Question: Some people state that logic is not valid, since at some point it breaks down, i.e. the law of the excluded middle does not hold true in cases of self-reference. (For example: This statement is false... no matter whether you agree or not the statement is both true and false.) What is the Objectivist position on this?

My argument in the discussion was: Truth means correspondence to reality. False means contradicting reality. But the statement has no relationship to reality, since it is not an identification of reality (since when you begin to make the statement it does not yet exist, hence there is nothing there to identify). It is therefore arbitrary, it can not be logically processed and has to be dismissed.

Answer: Your argument is correct. But I will elaborate a little to make clear the Objectivist view.

Objectivism holds that logic has its basis in the metaphysical axiom of identity. The three basic canons of logic are called identity (A is A), non-contradiction (A is not non-A), and the law of the excluded middle (X is either A or non-A). But all of these follow from the axiomatic fact that Ayn Rand called "identity," that whatever a thing is, it is that thing. Our great challenge in forming knowledge is to know the identity of things: what they are. And logic is the means by which we can do this. This is why Ayn Rand called logic "the art of non-contradictory identification." So the basis of logic is not an arbitrary human construct, but, while no doubt human, lies entirely in reality.
The puzzle you mention ("This statement is false") is only relevant to understanding logic in context. First, the canons of logic are for human conceptual thought. We formulate conceptual ideas, beliefs, and claims as propositions: conjunctions of concepts that meaningfully ascribe an identity to an existent. We express propositions as sentences and statements, but a sentence is defined by its grammatical structure and the classes of words it employs. Yet meaning is more than accordance with grammar. Thus, there are sentences with verb and noun that do not express meaning.
"This statement is false" is an arbitrary (i.e., meaningless) statement (although a specious one) because it ascribes the identity "false" to an existent that need not possess either truth or falsity. Truth and falsity are traits of propositions and refer to whether the proposition corresponds to reality. The sentence "this statement is false" however, does not refer to a proposition. One might as well say something like "Jim has a high frequency" (frequency being a trait of waves, not humans), and make as much sense. The identity ascribed is not appropriate to the existent. It happens that "this statement" is not the kind of statement that expresses a meaningful proposition.

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