Stephen Davis’s objection to the second ontological argument

Abstract

Stephen Davis has argued that the second ontological argument (OA) fails as a theistic proof because it ignores the logical possibility of what he calls an ontologically impossible being. By an “ontologically impossible being” (OIB) he means a being that (a) does not exist, (b) logically-possibly exists, and (c) would exist necessarily if it existed. In this brief essay, I argue, first, that even if an OIB is logically possible, its logical possibility is irrelevant to the OA at issue; and second, that an OIB is in fact logically impossible, because the predicates which define it are inconsistent. The concept of an OIB may be coherent if necessity is understood as ontological self-sufficiency, but even so the OIB is irrelevant to the OA.

Introduction

The purpose of this brief essay is to defend a version of the ontological argument (OA) against an objection raised by Davis in God, Reason & Theistic Proofs (1997, pp. 139–144). The argument at issue is a theistic proof that seems to occur in St. Anselm’s Proslogion (in Ch. 3), though as Davis points out (p. 139), it was not understood as a distinct argument until the twentieth century, when Barth (in Anselm: Fides Quaerens Intellectum II.B.2, 1960) and Malcolm (in “Anselm’s Ontological Arguments”, 1960) independently explained it. Whether the argument really occurs in the Proslogion is another question. But even if this “second OA” is the discovery of Anselm’s interpreters rather than of Anselm, it is Anselmian in that it rests on the definition or conception of God as the greatest conceivable being (GCB), and it is an OA in that it begins entirely with putative a priori truths and ends with a positive existential conclusion, namely, that God exists. The argument can be formulated as follows:

• 1. It is not logically possible for the GCB to exist contingently.

• 2. For any $x$, if it is not logically possible for $x$ to exist contingently, then $x$ is either a necessary being or a logically impossible being.

• 3. The GCB is either a necessary being or a logically impossible being. (1, 2)

• 4. The GCB is not a logically impossible being.

• 5. The GCB is a necessary being. (3, 4)

• 6. The GCB exists. (5)

This argument rests on two intuitive propositions, both of which are supposed to be a priori truths. The first is that God is logically possible; he either exists or could have existed. The second is that a contingently existing being is, as such, unworthy to be called God or the GCB: any existing being worthy of that name, title, or description—any existing being that exemplifies that concept—will exist necessarily, not contingently. The conjunction, however, of “It is logically possible that the GCB exists” and “If the GCB exists, then the GCB is such as could not have failed to exist” (Anselm says that the GCB cannot be conceived not to exist) seems to entail that the GCB does exist. This is because the categories of necessary being, contingent being, and logically impossible being seem exhaustive, and each seems exclusive of the others, so that a being which is neither contingent nor logically impossible seems bound to exist.

It is precisely this point that Davis resists. His objection to the second OA is that proposition (2) commits a logical error by ignoring the logical possibility of what he calls an ontologically impossible being. By an “ontologically impossible being” (OIB) he means

$(\hbox {OIB}=_\mathrm{df})$ A being that (a) does not exist, (b) logically-possibly exists, and (c) would exist necessarily if it existed.

If this concept of ontological impossibility is coherent, then an OIB is a logically possible being that is neither necessary nor contingent. A necessary being is a being whose nonexistence is logically impossible: an OIB is not a necessary being, because its nonexistence is actual. A contingent being is one that contingently exists if it exists and contingently fails to exist if it fails to exist: an OIB is not a contingent being, because if it exists, it exists necessarily, not contingently. According to Davis, an OIB is what many atheists have in mind when they conceive of God: that is, many atheists agree that God could have existed and that if God had existed, he would have been a necessary being; but the fact of the matter, they want to add, is that God does not exist. If this concept of an OIB is coherent, then it seems that (2) must be replaced with

${2^\prime }$.:

For any $x$, if it is not logically possible for $x$ to exist contingently, then $x$ is a necessary being, a logically impossible being, or an OIB,

and it seems that (5) must be replaced with

${5^\prime }$. The GCB is either a necessary being or an OIB.

Since (${5^\prime }$) does not entail that the GCB exists, the second OA seems (so Davis argues) to fail as a theistic proof.

In order to rescue the second OA from this objection, the ontological arguer must show that the concept of an OIB does not invalidate the inference from (1) and (4) to (5). In this essay I will argue, first, that even if an OIB is logically possible, (2) is true and the inference to (5) is unaffected; and second, that an OIB is in fact logically impossible, because the predicates which define it are inconsistent. We can perhaps rescue the OIB by distinguishing between two senses of the term “necessary being”, but the new OIB that emerges is, like the first, irrelevant to the second OA.

The OIB is irrelevant

An OIB is defined as a being that (a) does not exist, (b) logically-possibly exists, and (c) exists necessarily if it exists. The first thing to notice is that even if the concept of an OIB is coherent, the proposition that something is necessarily an OIB is logically impossible: if something is necessarily an OIB, then it necessarily fails to exist; and if something necessarily fails to exist, then it does not logically-possibly exist, as it would if it were an OIB. In order for the logical possibility of an OIB to invalidate the second OA in the way that Davis claims it does, the logical possibility that something is contingently an OIB must be inconsistent with (2).

It is not. If something is contingently an OIB, then it is logically-possibly not an OIB. From the definition of an OIB, it is supposed to follow that necessary beings, logically impossible beings, and contingent beings are not OIBs, but if something is logically-possibly a necessary being, logically-possibly a logically impossible being, or logically-possibly a contingent being, then it is a necessary being, a logically impossible being, or a contingent being, in which case it is not an OIB at all. In order to make sense of the proposition that something is contingently an OIB, we must introduce a fifth concept, that of an ontologically necessary being. By an “ontologically necessary being” (ONB) I mean

$(\hbox {ONB} =_\mathrm{df})$ A being that (d) exists, (e) logically-possibly fails to exist, and (f) necessarily fails to exist if it fails to exist.

I will argue later that this concept of an ONB, like that of an OIB, is incoherent. But if the concept is coherent, then an ONB is a logically possible being that contingently exists if it exists and necessarily fails to exist if it fails to exist. So for any $x$, if it is logically possible for $x$ to be an ONB, then it is logically possible for $x$ to exist contingently. This entails that if $x$ is contingently an OIB, then it is logically possible for $x$ to exist contingently. The logical possibility of an OIB is therefore consistent with (2): the concept of an OIB does not invalidate the second OA, even if the concept is coherent.

The OIB is logically impossible

If the concept of an OIB is coherent, then the following is a consistent triad of propositions:

• A. $x$ does not exist.

• B. It is logically possible for $x$ to exist.

• C. If $x$ exists, then $x$ exists necessarily.

Now, if some proposition or state of affairs is logically possible and if it entails another proposition or state of affairs, then that other proposition or state of affairs is also logically possible. Therefore, if the existence of $x$ is logically possible (as (B) states) and if the existence of $x$ entails the necessary existence of $x$ (as (C) states), then

• D. The necessary existence of $x$ is logically possible.

Another way of deriving (D) is by inferring that if (C) is true, then it is not logically possible for $x$ to exist contingently, and if (B) is true, then it is either logically possible for $x$ to exist contingently or logically possible for $x$ to exist necessarily. How can it be otherwise? How can it be that $x$ can exist but cannot exist contingently or necessarily? It follows, then, that the conjunction of (B) and (C) entails (D). (D), however, entails

$\sim \hbox {A}.$ :

$x$ exists.

This is because for any $x$, the logical possibility that $x$ is a necessary being entails that $x$ is a necessary being, and if $x$ is a necessary being then $x$ exists. The concept of an OIB is incoherent because the predicates that define it are incompatible. The conjunction of (B) and (C) entails the negation of (A).

One possible response to this argument (as well as the argument of the previous section) is to reject the assumption that the logical possibility of necessity entails necessity. There is a way, however, of arguing that the OIB is logically impossible without presupposing this principle. Taken together, (A) and (C) state that $x$ does not exist but would have been a necessary being had it existed. This entails that the proposition “$x$ does not exist”, which is true and therefore logically possible, would have been necessarily false and therefore logically impossible had $x$ existed. If, in addition to (A) and (C), $x$ is logically possible (as (B) states), then the set of logically impossible propositions could have been larger than it is. But the set of logically impossible propositions could not have been larger than it is. Therefore, if (A) and (C) are true, then (B) is false. Therefore, the concept of an OIB is incoherent and the OIB is logically impossible.

It seems clear that the set of logically impossible propositions could not have been smaller than it is. That is, a logically impossible proposition could not have been logically possible instead. If some proposition $p$ could be logically possible, then it could be the case that $p$ could be true. But if it could be the case that $p$ could be true, then $p$ could be true. In other words, the logical possibility that $p$ is logically possible entails that $p$ is logically possible. So if $p$ is logically impossible, then $p$ could not have been logically possible instead. The same conclusion seems to follow from the fact that if $p$ is logically impossible, then (i) $p$ is false, (ii) $p$ does not simply happen to be false (as if by accident or mere chance), and (iii) the truth value of $p$ does not depend on any contingent truth or state of affairs. It is very hard to see how it could have been the case that such a proposition does simply happen to be false or that its truth value does depend on something contingent. That would seem to imply a logically possible difference in the necessary truths of logic or in the very nature of the proposition or in the nature of logical possibility. By reductio ad absurdum, we can conclude that the logical impossibility of a proposition cannot be contingent. If $p$ is logically impossible, then it is necessarily logically impossible, and necessarily necessarily logically impossible, and so on. The regress is benign, and it entails that the set of logically impossible propositions could not have been smaller than it is.

It also entails that the set of logically impossible propositions, and with it the set of necessary propositions, could not have been larger than it is. The reason is simple: if a logically possible proposition like “The OIB does not exist” were logically impossible instead, then its logical impossibility would be contingent, not necessary. No sense can be made of the suggestion that a logically possible proposition could have been logically impossible without being logically-possibly logically possible. But if a proposition is logically-possibly logically possible, then it is logically possible. So no sense can be made of the suggestion that a logically possible proposition could have been logically impossible instead—and hence no sense can be made of the suggestion that something not necessary could have been necessary instead.

Similarly, no sense can be made of the suggestion that an actually true proposition like “The OIB does not exist” could have been false without being logically-possibly true. Nothing can change the fact that the actual truth is logically possible, just as nothing can change the fact that the actual logical possibilities are logically-possibly logical possibilities: the logical possibility of a proposition (or being or state of affairs) is always a matter not only of fact but of necessity. Whether some proposition is true may and often does depend on contingent factors, but whether the proposition is logically possible does not depend on anything (except in the sense that the logical possibility of a proposition depends on the necessary truths of logic or the nature of the proposition itself or the nature of logical possibility). Even though the world could have been different from how it actually is, no logically possible difference would entail that the actual world is (or that certain existent beings or true propositions or obtaining states of affairs are) logically impossible. This is because the logical possibility of something does not at all depend on the actuality of anything, and so the question which world (or being) is actual has no bearing at all on the question which worlds (or beings or propositions or states of affairs) are logically possible. Yet this is all false if an OIB is logically possible. The logical possibility of an OIB entails that the world could have been different in such a way as to make an actually true proposition logically impossible instead.

This second way of arguing against the logical possibility of an OIB can be explained in the following terms. Using the OIB as a model, we can define an ontologically necessary state of affairs (ONS) as one that (d) obtains, (e) logically-possibly fails to obtain, and (f) would be logically impossible if it failed to obtain. We can then say (i) that for every OIB there is a corresponding ONS, namely, the state of affairs of the OIB’s failure to exist; (ii) that the logical possibility of the OIB stands or falls with that of the ONS; (iii) that the concept of an ONS suggests the incoherent notion that something logically possible is logically-possibly logically impossible; and (iv) that both the concept of an OIB and that of an ONS are incoherent.

This being the case, the concept of an OIB has no bearing on the inference from the noncontingency affirmed by (1) and the logical possibility affirmed by (4) to the necessity affirmed by (5). The concept of an OIB does not invalidate the inference, because an OIB is logically impossible.

A second OIB

If the concept of an OIB has some intuitive appeal, it might be because there are at least two logically distinct senses in which a being can be called necessary. In one sense, a necessary being is

(Logically necessary being =$_\mathrm{df})$ A being whose nonexistence is logically impossible.

This is the concept of necessity that I have been presupposing in this essay, and in the remaining paragraphs I will distinguish it as logical necessity. In his discussion of the OA, Davis himself defines a necessary being as one that, among other things, “cannot not exist” (p. 30). But Davis also says that a necessary being “depends for its existence on no other being” (30, 155). Though this second predicate has an intuitive connection to the first, these two features of Davis’s necessary being seem to be logically distinct. They can therefore be uncoupled. Accordingly, a second sense of necessity is ontological self-sufficiency. An ontologically self-sufficient being is

(Ontologically self-sufficient being =$_\mathrm{df})$ A logically possible being that does not depend on any other being for its existence.

It seems reasonable to hold that logical necessity does not entail ontological self-sufficiency. If God is logically necessary and if God logically-necessarily sustains the existence of some eternal being, then that eternal being is logically necessary too. But even on this hypothesis that eternal being is not ontologically self-sufficient.

Less clear is whether ontological self-sufficiency entails logical necessity. One reason why someone might think it does is that an ontologically self-sufficient being cannot have a reason for not existing. Usually, if a being does not exist but logically-possibly exists, we can ask and in some sense explain why it does not exist. If unicorns are logically possible and we want to explain why they are not actual, we might note that the initial conditions of the universe were not such as to guarantee the emergence of unicorns, and the complex process that is responsible for the diversification of life on earth has not produced unicorns, and human beings have not engineered them, and God has not created them, and so on. But such explanations are impossible for an ontologically self-sufficient being. This is because nothing that is external to such a being has any bearing on its existence: there is no state of affairs whose failure to obtain explains the being’s failure to exist; it simply does not depend on initial conditions or natural processes or intelligent design—or anything. For this reason it may seem logically impossible for an ontologically self-sufficient being not to exist, the way (roughly) in which it may seem logically impossible for something to emerge from nothing.

Despite this consideration, perhaps it is logically possible for a self-sufficient being not to exist. If an OIB is a being that (a) does not exist, (b) logically-possibly exists, and ($\mathrm{c^\prime }$) does not depend on any other being for its existence, then perhaps the concept of an OIB is coherent after all. The proposition that something is logically-necessarily an OIB is still incoherent, but the proposition that something is contingently an OIB may be logically possible. If it is, then this second OIB is an ontologically self-sufficient being that simply happens not to exist. Such a being may be what Davis has in mind.

Is its logical possibility (if it is in fact logically possible) a problem for the second OA? The answer depends on how we should understand

• 1. It is not logically possible for the GCB to exist contingently.

Just as we can distinguish logical necessity from ontological self-sufficiency, we can distinguish logical contingency from ontological dependence. A being logically-contingently exists just in case it both exists and logically-possibly fails to exist. A being ontologically-dependently exists just in case it both exists and depends on another being for its existence. The first premiss of the second OA can be understood as

• 1a. It is not logically possible for the GCB to exist logically-contingently,

• 1b. It is not logically possible for the GCB to depend on another being for its existence,

or the conjunction of (1a) and (1b). In this essay I have been presupposing (1a). Since Davis (in his book on theistic proofs) tends to consider logical contingency and ontological dependence together, he may understand (1) as the conjunction of (1a) and (1b). What the logical possibility of the second OIB shows is that the second OA requires at least (1a). But if (1) is understood as (1a), then the second OIB is, like the first OIB, irrelevant. The reason is that the second OIB would exist logically-contingently if it existed: its nonexistence would be logically possible, because it is even now actual. True, the second OIB would exist independently of all other beings, but its existence would be logically contingent nonetheless.

So even when the possible necessity of the OIB is specified as ontological self-sufficiency, the logical possibility of the OIB (if the second OIB is logically possible) does not invalidate the second OA. The defender of Davis’s objection must go on to show that (1a) is false. In general, the critic of the second OA should try to refute (1a) or (4), either by arguing that logical contingency, unlike ontological dependence, has no bearing on the greatness of a being or by arguing that the concept of a GCB, like that of the first OIB, is incoherent.