This question relates to exercise 10.4 in Nielsen and Chuang.
For syndrome diagnosis, the textbook provides an example where one has four projectors, by which, you can identify where a one qubit error has occurred. In this scheme, the syndrome measurement does not cause any change to the state.
The exercise asks you to write down the projectors for the eight computational basis states, which is easy enough. $|0/1 \,\, 0/1 \,\, 0/1 \rangle \langle0/1 \,\, 0/1 \,\, 0/1 |$.
I think the point of the question is for one to say that when you project these onto your one qubit corrupted state, you can tell where the one qubit error has occurred. However, I thought projective measurement needs to be repeated multiple times to be able to reconstruct your full state (for example, if my state is $\alpha |000\rangle + \beta|111\rangle$, in order for me to actually find out what $\alpha$ and $\beta$ are, I need to repeat projection of the state on to $|000\rangle$ and $|111\rangle$ multiple times to form a statistic.
So, the bottom line is, I don't see how projections on to the eight computational basis states can be used to diagnose the error syndrome. Since projecting your state onto any of the eight computational basis states will mess up your state.
(Even before getting to the question above, how does one perform the syndrome measurement without causing any change to the state? On page 428, it's claimed that if the corrupted state is $\alpha |100\rangle + \beta|011\rangle $, it remains $\alpha |100\rangle + \beta|011\rangle $ even after the syndrome measurement. I'm having a hard time wrapping my head around the idea of constructing such syndrome projectors in terms of designing an actual experiment.)