Positivism asserts that any statement that cannot be empirically tested is meaningless. But obviously, this is a self-refuting statement, because it can't be empirically verified as well.
How do the logical positivists, or the descendants ( modified or not) of logical positivists defend against such an argument?
What you call "Positivism" there is but a crude description of a principle that has been associated with them. Let's consider a more faithful description of the so-called empiricist criterion of meaningfulness:
Criterion. (Hempel 1965b) A sentence makes a cognitively significant assertion, and thus can be said to be either true or false, if and only if either:
it is analytic or contradictory, or
it is capable, at least potentially, of test by experiential evidence.
In the first case, the sentence is said to have a purely logical meaning; in the second case—an empirical meaning or significance. Hempel calls this principle the testability criterion of empirical meaningfulness, and ascribes it not only to empiricism, but also to operationalism and pragmatism (appropriately or not isn't the topic of our discussion here). The criticism as applied to Criterion would be
Criticism. (Graviton) Criterion is self-refuting, because it cannot be empirically verified.
Of course, Hempel's version wasn't the one being criticized. I welcome Graviton to modify the critique to this particular case, if it too suffers from the same flaw. Until then, we should start by noting that the Criterion is of a disjunctive form: either condition (1) is met or (2). It is obvious that Criterion, like the claim "0 ≠ 1", is incapable, even potentially, of being tested. The arithmetical claim is analytic (logically true with respect to Peano's Axioms, say). But so is Criterion:
Defense. Criterion is analytic, and is thus: meaningful according to Criterion.
Neither is capable of being tested. You can reject Peano's Axioms so that you won't have to accept that 0 ≠ 1. You can also reject Criterion so that sentences like "John's aura is powerful" don't become cognitively insignificant. Tolerance (another positivist 'doctrine') is the key here. Consider the definition:
Criterion 2. A binary relation R is called 'symmetric' if and only if for all x, y, xRy implies yRx.
Is Criterion 2 meaningful? Yes. Is it capable of empirical testing? No. Is it analytically true? Yes. It is, of course, possible to say that definitions like Criterion and Criterion 2 are not the sorts of things that can be true or false, but I'm hesitating to commit to that stronger view of definitions to keep things simple.
The topic has spawned a lot of literature over the last (half a) century. It would be good news to some if positivism was dead, for then, there would arise the opportunity to skip reading stuff from Russell, Wittgenstein, Carnap, Ayer, Hempel, Reichenbach, Menger, Hahn, and others on all sorts of interesting topics. But if we're going to bury positivism, the least we can do is shove the right body into the coffin.
References
Lewis, D. (1988) "Ayer's First Empiricist Criterion of Meaning: Why Does it Fail?"
Hempel, C.G. (1965a) Scientific Explanation: Essays in the Philosophy of Science.
Hempel, C.G. (1965b) "Empiricist Criteria of Cognitive Significance: Problems and Changes".
