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:
http://www.spiegel.de/international/germany/scientists-use-computer...
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.
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Axiom 2 is very dubious. It could be that two positive properties are inconsistent with each other. Surely being beautiful is a positive property, for example - but you could say that red is beautiful and green is beautiful, but something can't be red and green at the same time (in the same place).
Power and innocence are both positive properties - but power always involves some guilt, since people with power often have to choose the lesser of two evils. Their choices do hurt people. So power and innocence are inconsistent.
Axiom 2 is very dubious. It could be that two positive properties are inconsistent with each other.…something can't be red and green at the same time (in the same place).
You might want to choose a different example since color is a property observed and not necessarily intrinsic to the object itself. One observer might see an object as green and another as shifted toward red.
As you do, I believe there are problems with the axioms. Axiom 1 seems far too strong and it might be possible to find two positive properties that give rise to a contradiction. Axiom 3 appears to lead to an impredicative definition of the set of positive properties and might provide a paradox.
An excellent article on Gödel's argument by Christopher Small is available online:
http://sas.uwaterloo.ca/~cgsmall/Godel.final.revision.PDF
He provides enough background to make it an easy to read introduction to modal logic and the argument.
PS That two properties can be positive, yet inconsistent with each other, shows that Axiom 2 is wrong. Red for example entails "not green" but "not green" isn't usually considered a positive property. And being powerful entails guilt, but guilt isn't considered a positive property.
The authors of the recent formalization provide the much better version due to the logician Dana Scott, to whom Gödel showed the argument in 1970:
A1 Either a property or its negation is positive, but not both:
A2 A property necessarily implied by a positive property is positive:
T1 Positive properties are possibly exemplified:
D1 A God-like being possesses all positive properties:
A3 The property of being God-like is positive:
C Possibly, God exists:
A4 Positive properties are necessarily positive:
D2 An essence of an individual is a property possessed by it and necessarily implying any of its properties:
T2 Being God-like is an essence of any God-like being:
D3 Necessary existence of an individual is the necessary exemplification of all its essences:
A5 Necessary existence is a positive property:
T3 Necessarily, God exists:
As given in your version, Axiom 1 makes no sense at all as stated: first of all it is given as a definition rather than an axiom and as such, it defines positive in terms of negative, which is undefined and never mentioned again.Wording in mathematics and logic is often a subtle matter. The correct version would be:
A property is positive if and only if its negation is not.
Whoever did this version did not understand modal logic or what was being done. Where did this come from? Some of the other wordings are ambiguous as well.
I forget the site I found this on, but supposedly it was used to "prove" atheism.
There's a great deal wrong with it. The definition of 'essence of x' is incorrect and so is the definition of necessary existence. It's really jumbled up as it stands.
color is a property observed and not necessarily intrinsic to the object itself. One observer might see an object as green and another as shifted toward red.
? Surely, you can define the color objectively in a fair way, and it's intrinsic to a substance, if only the paint on the object.
Even an objective definition of color as the wavelength of reflected light has the same problem that different observers may register different wavelengths if they are in motion relative to the light. Light may be red-shifted or blue-shifted. You can specify that the observer must not be in motion relative to the object observed, but then the property is not intrinsic—it depends on a condition imposed on the observer.
A positive property is always positive relative to some observer. There has to be someone to be positively affected in some way.
What Gödel had in mind, I suppose, was a logical notion of a positive property and it's just possible that he thought the five axioms might be uniquely satisfied by a set of properties not otherwise defined. At one point he said something about positive being positive from a moral or aesthetic perspective.
It doesn't seem probably to me that he was thinking of the effect of a property on observers, but we can't now know what he was thinking—it isn't likely new material will surface after all these years.
My own intuition is that it is impossible to single out a set of properties as positive without somehow running into paradox or contradiction, but I don't have an argument to support that conjecture.
My own intuition is that it is impossible to single out a set of properties as positive without somehow running into paradox or contradiction, but I don't have an argument to support that conjecture.
I already argued that his axiom 2 doesn't work for what is usually considered positive.
And how is "necessarily existing" a positive attribute? There are lots of things you DO NOT want to necessarily exist.
I always thought the Anselm ontological argument was totally bogus, enough so that Goedel's supposed rigorization of it is of little interest to me.
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