A Gödelian Ontological Argument Defended
Alexander R. Pruss
Oppy (1996) has attempted to refute the Gödelian ontological argument by constructing a parody argument that shows that the atheist has equally good grounds for accepting the premises of a parallel argument that shows the necessary existence of a being less than God. I shall argue that Oppy’s parody fails, but how it fails depends on whether the following additional axiom holds:
where, a property is negative provided that its negation is positive. It will turn out that Axiom E is incompatible with Oppy’s parodic axioms though it appears to be compatible with the axioms of Gödel’s argument. Insofar as Axiom E is plausible, this means that Oppy’s axioms are less plausible. But things get more interesting. If Axiom E fails to hold, then the axioms of the Gödelian argument cannot all be true. However, in such a case, a generalization of the Gödelian argument still works assuming S5. Moreover, the only case in which Oppy’s parody works is the one where the parody is equivalent to this generalization, and where its conclusion is the same as that of the generalization. Hence, the parody can’t work if Axiom E holds, and if Axiom E doesn’t hold, the only case in which it works isn’t a parody.
Gödel’s argument is as follows, taken verbatim from Oppy, including the remarks in brackets:
Definition 1: x is God-like iff x has as essential properties those and only those properties which are positive.
Definition 2: A is an essence of x iff for every property B, x has B necessarily iff A entails B.
Definition 3: x necessarily exists iff every essence of x is necessarily exemplified.
Axiom 1: If a property is positive, then its negation is not positive.
Axiom 2: Any property entailed by [= strictly implied by] a positive property is positive.
Axiom 3: The property of being God-like is positive.
Axiom 4: If a property is positive, then it is necessarily positive.
Axiom 5: Necessary existence is positive.
Theorem 1: If a property is positive, then it is consistent [= possibly exemplified].
Corollary 1: The property of being God-like is consistent.
Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.
Theorem 3: Necessarily, the property of being God-like is exemplified.
(Note: In Definition 2, the “necessarily” presumably has narrow scope: “Necessarily(x has B) iff A entails B. We do not want to require essences to entail non-essential properties.)
With Oppy (1996, p. 226), I assume that the argument works given an appropriate modal logic and “a sufficiently generous conception of properties”. The proof of the generalized version in my Appendix can be made to work for this case. Oppy’s parody then is to take a list of positive properties P1,…,Pn and change Definition 1* and Axioms 3* and 5* as follows:
Definition 1*: x is God*-like iff x has as essential properties those and only those properties which are positive, except for P1,…,Pn.
Axiom 3*: The property of being God*-like is consistent.
Axiom 5*: Necessary existence is positive, and distinct from each of P1,…,Pn. (Oppy, 1996, p. 227)
Oppy claims that Theorem 3 will hold with “God*-like” in the place of God-like. But if the original argument was acceptable, so should this one be. And by choosing different lists of excepted positive properties, we get many distinct necessary beings. This is the parody.
I will now show that Axiom E is incompatible with the conjunction of Oppy’s Axioms 1, 2 and 3* and the claim that P1,…,Pn are positive properties. For, by Definition 1*, being God*-like entails not having E(P1), where as before E(A) is the property of having A essentially. By Axiom 3*, being God*-like is positive, and hence so is not having E(P1). Therefore, E(P1) is a negative property. But by Axiom E, E(P1) is a positive property since P1 is. By Axiom 1, no property is both positive and negative, and a contradiction ensues.
Insofar as Axiom E is plausible, the original Gödelian argument is rationally preferable to Oppy’s. Hence if Axiom E is plausible, it is false that the atheist has equally good grounds for accepting the parody as for accepting the original argument.
Next, note that if Axiom E is not true, then the original Gödelian argument is unsound. For supposing that Axiom E is false, let P be a positive property such that E(P) is negative. Then, being God-like entails having E(P). But by Axiom 3, being God-like is positive, and by Axioms 1 and 2, a positive property can’t entail a negative one.
Thus, if Axiom E holds, Oppy’s parody fails, and if Axiom E fails, the original Gödelian argument fails. In one case the parody fails and in the other it is de trop.
I shall now argue that a modification generalization of the Gödelian argument works even if Axiom E fails. First add the following definition:
Definition. A property A is strongly positive provided that E(A) is positive. A property A is weakly positive provided that it is positive but not strongly so.
By Axiom 2, a strongly positive property is positive, since E(A) entails A. Now modify Definition 1 to say:
Definition 1**. x is God**-like iff x has as essential properties those and only those properties which are strongly positive.
Replace Axiom 3 by:
Axiom 3**. The property of being God**-like is positive.
If Axiom E holds, being God-like and being God**-like are equivalent, and so the argument is a generalization of Gödel’s. As before (see the Appendix), one can prove a version of Theorem 3, that necessarily being God**-like is exemplified. This argument works whether or not Axiom E holds.
In fact, I think there is some reason to think Axiom E does not hold. For consider the property of creating a horse ex nihilo. This seems to be a positive property. But it would not be an unqualifiedly good thing to essentially have that property, since having that property essentially would imply not being able to refrain from creating a horse. So creating a horse ex nihilo is positive, but doing so essentially is limiting and hence not positive.
Now, let us return to Oppy’s parody. Observe that Definition 1** is a version of Definition 1* in the special case where P1,….,Pn is a complete list of all and only the weakly positive properties. Hence, in the case of that list, the Oppy argument works, but simply yields the existence of a God**-like being, and that is no parody.
I will now show that that is the only case in which the Oppy argument works. To do this, I will first show that in Oppy’s argument, P1,…,Pn must contain all the weakly positive properties, and then show that none of P1,…,Pn can be strongly positive, in both cases on pain with inconsistency among Oppy’s assumptions.
On to the argument. Let Q be a weakly positive property. For a reductio, suppose Q is not one of P1,…,Pn. Then by Definition 1*, necessarily, any God*-like being has Q essentially, and hence has E(Q). Thus, God*-likeness entails E(Q), and by Axioms 2 and 3*, E(Q) is positive, contrary to the assumption that Q was weakly positive. Hence, every weakly positive property is in the list P1,…,Pn.
Suppose now for a reductio that the list P1,…,Pn contains some strongly positive property, say Q. Then being God*-like entails not having E(Q), since a God*-like being doesn’t have any of the items on the list P1,…,Pn essentially. Hence, being God*-like entails ~E(Q). By Axioms 2 and 3*, ~E(Q) is positive. But Q was strongly positive, so E(Q) is positive. Hence, E(Q) and ~E(Q) are both positive, contrary to Axiom 1.
We have thus shown that Oppy’s parallel argument works only in the case in which Definition 1* reduces to Definition 1** in the generalized Gödelian argument. The generalized Gödelian argument works whether or not Axiom E holds, and is not subject to Oppy’s parody. The original Gödelian argument worked only if Axiom E held, but given Axiom E was not subject to Oppy’s parody.
We now prove that God**-likeness is necessarily exemplified given Axioms 1, 2, 3**, 4 and 5. I shall assume S5.
Theorem 1. If Axioms 1 and 2 hold, then every positive property is consistent.
Proof: For a reductio, let A be an inconsistent positive property. Then, necessarily, nothing exemplifies A. Thus, necessarily, everything exemplifies ~A. Thus, every property entails ~A. In particular, A entails ~A. But A is positive. Hence, ~A is positive by Axiom 2. But ~A is not positive by Axiom 1, and so ~A is both positive and non-positive, which is absurd.
Corollary 1**. If Axioms 1, 2 and 3** hold, then being God**-like is consistent.
Lemma 1. If Axiom 4 holds, then if A is strongly positive, then necessarily A is strongly positive.
Proof: If A is strongly positive, then both A and E(A) are positive, and by Axiom 4, both necessarily are positive. Hence necessarily A and E(A) are positive, and so necessarily A is strongly positive.
Lemma 2**. If Axioms 1, 2, 3** and 4 hold, then if something is God**-like, then it is essentially God**-like.
Proof: Suppose for a reductio that x is in fact God**-like, but possibly x is not God**-like. If possibly x is not God**-like, then, possibly, x fails to essentially have a strongly positive property or else, possibly, x essentially has something other than a strongly positive property (this uses the claim that if M(p or q), then Mp or Mq). Neither of these will work. Suppose that, possibly, x fails to essentially have a strongly positive property. Using S5 together with Lemma 1, we conclude that x in fact fails to essentially have a strongly positive property, contrary to x’s being God**-like. Suppose, on the other hand, that possibly x essentially has something other than a strongly positive property. By exactly the same reasoning, using S5 and Lemma 1, we conclude that actually x essentially has something other than a strongly positive property, again contrary to x’s being God**-like.
Lemma 3. If Axioms 2 and 5 hold, then necessary existence is strongly positive.
Proof: Necessary existence entails E(necessary existence), since if one exists necessarily, then necessarily one exists necessarily by S4, and so if necessary existence is positive (Axiom 5), so is E(necessary existence) by Axiom 2.
Theorem 3**. If Axioms 1, 2, 3**, 4 and 5 hold, then God**-likeness is necessarily exemplified.
Proof: By Corollary 1**, God**-likeness is possibly exemplified. Thus, possibly, there is a being that has God**-likeness. Thus, possibly, there is a being that essentially has God**-likeness, by Lemma 2**. Thus, possibly, there is a being that essentially has God**-likeness and necessary existence, by Lemma 3 and Definition 1**. By S5, this being actually has necessary existence and has God**-likeness essentially. Q.E.D.
 There is a slight difference between my argument and Oppy’s in that case, in that Oppy’s replaces Axiom 5 by Axiom 5*. But in the special case where P1,…,Pn are all the weakly positive properties, the extra claim in Axiom 5* comes for free from Lemma 3 in the Appendix..