If you wish to understand this post, you should probably read my exposition on modal logic first. You may also wish to read my analysis on a much simpler ontological argument.
To briefly review the simpler ontological argument: The argument says that we can define God as a necessarily existing being. Therefore, by definition, God exists. However, this takes the power of definition too far. The most we can say is that if there exists an object which we can properly call God, then that object, by definition, exists. If God exists, then God exists.
The problem with the argument is that it's rather useless to define God as a being which exists. But hold on! I never defined God as a being which exists. I defined God as a being which necessarily exists. This definition is not quite as useless. We can say that if there exists an object which we can properly call God, then that object, by definition, necessarily exists. If God exists, then God necessarily exists.
By definition of God: g gThe letter "g" represents the specific statement "God exists". Note, however, we did not completely define God. In fact, we could replace God with all kinds of absurd objects, and the proof would still hold. This is actually quite a problem for the ontological argument (as well as many other proofs of God). The proof is valid for any statement p as long as pp is necessarily true. For instance, we could replace the object God with "the invisible pink unicorn which necessarily exists" or "a left shoe which necessarily exists".
Nonetheless, let's continue onward. I'm going to go through the proof, step by step.
- Theorem 1: ¬g ¬g (The contrapositive of the definition of g)
- Theorem 2: ¬g ¬g (Using the definition of to substitute into Theorem 1)
- Theorem 3: (¬g ¬g) (Application of Axiom N to Theorem 2)
- Theorem 4: ¬g ¬g (Application of Axiom K to Theorem 3)
- Theorem 5: ¬g¬g (Axiom 5 applied to ¬g)
- Theorem 6: ¬g ¬g (Combining Theorems 4 and 5)
- Theorem 7: ¬¬g ¬¬g (The contrapositive of theorem 6)
- Theorem 8: gg (Using the definition of to substitute into Theorem 7 twice)
- Theorem 9: g g (Combining Theorem 8 with Axiom T)
And here is where the problems begin. All the previous work was purely logical manipulation, and is necessarily true if you accept the axioms of modal logic. I thought the axioms were pretty reasonable, and rejecting them would be too high a price to pay. However, it seems we need another premise, g. To support this premise, lots of arguments have been offered by various people, but I don't think they're nearly as fun or rigorous as the modal logic section.
One common argument for g is that g is self-consistent. I can conceive of a God without having any contradictions. Based on my current knowledge, it is entirely possible that God exists. It's possible that God exists, therefore God exists.
The problem is that this same argument seems to apply to the statement ¬g. ¬g is a self-consistent statement. I can conceive of a world in which God does not exist without having any contradictions. Indeed, I can conceive of a world where nothing at all exists, where there could not possibly be any contradictions since there is nothing around to contradict. Based on my current knowledge, it is entirely possible that God does not exist. Therefore we can take the premise ¬g. By Theorem 2, we conclude ¬g: God does not exist.
Obviously, the premises g and ¬g cannot both be true. Either God exists in all possible worlds, or God exists in none of them. Since we used the same argument for both both g and ¬g, that argument must be fallacious. But where exactly did it go wrong?
The error, I think, is in the concept of . g does not quite mean "g is possibly true." In fact, it means "Among all possible worlds, there exists at least one in which g is true." The concept of "All possible worlds" was never exactly defined. In fact, the definition is arbitrary. I could have declared "all possible worlds" to be our world and our world alone, and you never would have been able to prove me wrong from the axioms.
I also could have declared "all possible worlds" to be the set of worlds which are metaphysically possible (a rather complex philosophical concept). Under this definition, we would not be able to prove g or ¬g.
However, if I had declared "all possible worlds" to mean the set of all worlds which are self-consistent, then we would run into problems. Because under this definition, both g and ¬g appear to be true (unless the concept of God is inconsistent). And they contradict each other.
Similarly, if I had declared "all possible worlds" to mean the set of all worlds which are epistemologically possible (meaning, it may be true, for all we know), then we would have the same contradiction.
And what happens if we construct a pathological definition of "all possible worlds" such that g is true and ¬g is not true? Then I might question Axiom T, since it is no longer obvious that our world is included among this so-called set of "all possible worlds."
And it goes on and on. Well, wasn't it clever, at least at first? I think so. It seemed for a moment that we arrived at a paradox, a sort of 1=2 moment. Ontological arguments tend to be that way. Most people immediately recognize that it is a little too clever, that it proves a statement which is a little too strong to be true. Similarly, most philosophers think that ontological arguments fail, though they may disagree on exactly why they fail.
In my opinion, ontological arguments are merely interesting philosophical curiosities. It's rather silly when an apologist actually tries to use one as a serious argument.