The Validity of Plantinga’s Ontological Argument


By Prayson Daniel


Ontological Argument

1. It is possible that a greatest conceivable being exists.
2. If it is possible that a greatest conceivable being exists, then a greatest conceivable being exists in some possible world.
3. If a greatest conceivable being exists in some possible world, then it exists in every possible world.
4. If a greatest conceivable being exists in every possible world, then it exists in the actual world.
5. If a greatest conceivable being exists in the actual world, then a greatest conceivable being exists.
6. Therefore, a greatest conceivable being exists.

The nonmodal propositional and quatificational inference rules of 2QS5(fromGustason and Ulrich)

Existential Generalization (EG), Universal Instantiation (UI), Modulas Ponens (MP), Equivalence (Equiv), Simplification (Simp), Commutation (Com),  Modal Equivalence (ME), Double Negation (DN), and Necessity Elimination (NE)

Let:
Ax = dfx is maximally great
Bx = dfx is maximally excellent
W(Y) = df Y is a universal property
Ox = df x is omniscient, omnipotent, and morally perfect

Deduction:

1   ◊(Ǝx)Ax                                                                       pr

2   ☐(x)(Ax ≣ ☐Bx)                                                            pr

3   ☐(x)(Bx ⊃ ☐Ox)                                                           pr

4   (Y)[W(Y) ≣ (☐(Ǝx)Yx ⋁ (☐∼(Ǝx)Yx)]                                 pr

5   (Y)[(ƎZ)☐(x)(Yx ≣ ☐Zx) ⊃ W(Y)]                                     pr

6   (ƎZ)☐(x)(Ax ≣ ☐Zx)                                                     2, EG

7   [(ƎZ)☐(x)(Ax ≣ ☐Zx) ⊃ W(A)]                                       5, UI

8   W(A) ≣ (☐(Ǝx)Ax ⋁ (☐∼(Ǝx)Ax)                                     4, UI

9   W(A)                                                                          6, 7 MP

10  W(A) ⊃ (☐(Ǝx)Ax ⋁ (☐∼(Ǝx)Ax)                                   8, Equiv, Simp

11   ☐(Ǝx)Ax ⋁ (☐∼(Ǝx)Ax)                                              9, 10 MP

12  ∼◊∼∼(Ǝx)Ax ⋁ (☐(Ǝx)Ax)                                          11, Com, ME

13  ◊(Ǝx)Ax ⊃ ☐(Ǝx)Ax                                                   DN, Impl

14  ☐(Ǝx)Ax                                                                   1, 13 MP

15  ☐(x)(Ax ≣ ☐Bx) ⊃ (☐(Ǝx)Ax ⊃ (☐(Ǝx)☐Bx)                 theorem

16  ☐(Ǝx)☐Bx                                                                14, 15 MP (twice)

17  ☐(x)(Bx ⊃ ☐Ox) ⊃ (☐(Ǝx)☐Bx ⊃ (☐(Ǝx)☐Ox)             theorem

18  ☐(Ǝx)☐Ox                                                               16, 17 MP (twice)

19  (Ǝx)☐Ox                                                                 18, NE

More detailed in The Blackwell Companion To Natural Theology p 590(Robert E. Maydole). Click on the argument for explanation

For every substitution instance of p and q

Necessity Elimination (NE)                       ☐p / ∴p

Reading the Symbols(Operators) and example:

Symbol-(Name)-Logical function-(Used to translate)

~ (tilde) negation (not, it is not the case that)

~ A It is not the case that A

⋁ (wedge) disjunction (or, unless)

A ⋁ B Either A or B

⊃ (horseshoe) implication (it… then …, only if)

A ⊃ B If A then B

thus (x)(Ax ⊃ Bx) for any x, if x is a A, then x is also a B

≣ (triple bar) equivalence (if and only if)

A ≣ B A if and only if B

(Ǝx) existential quantifier; there exists an x such that …

◊(Ǝx)Ax   It is possible there exists an x such that x is a A

Ax = dfx is maximally great

Thus: It is possible that a greatest conceivable being exists.

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