Gödel's ontological proof

Gödel's ontological proof is a formalization of Saint 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 that than which a greater cannot be thought. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist." A more elaborate version was given by Gottfried Leibniz; this is the version that Gödel studied and attempted to clarify with his ontological argument.

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Proof Illustrated

While Gödel was religious, he never published his proof 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; it was published in 1987, nine years after his death. An outline of the proof follows.

Modal logic

The proof uses modal logic, which distinguishes between necessary truths and contingent truths.

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 each 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 has only 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.

Axioms

We first assume the following axiom:

Axiom 1: It is possible to single out positive properties from among all properties. Gödel defines a positive property rather vaguely: "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)... It may also mean pure attribution as opposed to privation (or containing privation)." (Gödel 1995)

We then assume that the following three conditions hold for all positive properties (which can be summarized by saying "the positive properties form an ultrafilter"):

Axiom 2: If P is positive and P entails Q, then Q is positive.

Axiom 3: If P₁, P₂, P₃, ..., P_n are positive properties, then the property (P₁ AND P₂ AND P₃ ... AND P_n) is positive as well.

Axiom 4: If P is a property, then either P or its negation is positive, but not both.

Finally, we assume:

Axiom 5: Necessary existence is a positive property (Pos(NE)). This mirrors the key assumption in Anselm's argument.

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 God.

Derivation

From axioms 1 through 4, Godel argued that in some possible world there exists God. He used a sort of modal plenitude principle to argue this from the logical consistency of Godlikeness. Note that this property is itself positive, since it is the conjunction of the (infinitely many) positive properties.

Then, Gödel defined 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 necessarily exists if for every essence P(x) the following is true: in every possible world, there is an element y with P(y).

Since necessary existence is positive, it must follow from Godlikeness. Moreover, Godlikeness is an essence of God, since it entails all positive properties, and any nonpositive property is the negation of some positive property, so God cannot have any nonpositive properties. Since any Godlike object is necessarily existent, it follows that any Godlike object in one world is a Godlike object in all worlds, by the definition of necessary existence. Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

From these hypotheses, it is also possible to prove that there is only one God in each world: by identity of indiscernibles, no two distinct objects can have precisely the same properties, and so there can only be one object in each world that possesses property G. Gödel did not attempt to do so however, as he purposely limited his proof to the issue of existence, rather than uniqueness. This was more to preserve the logical precision of the argument than due to a penchant for polytheism. This uniqueness proof will only work if one supposes that the positiveness of a property is independent of the object to which it is applied, a claim which some have considered to be suspect.

Critique of definitions and axioms

There are several reasons Gödel's axioms may not be realistic, including the following:

It may be impossible to properly satisfy axiom 3, which assumes that a conjunction of positive properties is also a positive property; for the proof to work, the axiom must be taken to apply to arbitrary, not necessarily finite, collections of properties. Moreover, some positive properties may be incompatible with others. For example mercy may be incompatible with justice. In that case the conjunction would be an impossible property and G(x) would be false of every x. Ted Drange has made this objection to the coherence of attributing all positive properties to God - see this article for Drange's list of incompatible properties and some counter arguments. For these reasons, this axiom was replaced in some reworkings of the proof (including Anderson's, below) by the assumption that G(x) is positive (Pos(G(x)).
It was argued by Jordan Sobel that Gödel's axioms are too strong: they imply that all possible worlds are identical. He proved this result by considering the property "is such that X is true", where X is any true modal statement about the world. If g is a Godlike object, and X is in fact true, then g must possess this property, and hence must possess it necessarily. But then X is a necessary truth. A similar argument shows that all falsehoods are necessary falsehoods. C. Anthony Anderson gave a slightly different axiomatic system which attempts to avoid this problem.

In Anderson's system, Axioms 1, 2, and 5 above are unchanged; however the other axioms are replaced with:

Axiom 3': G(x) is positive.

Axiom 4': If a property is positive, its negation is not positive.

These axioms leave open the possibility that a Godlike object will possess some non-positive properties, provided that these properties are contingent rather than necessary.

External links

References

C. Anthony Anderson, "Some Emendations of Gödel's Ontological Proof", Faith and Philosophy, Vol. 7, No 3, pp. 291-303, July 1990
Kurt Gödel (1995). "Ontological Proof". Collected Works: Unpublished Essays & Lectures, Volume III. pp. 403-404. Oxford University Press. ISBN 0195147227
A. P. Hazen, "On Gödel's Ontological Proof", Australasian Journal of Philosophy, Vol. 76, No 3, pp. 361-377, September 1998
Jordan Howard Sobel, "Gödel's Ontological Proof" in On Being and Saying. Essays for Richard Cartwright, ed. Judith Jarvis Thomson (MIT press, 1987)