Kurt Gödel's Ontological Argument - Part III


Gödel's positivity operator


Rembrandt's Descent from the Cross

To exist, to be happy, to be sad, to be wise, to be beautiful, to have extension in time and space, to be loving, to be filled with hate.

For the logician, these are all properties of individuals, or predicates to use a technical term. In first order predicate logic, they all have much the same status: a predicate is a unary operator providing a truth-functional assignment to individuals. But as human beings, we know that dry logic may range over the greatest good or the greatest evil, the deepest insights or the most ridiculous of follies.

To say, as Anselm did, that God is great, is to distinguish among properties. Wisdom is better than folly, and it is better to exist than not to exist. Personally, I think that it is better to love than to hate. Better to be at peace than at war. You make up your own mind. Most of us are egotistical enough to consider ourselves greater than the lowly earthworm. Why we believe so is not completely obvious, and it is hard to get the earthworm's perspective on the issue. But presumably, we humans think we are greater than earthworms because we have some positive properties that earthworms lack. Some might argue that our ability to contemplate the origins of the universe in the Big Bang, eons ago, is such a positive property. The earthworm might well counter this argument by pointing out that it is a much better custodian of this planet's soil than any human is.

To formalize the idea of a positive property, Gödel introduced a positivity operator. Just as a predicate or property provides a truth-functional assignment to individuals (i.e., Rx, where "x=Santa Claus" and R="wears a red suit"), so the positivity operator Pos provides a truth-functional assignment to properties themselves. We say that Pos(F) is true if F is a positive property.

Gödel suggested that a property could be said to be positive in a moral-aesthetic sense or in a sense of pure attribution. While many of us would differ over the details, a moral-aesthetic interpretation of Pos(F) is reasonably clear. The interpretation of Pos(F) as signifying "pure attribution" is far from obvious. Gödel interpreted the negation of "pure attribution" as "privation", i.e., a lacking in certain elements of being. For example, if F is the property of being present at the Eiffel tower on May 17 at 9:35 a.m. then we might be willing to accept that Pos(F) is true. This is not to say that being somewhere else at the same time is also not a form of pure attribution. If to be present at one location means to be absent from another location, then the participation in an aspect of being could not be said to be pure. So God could be said to be present at the Eiffel tower in the sense of pure attribution. Are there many other types of pure attribution? Probably, both sublime and ridiculous. One might argue that F="knows the capitals of all the states of the United States" is a form of pure attribution. Certainly, the property ~F seems to indicate a type of educational privation that most people possess, myself included. I would expect that God would get full marks in a quiz on this topic. Pure attribution may also require that a property has "fullness of being," although this idea is itself unclear to me.

Is there any relationship between positivity as a moral-aesthetic concept and positivity as pure attribution? I would like to think that the answer is yes. That which is moral or aesthetic typically enhances or deepens being. Alternatively, one could say that those things which are moral or aesthetic are things which affirm the creative over the destructive. A resolution of this issue is, fortunately, unnecessary to the argument which follows.

The only other indication that Gödel gave for his intensions here is to say that positivity is "independent of the accidental structure of the world". I would interpret this as meaning

Pos(F) Pos(F)

As the right-hand side implies the left-hand side, it is sufficient to state that

Axiom G0: Pos(F) Pos(F)

It is no coincidence that there is a similarity between this axiom and Axiom 2 of Anselm's argument. Just as God is conceived to be independent of the accidental structure of the world, so all positivity is conceived to be this way.

Now let us formalize some other axioms about positivity. The particular version of these axioms that I shall use is due to C. Anthony Anderson. See Some emendations of Gödel's ontological proof by C. A. Anderson, Faith and Philosophy 7 (1990) 291-303. The first axiom states that for any property F

Axiom G1: Pos(F) ~Pos(~F)

That is, if any property is positive, it's negation is not positive. The idea behind this should be clear from the discussion above.

For the next axiom, we shall need some notation. Let F and H be two properties. We shall write F => H if

(x)[Fx Hx]

and we shall say that the property F entails the property H.

This brings us to our next axiom, which states that a property entailed by a positive property is itself positive.

Axiom G2: Pos(F) [(F => H) Pos(H)]

Another concept that we shall need is the consistency of a property. This is not to be confused with logical consistency. We shall say that a property F is consistent if it is possibly exemplified, i.e., if (x)Fx is true. Note that if F is an inconsistent property, than F => ~F, that is, F entails its negation. The proof of this is elementary in quantified modal logic and is left to the reader.

With these ideas defined, we can now proceed to our first theorem, namely, that all positive properties are consistent.

Theorem G1: Pos(F) (x)Fx

Proof: Let Pos(F) be true. Suppose that F is inconsistent. We shall prove a contradiction. If F is inconsistent, then F => ~F. So Axiom G2 implies that Pos(~F) is true. However, from Axiom G1 and Pos(F), we get ~Pos(~F), which is a contradiction. Q.E.D.

Theorem G1 is a remarkably cheery theorem, which tells us that everything that is positive is also possible. We need both Axiom G1 and Axiom G2 to prove it. Of the two axioms, G2 looks less obvious than G1. The fact that we can prove Theorem G1 from it suggests that we should look closely at Axiom G2. I cannot help but feel that the idea of Axiom G2 in Gödel's proof owes something to Gottfried Leibniz (1646-1716), with his idea that God, conceived of as the greatest good, has created the best of all possible worlds. Axiom G2 is not quite as radical as this. It does not deny the existence of evil, but only asserts that it can never be entailed by pure goodness. Thus Axiom G2 is far from a trivial observation. If God, whose essence is "independent of the accidental structure of the world", can have created a world of good and evil, then the evil of this world can only be accidental and never necessary. To say otherwise would contradict Axiom G2. Those people who believe in the existence of evil as fundamentally as they believe in the existence of good, will probably have to disagree with Gödel and Leibniz on this point.

Surprisingly, Theorem G1 supports a principle that Immanuel Kant elaborated, despite the fact that Kant was a vigorous opponent of the Ontological Argument. Kant's position can be summarised by the aphorism that "ought implies can," namely that if one has an obligation to do something, then it must be possible to do so. While Kant was referring to the idea of human action in the world and its attendant moral obligations, there seems to be an interesting affinity with Theorem G1 which declares that the Good (i.e., positive states of affairs) are always possible.

The Basic Definitions

The next stage of Gödel's argument introduces some "God-language" into the discussion. We now introduce three concepts to the discussion. These are

A technical note: The particular "language" that is necessary for the task needs to be richer than the first-order modal logic that we have used up to now. From now on, we will have to quantify over predicates themselves. Anderson has proposed that Cocchiarella's semantics for second-order modal logic will suffice. See Nino B. Cocchiarella, "A completeness theorem in second order modal logic," Theoria 35 (1969), 81-103.

The argument now proceeds as follows. An individual x will be said to be God-like, that is, Gx will be said to be true, if every essential property of x is positive and if x has every positive property as an essential property. Formally, this is

Definition G1: Gx =df (F) [ Fx Pos(F) ]

Note that Gödel carefully distinguishes between existence proofs and uniqueness. There is nothing in Definition G1 which says that there is at most one God-like individual (monotheism). He carefully sets his signs on the existence part of the argument and leaves out the uniqueness issue. This is more a matter of logical precision than any dalliance with polytheism. When we were discussing the existence of Santa Claus earlier, we made the implicit and unwarranted assumption that there could be only one Santa Claus. However, there is no logical reason why two Santas could not exist, perhaps working as partners to ensure that all the presents get delivered on time. A more carefully constructed question would have been whether there exists a Santa Claus-like individual. As any mathematician would agree, having established the existence of a Santa Claus-like individual, one is in a position to try to prove that there is only one.

The next point to be made here is that Definition G1 does not assume that every property of God is positive. Only those properties of God which are essential are required to be positive according to this definition. Gödel's original definition of Gx was not exactly that given above, but rather

Gx =df (F) [Pos(F) Fx]

That is, an individual x is God-like if x possesses every positive property. This differs from our definition because it uses material implication rather than material equivalence, and because the modal operator does not appear.

Finally, note that the operator appears inside the quantification over F in Definition G1. We have to tread carefully here. Up to now, our quantifiers have been to the right of the modal operator. Putting a quantifier to the left of means that we have to be able to interpret properties between possible worlds, in the sense of a possible world semantics. In Cocchiarella's semantics a (singular) property is a function from the set of all possible worlds into a set of possible individuals.

The idea behind Definition G1 is to define a God-like individual as one having perfection (i.e., maximally positive properties) as the individual's essence. We can formalize an essence of an individual by saying

Definition G2: F Ess x =df (H) [Hx (F => H)]

On the left-hand side here, we read that F is an essence of x. There is nothing in this definition which formally requires that a property which is an essence of x is unique. However, a form of uniqueness is evident in the definition. If F Ess x is true and H Ess x as well, then it follows that F <=> H in the sense that F => H and H => F. Thus F and H will be realized in common or fail to be realized in common, and can be formally identified.

The question of whether every individual has an essence was at the heart of Jean-Paul Sartre's philosophy of existentialism. While he agreed that ordinary individuals such as rocks, trees, dogs and cats have essences, and that essence precedes existence for such things, Sartre argued that for human beings, existence precedes essence in the sense that we exist first and define ourselves secondly.

The third concept that we shall introduce in this section is the property of necessary existence. Although predicate logics have existential quantifiers, these quantifiers are not properties (as Kant reminded us). However, we can introduce the concept of necessary existence as a property by making it derivative from the notion of essence. We say that an individual x exists necessarily if every property which is an essence of x is necessarily realized in some individual. Formally,

Definition G3: NE(x) =df (F) [F Ess x (y) Fy]

The Rest of the Argument

In view of Definition G2, we might say that a God-like individual is one whose essence is saturated (in the mathematical sense) with respect to all positive properties. It would seem reasonable that it should follow from this that the property of being God-like is itself positive. However, we cannot deduce this from the axioms and definitions given above. The reason for this is because we cannot prove that positivity of properties is preserved under aggregation. For example, let F and H be properties. Define

(F&H)x =df Fx & Hx

That is, F&H is the property of having both F and H. Then we would expect that

[Pos(F) & Pos(H)] Pos(F&H)]

However, this cannot be proved from the axioms and definitions above. If we cannot prove this, it is no wonder that we cannot prove that the aggregation of all positive properties is positive. We could add this as an axiom, together with some notation describing aggregation over classes of properties. However, the route through this would be tedious and circuitous, since we only wish to prove that the property of being God-like is a positive property. It is simpler to just add

Axiom G3: Pos(G)

to our list of axioms. Readers who are unhappy with this shortcut are invited to write it all out in terms of arbitrary aggregations of properties.

An immediate consequence is that the property of being God-like is consistent, i.e., possibly exemplified.

Corollary G1: (x) Gx

Proof. Theorem G1 and Axiom G3. Q.E.D.

Corollary G1 is very close to being the statement that we called Axiom 1 in Anselm's proof. I claimed that Axiom 1 was the least supported statement in argument; so we have arguably made some progress. The thing to do now is to finish this proof off in a typically Anselmian fashion. Anselm argued that a being that exists necessarily is greater than a being that exists accidentally or one that does not exist at all, everything else being equal. In terms of the language of Gödel's positivity operator, this means that necessary existence is a positive property. So we have the Anselmian axiom:

Axiom G4: Pos(NE)

Obviously, for Axiom G4 to be true, we need to interpret the positivity operator as "pure attribution." There we have it. That's what we need. Now we can starting proving things using the full power of the modal logic S5. First we need the result that if an individual is God-like, then being God-like is the essence of that individual. Symbolically, this is:

Theorem G2: Gx G Ess x

Proof: Suppose Gx is true and that x necessarily has property H. That is, suppose that Hx is true. Then by Definition G1, we have Pos(H). That is, H is a positive property. But

[ Pos(H) (y)(Gy Hy) ]

which can be deduced from Definition G1 and the fact that anything which has a property necessarily must have the property. However, by Axiom G0, any property which is positive is necessarily positive. Thus Pos(H) is true. By an application of modal modus ponens, we deduce that

(y)(Gy Hy)

Thus we have proved that if x has any property H essentially, then that property is entailed by the property G. That is, G => H.

Conversely, suppose that Gx is true and that G => H. Then by Axioms G2 and G3, we must have Pos(H). It follows that a God-like individual x has property H necessarily, by Definition G1. That is, Hx is true.

Putting the two directions of the argument together, we see that G Ess x. Q.E.D.

Our final conclusion is that necessarily a God-like individual exists. Equivalently, we can say that necessarily the property of being God-like is exemplified. In symbolic terms, this is

Theorem G3: (x) Gx

Proof: If Gx were true, then by Definition G1, x has every positive property necesarily. But Axiom G4 tells us that necessary existence is a positive property. So it follows that NE(x) is true, i.e., that x would exist necessarily. But by Theorem G2, if Gx were true, then G Ess x would be true. Using Definition G3 we deduce that if any individual x is God-like, then the property of being God-like is necessarily exemplified. This can be written symbolically as

(x)Gx (x) Gx

As mentioned above, this follows from Definition G1, Axiom G4 and Theorem G2. We can recognize this as Axiom 2 from Anselm's argument. Since this last statement has been proved, the necessitation axiom of modal logic implies that

[ (x)Gx (x)Gx ]

Now a theorem of modal logic that can be proved is that (pq) (pq). Combining this with the formula above, we get

(x)Gx (x)Gx

But Corollary G1 tells us that (x)Gx is true. So it follows that

(x)Gx

is also true. But a particular theorem of modal logic S5 is that p p. (If you don't believe it, take the contrapositive of this statement!) So the conclusion follows. Q.E.D.


So that's it. The obvious question to ask is whether Gödel's proof is correct. I find myself agreeing with C. Anthony Anderson when he says the following:

Consideration of the axioms, especially ... [Axiom G2], may tend to dampen one's confidence in ... [Axiom G3] and ... [Axiom G4] -- 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.

To my way of thinking, this adds up to saying that Gödel's proof is really an argument, because the axioms (particularly those mentioned by Anderson) are not sufficiently self-evident to warrant calling the whole thing a proof.

Does that mean that we are back where we started? Far from it. Gödel's argument suggests an interesting via positiva (or via affirmativa) to understanding the idea of God. For example, it is interesting to compare the idea of a positive property in Gödel's sense with those aspects of this world that Calvin described as "the sparks of God's glory." According to Wordsworth this is

"A presence that disturbs me with the joy
Of elevated thoughts; a sense sublime
Of something far more deeply interfused,
Whose dwelling is the light of setting suns"

While nothing has been convincingly demonstrated, the argument may appeal to those who, like the philosopher Spinoza or the theologican Tillich, see God as in some sense the ideal aggregation of certain fundamental and essential aspects of being. It raises metaphysical questions about the determination of those essential aspects of being which must be present whenever something is said to exist. It raises the possibility of exploring the idea of God by a metaphysical enquiry into precisely those properties which may be determined to be positive.

Even if the argument is right in all its aspects, we still may not have a clear determination of what "pure attribution" is. It is possible that the God that will be proved to have necessary existence will be some blind equation that governs the quantum fluctuations of reality, or it is possible that the God that is proved to exist may be the Judeo-Christian God or the Tao of eastern mysticism. We simply do not know. However, Gödel's argument would then provide a signpost for the search for an answer into the essence of God. Is being conscious an example of pure attribution? The blind mathematical God of the physicist searching for fundamental laws may be an inadequate idea if it is determined that consciousness is a basic constituent of reality. If, as Berkeley said, esse est percipi, then the implications for our understanding of God would be radically transformed. Recently, a number of physicists have proposed that any fundamental description of quantum-mechanical reality may require an understanding of the nature of consciousness. Who knows? Gödel's proof deserves some thought in the context of this difficult problem.

Well, there it is. I should remind the reader again that the version of Gödel's argument that I have presented is not Gödel's original argument. When I first read Gödel's notes on his ontological argument, I, like many others, found the axioms confusing and difficult to accept. So I turned to the emended version of the argument due to Anderson. This version has now been widely discussed. But the more I fuss around with this version the more uncomfortable I become with it. I went back to Gödel's original version and found that it was far better than I had originally understood. If you are interested in studying this argument, and the reasons why I prefer it to Anderson's emendation, you can click to obtain a PDF file entitled Reflections on Gödel's Ontological Argument.

Finally, if you are looking for a different perspective on ontological arguments, I recommend that you explore Graham Oppy's work. Graham Oppy raises some important objections to ontological arguments which are well worth mulling over. I would strongly agree with Graham Oppy that the argument has enough uncertainty attached to it as to leave its conclusion in doubt. Readers of this essay who wish to raise objections to Gödel's argument or debate the point are encouraged to contact me.



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