Godel’s Ontological Argument

This page is an attempt at simplification of the material on Christopher Small’s website, at http://www.stats.uwaterloo.ca/~cgsmall/ontology2.html If you want to go through the complete agonizing detail, I recommend you check it out there. Ok, to business.

What does it mean to say “God is great” or “greatest”? It is to judge between some attributes, and call them positive.

Kurt GodelKurt GodelGodel invented a notation for this as follows:

Pos(x) means that “x” is a positive attribute. What does this mean? Well we could judge this on some sort of aesthetic basis, looking for beauty or moral goodness. Godel choose for his ontological argument to keep it in the context of “pure attribution”. By this he seems to mean things that ADD TO BEING, rather than being a lessening of being.

Pure positive attributes are also absolute- and don’t depend on the changing nature or accidental structure of the world. In modal language:

Pos(F) >> #Pos(F) (for an attribute to be positive implies that it is NECESSARILY positive)

Some more details about positive attributes:

Axiom G2: Pos(F) >> ~Pos(~F) (if F is positive, it’s opposite is negative)

This contradicts Brian, who insisted that, for example, God would have to be both absolutely good and absolutely evil to be the “greatest”. In Godel’s proof, only attributes with add to being or indicate MORE BEING are positive, and their opposites are absolutely negative.

Godel next proved, through a process I won’t repeat here, but which you are welcome to pursue on the page I listed, that positive properties had to be “consistent” – and by that, he means “possible”. It makes no sense to claim that an impossible property is positive, and Godel proved that it would involve a logical contradiction. He wrote this as follows:

Axiom G1: Pos(F) >> * (Ex) Fx “For every positive property, it is possible that there exists an x which has that property)

Definitions: with this in mind, Godel defines formally what he means by “God”. A Godlike individual, says Godel, would be one for whom every essential property is positive, and every positive property is essential. Christopher Small translates this as:

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

“In a Godlike individual x, for every property F which is positive, it is necessary that x have that property”

As a note, this does not say that God can’t have accidental properties that are not positive.

I will skip over some of the intermediate work of definitions and cut to the chase, since we’ve lost most people by now.

Axiom G3: Pos(G) “The quality of being God-like is positive”. There is considerable verbiage devoted to this axiom on the website, but I think most of us can simply nod.

Corrolary G1: * (Ex) Gx “It is possible that a Godlike individual exists” (Proof: Theorem G1 and Axiom G3)

Axiom G4: Pos(NE) “Necessary existence is a positive property” (Proof: already discussed previously. Necessary being would be greater being)

Theorem G2: Gx >> G Ess x “x being a Godlike individual implies that Godhood is the essence of x” (I skipped the disscussion of essence. Read it if you like. Basically it implies a property that isn’t accidental or contingent – but essential. The proof involves theorems I skipped. Again, I think this one’s pretty obvious. God isn’t just simply God by a strange quirk of chance 😉

Theorem G3: #(Ex) Gx “It is necessary that there exists an individual who is a godlike individual”

This is basically the QED – it is necessary that God exists, so I’ll try to show the proof.

We stated in definition G1 that being Godlike involved having every positive property necessarily. We stated in G4 that necessary existence is a positive property (and hence the godlike being would have it necessarily) We stated in theorem G2 that if there is a godlike individual, godhood is his essence. From the definitions of necessity (which we skipped) we conclude that if any individual x is God-like, then the property of being God-like is necessarily exemplified. This can be written symbolically as:

(Ex)Gx >> # (Ex)Gx “if there exists an x, and x is godlike, it is NECESSARY that there exists an x who is godlike”

Now it’s just cranking the logical equations…

  1. [(Ex)Gx >> # (Ex)Gx] (necessitation axiom of modal logic) “it is necessary that if there exists an x, and x is godlike, it is NECESSARY that there exists an x who is godlike”

Brief pause to introduce a new theorem of modal logic:

  1. [p>>q] >> (*p >> *q) (let’s call it the possibility theorem) “if it is necessary that p implies q, then the possibility of p implies the possibility of q”

Applied to the above, this gives us:

  • (Ex)Gx >> *#(Ex)Gx “the possibility that their exists a godlike individual implies the possibility that the existence of a goodlike individual is necessary”

But Corollary 1 told us * (Ex) Gx “It is possible that a Godlike individual exists”… therefore:

  • #(Ex)Gx “it is possible that it is necessary that a godlike individual exists”

but, according to theorem S5 of modal logic…

  • #p >> #p “if it is possible that it p is necessary, then p is necessary”

Applied to the above,

  1. Gx – it is necessary that God exists.
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