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Gödel's Modal Proof
(Sometimes referred to as his ontological argument)

This proof is the most complicated so far, including much notation. This is a somewhat shortened version of an online page for Gödel's proof, which can be visited HERE if you prefer to delve in. A nice neat .pdf file with the complete proof is found at this link.

Postulates:
N(a)=>a
a=>P(a)
N(a=>b) =>[ N(a)=>N(b)]  (modal modus ponens)
N(a) is true whenever a is provable, (Necessitation postulate) 

N(a)=>N(N(a))  Beckers Postulate S4

P(a)=>N(P(a))     Beckers postulate  S5

Definition
F is an essence of x if for every property G of x,  N(G(x)) ó N(Fx=>Gx)
I.e. N(Fx), F is an essential property of x

Anselm's Axiom 1
P(a) 

Anselm's Axiom 2
a=>N(a)


Axiom G0
Pos(F)=>N(Pos(F)),

I.e. positivity is free from contingency, And a form of 'pure attribution'. That is, what is not necessary could not be positive.

Axiom G1
Pos(F)=>¬pos(¬F) 

Negation of a positive property is privation, i.e. its lack could not be positive.

Definition
F => H   if   N(given x)[Fx=>Hx]

and we say F entails the property of  H necessarily in x, that is, x necessarily has H if it has F.

Axiom G2
Pos(F)=>[(F=>H)=>Pos(H)]

A property entailed from a positive is itself positive. Privation of H means lack of F.

Definition
F is consistent if it is possibly exemplified, ie,  P($x)F(x) is true.

Theorem G1 : Pos(F)=>F is consistent

Let Pos(F) be true, if F is inconsistent, then F=>¬F. So G2 implies Pos(¬F) is true, but from G1 and Pos(F), ¬pos(¬F), a contradiction, QED.

Definition G1
x is 'God-like', ie G(x) is true, if all essential properties of x are positive, and all positive properties of x are as essential properties, ie
G(x) ó ("F):[N(Fx) ó Pos(F)]

Definition G2
We formulate an essence of an individual as follows,
F Ess x = ($F)("H) [ N(Hx) ó (F=>H) ]

Thus F and H will be realized in common or fail to be realized in common as essences.

Definition G3
NE(x) ó ("F):[F Ess x => N($y)Fy]
We say that an individual x exists necessarily if every property which is an essence of x is necessarily realized in some individual

Axiom G3
Pos(G)
Being God-like is positive.

Corollary G1
P($x)(Gx)
Proof : theorem G1, Axiom G3

Axiom G4
Pos(NE) necessary existence is positive

Theorem G2
If a being is God-like, then being God-like is an essence of the individual.
Gx  =>  G Ess x

Proof
Suppose Gx is true and x necessarily has property H., Ie N(Hx) is true. Then by definition G1, we have Pos(H), but N[Pos(H) => ( $y)(Gy => Hy)] from definition G1 (Recall that Gx is true) and that if something necessarily has a property then it has the property. However, by Axiom G0, N(Pos(H)) is true,
by modal modus ponens, we have N($y)(Gy=>Hy). So if x has any property H necessarily, then that is entailed by the property G, so G=>H.
Conversely, suppose Gx is true and G=>H then by axioms G2 and G3, we have Pos(H) ). It follows that a God-like individual x has property H necessarily, by Definition G1. That is, N(Hx) is true

Putting both sides together, G Ess x

Theorem G3
Necessarily the property of being Godlike is exemplified
N($x)Gx

Proof
IF Gx were true, then by Definition G1 x has every positive property necessarily. Axiom G4 states necessary existence is positive, so NE(x) is true. But by theorem G2 if Gx were true, G Ess x is true by Definition G3, if any x is godlike then 'godlike' as a property is necessarily exemplified, ie
($x)Gx => N($x)Gx
this follows from Definition G1, Axiom G4 and Theorem G2. This is axiom 2 of Anselms argument.
The necessitation axiom of modal logic gives
N[($x)Gx =>N($x)Gx]
A theorem of modal logic (provable : consider (a=>b) = ¬N(a&¬b)) is that; N(p=>q) => (N(p)=>N(q))
Combining this,
P($x)Gx => P(N($x)Gx)
By corollary G1, ie P($x)Gx is true, it follows that P(N(($x)Gx)) is also true.
But by a theorem of S5, (simple contrapositive shows true) P(N(x))=>N(x)
Thus N($x)Gx

So it is necessary that such an individual exists who has the property of being 'God-like'.

Which was what was wanted. //


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