards P7, I should like to note an important difference that it bears to its parallel premise in MODAL, namely P3***. P3*** describes the property of being an actually necessary being as an essential property, whereas I have avoided that description in P7 in favour of describing that property as being constitutive of a being satisfying the concept God. It will be enlightening, then, to explain what I take a constitutive property to be, and how it might differ from an essential property. Philosophers have understood the notion of an essential property in different ways. Part of the reason for describing the kind of property I am concerned with as ‘constitutive’ rather than ‘essential’, is to prevent confusing it with one particularly well-known of those distinct ways of understanding ‘essential’. That way that essential properties have been understood that I especially want to contrast with what I am calling ‘constitutive properties’, is that an essential property of a thing or kind, is a property that is de re necessary of that thing or kind. That is, a property that a thing has in every possible world that it exists, or a kind of thing has in every possible world (and every instance of it in that world) where that kind of thing exists.²⁵ Whatever the merits of this way of understanding essential properties, it places no constraints on what things are like at impossible worlds. For the purpose of my argument, and perhaps for purposes more general, this clearly will not do.

The notion of a constitutive property that I am using is stricter than that of an essential property understood in the sense just described. A constitutive property of a thing or kind of thing, x, is a property of x, such that x would not be that thing or kind of thing respectively, if it lacked that property. So, to give an example in the case of a kind, a triangle would not be a triangle if it lacked three angles. It is thus constitutive of a thing being a triangle that that thing has three angles. And to give an example in the case of an individual, Socrates would not be Socrates if Socrates lacked the property of being identical to Socrates. Likewise, when P7 states that the concept God is such that it is constitutive of a being satisfying that concept that it possess the property of being an actually necessary being, if that being lacked that property, it would be a different thing and a different kind of thing. Since this notion of being a constitutive property is quite understandable, it seems to present no further difficulties to our ability to conceive of a being satisfying our concept God—in the weak sense of conceivable descried earlier. Accordingly, the objector would do well to find some other point of dispute.

There is perhaps an ambiguity in the concept God, as I have being using it, that might have here occurred to the reader, between it being a concept of an individual entity or of a kind of entity. But whether it is a concept of God, the individual, or God, the kind, should make little difference to our argument. However, I am not sure that I can make much sense of the former interpretation. And in so far as I do understand it, I worry, and hope that others might share this worry, that it presupposes the existence of God, the individual. For this reason, it is the latter interpretation of the concept God, as a concept of a kind of entity, which I have intended throughout this essay. This means that if the concept God were satisfied at an impossible world, that satisfying entity would possess the property of being an actually necessary existent at that world, lest it not be a God.

As already noted, from P1*—it is conceivable that a being satisfies the concept God—and P6—conceivability entails either possibility or impossibility—we arrive at I-C1: It is either possible or impossible that a being satisfies the concept God. If we can show that C2—there actually exists a being that satisfies the concept God—follows from one disjunct of I-C1, whilst the other disjunct leads to a contradiction, then we can validly infer C2 via disjunctive syllogism. Consider then the sub-argument from the first disjunct of I-C1, namely:

• A1. It is possible that a being satisfies the concept God.

• P7. It is part of the concept God that it is satisfied by a being, x, only if x has the constitutive property of being an actually necessary existent.

• P4*. If it is possible that an individual, x, exists and has the property of being an actually necessary existent, then x actually exists.

• From A1, P7, and P4*, via modus ponens:

• I-C2. There actually exists a being that satisfies the concept God.

When combined with P4* and P7, we can validly infer IC-2 from A1. Since the conjunction of A1 and P7 gives us a particular case of the antecedent of P4*, and accordingly we are allowed to infer the corresponding particular case of the consequent of P4*, namely I-C2. Our justification for accepting P4* has already been given with respect to MODAL. Indeed, the reasoning from A1 to I-C2 is the same as that employed in that modal ontological argument. This was found to rest upon some quite plausible assumptions. It is likely then that the objector will reserve their contention for the inference from the second disjunct of I-C1 to I-C3.

Let us consider, then, the sub-argument that runs from the second disjunct of I-C1:

• A2. It is impossible that a being satisfies the concept God.

• P7. It is part of the concept God that it is satisfied by a being, x, only if x has the constitutive property of being an actually necessary existent.

• P8. If it is impossible that a being satisfies the concept God, then there is an impossible world, w, such that a being in w satisfies the concept God.

• P9. If some individual, x, exists at an impossible world, and x has the property of being an actually necessary existent at that world, then x actually exists.

• From A2, P7, P8, and P9, via reductio ad absurdum:

• I-C3. It is not the case that it is impossible that a being satisfies the concept God.

When combined with P7, P8 and P9 we can validly infer I-C3 via a reductio ad absurdum on A2. This can be derived by first inferring the first conjunct of the antecedent of P9 from premises A2 and P8 via modus ponens. We can then combine P7 together with the consequent of P8 to fulfil a particular case of the antecedent of P9. Again, via modus ponens this allows us to infer the corresponding particular case of the consequent of P9. But since the consequent of the particular case of P9 is that there actually exists a being that satisfies the concept God, it must be possible that a being satisfies the concept God. This is of course assuming that the actual world is accessible from the actual world, and as such is possible relative to the actual world. But this assumption is hardly contestable; if anything is actually possible, then surely what is actually the case must be; we are dealing with an alethic modality here. However, the claim that it is possible that a being satisfies the concept God, contradicts our initial assumption, A2, that it is impossible that a being satisfies the concept God. And as such, we can reject A2 on the grounds that it leads to a contradiction when combined with other plausible premises.

The justification for P8 crucially depends on a commitment to impossible worlds. I gave some motivations for believing in impossible worlds in the previous section. The thought is that, if it is impossible that a being satisfy the concept God, as per assumption A2, then there must be an impossible world where a being does satisfy the concept God. This just falls out of the commitment to impossible worlds. But for it to be a God at that impossible world, that entity must possess the constitutive property of being an actually necessary being, in accordance with P7, lest it be some other kind of entity besides a God. However, if there were no impossible world where a being satisfied the concept God, then it would also not be the case that it is impossible for a being to satisfy the concept God, contrary to A2. Yet, now it seems that the inference from a God’s having the property of being an actually necessary existent in some world, to a God actually existing, is no more problematic where that world is of the impossible variety than if it were of the possible variety. It relies merely on the very weak modal logic system T, plus the meaning of the predicate ‘…is an actually necessary existent’. Accordingly, P9 also appears to be very plausible.²⁶ Again, it appears that an objector would make better use of their time looking elsewhere in the argument for weaknesses. From this we can then derive A1 via disjunctive syllogism, and reiterate the reasoning that brought us to I-C2. But, this time I-C2 does not merely rest on the assumption that A1 is true. A1 has been deductively derived, and as such, we can assert C2 without assumption.

Alternatively, if you dislike disjunctive syllogism as an inference rule,²⁷ the same result can be achieved by showing that C2 follows from each disjunct of I-C1, applying the uncontroversial inference rule of constructive dilemma. Here we follow the same line of reasoning from the A2 disjunct of I-C1, but instead of going on to show that A2 leads to a contradiction, we merely stop at the derivation of P9’s consequent. One corollary of adopting this alternative route to C2 is that, although we arrive at our destination—the conclusion that there actually exists a being that satisfies the concept God—we do not demonstrate along the way that,

Antitheist: There actually does not exist a being that satisfies the concept God.

is false. Since those who do not like disjunctive syllogism may not think that we can move from the truth of C2—that there actually exists a being that satisfies the concept God—to the negation of Antitheist, it remains an option to accept both C2 and Antitheist. This, however, would be a deeply unsettling result.

I suspect that few, if any, would be happy with this consequence as a result of accepting the seemingly reasonable premises of our argument. It is one thing to accept that something does and does not exist in an impossible world, and quite another to accept that something actually does and does not exist. At the very least, a deeply unattractive consequence of our premises would have been drawn out. But I think that most would think it quite reasonable to accept that things cannot both actually exist and actually not exist (at the same time). And this can be added as a further premise if need be. For the typical reasons given for rejecting the law of non-contradiction—the underlying reason for rejecting disjunctive syllogism—do not support the possibility of something actually existing and actually not existing (at the same time). We have then, here, a justification for the premises of the argument. And overall we have found that, at least prima facie, the premises seem rather plausible. Let us dig deeper and make a few critical comments in the final section on some potential responses to the argument.

Mission IMPOSSIBLE

I imagine that the instinctive reaction of many to IMPOSSIBLE will be that, if successful, it would open the gates that hold back the impossible, and allow us to postulate the actual existence of many other intuitively impossible beings.²⁸ This is clearly an unwanted conclusion for a theist hoping to defend IMPOSSIBLE, and mirrors Gaunilo’s famous ‘Perfect Island’ objection to the traditional ontological argument. If the theist were to defend IMPOSSIBLE, they would need some way of restricting the variety of molds subject to the argument form so that it can only be employed to demonstrate the existence of a being satisfying the concept God, and not the existence of the intuitive impossibilia.

St. Anselm relied on the concept God as that of the greatest conceivable being to make his case for treating the concept God differently from other concepts. The superlative nature of any being satisfying that concept ensured that the property existence was essentially instantiated by anything satisfying that concept. But unlike other superlative concepts, it was conceivable alone of the concept God that it be satisfied. It was thought of our other superlative concepts, like that of greatest island or greatest painting, that it be inconceivable that they be satisfied, since all potential satisfiers of those concepts are superable without limit; or in St. Anselm’s terminology, they have no ‘intrinsic maxim’. For these other kinds of superlative concepts, there simply is no greatest that is conceivable; there could always be a greater one conceived. It was claimed that this was not so for our concept God. Though, it is in fact hard, at least for me, to understand why the concept God alone avoids this charge.

However, the construal of the concept God as that of the greatest conceivable being was earlier found to be problematic. The superlative interpretation of the concept God was found to be, at best, of no real help to the theist, since it is unclear what it takes for one being to be greater at being than another. At worst, the superlative conception was found to be semantically incomplete; the notions of ‘greatest’, ‘best’, ‘perfect’, ‘supreme’ etc. all require some determinate to fix the subject of the relative ordering. Whilst someone can be the best gladiator, or the greatest poet, whereby the determinates, here, are ‘gladiator’ and ‘poet’ respectably. But the notion of something being the best, greatest, or perfect simpliciter is nonsensical.

Let us not concern ourselves with the difficult task of trying to distinguish the concept God alone as that which has constitutively built into it the condition of being satisfied only by a being that is actually necessary. I leave this difficult task to those with a greater inclination. Nevertheless, an objector who criticises IMPOSSIBLE merely on the grounds that it sets a precedent for arguing many apparent impossibilia into existence, will have missed the point of the exercise entirely. All they would have established is that IMPOSSIBLE is not a sound argument. But this was freely admitted. The difficulty arises from pinpointing where exactly the argument errs. As such, this objector’s response offers us little headway on the real issue. Yet, perhaps it does lend some support to the idea that rejecting P1* is not the way to go. For even if the concept God is incoherent, a parallel argument might potentially be employed using the concept of some other kind of being for which it is constitutive of that kind of being that it is an actually necessary existent.²⁹

Notably, IMPOSSIBLE will not be a problem for those accounts of modality that do not invoke impossible worlds. For this reason, P8 stands out to me as the most vulnerable premise. And accordingly, it does not seem unreasonable, for someone to present IMPOSSIBLE as a case for favouring, ceteris paribus, those accounts of modality that do not camber one towards a commitment to impossible worlds.³⁰, ³¹ But let us not be too hasty here. There are perhaps some alternative responses for those interested in defending impossible worlds. One interesting response would be to claim that I have overestimated the scope of modal space. Underlying P6, the claim that conceivability entails possibility or impossibility, is the following two principles:

Proposition: Conceiving is a propositional attitude, such that there is some propositional content to that which is conceived, which is capable of being true or false.

Ubiquity: For every conceivable proposition p, p represents something that is either possible or impossible.³²

Both seem to be very plausible. But if one were to be denied, P6 would be false, and our justification for believing I-C1 would be in jeopardy. Proposition seems to me to be quite plausible.³³ However, I suspect that Ubiquity might become a target for some objectors. They would likely claim that, despite our ability to conceive of a being satisfying the concept God, the proposition that there is something that satisfies the concept God represents a situation that has no modal status, it is amodal—the existence of a God is neither possible nor impossible.³⁴ As things stand, Ubiquity is deeply integrated into our modal logic, such that, if a proposition, p, is not true at any possible world, then p’s truth is impossible. And under the impossible worlds assumption, Ubiquity demands that there are impossible worlds at which propositions that cannot be true, are true. The impossible worlds assumption makes a distinction between that which is not possible, and that which is impossible. That is, impossibility is not given a relational or negative characterisation, as the absence of possibility, but rather is given a non-relational or positive characterisation, namely, being so at an impossible world.³⁵ Though it is typical, and indeed natural, to accept, in accordance with Ubiquity,³⁶ that not being possible entails being impossible, given the positive characterisation of the impossible, there opens up conceptual space for a view that denies this.³⁷

However, given the sheer intuitiveness of Ubiquity, there would need to be sufficient justification for denying it. Indeed, it sounds incredibly odd, if not straightforwardly absurd, to say that something is not possible, but it is not impossible.³⁸ Though, the awkwardness of such a statement perhaps could be explained in terms of it being a category mistake to apply any modal notions at all to that represented by a certain proposition. So let us look briefly at how you might go about trying to undermine Ubiquity. One line of argument would draw its influence from certain a posteriori necessities. It would be argued that we can conceive of Hesperus being non-identical with Phosphorus, but that this situation is neither possible nor impossible. Since, the scenario is conceivable, there must be some proposition that represents the content of that which is conceived (from Proposition). But if what is represented by that proposition is neither possible nor impossible, as per this line of argument, then Ubiquity must be false.

The case assumes that the identification of Hesperus and Phosphorus is stronger than necessity. More would need to be said to justify this claim and spell out just what level of strength this identity relation has.³⁹ Still, prima facie, this seems to be an interesting case to explore. It is, nevertheless, not at all obvious to me that the ensuing investigation would be successful in disposing of P6. There appears to be two ways of interpreting the case in question:

• 1. Conceptual: This interpretation states that we can conceive of our concept Hesperus being satisfied by something distinct to that which satisfies our concept Phosphorus.⁴⁰

• 2. Metaphysical: This interpretation states that we can conceive of the individual actually picked out by our concepts Hesperus and Phosphorus, being distinct from itself.

Now, the former interpretation appears to be conceivable. This explains how prior to the empirical discovery that Hesperus and Phosphorus are actually numerically identical to Venus, people were able to comprehend their distinctness. However, interpretation 1 presents no problem to Ubiquity, since there accordingly appears to be no good reason to think that the concepts be applicable to distinct things in some (possible or impossible) world. The second interpretation, on the other hand, would seem to challenge Ubiquity, if it were at all conceivable. But why should we believe that we can conceive of that? The absurdity of the situation that interpretation 2 describes prevents our being able to conceive of it. Just as I cannot conceive of square circles, I cannot conceive of an individual being distinct from itself.⁴¹ It seems plausible that those who think they can conceive of 2, are really conceiving of 1, and mistaking it for 2. The objection equivocates between discussing the concepts in the conceivability claim—that we can conceive of our concept Hesperus being satisfied by something distinct from that which satisfies our concept Phosphorus—and the individual that satisfies those concepts in the propositional claim—that the proposition (assuming on behalf of the objector that it is a proposition) that the individual that satisfies our concepts Hesperus and Phosphorus is distinct from itself is not true in any (possible or impossible) world.

Yet, even if this last avenue were to reveal something suspicious about P6, there would still be the further task of showing that the reasons for rejecting that conceivability entails possibility or impossibility in any suggested counterexamples are transferable reasons for believing that our conceiving of a being satisfying the concept God does not entail that the satisfying of the concept God is either possible or impossible. And it would also need to be shown where the weaker argument, that bypasses P1* and P6, and instead begins at I-C1, goes wrong. Nevertheless, it would be both a significant and surprising result if we were to discover that conceivability did not entail possibility or impossibility. One consequence of this outcome would be that there would then be counterpossible conditionals where the antecedent is neither possible nor impossible, such that the impossible worlds semantics for counterfactuals would either be unable to deal with, or produce a counter-intuitive result for, them. This would undercut one of the main motivations for believing in impossible worlds in the first place.⁴², ⁴³

Another objection that initially seems promising, but ultimately goes nowhere, targets P9. This objection states that one way that a world can be impossible is by misrepresenting modal space. So from the perspective of the impossible world we can follow the sub-argument that proceeds from A2, to the conclusion that at some impossible world, w, it is true that a being satisfying the concept God actually exists, but still maintain that it is not the case that at the actual world, @, that a being satisfying the concept God actually exists. The proposition a being satisfying the concept God actually exists is in fact world-indexed. Though such world-indexed propositions, if true at some possible world, are necessarily true,⁴⁴ this objector claims that there are some impossible worlds where world-indexed propositions are not true, despite being true at every possible world. Importantly, this result would show that iterated world-indexing is not redundant (contrary to Yagisawa 2010, Sect. 3.5), at least if we are allowing impossible worlds into our modal space. Again, this would be a significant result.

However, I do not think that I can make any sense of this proposal. The proposition that a being satisfying the concept God actually exists is about how the actual world is. It is constitutive of a world that it is the way that it is. Worlds are defined by their members.⁴⁵ If things were different at a world, it would not be that world. It would be some other world. But then, if a being satisfying the concept God does not exist at the actual world, then the proposition that a being satisfying the concept God actually exists is surely not going to be true at any world, lest the proposition not be about the actual world. Modal space, it would seem, cannot be misrepresented in the suggested way, even at impossible worlds. The confusion here comes about because when we conceive of things actually being different to how they are, the ‘actually’ here is the non-rigid ‘actually’. We thus understand things being actually different to how they are by “looking” at how things are at some other world where things are that way (cf. Yagisawa 2010, Sects. 8.10, 8.11.2, and 8.11.3). But I have explicitly stated that the ‘actually’ I have been using is the rigid notion of actually.

Yagisawa (2010, Sect. 8.10) in fact proffers a way to understand things being (rigidly) actually different from how they (rigidly) actually are.⁴⁶ According to Yagisawa, to do so we must consider how things are with respect to some ‘non-local metaphysical space’, where the modal facts are different: ‘Since in our metaphysical space, Twin Earth does not exist at @ (as we assume), we need an alternative metaphysical space, a metaphysical space in which Twin Earth exists at @. For any such metaphysical space M, it is true to say of any world w in M that w is _m such that Twin Earth exists at @ in M.’ (Ibid., p. 220).⁴⁷ Let us grant Yagisawa for the sake of argument these alternative metaphysical spaces. IMPOSSIBLE concerns what is actually impossible, not what is impossible in some alternative metaphysical space. So Yagisawa’s way of understanding things being (rigidly) actually different from how they (rigidly) actually are, helps our fictional objector not.

Likewise, given P7—that it is part of the concept God that it is satisfied by a being, x, only if x has the constitutive property of being an actually necessary existent—if the entity in the impossible world were not an actually necessary existent, then it would not satisfy our concept God. That is, if, from the perspective of our world, i.e. the rigidly actual world, the proposition that a being satisfying the concept God actually exists is false, then any entity we care to arbitrarily select at an impossible world would not satisfy the concept God. This is regardless of whether or not there are other modal spaces, where at the actual world in one of those modal spaces, a being satisfying the concept God actually exists. This is because when we describe an actually impossible world, we use our actual concepts, not the concepts we would have if we were at that impossible world. Even if the actual world is different in alternative modal spaces, our use of the term ‘actual’ picks out our actual world, in our modal space. It should be clear then that this line of objection to IMPOSSIBLE does not hold up under scrutiny.

This response should now put us in good stead to reply to another objection to P9. It has been suggested to me that P9 will only work to establish the existence of possible beings, and not impossible beings, that exist at an impossible world. On the one hand, this is obviously true, but merely supports IMPOSSIBLE. That argument shows that a being that satisfies our concept God, is not an impossible being, since it is constitutive of a being that satisfies that concept that it instantiates the property of being an actually necessary existent. As was earlier explained, if a being does instantiate the property as being an actually necessary existent, it instantiates that property in virtue of being actually necessary. And since the actual world is an actually possible world, that being must also be an actual being. Eo ipso, it must be a possible being, i.e. it will not be an impossible being.

But there is a further element to this objection. An impossible being may well instantiate incompatible, or even inconsistent, properties. The thought seems to be that the properties of actually existing and of actually not existing can be jointly instantiated by a being at an impossible world. This objector claims that what IMPOSSIBLE establishes is that a being satisfying the concept God both actually exists and actually does not exist. Which are we to believe; that a being satisfying the concept God actually exists, or that it actually does not exist? The objector would claim that IMPOSSIBLE moves us neither one way nor the other, and uses this contradictory result to suggest that nothing has been established regarding the actual existence of a being satisfying the concept God.

However, as we have just seen in our response to the previous failed objection to P9, worlds appear to be constrained in their representation of modal space. It was shown that if a world, w, be it impossible or not, represented an individual, x, as (rigidly) actually existing, as the antecedent of P9 demands, given the proviso that the actual world is one of the actually possible worlds, then the actual world must be such that x does in fact exist. Likewise, if that same world, w, further represents that same individual, x, as also (rigidly) actually not existing, then the actual world must be such that it is not the case that x exists. But how can this be?

One way of interpreting the result being proposed here is just that option, discussed at the end of section three, of accepting both C2, that there actually exists a being that satisfies the concept God, and Antitheist, that there actually does not exist a being that satisfies the concept God. It was noted there that the negation of Antitheist, cannot strictly be proved from the premises without the law of non-contradiction: ‘¬ (p & ¬p)’. And that those accepting impossible worlds may well be motivated to deny that logical truth. Yet, since we are able to show via IMPOSSIBLE that C2 follows from the premises, C2 itself can be used as sufficient reason to reject Antitheist. To reiterate, it is one thing to accept that something does and does not exist in an impossible world, and quite another to accept that something actually does and does not exist. And the objector’s proposal would have us accept the latter. However, most, I think, would not entertain this result.

An alternative, and more interesting way of interpreting the result, is to offer a coherent interpretation of an entity in an impossible world both actually existing and actually not existing without there being an entity at the actual world that actually does and does not exist. This strategy relies on the contingent identity thesis, and serves to highlight a real hole in the argument that we shall shortly plug. According to Gallois (1998, Chap. 6), things can be identical in one world but non-identical in another. I shall not delve into the complicated details of the account, since this would take up too much space. The reader is referred to Gallois’s excellent book for further details.

Importantly, one consequence is that, if entities are identical at a world, they share all their properties, including modal properties, at that world. This follows from the following restricted version of the principle of the indiscernibility of identicals: ‘(LL_w): (x) (y) (w) [in w: x = y → (in w: Φx → in w: Φy)]’ (Ibid., p. 145). This means that if the thing that stands on one side of the identity relation at some impossible world, w, actually exists, whilst the thing that stands on the other side does not actually exist, as the contingent identity theory permits, then a being existing at w can instantiate at w both properties of actually existing and actually not existing, since they are not contradictories under the contingent identity thesis. The problem this raises is that, the being in the impossible world that satisfies the concept God may actually exist at that world, not in virtue of satisfying the concept God—that thing does not actually exist—but in virtue of the kind of entity that stands on the other side of the identity relation.

Accordingly, there may be some philosophers who are inclined to find the argument less startling than it initially appears to be. Consider the following criticism that Plantinga offers against Hartshorne and Malcom’s version of the modal ontological argument:

As it is stated, however, there is one fairly impressive flaw: even if an essence entailing is maximally great in W is exemplified, it does not so far follow that this essence entails is maximally great in α. For all we have shown so far, this being might be at a maximum in some world W, but pretty insignificant in α, our world. So the argument does not show that there is a being that enjoys maximal greatness in fact; it shows at most that there is a being that in some world or other has maximal greatness. (1974, p. 213)

It would seem that IMPOSSIBLE, as it presently stands, is vulnerable to an analogous version of this ‘fairly impressive flaw’. The conclusion of IMPOSSIBLE, C2, is that there actually exists a being that satisfies the concept God. But all that the argument so far establishes is that an actually existent being satisfies the concept God in some world or other, be it possible or impossible. Perhaps in the actual world, that being fails to satisfy the concept God. Maybe, I satisfy the concept God in an impossible world. Yet, if this were so, it would be less than clear that I actually deserve homage for my otherworldly deeds or Godliness. What is needed to captivate this objector is not C2, but the stronger claim:

• C3. There actually exists a being that actually satisfies the concept God

That is, what is demanded by this objector is that the satisfaction of the concept God be actual satisfaction, the satisfaction of the concept God in the actual world. For those with resolute dispassion for anything less than C3, fear not. We need only insist that,

• P10. It is part of the concept God that it is satisfied by a being, x, only if x has the constitutive property of satisfying the concept God in the actual world.⁴⁸

But once this has been granted, it should be clear that, C2—there actually exists a being that satisfies the concept God—can be combined with P10 to yield the desired claim, C3. Indeed, this stratagem is in fact analogous to the one that Plantinga’s himself employed to resolve this issue. And neither is it hard to find support for P10. I follow Plantinga in drawing attention to the remarks of Findlay in support of something close to P10:

Not only is it contrary to the demands and claims inherent in religious attitudes that their object should exist “accidentally”: it is also contrary to those demands that it should possess its various excellences in some merely adventitious or contingent manner. It would be quite unsatisfactory from the religious stand point, if an object merely happened to be wise, good, powerful, and so forth, even to a superlative degree, […] And so we are led on irresistibly, by the demands inherent in religious reverence, to hold that an adequate object of our worship must possess its various qualities in some necessary manner. (1948, pp. 180–181)

We also find in this passage additional support for P7. And yet I see no obvious additional worries that would threaten P10 that has not yet been addressed elsewhere in this essay.

This issue now resolved, an alternative line of response to IMPOSSIBLE that might be suggested is to question whether we can really conceive of an entity, or kind of entity, such that, if it were not actual, or there were no actual instance of that kind, then we could not have conceived of that thing, or kind. Here it seems that our ability to conceive is constrained not merely by internal cognitive states, processes, and events, but also, by how the actual world is, external to our mind. And as such, the argument might appear to assume an unwarranted externalism about conceptual content, and should be rejected on those grounds.

But this will not have got things quite right. Firstly, the property in question is that of being an actually necessary existent. This property does not in itself commit you to the actual existence of a being satisfying the concept God, but does so only under the condition that there is an actual world and that it is accessible to itself. There does not appear then to be any commitment to externalism about conceptual content here. There still remains conceptual room, despite its diminutive size, for our conceiving of a being satisfying the concept God not entailing the actual existence of any such being. A fortiori, this line of criticism gets things backwards. We arrive at the actual existence of a being satisfying the concept God in virtue of the conceivability of such a being; a move from the conceptual to the actual, not from the actual to the conceptual.

Finally, a related form of objection to IMPOSSIBLE would be to argue from an externalism about conceptual content. According to this view, our having a concept would depend partly on the way reality is. The objection would run such that if a being satisfying the concept God were not actual, then we could not possess that concept God. And since we do not know whether a being satisfying the concept God actually exists, we do not know whether we possess the concept God—or at least not the particular concept God that has been employed in IMPOSSIBLE. As such, the objector would claim that P1* begs the question, since our ability to conceive of a being that satisfies the concept God would presuppose our possessing the concept God, and as a result presupposes the very point of contention, that a being satisfying the concept God actually exists.

I have very little to offer in response to this objection other than to state that this externalism about conceptual content seems clearly false. We seem able to conceive of all sorts of things that are not reflected in the structure of reality. Nor does it seem plausible to suggest that I do not know what it is that I am conceiving of. I may sometimes be unable to describe what I conceive of, or be confused about just what it is that I am conceiving of. But that is another matter entirely. My concept God certainly seems clear enough that I am able to explicate the important characteristics of the concept that are required to get the argument off the ground. So once again, the response looks to be quite counter-intuitive, and would be very surprising if it were right.

It should now be clear, IMPOSSIBLE permits no easy refutation. It has shown itself to be unusually resilient. But despite this resiliency, we should not let ourselves get carried away and think that the argument is sound. At the very least, I think that I have shown that the result of rejecting one of the premises promises to be a significant, interesting, and perhaps surprising outcome. Your mission, should you choose to accept it, is to reveal the indiscretion in my argument, and settle this matter in whatever way you deem fit.