You should read the argument yourself at the linked page, but as I understand it, it goes as follows:
- If the existence of God were a logically necessary truth, then any statement that follows logically from God's existence would also be a logically necessary truth.
- Thus the negation of any such statement would be logically incoherent.
- When we look at these negated statements, they are not obviously incoherent.
- Therefore, the existence of God must not be a logically necessary truth.
Let's try out the same argument on a different topic. Let us suppose that the basic theorems of arithmetic are logically necessary. Now, take any statement that follows from the basic theorems: Fermat's Last Theorem, for instance. The negation of that statement is, of course, false. Here is the negation:
The equation an + bn = cn has a solution for some integers a, b, and c, and some integer n greater than 2.Now, that statement is certainly not obviously incoherent. Indeed, no one knew whether it was true or false for over 250 years. By Swinburne's argument, then, the basic truths of mathematics must not be logically necessary.
So, why does Swinburne think that the negation of a necessarily true statement should be obviously incoherent? It beats me.
Intuitively, it seems unlikely that any such a sweeping argument against deductive proofs of God's existence will be successful, any more than deductive disproofs of God's existence.