What is the Verifiability Criterion a Criterion of?

As my title implies, I think the verifiability criterion is indeed a criterion of something. I do not intend, therefore, merely to commemorate it. On the other hand I am not sure that those who put it forward in its more liberal forms as a criterion of ‘factual significance’ or ‘literal meaningfulness’ were right in what they identified as the consequence of a sentence's failing to satisfy it. What I want to argue for, in a somewhat reductionist spirit, is a resurrected version of the ‘weak’ verifiability criterion. My resurrected version will certainly appear more rarefied, in so far as it is independent of (and does not therefore require to be embodied in) empiricism. It will, I hope, also be purified of some of the mortal blemishes from which the criterion, as construed by members of the Vienna Circle, seems not to have recovered.