This is the Logical Positivist skeptical argument, as understood by Richard Swinburne:

(1) If the sentence “God exists” expresses a coherent statement, then the sentence “God exists” expresses either an analytic proposition or else it expresses a synthetic proposition.
(2) The sentence “God exists” does not express an analytic proposition.
(3) The sentence “God exists” does not express a synthetic proposition.
Therefore,
(4) It is not the case that “God exists” expresses a coherent statement.

The key question about this argument is this: Is premise (3) true or well supported?

First, we need to understand what is meant by a “synthetic proposition”. For Swinburne, a “proposition” is a coherent statement, and a “synthetic proposition” is a coherent statement whose negation is also a coherent statement. Some examples will help to clarify this concept.

(5) All triangles have three sides.

(6) Water boils at 212 degrees fahrenheit.
(7) The moon is made of cheese.

All three sentences above express coherent statements. A coherent statement is

…one which it makes sense to suppose is true; one such that we can conceive of or suppose it and any other statement entailed by it being true; one such that we can understand what it would be like for it and any statement entailed by it to be true. (COT, p.12-13)

It makes sense to suppose the (5) is true. It makes sense to suppose that (6) is true. It makes sense to suppose that (7) is true. Of each statement we can conceive of that statement as being true and any other statement entailed by it being true.

However, (5) is not a synthetic proposition, because although we can conceive of (5) being true, we cannot conceive of the negation of (5) being true:

(8) It is not the case that all triangles have three sides.

Since (8) is the negation of (5), and since it makes no sense to suppose (8) to be true, that is to say, we cannot conceive of (8) being true, (5) is not a synthetic proposition, but rather is an analytic proposition. We can conceive of it being the case that (5) is true, but we cannot conceive of it being the case that (5) is false. There is a logical contradiction involved in supposing (5) to be false or in supposing the negation of (5) to be true.

Both (6) and (7) are synthetic propositions because (a) they are coherent statements and (b) the negations of these propositions are also coherent statements:

(9) It is not the case that water boils at 212 degrees fahrenheit.
(10) It is not the case that the moon is made of cheese.

(6) is true and its negation (9) is false, but both sentences express coherent statements. (7) is false and (10) is true, but both sentences express coherent statements. There is no logical contradiction involved in any of these statements nor in their negations.

The Logical Positivist claim in premise (3) of the skeptical argument above is based on a criterion for determining whether a sentence expresses a factual claim (i.e. a synthetic proposition). Swinburne uses couterexamples to refute three Logical Positivist candidates for such a criterion: the Strong Verificationist Principle (SVP), the Strong Falsificationist Principle (SFP), and the Strong Verification-or-Falsification Principe (SVFP).

Next Swinburne considers the Weak Verification-or-Falsification Principle (WVFP):

q is a factual statement if and only if:
(a) q is a statement, and
(b) either q is an observation statement or there are observation statements which, if true, would confirm or disconfirm q.

This is the principle advocated by Ayer, according to Swinburne. No counterexample is put forward to refute WVFP. Instead, Swinburne argues
for the weaker objection that there is no good reason to accept this principle. Since premise (3) of the Logical Positivist skeptical argument is based on WVFP, we have no good reason, according to Swinburne, to accept premise (3).

Swinburne examines two arguments for WVFP, and argues that both arguments are unsuccessful.