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.