Those who claim to distinguish a posteriori and a priori propositions commonly maintain that certain truths (the synthetic, factual ones) are "empirically falsifiable," whereas others (the analytic, logical ones) are not. In the former case, it is said, once can specify experiences which, if they occurred, would invalidate the proposition; in the latter, one cannot. For instance, the proposition "Cats give birth only to kittens" is empirically falsifiable" because one can invent experiences that would refute it such as the spectacle of tiny elephants emerging from a cat's womb. But the proposition "Cats are animals" is not "empirically falsifiable" because "cat" is defined as a species of animal....
Observe the inversion propounded by this argument: a proposition can qualify as a factual, empirical truth only if man is able to evade the facts of experience and arbitrarily ... invent a set of impossible circumstances that contradict these facts; but a truth whose opposite is beyond man's power of invention, is regarded as independent of and irrelevant to the nature of reality, i.e., as an arbitrary product of human "convention." [IOTE, 117-118]
This passage provides a nice brief summary of what is wrong with Peikoff's essay. His objections are ludicrous. They seem founded on little more than an animus against non-Objectivist positions coupled with an inability to understand and appreciate any view not sanctioned by Rand. The distinction between empirically falsfiable and non-falsifiable statements is not made so that individuals can "evade facts of experence and arbitrarily invent" impossible circumstances. To even assume such a thing proves Peikoff's cluelessness about not only the analytical-synthetic dichotomy, but falsifiability as well. The reason for prefering falsifiable to non-falsifiable propositiosn has to do with the issue of testability. Empirically falsifiable proposition are testable; non-empirically falsifiable views are not testable. Analytic propositions are not testable because they are true by definition: no experience can refute them; they tell us more about word usage than reality.
The whole point of the analytic-synthetic dichotomy is to note this duality in propositions: some propositions are self-referential. If they are true, it is because they are "true by definition." Such "truth" is not all that significant because it is lacking in empirical content. Other statements are rich in empirical content. These statements are much more significant. While it may be true that the ASD tends to draw this distinction a bit too tightly, nevertheless there are statements that are more or less analytic and untestable on the one side, and statements that are more less synthetic and testable on the other.
Now Rand over and over again insisted that she sought to establish the connection of man's knowledge to reality. Even if we are willing to entertain Rand's suggestion that this connection had been compromised or severed by "evil" philosophers such as Kant and Hegel, her Introduction to Objectivist Epistemology does little to help re-establish the connection. Rand decided that the problem of knowledge is largely a problem of conceptualization. For Rand, men are led astray, cognitively speaking, not by poorly tested theories, but by "improper" conceptualization. The process of forming concepts, Rand declared, is not automatic. Men must discover a method of concept formation, which must be consciously applied (or consciously programmed into the "subconscious"). But as I have repeately noted, this approach leads to an over-emphasis on the process by which knowledge claims arise. Since, as cognitive science has discovered, much of this process is unconsious, hidden from the watchful eye of consciousness, it is futile to emphasize how knowledge claims arise. We can't actually evaluate the rationality of how conclusions arise, since those processes are (mostly) invisible to us; but we can evaluate the claims themselves by subjecting them to criticism and tests. The path to the very sort of rationality Rand claims to support involves concentrating on testing and criticizing our knowledge claims after they have been made, rather than worrying about how those claims were originally formed. However, before we can test a conclusion, it must in fact be testable. Hence the usefulness of the analytic-synthetic dichotomy. If we wish to achieve whatever degree of rationality that human beings are capable of attaining, then we should encourage the development of testable (i.e., largely synthetic), rather than non-testable (i.e., largely analytic) knowledge claims. The lingua franca of empirically responsible philosophy is testable conclusions. Non-testable conclusions are the province of rationalizers, ideologues, deceivers, and cult-mongers. The philosopher who passes off non-testable "analytic" propositions as the chief "axioms" of his philosophy is not seeking truth; he is, rather, looking for a place to hide, so that he can safely evade any potentially effective criticism which might come his way.