It is the real world, not logic, which makes a thing true. Facts, nature, reality constitute the standard of truth, not logic. I would also note that, while there exists an infinite number of logical expressions (after all, every mathematic equation is a logical expression, and there are an infinite number of such expressions), only a small fraction of those will find exemplification in existence. Logical validity is therefore no warrant of truth.
Peikoff's attack on the analytic-synthetic dichotomy is primarily an attack on the distinction, central to "metaphysical" realism, that facts are empirical and logic is ideal. In order to carry out his attack, Peikoff draws inferences from the analytic-synthetic dichotomy that only the most doltish philosophers would ever dream of accepting:
Analytic truths ... are non-empirical -- because they say nothing about the world of experience. No fact can ever cast doubt on them, they are immune from future correction -- because they are immune from reality....
Synthetic propositions, on the other hand, are factual -- and for this, man pays a price. The price is that they are contingent, uncertain and unprovable.
The theory of the analytic synthetic dichotomy presents men with the following choices: If your statement is proved, it says nothing about that which exists; if it is about existents, it cannot be proved. If it is demonstrated by logical argument, it represents a subjective convention; if it asserts a fact, logic cannot establish it. If you validate it by an appeal to the meanings of your concepts, then it is cut off from reality; if you validate it by an appeal to your percepts, then you cannot be certain of it. [IOTE, 93-94]
Here we find a choice example of a failure to get the point. When Wittgenstien wrote, "The propositions of logic all say the same thing: that is nothing," this was not meant as an attack against logic or truth; it was meant as an attack against rationalistic speculation. Logic, by it's own devices, can only insure that the conclusion of an argument are consistent with its premises (i.e., that it says "the same thing," as Wittgenstien puts it). It's not the function of logic to determine whether the premises or the conclusion are true. While logic can be a very useful tool in testing and arriving at truth, it is not itself true, nor is it a fail-safe test of truth.
If there's a flaw in analytic synthetic dichotomy (at least in the version presented by Peikoff), it arises from the expression "analytic truth." There is, properly speaking, no such thing. All truth is empirical (i.e., synthetic). Now this does not mean, as Peikoff incorrectly infers, that no analytical propositions can be empirically true. Many undoubtedly are. But they are not true for logical reasons; they are true because they serve as appropriate descriptions of reality. 2 + 2 = 4, if taken as an analytical statement, is not true, it's merely correct. As an empirical statement, it may be true in some instances and false in others. As Karl Popper has pointed out, if you place two plus two drops in a dry flask, you will never get four drops out of that flask. A logical argument, assuming it's valid, is correct; it may also be true; but if true, the truth comes from reality, not from logic. Logic tests the validity (i.e., the correctness) of arguments. It is not a test of truth (although it can be used to falsify universal statements). Empirical testing, rather than logic, is the test of truth. Logic may play a critical role in the empirical testing; but one should not confuse the method with the result. Grammar is the method of speaking and writing correctly; but grammar isn't true of reality, nor is grammatical speech even necessary for making oneself understood.
Another misunderstanding that emerges from Peikoff's analysis of the ASD involves a failure to distinguish between significant and trivial truth. Modern philosophers object to true analytical propositions, not because they wish to deny A is A or existence exists, but because they find such propositions to be trivial and unworthy of any philosopher who is attempting to convey significant insights about the real world. Such propositions are trivial because nothing specific can ever be deduced from them. From a tautological premise, only tautological conclusions can be deduced. A philosophically literate person should know this, but, as David Stove explains, philosophers sometimes forget this important truth:
Just so, the fact that philosophers all know that necessary truths have only necessarily-true consequences, and tautologies only tautological ones, is no guarantee that they will always bring this knowledge to bear in cases where they should. It is perfectly possible that they will mistake a particular argument, say, from a tautological premise to a contingent conclusion, for a valid one.
Not only is this possible: it is a temptation to which everyone, including philosophers, is constantly exposed. For we all want, as it is perfectly reasonable to want, our conclusions to be as interesting as possible, our premises to be as certain as possible, and our reasoning to be as conclusive as possible. And who cannot see that this threefold want, if it is not restrained by our own better knowledge, will sometimes lead us to imagine that these three desiderata have all been maximally satisfied at once: for example, that some non-tautological conclusion has been rigorously derived from a tautological premise?
It is in just this way that millions of people, some of them philosophers, have concluded that all human effort is ineffectual, from the premise that whatever will be, will be; have concluded that everyone is selfish, from the premise that a man's desires are necessarily his desires; have concluded, from the tautology that every effect has a cause, that every event has a cause. Shall I go on? Shall I mention … Hegel, fixing the number of planets, with no other premises than truths of logic? Bentham, whose only real argument from "the principle of of utility" was, that "it is but a tautology to say, that the more consistently pursued, the better it must ever be for mankind?"
Tedium forbids that this list be lengthened: we have all heard this complaint far too often. Ever since Francis Bacon, a hurricane of complaint has been blowing, to the effect that philosophers have tried to do "a priori," or "deductively," or "speculatively," what can only be done "inductively" or "a posteriori," or "naturalistically," if it can be done at all…. This hurricane of complaint is blowing still…. But it is, and always has been, essentially right….
… it must be admitted that this temptation to produce non tautological rabbits out of tautological hats is especially strong for philosophers. For consider. Rigorously valid reasoning is both part of philosophers' subject-matter, and one of their boasts; philosophers' conclusions are, characteristically, extremely general and interesting, not to say amazing; and yet, except per accidens, they have no more knowledge of any contingent matter [i.e., matter of fact], that could serve as the premises of their reasonings, than the next man has. Any special expertise which they possess is all confined to "the realm of essence" [i.e., the meanings of words]. Clearly, therefore, it must be an occupational hazard for philosophers to be tempted to milk interesting results out of tautologies; even though they are also the very people who best know that the thing cannot be done. [Idols of the Age, 156-157]
When David Stove declares that all philosophers know that only tautological conclusions can be drawn from tautological premises, he obviously betrays ignorance of Rand and her disciples. For in Objectivism we discover philosophers who appear entirely innocent of the empirical vacuity of tautologies. And it is largely in the service of perserving this rather quaint philosophical illiteracy that Peikoff trots out his "refutation" of the analytic-synthetic dichotomy. While Objectivists are opposed to rationalism in others, they are not opposed to rationalism in their own speculations; and that is why they must attack the distinction between tautological (i.e., analytical) propositions and synthetic propositions (i.e., statements about matters of fact). Objectivism cannot be an empirical philosophy for the simple reason that too many of its core doctrines fail to accord with reality. Man is not a blank slate governed by general premises; physical reality is not "logical"; definitions do not define concepts; morality cannot be founded solely on reason; laissez-faire capitalism is not a politically feasible system. These are all assertions about matters of fact that cannot be defended on the basis of empirical research and testing. They can only be defended speculatively, on the basis of tautologies and empirically vacuous generalizations. Peikoff's attack on the ASD is an attack on empiricism from a rationalistic perspective.