Before we look at the a posteriori arguments that Swinburne presents and evaluates in The Existence of God (EOG), I should briefly describe his views on a priori arguments for and against the existence of God.
In Chapter 1 of EOG, Swinburne mentions an assumption that his case for God makes:
In reaching my final conclusion about how probable it is that there is a God, I assume that no a priori arguments of either species, and no a posteriori arguments other than those that I discuss, have any significant force. (EOG, p.9)
In EOG, Swinburne ignores a priori arguments for and against God:
I shall not discuss a priori arguments–these are arguments in which the premisses are logically necessary truths–namely, propositions that would be true whether or not there was a world of physical or spiritual beings. Among logically necessary truths are the truths of mathematics or logic. Hence I shall not discuss the traditional ontological argument for the existence of God, or any variants thereof. Nor shall I discuss arguments against the existence of God that claim that there is something incoherent or self-contradictory in the claim that there is a God. (EOG, p.8-9)
So, the arguments that Swinburne presents and evaluates in EOG are all a posteriori arguments, arguments based on premisses that “report what are (in some very general sense) features of human experience…” (EOG, p.8).
How can Swinburne be justified in making the assumption that there are no a priori arguments which have significant force for or against the existence of God? For one thing, Swinburne mentions the views of leading philosophers of religion:
The greatest theistic philosophers of religion have on the whole rejected ontological arguments and relied on a posteriori ones. (EOG, p.9)
He also points out that the majority of philosophers are in agreement about the traditional ontological argument:
…almost all philosophers argue that it is not a good argument. (The Coherence of Theism, revised ed., p.273)
But Swinburne does more than just appeal to the authority of philosophical experts. The first book of his trilogy on theism, The Coherence of Theism, focuses on various a priori arguments against the existence of God, and it also includes a brief critique of the traditional ontological argument for the existence of God. His main objection to the ontological argument has general implications for other a priori arguments for the existence of God.
So, Swinburne’s justification for assuming that there are no a priori arguments for or against the existence of God which have significant force is that he has previously (in The Coherence of Theism) answered many a priori arguments against the existence of God, made a positive case for the meaningfulness and the coherence of the claim ‘God exists’, and presented a strong objection to the traditional ontological argument, an objection that applies to any argument which attempts to show that the claim ‘God exists’ is a necessary truth.
Here, for your personal enjoyment and edification, is Swinburne’s objection to the traditional ontological argument:
Nevertheless, it is, I think, easy enough to show fairly conclusively that ‘God exists’ is not logically necessary… .For to say this is, as we saw, to say that (s) ‘there exists a personal ground of being’ is logically necessary. But if this were so, any statement entailed by (s) would also be logically necessary. (s) entails such statements as the following: ‘it is not the case that the only persons are embodied persons’, ‘it is not the case that no one knows everything about the past’, ‘it is not the case that no one can make a weight of more than ten million pounds rise into the air’. Hence, if (s) is logically necessary the negations of these latter statements will be incoherent. But fairly obviously they are not. Fairly obviously ‘the only persons are embodied persons’, ‘nobody knows everything about the past’, and ‘no one can make a weight of more than ten million pounds rise into the air’ are coherent claims, whether false or true. Hence (s) is not logically necessary. (The Coherence of Theism, revised ed., p.274-275)
Since the traditional ontological argument is an attempt to prove that the sentence ‘God exists’ expresses a logically necessary truth, this is a counter-argument to the traditional ontological argument, and to any other a priori argument which attempts to show that ‘God exists’ is a necessary truth.