The Validity of Plantinga’s Ontological Argument

by Max Andrews

In 1974 Alvin Plantinga developed a modal version of the ontological argument, which is as follows:

  1. The property of being maximally great is exemplified in some possible world.
  2. The property of being maximally great is equivalent, by definition, to the property of being maximally excellent in every possible world.
  3. The property of being maximally excellent entails the properties of omniscience, omnipotence, and moral perfection.
  4. A universal property is one that is exemplified in every possible world or none.
  5. Any property that is equivalent to some property that holds in every possible world is a universal property.
  6. Therefore, there exists a being that is essentially omniscient, omnipotent, and morally perfect (God).


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]).

9 Comments to “The Validity of Plantinga’s Ontological Argument”

  1. Hey there, Max! I love your posts – I read all of them as they come out. I’m an undergraduate philosophy major – so reading your work is very inspiring.

    About the ontological argument: I know it uses modal logic, right? And modal logic a type of symbolic logic?

    Can you recommend a book or online introduction to this type of logic? Either modal or symbolic? I’m really interested in trying to learn this material outside of school.
    I’m taking a formal logic class next semester, but I would love to get a head start.

  2. Michael,
    You might look into Kenneth Konyndyk’s book:
    He introduces the various systems of modal logic.

  3. I appreciate both of those responses! I’ll definitely be looking into those. Thanks!

  4. I would say Plantinga’s books themselves, like The Nature of Necessity, are excellent resources on modal logic. He did a lot of groundwork. As an undergrad, you should also hopefully have access to journals for free through your library. If you go to a resource, say the Stanford Encyclopedia of Philosophy, and look at their references, you should get some good articles. One article I found interesting was by Michael Tooley called Plantinga’s Defence of the Ontological Argument (Mind, 1981). In it, Tooley gives some reasons why this argument may not work due to reliance on certain questionable factors in modal logic. I’d get your feet wet in modal logic before reading it, though.

    Hope that helps.

  5. What about that objection that his premise is the same as his conclusion?

    • Where do you see that? 6 is the conclusion and I don’t see how that is equivalent to any of the preceding premises because that would certainly be an issue.

    • That would be assuming the conclusion, wouldn’t it. But I don’t think that Plantinga is doing this, for several reasons. First, Premise One and the conclusion are different in that Premise one only establishes that the existence of the property of being maximally great is logically possible – that is, that it exists in a possible world. The conclusion says that, as opposed to existing in only a possible world, the property of being maximally great exists in the actual world. Premise Two might be another candidate for this objection, but I don’t think that it succeeds. It’s simply defining the term, “property of being maximally great”, and seems to me to be an accurate one. For the property of being maximally great to be a coherent term, it would need to have the greatest conceivable characteristics. For example, a state being that only exists in some possible worlds would be, rather self-evidently, inferior to a state of being that exists in all possible worlds. From those premises, the rest follows rather well. The only way to actually critique this form of the ontological argument that I can see is to demonstrate a contradiction or impossibility within the first two premises. I haven’t seen a good one to date, but there may exist one. And that piteous canard about a “state of being that is maximally great while not existing” is not even close to bad or illogical; it does not even rise to the dignity of error.

      And the last reason to think that Plantinga does not assume the conclusion is, well, he’s Alvin Plantinga, baby. 🙂 I find it unlikely that a philosopher of his calibre would make such a puerile mistake. He’s rather too meticulous for that. So while it’s possible that that’s the case, it’s probably about as likely as a giant oyster materializing in the English Channel due to quantum fluctuations. Though, depending on how ones world view functions, said oyster might conceivably, at any moment, materialize out of the “nothing” of the quantum vacuum.

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