Philosophy
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It is now generally admitted, at any rate by philosophers, that the existence of a being having the attributes which define the god of any non-animistic religion cannot be demonstratively proved. To see that this is so, we have only to ask ourselves what are the premises from which the existence of such a god could be deduced. If the conclusion that a god exists is to be demonstratively certain, then these premises must be certain; for, as the conclusion of a deductive argument is already contained in the premises, any uncertainty there may be about the truth of the premises is necessarily shared by. But we know that no empirical proposition can ever be anything more than probable. It is only a priori propositions that are logically certain. But we cannot deduce the existence of a god from an a priori proposition. For we know that the reason why a priori propositions are certain is that they are tautologies. And from a set of tautologies nothing but a further tautology can be validly deduced. If follows that there is no possibility of demonstrating the existence of a god.

A. J. Ayer (from Language Truth and Logic, 1936)

[Ayer is right in claiming that a transcendent god cannot be demonstratively proved. He is wrong, however, in implying that such a fact is significant. Indeed, he is inconsistent with the whole thesis of his book from which the above passage is taken. The burden of that thesis is to deny legitimacy to all metaphysical statements by claiming that such statements are neither true nor false, but meaningless. He wrote:

As for the agnostic, although he refrains from saying either that there is or that there is not a god, he does not deny that the question whether a transcendent god exists is a genuine question. He does not deny that the two sentences “There is a transcendent god” and “There is no transcendent god” express propositions one of which is actually true and the other false. All he says is that we have no means of telling which of them is true, and therefore ought not to commit ourselves to either. But we have seen that the sentences in question do not express propositions at all. And this means that agnosticism also is ruled out.

Ayer was wrong, however, to attach any importance to the fact that God’s existence cannot be proved because any philosophical “proof” begins from beliefs which precede logic. For as Bertrand Russell remarked, ‘All knowledge must be built up upon our instinctive beliefs, and if these are rejected, nothing is left.’ In the preface to a subsequent edition written ten years after Language Truth and Logic first appeared, Ayer made a very weak admission that his book attempted to do the very thing that he had said the philosopher must not do, namely, to present a picture of reality based on a metaphysical principle.]

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