Gödel's ontological proof

Gödel's ontological proof is a variation on St. Anselm's ontological argument for God's existence by the mathematician Kurt Gödel.

St. Anselm's ontological argument, in its most succinct form, is as follows: "God, by definition, is absolute perfection. Non-existence is an imperfection. Therefore, God exists." A more elaborate version was given by Gottfried Leibniz; this is the version that Gödel studied and attempted to clarify with his ontological argument.

While Gödel was deeply religious, he never published his argument because he feared that it would be mistaken as establishing God's existence beyond doubt. Instead, he only saw it as a logical investigation and a clean formulation of Leibniz' argument with all assumptions spelled out. He repeatedly showed the argument to friends around 1970 and it was published after his death in 1987. An outline of the proof follows.

The proof uses modal logic which distinguishes between necessary truths created by definitions and contingent truths inferred from observations of a world.

A truth is necessary if it cannot be avoided, such as 2 + 2 = 4; by contrast, a contingent truth "just happens to be the case", for instance "more than half of the earth is covered by water". In the most common interpretation of modal logic, one considers "all possible worlds". If a statement is true in all possible worlds, then it is a necessary truth. If a statement happens to be true in our world, but is not true in all other worlds, then it is a contingent truth. A statement that is true in some world (not necessarily our own) is called a possible truth.

A property assigns to every object in every possible world a truth value (either true or false). Note that not all worlds have the same objects: some objects exist in some worlds and not in others. A property only has to assign truth values to those objects that exist in a particular world. As an example, consider the property

P(x) = x is grey

and consider the object

s = my shirt

In our world, P(s) is true because my shirt happens to be grey; in some other world, P(s) is false, while in still some other world, P(s) wouldn't make sense because my shirt doesn't exist there.

We say that the property P entails the property Q, if any object in any world that has the property P in that world, also has the property Q in that same world. For example, the property

P(x) = x is taller than 2 meters

entails the property

Q(x) = x is taller than 1 meter.

As an axiom, we now assume that it is possible to single out positive properties from among all properties, and that the following three conditions hold:

If P is positive and P entails Q, then Q is positive.
If P₁, P₂, P₃, ... are positive properties, then the property (P₁ AND P₂ AND P₃ ...) is positive as well.
If P is a property, then either P or its negation is positive, but not both.

(These can be summarized by saying "the positive properties form an ultrafilter".)

There are several reasons these assumptions are not considered by some to be realistic. It is not apparent that "positive" and/or its negation is defined for all properties in any world, though this does not have to be the case as far as the proof is concerned - all that is required is that some properties may be singled out as positive. Second, the question may be raised, "why is a property that is positive for one object positive for all objects?" Ultimately though, for the sake of this proof, all that is required is that the property is positive for some or even one object. An objection to the second proof may be made in the form "a frequently noted feature of human affairs is that a combination of positive qualities in a powerful leader is sometimes disastrous". This objection fails though, because it assumes that a set of properties is the same thing as an object or that the given set of properties must completely define the object, and the axiom specifically refers to all sets of properties, not just those sets of properties which completely describe objects. The strongest opposition by far is to the third axiom - there may be no justification for imposing a binary value system on the world.

Now we define a new property G: if x is an object in some possible world, then G(x) is true if and only if P(x) is true in that same world for all positive properties P. G is called the "God-like" property. An object x that has the God-like property is called a god, although any resemblance to the God of any actual religion is dependant on the properties one defines as good.

The final axiom is that necessary existence is a positive property. This was also the key innovation in Anselm's argument. The usual notation for this is Pos(NE).

Subject to the assumptions, it is asserted that one can now already show that in some world there exists a god. But we want more: we want to show that necessarily, in every world there exists a unique God.

In order to do this, Gödel first defines essences: if x is an object in some world, then the property P is said to be an essence of x if P(x) is true in that world and if P entails all other properties that x has in that world. We also say that x strongly exists if for every essence P of x the following is true: in every possible world, there is an element y with P(y). He adds one last axiom: the property of "strongly existing" is positive.

From these hypotheses, it is now possible to prove that there is one and only one God in each world. God necessarily exists.

It was pointed out by Sobel that Gödel's axioms are too strong: they imply that all possible worlds are identical. Anderson gave a slightly different axiom system which attempts to avoid this problem.

All variants of this "proof" equally prove the nonexistence of any god. This requires only two changes:

Substitute "incompatible with a god" for "positive" as the attribute of an acceptable property.
G is the "no-god" property and an object x that has any no-god property prevents the existence of a god.

The same derivation proves there is no god in any world.

The assumptions for the no-god version are exactly as valid and reasonable as those for the God version. Start with the right definitions and you can prove anything.

References:

Jordan Howard Sobel, "Gödel's Ontological Proof" in On Being and Saying. Essays for Richard Cartwright, ed. Judith Jarvis Thomson (MIT press, 1987)
C. Anthony Anderson, "Some Emendations of Gödel's Ontological Proof", Faith and Philosophy, Vol. 7, No 3, pp. 291-303, July 1990
A. P. Hazen, "On Gödel's Ontological Proof", Australasian Journal of Philosophy, Vol. 76, No 3, pp. 361-377, September 1998