So this is the physicalist's thesis: The organization and special functions of the nervous system interacting with the body as a whole and also with the environment produce what we call mental functions. Note the adjectival form mental.
So what about the non-reductive part? I said above that the main issue is causation. To say that the mental level of functioning is not reducible is to say that a complete causal account of what goes on in the world has to take it into account.
Now it may be difficult, surprisingly so, to give an argument for the causal difference that consciousness makes. Philosophers fool around with thought experiments -- couldn't we all be zombies and be doing just exactly the same things that we do and would anybody ever know the difference.
But the causal difference made by information processing is totally manifest. For example, the rabbit jumps up when the light is shown on it and the pound of hamburger doesn't. So it's very clear that the rabbit is processing information, and we can just tell that by watching what it does.
In my next lecture, I'm going to talk about downward causation of what we usually think of as the mental ideas, reasons, etcetera. But I'm going to put that off until the next lecture because I want to do that in conjunction with talking about free will.