[FOM] The characteristic S5 axiom and the ontological argument

laureano luna laureanoluna at yahoo.es
Thu Apr 9 08:30:23 EDT 2009

Alex Blum wrote:

>I wonder if Godel noticed that the characteristic S5 axiom:
>'If Pos Nec p then Nec p',
>will give him straightaway the skeleton of the ontological proof for the 
>existence of God; and that what remains is the not easy task to prove 
>the consistency of 'God exists'.
>For what the skeleton of the argument proves is that: if it is possible 
>that God exists necessarily then God exists necessarily, which is but an 
>instance of the characteristic S5 axiom.
>It could very well be that his suspicion of the characteristic S5 axiom 
>held him back.

I guess Godel did notice that.

If you think that, having realizing that, Godel could have solved that part of the argument with just one stroke, note that simply "if pos nec p, then nec p" is not sufficient for the argument; it requires establishing "nec (if p, then nec p)" to derive "if pos p, then pos nec p". Where "p" is, of course, "God exists".

Leibniz in Modales Probationes gave a modal argument not involving S5 that may be of your interest. Roughly:

God is defined as a necessary being (inter alia).
The proposition: "God does not have necessary existence" is self-contradictory, given the definition of God. 
Call that proposition P. 
Self-contradictory propositions can be true if and only if they involve impossible concepts. For instance, the proposition "a square circle is no circle" is true.
So it suffices to prove that the concept of God is not impossible, to prove that P is false and that God has necessary existence.

As far as I see this argument is likely to fall prey to an analogue of the criticism to the non modal version put forward by Aquinas: existence as included in the concept of God is only the thought of an existence, not actual existence. Similarly, necessary existence as included in the concept of God is but the thought of a necesary existence. Hence P should be read as: "the being defined as possessing necessary existence (inter alia) does not actually possess necessary existence". Call it Q. 

Q could be true even if the concept of God is possible, unless you assume S5, for again you need S5 to conclude that any being defined as possessing necessary existence has necessary existence if only it possibly exists.


Laureano Luna


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