A Modal Ontological Argument for God

Next in my tour of arguments for the existence of God, a perennial favorite, the Ontological Argument for the Existence of God.

St. AnselmSt. AnselmThe argument is originally credited to St. Anselm, who probably developed the ideas as a meditative technique, rather than an actual argument. Since his time, a number of brilliant minds have both defended and attacked the argument. It’s proponents have included Leibniz, Godel and Plantigna. At the current time, my assessment is that the proponents of this argument have the upper hand over it’s detractors. The version I will give here is one that uses a form of logic called modal logic. There have been some attacks on modal logic, and there are forms of the argument that do NOT use it, but they are a bit complex, and the attacks on modal logic also seem to have petered out. If anyone is a real critic of modal logic, I can post the other version of the argument or links to it. First, I’ll explain the symbolism I’ll use for the logic and the basic axioms of modal logic.

Symbols of modal logic:

1. “>>” Material implication. Example x>>y is “x materially implies y” (usually written like a side-ways horse shoe)

2. “~” Negation. Example ~x is “Not x”

3. “*” Possibility. Example *x is “It is possible that x” (Usually written as a diamond)

4. “#” Necessity. Example #x is “It is necessary that x” (Usually written as a square)

5. “V” the “or” operator. Example x V y is “Either x OR y”

Basic axioms and postulates of modal logic:

1. #x >> x “If it is necessary that ‘x’ is true, then ‘x’ is true”

2. x >> *x (contrapositive) “If x IS true, then it is possible for x to be true”

3. #(x>>y) >> (#x >> #y) (called ‘modal modus ponens’) “If it is necessary that x being true implies y is true, and x is necessarily true it implies y is also necessarily true.”

4. #x is true if x is a proven axiom (the principle of necessitation) “If we can prove that x MUST always be true, then x is necessarily true”

5. #x V *~x (law of excluded middle) “Either it is necessary that x is true, or it is possible that x is NOT true”

6. #x >>##x (Beckers first postulate) “If x is necessarily true, then it is NECESSARY that x is necessarily true”

7. *x >> #*x (Becker’s second postulate) “If it is possible that x is true, then it is NECESSARY that x is possibly true”

The Modal Ontological Argument (g = God exists)

Axiom 1: *g (“it is possible that God exists”)

Axiom 2: *g >> #g (“if God exists, he exists necessarily (he is not a contingent being)”)

The proof

1. *g >>#g (axiom 2) “if God exists, he exists necessarily”

2. *~g >>#*~g (Becker’s second postulate) “if it is possible God doesn’t exist, it is necessary that it is possible God does not exist.

3. #g V *~g (excluded middle) “Either God necessarily exists or it is possible he does not exist”

4. #g V #*~g (substitute 2 into 3) “Either God necessarily exists or it is necessarily possible he does not exist”

5. *~g >> ~g (contrapositive of axiom 2) “If it is possible God doesn’t exist, God doesn’t exist”

6. #(*~g >>~g) (necessitation postulate on 5) “It is necessary that if it is possible God does not exist, God does not exist”

7. #*~g >> #~g (modus ponens on 6) “If it is necessary that it is possible that God does not exist, then it is necessary that God does not exist”

8. #g V #~g (substitution of 4 into 7) “Either it is necessary that God exists or it is necessary that he does not exist”

9. ~#~g (axiom 1) “It is not necessary that God does not exist”

10. #g (8 and 9) “It is necessary that God exists”

A Possible Worlds Version:

Assuming that the modal proof above was a tad hard to follow for some, let me give Plantigna’s “possible worlds” version, which I have used here before.


Maximal excellence: To have omnipotence and omniscience in some world

Maximal greatness: To have maximal excellence in every possible world.

Why is it maximally great to have maximal excellence in every possible world? Because this indicates that God’s greatness doesn’t depend on this or that particular circumstance. God MUST have maximal excellence, regardless of the possible world in which he is present. Nothing can prevent him from having maximal excellence.

1. There is a possible world (W) in which there is a being (X) with maximal greatness.

i.e. It is possible that God exists.

2. But X is maximally great only if X has maximal excellence in every possible world.

3. Therefore, X is maximally great only if X has omnipotence, omniscience and moral perfection in every possible world.

4. In possible world W, the proposition “There is no omnipotent, omniscient being” would be impossible – that is, necessarily false.

5. But what is impossible does not vary from world to world.

6. Therefore, the proposition, “There is no omnipotent, omniscient being” is necessarily false in this actual world too.

7. Therefore, there actually exists in this world, and must exist in every possible world, an omnipotent, omniscient being.

The last time we tried this, we got hung up a lot on point 5, with some suggesting that there are possible worlds where what is impossible DOES vary. This is a misunderstanding of what is meant by “possible worlds”. A “possible world” is a theoretical construction of a world that differs from ours only in it’s contingent details, NOT in the laws of logic. A world where the impossible is possible is not a “possible” world, but an “impossible” one. Without resolving the question of whether the laws of logic can in fact change, let’s just agree that for this example, we are considering only that set of worlds that obey the laws of logic – where that which is logically impossible does not vary.

What do these arguments tell us?

There are several trivial attacks against the ontological argument, and a few serious ones. One of the most significant attacks is to point out that the logic of these proofs can be reversed. You can prove with the same analysis that if it is POSSIBLE that God does not exist, then it is absolutely impossible for him to exist.

What these proofs REALLY tell us is that, contrary to what we might assume, God is not a mere possibility – something which may or may not exist. He is either absolutely necessary, or he is logically impossible. Any middle ground is an illusion based on not understanding the concepts involved. Which proposition, then, can marshall more support?

1. It is possible that God exists (and hence he is logically necessary)


2. It is possible that God does NOT exist. (and hence he is logically impossible)

I think the best support can be gathered for #1. It is easy to see that if God exists, he would be the fundamental creator and/or sustainer of every atom and every photon – an absolute necessity. In fact, as we saw in the cosmological proof – if our notions of cause and effect, and the principle of sufficient reason, have any application on the cosmic scale, God WOULD be absolutely necessary.

On the other hand, it is difficult to see why the existence of God should be a logical impossibility. For one thing, we can coherently form a conception of God – something we really can’t do of logical impossibilities. We can form coherent concepts of CONTINGENT impossibilities (like pink unicorns) but NOT of logical impossibilities (like square circles). I can get my mind around the concept “If God exists, he exists necessarily”. The contrary, that if God does NOT exist, his non-existence is a logical necessity – just doesn’t seem as convincing. And it it much harder to summon up the idea “There is a possible world W where God is logically impossible” than the contrary.

Secondly, millions of people claim to have had some kind of contact with God in mystical or religious experience. This should give the benefit to at least the possibility of God’s existence – and, as we have seen from the proof, if God is at least possible, then he necessarily exists.

Godel also formulated an argument for God’s possibility based on some very interesting principles, which I can introduce later if anyone’s interested.

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