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Gödel with a friend…

Ontological Arguments

Many thinkers have attempted to prove the existence of an all-powerful being (like the one religions use to call “God”). These attempts are interesting not because they prove something beyond the shadow of a doubt (there are indeed logicians who think they are correct, but there are also others who think otherwise), but because the show that logic can be a tool that leads to God.

Gödel’s ontological argument

One of the greatest logicians of all times, Gödel, has made such an ontological argument which you can find at the book “Types, Tableaus, and Gödel’s God” (1) (3).

The argument can be summarized as follows.

We first assume the following axiom:

  • Axiom 1: It is possible to single out positive properties from among all properties. Gödel defines a positive property rather vaguely: “Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)… It may also mean pure attribution as opposed to privation (or containing privation)” (Gödel 1995)

We then assume that the following three conditions hold for all positive properties (which can be summarized by saying “the positive properties form an ultrafilter”):

  • Axiom 2: If P is positive and P entails Q, then Q is positive.
  • Axiom 3: If P1, P2, P3, …, Pn are positive properties, then the property (P1 AND P2 AND P3 … AND Pn) is positive as well.
  • Axiom 4: If P is a property, then either P or its negation is positive, but not both.

Finally, we assume:

  • Axiom 5: Necessary existence is a positive property (Pos(NE)). This mirrors the key assumption in the respective Anselm’s ontological argument.

Now we define a new property G: if x is an object in some possible world, then G(x) is true if and only if P(x) is true in that same world for all positive properties P. G is called the “God-like” property. An object x that has the God-like property is called God.

With the above reasoning, Gödel argued that in some possible world there exists God. Then he went on proving that since a Godlike object exists in ONE possible world, then it necessarily exists in ALL OTHER possible world (since “necessary existence” is one of its positive properties).

Thus, God exists.

The symbolic summarization of the above logical syllogism can be seen in the picture below (2), where Ax refers to Axioms, Th to theorems and Df to definitions used in the syllogism.

Gödel’s ontological proof symbolic notation

There are numerous objections with the above argument, the main of which are summarized in the next section.

Criticism to Gödel’s proof

The logic of the argument is not easily refuted. Of course as in any other argument there are counter-arguments and then arguments which counter those counter-arguments (4).

However the Achilee’s Heel of the argument (as that of any argument per se) is its foundations. The axioms that are innevitably stated when formulating an argument are considered as true based on the opinion of the author of the argument and, thus, can be refuted by others as simply invalid.

Gödel’s proof v2.0

In an attempt to clear things out regarding the proof, I have made a small addition to the debate on the validity of Gödel’s axioms so as to solve the issue once and for all: If some people argue that Gödel had defined “positive” too vaguely or that Gödel’s definition of “positive” is wrong altogether, then why not just accept their objections?!

And by doing that let’s say for a second that “existence” is indeed a “negative” property (and not a positive one as Godel claims in his axioms). Having that as granted, then the problem of God might not be solved but another similarly important is: All people should stop worrying about dying, since “not existing” is something good (i.e. a positive property)!

In that way all great philosophical problems of humans will be solved in a strange way. Philosophy does work in mysterious ways…

The problem of the existence of God is then solved indirectly: Since non-existence is a good thing, the phrase “God does not exist” takes a weirdly positive effect that could puzzle the greatest of atheists…

Instead of a Conclusion…

All in all, one might disagree with that argument. But the critical point here is that some other logicians agree! So even though this argument has not solved the great mystery of them all, it has given us a great lesson: Logic is not a tool for atheism only, it is a tool for theism as well…

References

  1. Types, Tableaus, and Gödel’s God, Springer, Series: Trends in Logic , Vol. 12, Fitting, M., 2002, 196 p., Hardcover, ISBN: 978-1-4020-0604-3.
  2. Gödel’s ontological proof, Wikipedia article, retrieved from here on 2021-02-04.
  3. Graham Oppy, 1996, Godelian ontological arguments, Analysis 56(4), DOI: 10.1093/analys/56.4.226.
  4. Curtis Anthony Anderson, 1990, “Some Emendations of Gödel’s Ontological Proof”, Faith and Philosophy. 7 (3): 291–303. doi:10.5840/faithphil19907325.