This post at TGC by James Anderson is an excellent, nuanced answer to the question. It takes into account a number of important elements, including discussion of certainty, circularity, and whether an argument needs not just to be sound but also persuasive.
Here’s the upshot of his proposal:
We should think of proofs in terms of proofs for a particular person. In much the same way that mathematical proofs are system-dependent, so proofs of the existence of God need to be seen as person-dependent. The question “Can we prove the existence of God?” then becomes “Can we prove the existence of God to so-and-so?” My suggestion is that if we can show, without begging the question, that the existence of God logically follows from propositions that a person already accepts, or is willing on reflection to accept, then we have indeed proven the existence of God to that person. If they fail to see that the existence of God follows from what they already believe or take for granted, or if they prefer to abandon other beliefs rather than to affirm the existence of God, the problem doesn’t lie in the proof.
What does this mean for our test-case argument? If we understand proof along the lines I’ve suggested, the argument is indeed a proof for particular people, not necessarily for everyone. What’s more, on this understanding there are numerous proofs of God’s existence. There are many arguments that demonstrate the existence of God from beliefs or assumptions that people already hold. (Consult the resources listed below for examples.) Some of these proofs might be deemed more effective or more persuasive than others, depending on the target audience, but as we’ve seen, proof and persuasion are two distinct things.
But read the whole thing.
