Gödel's ontological proof

Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel. It is in a line of development that goes back to Anselm of Canterbury. St. Anselm's ontological argument, in its most succinct form, is as follows: "God, by definition, is that for which no greater can be conceived. 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.

The first version of the ontological proof in Gödel's papers is dated "around 1941". Gödel is not known to have told anyone about his work on the proof until 1970, when he thought he was dying. In February, he allowed Dana Scott to copy out a version of the proof, which circulated privately. In August 1970, Gödel told Oskar Morgenstern that he was "satisfied" with the proof, but Morgenstern recorded in his diary entry for 29 August 1970 that Gödel would not publish because he was afraid that others might think "that he actually believes in God, whereas he is only engaged in a logical investigation (that is, in showing that such a proof with classical assumptions (completeness, etc.) correspondingly axiomatized, is possible)." Gödel died in 1978. Another version, slightly different from Scott's, was found in his papers. It was finally published, together with Scott's version, in 1987.

Morgenstern's diary is an important and usually reliable source for Gödel's later years, but the implication of the August 1970 diary entry — that Gödel did not believe in God — is not consistent with the other evidence. In letters to his mother, who was not a churchgoer and had raised Kurt and his brother as freethinkers, Gödel argued at length for a belief in an afterlife. He did the same in an interview with a skeptical Hao Wang, who said: "I expressed my doubts as G spoke [...] Gödel smiled as he replied to my questions, obviously aware that his answers were not convincing me." Wang reports that Gödel's wife, Adele, two days after Gödel's death, told Wang that "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning." In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza."

Gödel left a fourteen-point outline of his philosophical beliefs in his papers. Points relevant to the ontological proof include

4. There are other worlds and rational beings of a different and higher kind.
5. The world in which we live is not the only one in which we shall live or have lived.
13. There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.
14. Religions are, for the most part, bad -- but religion is not.

The proof

Symbolically:

${\begin{array}{rl}{\mbox{Ax. 1.}}&P(\varphi )\land \Box \;\forall x[\varphi (x)\rightarrow \psi (x)]\rightarrow P(\psi )\\{\mbox{Ax. 2.}}&P(\neg \varphi )\leftrightarrow \neg P(\varphi )\\{\mbox{Th. 1.}}&P(\varphi )\rightarrow \Diamond \;\exists x\;[\varphi (x)]\\{\mbox{Df. 1.}}&G(x)\iff \forall \varphi [P(\varphi )\rightarrow \varphi (x)]\\{\mbox{Ax. 3.}}&P(G)\\{\mbox{Th. 2.}}&\Diamond \;\exists x\;G(x)\\{\mbox{Df. 2.}}&\varphi \;\operatorname {ess} \;x\iff \varphi (x)\land \forall \psi \lbrace \psi (x)\rightarrow \Box \;\forall x[\varphi (x)\rightarrow \psi (x)]\rbrace \\{\mbox{Ax. 4.}}&P(\varphi )\rightarrow \Box \;P(\varphi )\\{\mbox{Th. 3.}}&G(x)\rightarrow G\;\operatorname {ess} \;x\\{\mbox{Df. 3.}}&E(x)\iff \forall \varphi [\varphi \;\operatorname {ess} \;x\rightarrow \Box \;\exists x\;\varphi (x)]\\{\mbox{Ax. 5.}}&P(E)\\{\mbox{Th. 4.}}&\Box \;\exists x\;G(x)\end{array}}$

Modal logic

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

A truth is necessary if its negation entails a contradiction, such as 2 + 2 = 4; by contrast, a truth is contingent if it 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 false in some 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(s) = s is pink

and consider the object

s = my shirt

In our world, P(s) is true because my shirt happens to be pink; 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, Gödel 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 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 for 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)).

However the Compactness Theorem says that a set of sentences is satisfiable if and only if every finite subset is satisfiable.

And it shows if there exists an arbitrary large object of some type, then there must also be an infinite object of that type. It shows e.g. there is an infinite commutative group in which every element times itself gives the identity element. In our case also it means if there is a finite collection of positive properties then there must exist an infinite collection of positive properties. You need the set of positive properties to include existence. If you don't have that you don't have anything—literally. The only way to get to that is by the set itself having infinite positive properties.

• The set of all properties of any object a as a candidate for the set of all positive properties is always consistent with axioms 1–4 concerning positive properties, because the true statements P(a) form a class of statements closed under deduction. Any one property could be claimed to be positive, so long as it is not self-contradictory, with the right choice of a. Specifically, any property that can be possessed without contradiction is positive in some model of axioms 1–4, and any property that can be avoided without contradiction is non-positive in some model of axioms 1–4. Positivity of a property is as implicitly defined as anything can get. Why, then, should any one property (such as the one addressed in axiom 5) be assumed to be positive, given that no such statement is ever a tautology (although it can be a contradiction if the property is unsatisfiable)? Note that, with the right choice of axiom 5, all sorts of things could be proven (see also the objection below), an error common in some form to all ontological arguments. This problem with axiom 5 is a logically inescapable point, and is similar to the demonstration that, in the deontic logic of Ernst Mally, a statement is morally necessary if and only if it is true.
• 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.

See also

• Absolute Infinite
• Existence of God
• Modality
• Philosophy of religion
• Synthetic proposition
• Theism

References

• C. Anthony Anderson, "Some Emendations of Gödel's Ontological Proof", Faith and Philosophy, Vol. 7, No 3, pp. 291–303, July 1990
• John W. Dawson, Jr (1997). Logical Dilemmas: The Life and Work of Kurt Godel. Wellesley, Mass: AK Peters, Ltd. ISBN 1-56881-025-3.
• Melvin Fitting, "Types, Tableaus, and Godel's God" Publisher: Dordrecht Kluwer Academic ©2002, ISBN 1402006047 9781402006043
• 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)
• Wang, Hao (1987). Reflections on Kurt Gödel. Cambridge, Mass: MIT Press. ISBN 0-262-23127-1.
• Wang, Hao (1996). A logical journey: from Gödel to philosophy. Cambridge, Mass: MIT Press. ISBN 0-262-23189-1.

External links

• Kurt Gödel's Ontological Argument
• Graham Oppy. "Ontological arguments". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
• Annotated bibliography of studies on Gödel's Ontological Argument