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, (1033 - 1109). 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; (1646 CE to 1716 CE) 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 January 14, 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:

$\begin{array}{rl} \text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \to P(\psi) \\ \text{Ax. 2.} & P(\neg \varphi) \leftrightarrow \neg P(\varphi) \\ \text{Th. 1.} & P(\varphi) \to \Diamond \; \exists x[\varphi(x)] \\ \text{Df. 1.} & G(x) \iff \forall \varphi [P(\varphi) \to \varphi(x)] \\ \text{Ax. 3.} & P(G) \\ \text{Th. 2.} & \Diamond \; \exists x \; G(x) \\ \text{Df. 2.} & \varphi \text{ ess } x \iff \varphi(x) \wedge \forall \psi \left\{\psi(x) \to \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \\ \text{Ax. 4.} & P(\varphi) \to \Box \; P(\varphi) \\ \text{Th. 3.} & G(x) \to G \text{ ess } x \\ \text{Df. 3.} & E(x) \iff \forall \varphi[\varphi \text{ ess } x \to \Box \; \exists x \; \varphi(x)] \\ \text{Ax. 5.} & P(E) \\ \text{Th. 4.} & \Box \; \exists x \; G(x) \end{array}$

Modal logic

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

In the most common interpretation of modal logic, one considers "all possible worlds". A truth is necessary if its negation entails a contradiction, such as 2 + 2 = 4, and is true in all possible worlds. By contrast, a truth is contingent if it just happens to be the case, for instance, "more than half of the planet is covered by water". 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.

The proof can be summarized as:

IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists.

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 thus: "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)

If a property A entails a property B (ie in every possible world if an object has property A it must also have property B), and if A is positive, B must also be positive.

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

Axiom 2: For all properties A, either A is positive or "not A" is positive. Never both.

Axiom 3: The property of "being God-like", G is a positive property.

Axiom 4: If a property A is positive, then it is so in every possible world.

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.

Criticisms of Ontological arguments in general

St Thomas Aquinas rejected St. Anselm's ontological argument. Likewise, some Catholic theologians have rejected Gödel's revised version. Bertrand Russell noted: "The argument does not, to a modern mind, seem very convincing, but it is easier to feel convinced that it must be fallacious than it is to find out precisely where the fallacy lies." However, Russell was also known to say: "Great God in Boots! — the ontological argument is sound!" as a parody, including many others that parodied it, such as Gaunilo's Island. In Critique of Pure Reason, Immanuel Kant famously rejected existence as a property.

In David Hume's Dialogues Concerning Natural Religion, the character Cleanthes argues that no being could ever be proven to exist through an a priori demonstration:

[T]here is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.

There have been many other arguments against ontological proofs such as: Existence precedes essence; Gaunilo's island; Necessary nonexistence; Existence is not a predicate; and Problem of incoherence.

Criticism of Gödel's

C. Anthony Anderson has said:

Consideration of the axioms, especially ... [Axiom 2], may tend to dampen one's confidence in ... [Axiom 3] and ... [Axiom 4] — that is, if one harbors any real doubt about self-consistency. I don't say that the argument begs the questions of ... [God's possible existence]; the charge is too difficult to establish. but observe that one cannot just tell by scrutinizing a property what it entails; one might be surprised at a consequence.

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 Leibniz's law, the identity of indiscernibles: two or more objects are identical (are one and the same) if they have all their properties in common, and so, there would 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.

Gödel's Ontological Argument and Temporal Modal Logic

Temporal logic must be taken into account when considering the modal ontological argument. There are two theories of temporal logic, the "state and time" approach, and the "time interval" approach". For the purpose of the ontological argument, we will be focused on the "state and time" approach. In temporal logic, possible worlds are in a "state of the universe at some point in history" and are sometimes called "time-slices". Instead of simultaneous access, as in other modal logics, temporal logic accesses before or after. In traditional alethic logic, calculations are made to determine possibility or necessity of something, in temporal logic possibility and necessity is calculated either in the past, present or future. The symbols typically used to represent alethic possibility is ◊ and alethic necessity is □, however, these are replaced in temporal logic into two categories: possibility and necessity in the past and in the future. The symbols used to represent possibility in the past is (P) and necessity is (H), and the symbols used to represent possibiity in the future is (F) and necessity is (G). Therefore whenever (G) appears before another representative lowercase letter (subject), the statement would read, "It is always Going to be the case..." and whenever (F) appears before another subject, the statement would read, "At sometime in the Future it will be the case...." Whenever (H) appears before another subject, the statement would read, "It Has always been the case..." and whenever (P) appears before another subject, the statement would read, "At sometime in the Past it has been the case...."

Temporal Modal Logic Proof

(H ψ → (ψ • G ψ))

(T1) ~ (H ψ → (ψ • G ψ)) (n) NTF

(T2) H ψ (n) 1, PC

(T3) ~ (ψ • G ψ) (n) 1, PC

(T4) ~ψ (n) 3, PC

(T4)' ~G ψ (n) 3, PC

(T5) F~ ψ (n) 3, MN

(T6) nBk 5, FR

(T7) ~ψ (k) 5,6, FR

(T8) kBn 2, HR

(T9) ψ (k) 2,7, HR

MN = Modal Negation

FR = ◊R

HR = □R

Translation of Negating the Formula

(T1) = It is not true that if it always has been the case that a God-like being exists then a God-like being exists and it is always going to be the case that a God-like being exists in (n).

(T2) = It always has been the case that a God-like being existed in (n).

(T3) = It is not true that a God-like being exists and it is always going to exist in (n).

(T4) = A God-like being does not exist in (n)

(T4)' = It is not the case that a God-like being will always exist in (n).

(T5) = In sometime in the future it will be the case that a God-like being will not exist in (n).

(T6) = (n) occurred before (k).

(T7) = A God-like being does not exist in (k).

(T8) = (k) occurred before (n). (Reflexive Rule)

(T9) = A God-like being exists in (k).

The negation of the formula creates a contradiction.

The argument is logically valid, meaning that if the premises are true, then the conclusion is guaranteed to also be true.

See also

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

References

• 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. 
• Small, Christopher. "Reflections on Gödel’s Ontological Argument". University of Waterloo. http://www.stats.uwaterloo.ca/~cgsmall/Godel.final.revision.PDF. 

External links

• Kurt Gödel's Ontological Argument
• Ontological arguments entry by Graham Oppy in the Stanford Encyclopedia of Philosophy
• Annotated bibliography of studies on Gödel's Ontological Argument