I've read before that 'half adders' are called so, because two of them make up a 'full adder', and a 'full adder' is 'full' because you can get all the values $(0,0), (0,1), (1,0), (1,1)$.

But what irks me is that adding two bits would never get you $(1,1)$ in the first place, so why not call it a 2-bit adder, since that's exactly what it does. By extension, a 'full adder', in my mind, should be called a 3-bit adder.

If i think of full adders as '3 bit adders', would I ever run into any conceptual troubles? Same goes with '2 bit adders' and half-adders. Are there any historical insights as to why they are called such?

Cheers.