David Hume’s Dialogues Concerning Natural Religion is most fruitful as a source of material on the cosmological argument and the teleological argument. However, it also contains an argument against the possibility of an a priori proof of God’s existence. This argument is presented in Part IX by Cleanthes. It is used by Cleanthes as an objection to the kalam cosmological argument, but is surely better deployed as an objection to the ontological argument; the ontological argument (unlike the kalam cosmological argument) is entirely a priori.

Hume, or his character Cleanthes, puts the argument like this:

“I shall begin with observing, that there is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no Being, whose existence is demonstrable. I propose this argument as entirely decisive, and am willing to rest the whole controversy upon it.”

The argument in this passage, formalised, goes something like this:

(1) The only way to prove something a priori is if its opposite implies a contradiction.
(2) If something implies a contradiction, then it is inconceivable.
(2) Everything can be conceived not to exist.
(3) Nothing can be proved to exist a priori.

To find out whether a statement can be proved a priori, we try to imagine that it is false. If we are able to imagine that it is false, then we may infer that it cannot be proved a priori; empirical investigation will be necessary in order to discover whether the statement is true or false. If we are unable imagine the statement being false, then we may infer that the statement is true. This is because conceivability is a guide to possibility. What is impossible involves a contradiction, and what involves a contradiction is inconceivable, so what is impossible is inconceivable.

To find out whether God is a necessary being, therefore, we must try to imagine that he does not exist. As we are able to do so, his non-existence is possible. No amount of abstract reasoning will be able to establish his existence, therefore, because only necessary truths can be proved a priori. The conceivability of God’s non-existence shows that no a priori proof of his existence is possible.