# The Cartesian Ontological Argument

D1. God is the x such that for all attributes Y, if Y is a perfection, Y belongs to x.
P1. Necessarily existing is a perfection
P2. For all x, if it is not the case that x exists, possibly it is not the case that x exists.
C. God exists.

Let,

E!x ≝ x exists
P(Y)≝ Y is a perfection

g ≝ (ɿx)(∀Y)(P(Y)⊃ Yx)

1. P(☐E!) (premise)
2. (∀x)[~E!x ⊃ ♢~E!x] (premise)
3. ~E!g (IP)
4. (∃x){[(∀Y)(P(Y) ⊃ Yx) ∧ (∀y)[(∀Y)(P(Y) ⊃ Yy) ⊃ (y = x)]] ∧ ~E!x} (3 theory of descriptions)
5. [(∀Y)(P(Y)⊃ Yμ) ∧ (∀y)[(∀Y)(P(Y)⊃ Yy) ⊃ (y = μ)]] ∧ ~E!μ (4 EI)
6. ~E!μ ⊃ ♢~E!μ (2 UI)
7. ~E!μ (5 Simp)
8. ♢~E!μ (6,7 MP)
9. ~☐E!μ (8 MN)
10. (∀Y)(P(Y) ⊃ Yμ) ∧ (∀y)[(∀Y)(P(Y) ⊃ Yy) ⊃ (y = μ)] (5 Simp)
11. (∀Y)(P(Y) ⊃ Yμ) (10 Simp)
12. P(☐E!) ⊃ ☐E!μ (11 UI)
13. ☐E!μ (1,12 MP)
14. ☐E!μ ∧ ~☐E!μ (9,13 Conj)
15. E!g (3-14 IP)

In the definition, I am just setting down what I take g to mean… all I mean by g is that it is the something that, for any attribute, if that attribute is a perfection, then it has that perfect attribute. So God is the being that has all perfections (as I define God).  You might say that there is no definite description of a perfect being, i.e. there could be multiple perfect beings.  However, I would argue that there cannot be two omnipotent beings, since a simple reductio would rule out this possibility.  That is, if there are two omnipotent beings, then any power the one has would be limited by whether or not the other being wills to bring about a contradictory state of affairs.  Since they cannot both bring about contradictory states of affairs, they cannot both be omnipotent.  So there cannot be two beings that have all perfections, given that omnipotence is a perfection that implies uniqueness.  Descartes says that his idea of a supremely perfect being is clear and distinct, which in turn justifies is appeal to the definition (even incoherent stipulated definitions can be rejected).  Leibniz famously demanded a more rigorous proof that the definition is coherent, and sought to prove all perfections cohere.  I think it is a mistake to then interpret Leibniz’s ontological argument in terms of using God’s possibility to infer his necessary existence via S5 in modal logic.  Rather, I think he is doing what Descartes is doing, namely trying to show that the definition of God is self-consistent.

Leibniz’s proof for the self-consistency of the concept of a supremely perfect being is through an analysis of a perfection, which he says is simple, positive, and unlimited.  If any two perfections are inconsistent, one of them would have to be negative, or contain a part that is negative.  But a perfection cannot, by definition, be negative, or contain parts.  So any two perfections can cohere.  Leibniz reasons that if this is so, then all perfections cohere, and so a being that has all perfections is coherent.

P1 say necessary existence is a perfection because a perfection is any attribute that is of a simple kind that is positively complete. Omniscience is a perfection of because it is an attribute of the simple kind (knowledge) that is positively complete. Whatever has omniscience lacks nothing with respect to knowledge. So we recognize omniscience as a kind of perfection regarding knowledge. So necessary existence is an attribute regarding the simple kind “modes of existence” that is positively complete. Whatever exists necessarily exists in all possible situations, so it does not lack positive existence given any other state of affairs.

P2 is axiomatically true given that if something is necessarily true (system M of modal logic), then it is true. Assume P2 is false: ~(~E!x ⊃ ♢~E!x), this is logically equivalent to saying ~E!x ∧ ☐E!x (x does not exist and necessarily x exists). Given system M, ☐E!x implies E!x, so P2 cannot be false. In order to object to P2, you would have to say that some necessary truths are not actually true, which I think is a rather absurd position to take.