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I'm not sure if I'm allowed to post links, but I found a copy of Gödel's actual ontological argument published by professor Elke Brendel at Universitat Bonn, and can usually find it through Google.

There are four theorems in the proof, and I think I have the last three down alright, but the first one gives me trouble. The first theorem is, if a property is positive, then it is possible. I feel like I can think of some positive properties that aren't possible, and I don't understand his reasoning for this theorem (perhaps because I haven't taken a class in modal logic before).

I do think I understand the rest of his argument though, more or less. Once we understand that if a property is positive then it is possible, then we can then say the existence of something that has all positive properties is possible using a trivial axiom of Gödel's and modus ponens. I'm sure I'm skipping some stuff here, but at this point, to simplify things, I just use an axiom similar to his 5th one and say, "a positive thing existing in reality is a positive property" and can deduce the following - something that has all positive properties must exist in reality since a positive thing existing in reality is positive, and something that has all positive properties is positive.

This proof doesn't change my religious beliefs one way or another, but it's still interesting. I'm wondering if anyone can walk me through the proof of his first theorem though. Can anyone demonstrate to me Gödel's argument that a positive property is possible using his axioms and definitions? I can't follow some of his symbols, particularly the lambda symbol.

Brian M. Scott
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theboombody
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The proof itself is not a definitive proof that there is a god. It relies on some ideas which have no particular mathematical justifications. For example, positive properties are possible, or that necessary existence is a positive property.

I see Godel's argument as an attempt to formalize previous ontological arguments into modal logic. He's done that quite nicely, but it still relies on personal beliefs regarding what is a good property and so on.

After all, there was a good reason that we only learned of this proof after his death, and that he didn't want to publish it.

Asaf Karagila
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  • Yeah, but he did mathematically justify that positive properties are possible. That's one of his theorems, not one of his axioms. And that's the theorem I'm having trouble following the logic for. Necessary existence is positive was just an axiom though, as you state. – theboombody Jan 11 '15 at 17:35
  • Yes, you're right. Digging online I found these slides analyzing the proof, and on slide 13 (p. 44'ish of the .pdf) he explains how to formally derive this part. More or less. It's difficult since nowhere it says what is the modal system Goedel was using. – Asaf Karagila Jan 11 '15 at 18:14
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It's called vacuous truth. If a positive property were impossible - i.e. not apply to any object - it would imply every other property, including its own negation. In other words, it is true that every object that has that property also has its negation; think of the example Wikipedia provides: if there are no phones in the room, it is true that every phone in the room that's turned off is also turned on, because there aren't any phones in the first place. By Axiom 1, that should mean that the implied property - the negation of the 'impossible' property - is also positive. But by Axiom 2, only one of a property and its negation is positive. This is a contradiction. Therefore, there cannot be any 'impossible' properties.

Max
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I believe this is how he did it:

Axiom 1: If p is a good property, then any property q implied by p is also good.

Axiom 2: If p is a good property, then !p (that is, not p) is bad.

Assume p is a good but impossible property.

Then p implies all statements, since a false statement implies all statements.

Therefore p implies !p.

Therefore !p is good (by 1).

Therefore p is bad (by 2).

Tim
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