There are many problems with this idea. For every true statement, there is a corresponding negation. Assume that Anthony is standing, and so the sentence 'Anthony is standing' states something true. What about 'Anthony is not standing'? It cannot be true, for it cannot be true that Anthony is both standing and not standing. But it cannot be false, for if this 'objectivist' account is true, there are no false statements. Yet it seems to be a statement for all that. If I can meaningfully assert that Anthony is standing, why can I not equally deny that?
What about beliefs? Can I believe that Anthony is not standing, when he is standing? Apparently not, for the object of my belief is a state of affairs (Anthony-not-standing) that has no existence, and according to Rand, only existence exists. So I believe nothing. Only when Anthony sits down can I have such a belief.
What about questions? I ask 'is Anthony standing?'. The person who says 'yes' has agreed to something. The one who says 'no' has disagreed with the same thing, and so disagreed with something that exists. So you can say 'no' to a true statement, yet you cannot make the corresponding negation. You can say 'no' to 'is Anthony standing?', but you cannot say that Anthony is not standing. That is quite puzzling.
The Stoic philosophers resolved the problem by distinguishing between utterance (phone) which may be mere noise, e.g. 'arxas grexurgh', articulate speech (lexis) which may be meaningless, e.g. 'green is happy', and discourse (logos) which is meaningful speech. They also gave the name lekton to that which is signified by meaningful speech. Lekton is derived from the Greek verb legein, which signifies 'to mean' as well as 'to say' (somewhat like that Latin dico).
Sextus Empiricus* gives the most complete account of their theory:
The Stoics say that three things are linked together, that which is signified, that which signifies, and the existing thing. That which signifies is the utterance, e.g. 'Dion'. What is signified is the thing indicated by the utterance and which we apprehend as subsisting with our thought, but the barbarians [i.e. non-Greek speaking] do not understand, although they hear the utterance. The existing thing is that which exists outside, e.g. Dion himself. Of these, two are corporeal, i.e. utterance and the existing thing, while one is incorporeal, i.e. what is signified, i.e. the lekton, which is true or false.Thus we can distinguish between the state of affairs asserted by 'Anthony is standing', which some modern philosophers call the truthmaker, and the lekton, which some modern philosophers call the truthbearer. The truthmaker exists in reality, given that Anthony is standing. No truthmaker exists for 'Anthony is not standing'. The truthbearer, by contrast, is an immaterial, nonphysical entity, the meaning of 'Anthony is standing'. This has the value true. A truthbearer also exists for 'Anthony is not standing', but that has the value false.
There is nothing in Rand's theory that precludes the existence of such non-physical truthbearers or lekta. So long as such non-physical things exist, they are a form of reality (albeit a non-physical form). I don't know, of course, whether Rand's theory does preclude non-physical things. But if it does, it faces the difficulty of explaining statements which are false.
*Adv. Math. viii 11, 12.