In 1974 Alvin Plantinga developed a modal version of the ontological argument, which is as follows:
- The property of being maximally great is exemplified in some possible world.
- The property of being maximally great is equivalent, by definition, to the property of being maximally excellent in every possible world.
- The property of being maximally excellent entails the properties of omniscience, omnipotence, and moral perfection.
- A universal property is one that is exemplified in every possible world or none.
- Any property that is equivalent to some property that holds in every possible world is a universal property.
- Therefore, there exists a being that is essentially omniscient, omnipotent, and morally perfect (God).
Let,
Ax =df x is maximally great
Bx =df x is maximally excellent
W(Y) =df Y is a universal property
Ox =df x is omniscient, omnipotent, and morally perfect
1 ◊(∃x)Ax pr
2 (x)(Ax ≡ Bx) pr
3 (x)(Bx ⊃ Ox) pr
4 (Y)[W(Y) ≡ ((∃x)Yx ∨ (~(∃x)Yx)] pr
5 (Y)[(∃Z)(x)(Yx ≡ Zx) ⊃ W(Y)] pr
6 (∃Z)(x)(Ax ≡ Zx) 2, EG
7 [(∃Z)(x)(Ax ≡ Zx) ⊃ W(A)] 5, UI
8 W(A) ≡ ((∃x)Ax ∨ (~(∃x)Ax) 4, UI
9 W(A) 6, 7 MP
10 W(A) ⊃ ((∃x)Ax ∨ (~(∃x)Ax) 8, Equiv, Simp
11 (∃x)Ax (~(∃x)Ax) 9, 10 MP
12 ~◊~~(∃x)Ax ∨ ((∃x)Ax) 11, Com, ME
13 ◊(∃x)Ax ⊃ (∃x)Ax DN, Impl
14 (∃x)Ax 1, 13 MP
15 (x)(Ax ≡ Bx) ⊃ ((∃x)Ax ⊃ (∃x)Bx) theorem
16 (∃x)Bx 14, 15 MP (twice)
17 (x)(Bx ⊃ Ox) ⊃ ((∃x)Bx ⊃ (∃x)Ox) theorem
18 (∃x)Ox 16, 17 MP (twice)
19 (∃x)Ox 18, NE
This material is taken directly from Robert E. Maydole’s chapter “The Ontological Argument” in The Blackwell Companion to Natural Theology (553-592 [see 590 for step by step deduction]).