**A Gödelian Ontological Argument Improved**

Alexander R. Pruss

December 4, 2007

Department of Philosophy

Baylor University

One Bear Place #97273

Waco, TX 76798-7273

Alexander_Pruss@baylor.edu

I shall defend several versions of Gödel’s ontological argument that use weaker premises than the version of Anderson (1990), and that, like Anderson’s argument, avoid modal collapse. Moreover, the versions of the argument that I shall defend are not subject to Oppy’s (1996 and 2000) parody refutations.

The basic primitive notion in a
Gödelian argument is that of a *positive property*. We can understand
that in several different ways, each one giving rise to a different
interpretation of the argument. For instance, one might take a positive
property to be one that in no respect detracts from any respect of the
excellence (or greatness or value, depending on how we prefer to phrase it) of the
entity that has the property but whose negation does detract from some respect
of the excellence (or greatness or value) of the possessor. Or one might take
a positive property to be one that does not entail any limitation but whose
negation does. Or one might start with a somewhat Leibnizian structure of the
space of properties, on which there are some *basic* properties that are
mutually compatible (e.g., because they are logically independent of each
other), and then count a property as positive provided that it is entailed by
at least one of the basic properties.

Each of these interpretations makes plausible the following two “formal” axioms:

**Axiom F1.** If *A* is positive, then ~*A* is
not positive.

**Axiom F2.** If *A* is positive and *A*
entails *B*, then *B* is positive.

The correctness of F1 on the excellence, goodness, greatness
and no-limitation readings is clear, and on the Leibnizian interpretation F1
follows from the compatibility of the basic properties. Moreover, if a
property doesn’t detract from the excellence (or goodness or greatness) of an
entity, then anything it entails had better not detract from it either. On the
other hand, if a property detracts from the excellence (or goodness or
greatness) of an entity, so does any property that entails that property.
Hence if ~*A* detracts from excellence (etc.), and *A* entails *B*,
then ~*B* detracts from excellence (etc.), since ~*B* entails ~*A*
by contraposition. This yields Axiom F2 on the excellence, goodness and
greatness interpretations. Exactly the same reasoning shows that if a property
does not entail any limitation but its negation does, the same holds for any
property that it entails. And closure under entailment is trivial on the
Leibnizian interpretation, so F2 follows once again.

All of the Gödelian arguments I shall offer will make use of the “formal” axioms F1 and F2. In addition, all the arguments will make use of the following “non-formal” axiom:

**Axiom N1.** Necessary existence is positive,

where necessary existence (N.E.) is defined as follows:

**Definition D1.** x has N.E. iff $*F*[˙(*x* has
*F*) and ˙$*y*(*y* has *F*) and ˙"*y*(*y*
has *F* É *y*=*x*)].

Note that D1 makes necessary existence logically weaker than on Gödel’s original definition, since Gödel’s definition entailed that every property of a necessarily existing being was an essential property of it (Koons, 2005). Gödel’s analogue of N1 together with Axiom F2 will entail ourAxiom N1, so our axioms are still weaker than Gödel’s. I will call a being that has necessary existence a “necessary being”.

We need one more definition for our first ontological argument:

**Definition D2.** A property *A* is *strongly
positive* iff ˙(*having A
essentially* is a positive property).

I shall abbreviate the property of *having A essentially*
as *EA*. Then D2 says that *A* is strongly positive iff necessarily *EA*
is positive. Given Axiom F1, a strongly positive proposition is positive.

I shall assume a modal logic that includes S5. As is usual in modal logic, axioms, and hence theorems as well, are always taken to be necessary truths. Moreover, I shall assume that all properties exist in all worlds, i.e., that properties exist necessarily. Because of this, it is possible to interchange the order of modal operators and quantifications over properties.

At this point, we are ready to state the simplest ontological argument:

**Theorem T1.** Given Axioms F1, F2 and N1, if *A*
is any strongly positive property, then there exists a being that exists
necessarily and essentially has *A*.

It is rather surprising that T1 follows from F1, F2 and N1 without any further assumptions. Gödelian arguments typically assume additional axioms, e.g., the axiom that to have all and only positive properties as essential properties is a positive property (a claim stronger than F3, below). The proof of T1 is in the Appendix, but to assuage curiosity on the part of the reader, I will prove a crucial and simple lemma here:

**Lemma L1.** Given Axioms F1 and F2, if *A* and *B*
are positive, then *A* and *B* are compossible,

where:

**Definition D3.** A collection *C* of properties is
compossible iff possibly there is an entity that exemplifies all the members of
*C*.

The proof of L1 is very easy. If we assume for a *reductio*
that *A* and *B* are incompossible, then *A* entails ~*B*.
By F1, ~*B* is then positive, but this contradicts F2. Given L1, the
proof of T1, whose details are in the Appendix, depends on applying L1 to the
pair of positive properties *EA* and N.E. to conclude that possibly there
is a necessary being that essentially has *A*, and then using S5 to
eliminate the “possibly”.

How useful Theorem T1 is as a theistic argument depends on what kinds of strongly positive properties one thinks there are. The following non-formal axiom is plausible:

**Axiom N2.** Each of the properties of *essential
omnipotence*, *essential omniscience* and *essential perfect goodness*
is positive.

If it is an axiom that *EA* is positive, then
necessarily *EA* is positive, and hence *A* is strongly positive. Thus,
given N2, we immediately get the following result from T1:

**Corollary C1.** Given Axioms F1, F2, N1 and N2, there
exists an essentially omnipotent necessary being, and there exists an
essentially omniscient necessary being, and there exists an essentially
perfectly good necessary being.

It would be nice if we could prove
that one and the same necessary being is essentially omnipotent, omniscient and
perfectly good. But I do not think this can be shown from F1, F2, N1 and N2
alone, though of course the claim that the three existential claims in C1 are
each made true by the same being is *prima facie* compatible with these
axioms. However, C1 is already quite interesting. Presumably no atheist
accepts any of the conjuncts in the conclusion of C1. Moreover, if we can
further argue that there are logical connections between divine attributes, we
can get a little more. For instance, if a being is essentially omnipotent,
then at least it has the essential property of *being able to* figure out
the truth value of each proposition if it should want to, and so we might
conclude that there necessarily exists a being that is essentially omnipotent
and essentially able to figure out all facts that it wants to figure out.

To do better, we need a formal axiom about how positive properties can be conjoined. There are two options available. First, we might, not implausibly, suppose that the conjunction of all strongly positive properties is positive:

**Axiom F3.** The property of having all strongly
positive properties is a positive property.

Given F1, this axiom is weaker than the corresponding axiom
in the Anderson version of the Gödelian argument, which axiom stated that the
property of having all *and only* the positive properties as essential
properties is positive, since the latter property entails the property of
having all strongly positive properties that figures in F3. Moreover, F3 is a
bit more intuitive than Anderson’s axiom. It is not obvious, for instance,
that *lacking* any essential properties beyond the positive ones is
positive. Moreover, the full Anderson axiom is essential to the parodies in
Oppy (1996, 2000), while the present argument does not appear to be subject to
those parodies (though I have no argument that other parodies cannot be
constructed).

We now have the following result (see the Appendix for the proof):

**Theorem T2.** Given Axioms F1-F3 and N1, there is a
necessary being that essentially has all strongly positive properties.

One might, however, object to F3 on the
following grounds. Axiom F3 seems innocent until we realize the following
fact, which follows immediately from Lemma L1 in the special case where *A*=*B*:

**Lemma L2.** Given Axioms F1 and F2, any positive
property *A* is possibly exemplified.

Thus, F1, F2 and F3 entail that the conjunction of all strongly positive properties is possibly exemplified, and hence that all strongly positive properties are compossible. But the latter is quite a strong claim, and one might worry whether an intuition that so quickly entails it can be that plausible. One might also worry that F3 is more than just a formal claim about the logic of positive properties, since it seems to make the substantive claim that a particular property is positive, and hence is more akin to the non-formal axioms N1 and N2, and such non-formal axioms are likely more vulnerable.

For these reasons, it will be better to make use of the following “more formal” axiom:

**Axiom F4.** If *A* and *B* are strongly
positive and compossible, then their conjunction is positive.

As far as intuitive support goes, we could probably have gone with “positive” in place of “strongly positive” in the antecedent, but since putting “strongly positive” in the antecedent will yield a weaker set of axioms (since by F1 any strongly positive property is positive), and F4 is all I will need, I might as well go with F4. Observe that F3 together with F1 entails F4, so using F1, F2 and F4 will be using a weaker set of assumptions than just using F1, F2 and F3.

Now we have a result that, while not quite as satisfying as T2, is in practice about as useful to a theist:

**Theorem T3.** Given Axioms F1, F2, F4 and N1, if *U*
is any finite set of strongly positive properties, then there is a necessary
being that essentially has every member of *U*.

We can conclude from this:

**Corollary C2.** Given Axioms F1, F2, F4, N1 and N2,
there is a necessary being that is essentially omniscient, essentially
omnipotent and essentially perfectly good.

The same holds with F3 in place of F4, either because we then have a weaker set of axioms, or by using T2 instead of T3.

It would be nice if one could drop
the assumption of finiteness in T3, but I doubt this can be done without adding
an axiom like F4. The conclusion of T3 is at least *prima facie* compatible
with the stronger claim that there is a necessary being that essentially has
all strongly positive properties. Even though T3 may not give the theist all
that theist wants, there may be no harm in leaving to faith the claim that God has
*all* the strongly positive properties. And C2 is quite respectable,
surely, and more than sufficient to refute atheism.

The ontological arguments embodied
in T1, T2, T3, C1 and C2 are valid. Each of Theorems T1, T2 and T3 uses a
weaker (at least *prima facie*) set of axioms than the best previous
Gödelian argument, that of Armstrong. In particular, the problem of modal
collapse is avoided in these theorems just as it is in Armstrong’s work.

Whether the arguments are sound
will, I think, depend on a deeper analysis of the nature of positivity. But
the premises all appear at least somewhat *plausible*, are largely
independent of the premises of non-ontological arguments such as the
cosmological argument or the argument from religious experience, and hence the
Gödelian arguments should further lower the probability of atheism and increase
that of theism.

First we prove T1. To that end, we need two auxiliary results:

**Lemma L3.** If *a* has N.E., then ˙(*a* has N.E.).

*Proof of L3*: By S4, if ˙*p*,
then ˙˙*p*. The truth of L3 thus easily
follows from S4 and D1. ¨

**Lemma L4.** For any property *B*, if $*x*(*x* has N.E. and *x* has *B*),
then ˙$*x*(*x*
has N.E. and ŕ(*x* has *B*)).

*Proof of L4*: Suppose the antecedent of the claim in L4.
By existential instantiation, let *a* be such that *a* has N.E. and *a*
has *B*. Then, *a* has N.E. necessarily by L3. Moreover, by the
Brouwer axiom of modal logic, if *a* has *B*, then ˙ŕ(*a*
has *B*). Hence, ˙(*a* has
N.E. and ŕ(*a* has *B*)). It
follows from this that $*x*˙(*x* has N.E. and ŕ(*x* has *B*)). But if $*x*˙*Fx*,
then ˙$*xFx*,
and the proof is complete. ¨

*Proof of T1*: Let *P* be the conjunction of N.E.
and *EA*. By Lemma L4, possibly there is an *x* that satisfies *P*.
Thus, possibly, there is an *x* that has N.E. and *EA*. Now L4 is a
necessary truth, since it is a theorem of modal logic, so it follows from the
necessity of L4 that if ŕ$*x*(*x* has N.E. and *x* has *B*),
then ŕ˙$*x*(*x* has N.E. and ŕ(*x* has *B*)). Using S5 and
substituting *EA* for *B*, we conclude that there is an *x* that
has N.E. and that possibly essentially has *A*. But by S5, if one
possibly essentially has *A*, then one essentially has *A*
(accessibility is an equivalence relation given S5, so if *x* possibly
essentially has *A*, then *x* essentially has *A* in some
accessible world *w*, and hence *x* has *A* in every world
accessible from *w* at which *x* exists, which comes to the same thing
as saying that *x* has *A* in every accessible world at which *x*
exists). ¨

Theorems T2 and T3 both follow from T1 in a fairly easy way. First, we need the following result:

**Lemma L5.** Given F1, if *A* is strongly positive,
so is *EA*.

*Proof of L5*: *EA* entails *EEA*, given
axiom S4. If *A* is strongly positive, *EA* is necessarily positive,
and by F1 (which holds necessarily), so is *EEA*. Hence *EA* is
strongly positive. ¨

Now we can prove T2.

*Proof of T2*: Let *P* be the property of having
all properties of the form *EA* where *A* is a strongly positive
property. By L5, having every strongly positive property entails having *EA*
for every strongly positive property *A*. Thus, *P* is entailed by
the conjunctive property in F3, and hence *P* is positive by F1 and F3.
By T1, it follows that there is a necessary being that essentially has *P*,
and T2 follows. ¨

To prove T3, we first need the following result:

**Lemma L6.** Given F1, F2 and F4, the conjunction of any
finite set of strongly positive properties is strongly positive.

*Proof of L6*: It suffices to show that if *A* and
*B* are strongly positive, then the conjunction *A*&*B* is
strongly positive as well, since we can then repeat that argument to get the
strong positivity of the conjunction of all the members of the finite set. So
suppose *A* and *B* are strongly positive. Then *EA* and *EB*
are strongly positive by L5, and hence compossible by L1. By F4, we conclude
that *EA*&*EB* is positive, and since F4 holds necessarily, we in
fact conclude that *EA*&*EB* is necessarily positive. But *EA*&*EB*
entails *E*(*A*&*B*), and so by F1, we conclude that *E*(*A*&*B*)
is necessarily positive, and thus *A*&*B* is strongly positive. ¨

*Proof of T3*: Let *P* be the conjunction of the
properties in *U*. Then, *P* is strongly positive by L6, and the
conclusion of T3 follows from T1. ¨

**References**

Anderson, C. A.
1990. Some emendations on Gödel’s ontological proof. *Faith and Philosophy*
**7**: 291-303.

Koons, Robert C. Sobel on Gödel’s ontological proof. *http://www.phil.pku.edu.cn/documents/20050317075553_Sobel.pdf*

Oppy, Graham
1996. Gödelian ontological arguments. *Analysis* **56**: 226-230.

Oppy, Graham
2000. Response to Gettings. *Analysis*** 60**: 363-367.