I'm not that familiar with Carnap, so first I went here, for a quick crib-sheet. and followed a link to the dude himself - http://www.ditext.com/carnap/carnap.html

Logical positivism is out of favour by and large, and I am not the expert, but the game is to try and separate fact from fiction as it were, in an indubitable once and for all manner. Material statements are about the world - which is to say they are about what exists. So cats and mats come to mind. "The cat is on the mat" is material, and has ontological, or existential import. Carnap likes this kind of thing. Similarly, he has no problem with "there are two cats on the mat."

But then there are things like "2 is a prime number." which don't seem to be about cats or mats or anything material, but which don't seem to be quite fictional either. Hence 'abstract propositions'.

He's trying to formalise a scientific language devoid of ambiguity and fully rigorous in terms of being tied down to pure logic on one side, and pure observation on the other. I don't think he can do it, but I cannot articulate why not at the moment. Maybe head to the SEP for more detail but the distinction between real and ideal is a very old one that concerned Plato and continues to plague and confuse better minds than mine. I'm not sure where you should start.