To the editors:

I enjoyed reading Graham Oppy’s lucid and perceptive summary of the debate over Gödelian modal ontological-arguments.¹ While I agree with most of what he says, I want to outline how a defender of such arguments might respond to some of his main objections, and then suggest what I take to be the most persuasive distinct objections to such arguments.

Oppy rightly challenges the proponent of a modal ontological-argument (MOA) to offer an account of “positive” properties that delivers the conclusion that God exists. He is unpersuaded by Alexander Pruss’s attempts to fill in the analysis of a positive property in a way that unequivocally supports theism.² Still, the analysis of positivity that makes the most sense in the Anselmian tradition is the first of Pruss’s suggestions:

1. P is positive =_df. it is better to have P than to lack P.³

This is exactly what we should expect the Anselmian to say; the Anselmian believes that for all properties P, if it is better to have P than to lack it, then God has P.⁴ Let us proceed with this definition and see whether the Gödelian MOA is successful. I will consider two of Oppy’s objections, suggest replies, and then present two independent objections to the Gödelian MOA.

Oppy’s First Objection: Incoherence

Recall two relevant axioms:

• A1: If A is positive, then ~A is not positive.
• A2: If Δ is a set of positive properties, and Δ entails B, then B is positive.

Oppy writes:

Given (a), neither A1 or A2 seems plausible. For any F and G, F entails F∨G. If it is better to have F than not to have F, and better not to have G than to have G, then symmetry would suggest that F∨G is neither one nor the other.⁵

How might the theist respond to this argument?

A first reply would be to bite the bullet: affirm that F∨G is neither positive nor not-positive, by affirming that there is a truth-value gap here.⁶ Another reply would be to deny that there are such things as disjunctive properties.⁷ Sentences obviously entail disjunctions of which the sentence is a disjunct, but it does not follow that properties entail disjunctive properties of which the property is a disjunct. (Importantly, we are imagining that F and G each individually entail the property of disjunctively being F∨G, not that they either entail F or entail G (or both).)

Suppose that we do not want to bite the bullet and we nevertheless grant the existence of disjunctive properties. And suppose, as before, that F is positive and that G is not-positive. Then given A1-A2, F∨G is positive. Still, if we imagine that G is not-positive and entails F∨G, then that does not yet entail that F∨G is not-positive. We would need to add:

• A2*: If Δ is a set of not-positive properties, and Δ entails B, then B is not-positive.

Of course, A2* is unnecessary for the Gödelian MOA so far stated, so our theist can cheerfully decline to endorse it.

To reply, Oppy might argue that A2* is independently plausible, especially given A1 and A2. But I am not yet sure how plausible it is, nor am I sure about how much that alleged plausibility tells us. Philosophers of logic are quite familiar with the sometimes-puzzling results of taking standard first-order propositional logic literally, and asking whether the language we teach to first-year students really does match our English words “if,” “then,” and the rest.⁸ These debates should weaken our confidence that we can tell, simply by looking at A2* and surveying our immediate intuitions, whether A2* is true. Moreover, of course, if Oppy argues that A2* is plausible given A1 and A2, then he has already offered a rebuttal to that argument: his original criticism of (a) based on F, G, and F∨G. Our theist can simply report that A2* is no longer plausible when its logical consequences (from importing A1 and A2) are understood. Thus our theist could reject A2* and affirm that F∨G is positive.

One might argue independently that it is better to have F∨G than to lack it, since G might not be bad-making, whereas F is guaranteed to be good-making. Consider an analogy:

Suppose that envelope F contains $200 and envelope G either contains (with equal probability) $200, $100, or nothing. Suppose you have the option to pay $100 for an opportunity to randomly draw envelope F or envelope G, each at 50% probability. Intuitively, it is rational to pay the $100 to play the game, because the expected value of playing the game is $50: (½)(200) + (½)[(⅓)(200) + (⅓)(100) + (⅓)(0)] – 100.

By analogy, if F is positive and G is either positive, neutral, or “negative,” and the negativity would be of the same magnitude as the positivity (in G and in F), then it seems that a fully-rational, fully-informed cognizer (who is not extremely risk averse) would prefer the instantiation of the property F∨G to its non-instantiation.

To reply to these observations, Oppy might simply recast his example to add the following items, similar to suggestions that Pruss has made:⁹

• (a^–) P is negative =_df. it is worse to have P than to lack P.
• A1^–: If A is negative, then ~A is not negative.
• A2^–: If Δ is a set of negative properties, and Δ entails B, then B is negative.

In turn, Oppy could argue that it is then both better and worse (at the same time) to possess F∨G, when F is positive and G is negative. But as before, our theist has no reason yet to endorse (a^–), A1^–, and A2^–.¹⁰

Suppose someone is still unconvinced by these arguments. Then the theist could simply say that F∨G is of “indeterminate” valence, and add these axioms:

• A1**: If A is positive or indeterminate, then ~A is not positive nor of indeterminate valence.
• A2**: If Δ is a set of positive or indeterminate properties, and Δ entails B, then either B is positive or B is of indeterminate valence.

A2** still gives us the conclusion, modulo the other Gödelian assumptions, that God is possible. To see this, consider an analogue of the Gödelian argument Oppy considers for the possibility of God:

Suppose that Δ is a non-empty set of positive or indeterminate properties; and that the properties in Δ are not possibly instantiated. Anything that has all of the properties in Δ has, by D0, the negation of at least one. By A1**, the entailed property is not positive nor indeterminate, since it is the negation of a property that is positive or indeterminate; but, by A2**, the entailed property in Δ is positive or indeterminate, since it is entailed by a set of properties all of which are positive or indeterminate…¹¹

Upon first glance, that argument appears to be just as cogent as the original version, and has avoided Oppy’s objection from incoherence. Given that being God^†-like is positive, it follows that it is positive or indeterminate, and we reach the conclusion that being God^†-like is possible.

Oppy’s Second Objection: Circularity

Recall Oppy’s summary of the argument for the crucial lemma L1:

[The Possibility Argument]

Suppose that Δ is a non-empty set of positive properties; and that the properties in Δ are not possibly instantiated. Anything that has all of the properties in Δ has, by D0, the negation of at least one. By A1, the entailed property is not positive, since it is the negation of a property that is positive; but, by A2, the entailed property in Δ is positive, since it is entailed by a set of properties all of which are positive. This is a contradiction. The properties in Δ are possibly co-instantiated.¹²

Clearly, an ontological argument that used God’s possibility as a premise would be circular:

[Definition (a)] … can only support the claim that P is positive if possession of P is possible. If possession of P is impossible, then it is better to lack P then to have P, P detracts from excellence, and P entails being limited. The claim that satisfaction of (a) … supports the claim that P is positive assumes that possession of P is possible. But Gödel’s derivation is supposed to establish that possession of P is possible, and so cannot rely on the prior assumption that this is so.¹³

Therefore, Oppy concludes, to use (a) would produce merely a circular MOA.

One quick reply is to observe that the positivity of P is not intended to prove its possibility, so the argument is not really circular after all. Instead, the argument attempts to demonstrate the possible co-instantiation of God’s properties, from which it follows that each property is individually instantiable. Yes, the argument cannot be sound without P’s being possible, but no argument can be sound without its conclusion being true.

Another reply would be to argue that a property can pro tanto contribute to a being’s goodness (and thereby be positive) without its being possible. It is not good-making to be impossible, but it does not follow that it is not good-making to have some property such that the property is impossible. Suppose that an omnipotent, omniscient, perfectly evil being exists de-re-necessarily. Then it might be impossible to have the property “is a happy creature.” But it would not follow that the property “is a happy creature” is not positive. Similarly, suppose (as I have argued elsewhere)¹⁴ that omnipotence and omniscience are logically incompatible. It does not follow that it is not good-making to be omnipotent and omniscient.

Even if those replies do not work, the proponent of the MOA can go through the argument and replace every instance of positive with positive-if-possible, and modify the first two axioms:

• A1***: If A is positive-if-possible, then ~A is not positive-if-possible.
• A2***: If Δ is a set of positive-if-possible properties, and Δ entails B, then B is positive-if-possible.

Positive-if-possible would mean that such a property, if it is possible to possess at all, would pro tanto contribute to a being’s goodness. Compare:

[The Positive-if-Possible Argument]

Suppose that Δ is a non-empty set of individually positive-if-possible properties; and that the properties in Δ are not possibly instantiated. Anything that has all of the properties in Δ has, by D0, the negation of at least one. By A1***, the entailed property is not positive-if-possible, since it is the negation of a property that is positive-if-possible; but, by A2***, the entailed property in Δ is positive-if-possible, since it is entailed by a set of properties all of which are positive-if-possible. This is a contradiction. The properties in Δ are possibly co-instantiated.

[Therefore,]

L1***: Any non-empty set of positive-if-possible properties is possibly instantiated, i.e., it is possible for there to be something that has all of these properties.

If someone argues now that we do not know whether God’s properties are even possible to possess, then we have strayed out of the territory of evaluating the MOA and into a distinct discussion of divine-attribute arguments. I and other philosophers have written about such arguments before,¹⁵ but it does not strike me as an objection to the MOA in particular—rather than an independent argument against theism—to simply question whether these properties are individually possibly instantiated. In any case, the theist can simply report that God’s properties are prima facie conceivable and narrowly logically possible, which many or most philosophers will take to be prima facie evidence of metaphysical possibility.¹⁶

I have suggested some ways for the theist to respond to Oppy’s criticisms of Gödelian MOAs. Now let us turn to some independent objections to such arguments, objections that I believe to be more difficult to answer.

Denying the Compatibility-Possibility Link

One traditional objection to MOAs, one that Oppy himself essentially offers with his example of the Godless singularity, is that the atheist has no more reason to affirm the possibility of God than she has to affirm the possibility of atheism being true.¹⁷ The Gödelian possibility-lemma, L1, is intended to repair this flaw. Recall, once again, the Possibility Argument for L1:

Suppose that Δ is a non-empty set of positive properties; and that the properties in Δ are not possibly instantiated. Anything that has all of the properties in Δ has, by D0, the negation of at least one …. 

The underlying argument appears to require:

[The Compatibility-Possibility Link] Every set of properties is either possibly co-instantiated or narrowly logically incompossible.

Robert Maydole makes the Compatibility-Possibility Link explicit in a slightly different form:

If the essential properties of something are compatible, then it is possible that it exists.

He reports that the Compatibility-Possibility Link is self-evident to him.¹⁸

However, the atheist has good reason to deny the Compatibility-Possibility Link. The reason is the peculiarity of the subject of our argument: a de-re-metaphysically-necessarily-existing being that is arguably not conceptually nor narrowly logically necessary. Notably, Oppy’s gloss of the argument for L1 does not specify the sense of modality we are talking about. Relevant to our purposes, the modality might be metaphysical modality, or it might instead be narrow logical-modality. (Roughly speaking, something is metaphysically possible just in case it could have existed, given some set of physical laws or other. Something is narrowly logically possible just in case the denial of its existence is not a theorem of standard logic.)¹⁹ Therefore, we really have two Possibility Arguments to consider:

• Metaphysical
  
  (M1) Suppose that Δ is a non-empty set of positive properties; and that the properties in Δ are not metaphysically-possibly instantiated. [Therefore,] (M2) Anything that has all of the properties in Δ has, by D0, the negation of at least one…
• Narrow Logical
  
  (N1) Suppose that Δ is a non-empty set of positive properties; and that the properties in Δ are not narrowly-logically-possibly instantiated. (N2) Anything that has all of the properties in Δ has, by D0, the negation of at least one. By A1, the entailed property is not positive, since it is the negation of a property that is positive; but, by A2, the entailed property in Δ is positive, since it is entailed by a set of properties all of which are positive. This is a contradiction. [Therefore,] (N3) The properties in Δ are narrowly-logically possibly co-instantiated. (N4) Therefore, the properties in Δ are metaphysically possibly co-instantiated. (N5) Therefore, given S5, a metaphysically necessary God exists.

The problem with the Metaphysical argument is that (M2) does not follow from (M1). In the metaphysical modality sense, an object can be impossible in all sorts of ways that it cannot be in the narrow logical sense. For example, suppose the Anselmian God exists. Then arguably an instance of gratuitous evil would be impossible.²⁰ It would not be narrowly logically impossible, because there is no contradiction in the sentence, “There is an instance of gratuitous evil,” nor in “(∃x)(Ex & Gx).” Instead, it would be metaphysically impossible, because it would be incompossible with the existence of a certain de-re-necessarily existing Anselmian God.

In contrast, the problem with the Narrow Logical argument is that (N4) does not follow from (N3), and (N3) alone is not enough to prove that theism is metaphysically possibly true. As before, that inference would assume the Compatibility-Possibility Link. But the example of gratuitous evils is a counterexample to the connection between narrow-logical modality and metaphysical modality. And crucially, in the end, our theist wants to derive the metaphysical possibility of God’s existence. The atheist can grant that God is narrowly logically necessary without yet granting that God is actual; the former claim is simply the relatively uninteresting claim about which formulae are properly derivable from which other formulae in a given language of logic.

Last, someone might argue that narrow-logical possibility is good evidence of metaphysical possibility. My suspicion, although I do not have the space to develop it here, is that the atheist can ultimately make the same case for the metaphysical possibility of a Godless universe.

S5 Is Unsound

As noted in the previous section, it is possible to derive all sorts of interesting results in narrow-logical modality, without learning anything interesting about truth and falsity in the real world. For those conclusions, we need a story about semantics; we need to connect the derivations in our logic to the facts about reality. Therefore, we need to know whether S5 is sound: we need to know whether, given true premises, expressed in S5 modal logic, it is ever possible to derive a conclusion that is (semantically) false but validly derived in S5.²¹

Consider the notion of accessibility. Suppose that some world w* is accessible to w if and only if: if w were actual then w* would be possible. Thus accessibility can be imagined as a two-place relation that takes possible worlds as its relata. For S5 to be sound, this relation must be reflexive, symmetric, and transitive.²²

There are multiple reasons to reject transitivity.²³ Here is a version of the Modal-Sorites Argument:²⁴

• T1: Any normal human (in any possible world that contains humans) could have existed if 0.00001% of her DNA had been different from what it actually is.
• T2: Accessibility is transitive.
• T3: Therefore, any normal human in the actual world could have existed if all of her DNA had been different from what it actually is.

Premise (T1) is supposed to be intuitive; in the real world, ionizing radiation that mutated 0.0001% of your DNA would intuitively not ipso facto have killed you. And conclusion (T3) follows from (T1) and (T2); we just keep imagining further possible worlds in which another 0.0001% of the person’s DNA is changed. Thus: Some human exists in w₀. She could have existed in w₁, in which 0.0001% of her DNA is different. The w₁-human could have existed in w₂, in which a further 0.0001% of her DNA had been different from how it is in w₁, still. And given these two facts plus transitivity, the w₀-human could have still existed in w₂, and even (ultimately) in w_1,000,000. Yet (T3) will be implausible to many or most readers. Therefore, we should reject (T2).

Reina Hayaki offers interesting objections to the Modal-Sorites Argument. First, she argues (in effect) that the Modal-Sorites Argument cannot be successfully evaluated until we have a general theory of vagueness. Second, she argues that the entity in the distant possible world (say, the person with all of her DNA changed) is not actually existent at a possible world after all.²⁵ I reply to the first objection by noting that the argument appears compatible with all serious theories of vagueness; indeed, it does not depend on saying that a person’s identity changes at any particular point. (Vagueness is only interesting when we are forced to say that (e.g.) a non-heap becomes a heap, but the point of this Modal-Sorites argument is that given transitivity, the person’s identity does not change.)

I reply to the second objection as follows. Either Hayaki means that the distant possible world is impossible relative to the actual world or that it is impossible relative to all possible worlds, i.e., is intrinsically impossible. If she means the former, then that is exactly the point of the Modal-Sorites Argument: transitivity entails the possibility of this distant possible world relative to the actual world, so transitivity is false. If she means the latter, then there is no reason yet to believe this; there is no contradiction in describing a creature, 100% of whose DNA is different from mine.

As with transitivity, there are multiple arguments against symmetry.²⁶ One appeals to the somewhat-controversial thesis of the necessity of the past: in some worlds (e.g. the ones in which humans never existed), it is at least possible that Julius Caesar never existed (i.e. the world in which he never existed is accessible), but not in others.²⁷ The world in which Julius Caesar has existed is accessible from some worlds in which he has not (yet) existed (after all, he could have been born or conceived an hour later than he was), but the world in which he has not (yet) existed is inaccessible from the worlds in which he has existed.

The other argument against symmetry that I want to consider appeals to an intuitive premise that may be difficult to describe sufficiently precisely. It would look something like this:

What happens in other possible worlds, stays in other possible worlds.

That is, whether some actually-nonexistent being exists in some other possible world should have no bearing on what exists in the actual world. If we accept a version of this principle, then we find that in a sense, MOAs themselves are counterexamples to symmetry; they deliver the implausible conclusion that just because something is possibly necessary, it is actual.

To this, the proponent of an MOA might reply that our critic is begging the question. Reply: the friend and the foe of MOAs both inspect the structure and content of the MOA in question and come to different conclusions. To the foe, the mere fact that God could have existed necessarily is irrelevant to whether he exists in the actual world, and the MOA is a nice illustration of this point. Our critic’s position is that the MOA has led her to notice that symmetry is independently implausible. She might say honestly that she would have been led to the same conclusion even if the subject of the argument were not the Anselmian God but instead some other entity that was alleged to exist necessarily if at all, for example nunicorns, which are necessarily existing unicorns, or Kane’s “less-than-perfect necessary beings.”²⁸

These observations answer a reply that the proponent of MOAs might offer at this point:

Look; it is not my job to defend S5 (or transitivity, or symmetry) independently. S5 is less controversial than theism. To make gains for theism, I only need to show that given S5, theism is justified.

My suggestion is that, to the contrary, it is incumbent on the proponent of a MOA to defend symmetry, because that debate is, in a sense, the debate over MOAs themselves. It is a debate over—fundamentally—whether the possible existence of a necessary being tells us anything at all about the actual world.

Robert Kane offers three arguments for symmetry.²⁹ The first argument is based on an intuition that if something is not actual, then it could not possibly be necessary. The second is based on intuition that accessibility would be universal. The third is based on an intuition that if a proposition is actually true, then it cannot possibly be impossible. But these arguments are refuted by my examples above. It could have been necessary that Earth has been destroyed, even though it is not necessary, because Earth has not been destroyed yet. In the actual world, the trillionth human ever to be born does not exist, but it could have been impossible that the trillionth human being to be born never existed, e.g. if this human being had already been born. And it could have been necessary that unicorns exist. Upon considering these examples, I no longer have the intuition that accessibility is universal, nor that the conditionals Kane adduces are true.

I conclude, therefore, that S5 is probably unsound.

Conclusion: Further research

I have argued that the proponent of a Gödelian MOA can successfully derive the conclusion that the Anselmian God is narrowly-logically necessary, given S5. But S5 is questionable in ways that most proponents of MOAs have not tried to address. And the mere narrow-logical necessity of God is, at best, good evidence of God’s metaphysical possibility. Moreover, I suggested that it is probably not too difficult to construct an analogue of the Possibility Argument, but for the possibility of a Godless world.

Therefore, the theist ought to (i) defend symmetry and transitivity explicitly, to (ii) argue explicitly for an evidential connection between narrow-logical necessity and metaphysical necessity, and to (iii) refute an analogue of the Possibility Argument that attempts to confirm the possibility of a Godless world. If these tasks can be accomplished, then our theist may have discovered a cogent modal ontological argument.

Thomas Metcalf

Graham Oppy replies:

Thomas Metcalf offers a range of comments on my paper. These comments are of two kinds. Some are intended to defend Gödel’s ontological argument against the criticisms that I make of it. And some are intended to raise more challenging criticisms of Gödel’s ontological argument.

The more challenging criticisms that Metcalf develops are aimed at the logical system within which Gödel’s ontological argument is developed. Gödel’s logical system is a third-order classical predicate logic with S5 modal base and λ–abstraction. As I noted in my paper, there are various contestable elements in Gödel’s logical system. Some do not like higher-order logic. Some do not like modal logic. Some do not like λ–abstraction. Metcalf objects to S5; he thinks that a correct modal logic will be significantly weaker than S5. Moreover, he claims that, even granted S5, the most that Gödel’s argument establishes is that it is narrowly-logically necessary that God exists, whereas the conclusion that theists want is that it is metaphysically necessary that God exists.

In my paper, I argued that, even granting Gödel everything that his logical system requires, his argument fails. That leaves it entirely open whether we should grant Gödel everything that his logical system requires. Higher-order logic, modal logic, λ-abstraction, S5, and the alleged distinction between narrowly-logical modality and metaphysical modality all raise big, controversial philosophical questions.  But there is no prospect of giving an adequate discussion of any of these matters here. So I shall not try. Instead, I will just take up the comments in which Metcalf defends Gödel’s argument against the criticisms that I make of it.

“Incoherence”

In my paper, I say that, in the context of Gödel’s argument, a minimum requirement on any account of positive properties is that the following two principles are both endorsed:

• A1: If A is positive, then ~A is not positive.
• A2: If Δ is a set of positive properties, and Δ entails B, then B is positive.

The reason for this is that A1 and A2 are the premises in Gödel’s key lemma:

• L1: A1, A2 ⊢ Any non-empty set of positive properties is possibly instantiated

Metcalf claims that the proponent of Gödel’s argument can opt for the following account of positive properties:

• A is positive =_df. It is better to have A than not to have A.

Against this account, I objected that, if A is a property that it is better to have than not to have, and B is a property that it is better not to have then to have, then, while symmetry tells us that A∨B is a property that it is neither better to have nor better not to have, A2 wrongly entails that it is a property that it is better to have. (This is because A entails A∨B, and A is a property that it is better to have.)

Metcalf first suggests that this objection can be met by claiming that A∨B is neither positive nor not-positive (“by affirming a truth-value gap”). This reply is odd. Gödel’s premises require that A∨B is better to have. So a defense of Gödel would need to be an argument that A∨B is better to have. An argument that A∨B is neither better to have nor better to not have is not such an argument.

Metcalf next suggests that the objection can be met by denying that there are disjunctive properties. But, in the context of Gödel’s logical system, it is obvious that there are disjunctive properties (in the sense relevant to Gödel’s proof). Suppose that A(x) and B(x) are well-formed formulae in which the variable x has one or more free occurrences. In Gödel’s system, we can form the lambda abstracts λxA(x) and λx(B(x); informally, we can read these as “the property A” and “the property B.” And it is a theorem of the logical system that □∀x[λyA(y)]x) ⊢ □∀x{[λy[A(y)vB(y)]x}. That is to say, it is a theorem of the logical system that the property A entails the property A∨B.

Setting aside these two objections, Metcalf then suggests that, on the assumption that F is positive and G is not positive, the fact that G is not positive does not entail that F∨G is not positive. So, he suggests, we need a further axiom. Perhaps:

• A2* If Δ is a set of not-positive properties, and Δ entails B, then B is not positive.

This suggestion is also odd. It should be noted that my argument assumed, not that G is not better to have, but rather that G is better not to have. If A is a property that it is better to have then not to have, then, necessarily, for any x, it is better that Ax than that ~Ax; if A is a property that it is better not to have than to have then, necessarily, for any x, it is better that ~Ax than that Ax. There are many properties that satisfy neither of these conditions. Consider, for example, the property of being either good or a mass murderer. It is better than not that good beings have this property; it is not better than not that mass murderers have this property. The argument that F∨G is not better to have than not to have, given that F is better to have than not to have and G is better not to have than to have, does not depend upon A2, or A2*, or any similar principle. Rather, as I said, it is a straightforward appeal to symmetry: given that F is better to have than not to have, and G is better not to have than to have, it cannot be that F∨G is better to have than not to have, and nor can it be that F∨G is better not to have than to have.

Finally, Metcalf suggests that we might entertain some more complicated view about the valences of disjunctive properties: perhaps, for example, we might suppose that F∨G is of indeterminate valence.  And, he suggests, we might adopt axioms couched in terms of “positive or indeterminate” properties. Again, this suggestion seems strange to me. We have it quite generally that, given (1) if ϑ(A) then ~ϑ(~A), and (2) if, for all A ∈Δ, ϑA, and Δ entails B, then ϑ(B), it follows (3) that any non-empty set of ϑ-properties is possibly instantiated. Theists are interested in the case in which the ϑ-properties are the positive properties because they (allegedly) find it intuitive that the positive properties are the divine properties. Theists surely do not find it intuitive to suppose that the positive or indeterminate properties are the divine properties. But, in any case, in order to meet my objection, there is no point looking at alternative ϑ-properties; in order to defend the claim that, in Gödel’s argument, we can take the positive properties to be the properties that are better to have than not to have, we need to have a direct argument that it is better than not that Hitler had the property of being either good or a mass murderer. Best of luck with that.

“Circularity”

In my paper, I added a second line of criticism of (a). (This line of criticism applies to a wide range of attempts to define the positive properties.) Given A1 and A2, a property can only be positive if it is possibly instantiated. Given some account of what it is to be a positive property—e.g. that it is a property that is better to have than not—we are faced with the problem of assessing those axioms in Gödel’s argument that claim that particular properties are positive.

To set up the problem, consider someone who thinks that the only causal entity that exists of necessity is the initial singularity from which our universe emerges; every possible world starts with that initial singularity, and diverges from the actual world only because chance plays out differently. On this view, there are none but natural causes, involving none but natural properties; it is impossible for there to be gods, or other supernatural entities. While this person does not claim to know the essential properties of the initial singularity, they insist that the initial singularity does have essential properties. Consequently, they think that there is a set of essential properties of the initial singularity that is non-trivial and closed under entailment. Call the members of this set of essential properties of the initial singularity the €-properties.

Given these assumptions, that person accepts:

1. For all first-order properties P, if €(P) then ~€(~P); and
2. For all non-empty sets of first-order properties Δ, if, for all P∈Δ €(P), and Δ entails Q, then €(Q); and
3. For all first-order properties P, if €(P) then €(Necessarily P); and
4. €(Having all of the Ƥ-properties essentially); and
5. €(Necessarily Existing)

Moreover, they have the following result:

• Theorem N: (1), (2), (3), (4), (5) ⊢ There is a necessarily-existing initial singularity

By contrast, a theist who accepts that God exists of necessity and possesses all of the properties that it is better to have than not to have essentially accepts:

1. For all first-order properties P, if Ƥ(P) then ~Ƥ(~P); and
2. For all non-empty sets of first-order properties Δ, if, for all P∈Δ Ƥ(P), and Δ entails Q, then Ƥ(Q); and
3. For all first-order properties P, if €(P) then €(Necessarily P); and
4. Ƥ (Having all of the Ƥ-properties essentially); and
5. Ƥ(Necessarily Existing)

And they have the following result:

• Theorem T: (1), (2), (3), (4), (5) ⊢ There is a necessarily existing God-like being.

The question that I wish to raise is this: how do we collectively decide between the many different arguments of this form that we can construct? The key lemma is a perfectly general result. What follows from it depends upon which higher-order properties we take to satisfy the conditions of the lemma. We could suppose that, under some construal, positive properties fit the bill. We could decide that divine properties fit the bill. We could decide that the properties of the initial singularity fit the bill. But our taking any argument of the form of Gödel’s argument to be sound depends upon our deciding which higher-order properties make (3), (4) and (5) true.

Consider the claim that having all of the positive properties is positive, under the interpretation according to which the positive properties are those that it is better to have than not to have. If it is impossible for anything to have all of the properties that it is better to have than not to have, then it simply is not true that having all of the positive properties is positive. According to the view for which the only causal entity that exists of necessity is the initial singularity, it is plainly not true that having all of the positive properties is positive, if the positive properties include such properties as omnipotence, omniscience, perfect goodness, and so forth. If it is impossible for anything to be omnipotent, then it is better not to be omnipotent!

Metcalf takes me to be claiming that Gödel’s argument is circular. I do not think that is quite right. It is rather, as I suggested at the end of my article, that the argument should leave anyone who is not already a particular kind of theist cold. Even if all that is true of you is that you think it no less plausible that there is a necessarily-existing initial singularity than that there is a necessarily-existing God, you have no reason at all to think that Gödel’s argument is sound.

In his discussion of my second line of criticism, Metcalf claims that, even if it is impossible to have a property, it might nonetheless be the case that the property is positive. But, given the assumptions in play, that cannot be right. Suppose that it is impossible for anything to be F, and yet that F is a positive property. Given that F is a positive property, by the key lemma, it is possible that F is instantiated. But that just contradicts the claim that it is impossible for anything to be F.

Metcalf suggests that we might avoid this response by modifying the first two axioms:

• A1**: If A is positive-if-possible, then ~A is not positive-if-possible
• A2**: If Δ is a set of positive-if-possible properties, and Δ entails B, then B is positive-if-possible.

He claims that we can then derive a modified version of the key lemma:

• L1**: Any non-empty set of positive-if-possible properties is possibly instantiated

According to Metcalf, a property is “positive-if-possible” just in case, if it is possible to possess it, it is a good-making property for whatever has it. I take it that there are two kinds of “positive-if-possible” properties: those that are possible, and those that are not. Consider a “positive-if-possible” property that is not possible. Any property that is not possible entails every property. So, if even one “positive-if-possible” property is impossible, then the “positive-if-possible” properties are just all of the properties (i.e. every pair of contradictory properties falls under the “positive-if-possible” properties). But it is impossible for all of the properties to be instantiated. Clearly, in order to avoid this unwanted conclusion, we need to suppose that all of the “positive-if-possible” properties are possible. But that just takes us back to consideration of the positive properties.

Concluding Remark

I agree with Metcalf that there are challenging questions to ask about the logical system within which Gödel’s ontological argument is developed. However, unlike Metcalf, I think that those who reject the conclusion of Gödel’s ontological argument can reasonably find the argument unconvincing even while granting—if only for the sake of argument—that the logical system is perfectly in order. Whereas Metcalf thinks that resistance to the suggestion that the positive properties are those properties that it is better to have than not to have can be overcome, I demur; on closer examination, the reasons that he offers for thinking this turn out to be insufficient.