Kurt Gödel, the leading logician of the twentieth century, proved theorems in logic which astounded the mathematical community and changed the face of logic. In addition he showed how modal logic can be useful in proof theory He also presented a solution to the equations of general relativity that caused his close friend Einstein to have doubts about his own theory. He was without question one of the most brilliant minds of the last century or indeed any century.
Gödel was also a mystic and a theist and gave much thought to a modal logic version of the ontological argument for the existence of God which originated with Saint Anselm. His proof has recently been analyzed using several formalized computer logic programs and passed the tests. The online edition of Der Spiegel touted this achievement in an October 2013 article:
Should atheists be worried? No. Definitely not. If you examine the argument (Dana Scott's version drawn from conversations with Gödel around 1970) you see clearly that what is proved is the following:
There necessarily exists an entity which possesses every positive property.
What, you ask, are we to understand as a positive property? Gödel himself never said explicitly, but he seems to have had an intuitive notion that a positive property was one without any notion of privation in it. That is not much help. Gödel's argument is based on five axioms (assumptions) about positive properties:
Axiom 1. Either a property or its negation is positive, but not both.
Axiom 2. A property entailed by a positive property is positive.
Axiom 3. The property of possessing all positive properties is itself positive.
Axiom 4. Positive properties are necessarily positive.
Axiom 5. Necessary existence is a positive property.
Whether or not a notion of positive property can be defined which satisfies these axioms without giving rise to contradictions is a serious question and beyond that question lies the issue of whether positive properties are of theological interest. It is claimed that an additional argument is able to show a large number of such beings are required to exist, which might throw religion into a state of confusion greater than already exists.
For example, is omnipotence a positive property? Intuitively it would seem to be, but the definition is power without limits which has a somewhat negative cast. The notion of positivity may in fact be incoherent. (Axiom 3 could be a source of incoherence.) Which is the positive property, being caused or its negation, being uncaused? Logicians are interested in Gödel's argument for non-theological reasons—it shows the power of modal logic. There does not seem to be much theological interest, perhaps because theologians are for the most part poor at logic.
Gödel and Einstein took long walks together in the woods on the grounds of the Institute for Advanced Study in Princeton. They were comfortable in each other's company because they could speak German together. After Einstein died, Gödel was alone intellectualy and his mental stability began to crumble. He thought he was being poisoned and would eat only food prepared by his wife. When she fell ill, he stopped eating altogether and starved to death—a bizarre and tragic end to a life of brilliant achievement.
Positive, as used in this context, will have to be defined as a logical term and not left to intuition. What one person considers positive may not be to another and in ordinary discourse there are properties that are neutral and those are not allowed here.
Necessarily existing in this context means—using the possible worlds semantics for the modal logic S5—existing in all possible worlds. It may have a slightly different meaning in Kripke semantics.
Gödel's argument represents rather more than a formalization of Anselm's, which it transfers in a much refined form into modal logic. Aside from theological concerns, the argument is interesting in itself. The recent tests with computer formalizations and automations says that the argument is valid if the premises hold. That is the reason for the interest in the axioms: they constitute the premises.
I wonder if Goedel's argument works if it's impossible in principle to prove that anything necessarily exists. Is a being with an attribute that's impossible to possess, a greater being than a being without that attribute?
The claim to prove that something necessarily exists in all possible worlds, is SUCH a grand claim. Physicists would be stunned.
Probably there are many holes of that sort.
Do you think Anselm's argument is silly? I do.
Anselm's argument was firmly line with the metaphysical presuppositions of his eleventh century lifetime—namely, that you could by pure speculative reason discover real facts. That kind of platonic reasoning lasted a good deal longer.
It was Kant who dismissed the proofs of the existence of God, in particular the ontological proof, by asserting that existence is not a predicate. Frege later modified this to existence is a second order predicate.
Since Anselm's time the path to factual discovery has been more and more restricted to scientific experiment with the result that his ideas seem quite foreign to us now.
Possible worlds semantics is merely a device for clarifying statements in modal logic by phrasing them in ordinary language. A statement is necessarily true if it is true in all possible worlds, and possibly true if it is true in at least one possible world.The theorems of mathematics, which are unconditional, are necessarily true, while statements of historical fact are only possibly true.
In logic necessary truths are the tautologies, statements which are true regardless of the truth values of their arguments.
However, possible worlds are not real worlds.
However, possible worlds are not real worlds.
Anselm was trying to make a statement about the real world though - not an abstraction like mathematics. God is supposed to be part of the real world, not an abstraction. It seems impossible to prove that anything necessarily is part of any real world.
Also, suppose the "world" is the natural numbers, and the positive properties are being big - and "necessarily existing". What is the number such that you can conceive of no bigger number, and does "necessarily existing" being a good thing make that number "really exist"?
Positive, as used in this context, will have to be defined as a logical term and not left to intuition.
If you abstract it too much, whatever is proved will have nothing to do with the concept of God.
At least with the concept of God in the minds of most believers, and it is an excellent point. However, no one seems to take Gödel's argument as serious theology.
One could present it without using the word God at all. It is interesting as an argument because although no one takes it seriously from a theological point of view, it is not at all clear where in the logic the fault lies. That is the question of interest. In view of the recent computer tests, the axioms seem to be the likely source of error.
At least with the concept of God in the minds of most believers,
Not only that - but what actually is proved is likely to be something that nobody would recognize as "God".
Similarly to the "first-cause" arguments of theists. People argue that this universe must be "caused" in some sense. I suppose theoretical physicists would agree that there has to be some reason for why the laws of physics are what they are. Part of that reason is likely that "we are here to observe the laws of physics, therefore the laws of physics enable us to observe them". And perhaps that many different variations on the laws of physics happen in different universes or even in different widely separated parts of our own universe.
This reason has nothing to do with anyone's concept of God.
Similarly, if you made a toy world and applied Goedel's argument to it, what the argument comes up with as "God" is likely some piece of mathematics that no sensible person would call God - it might be some kind of limit defined in the mathematical system.