terminology: half adders, full adders I'm very confused about the reasoning for these circuits being called 'full adders' and 'half adders'

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.
A "half adder" only does half the job: It adds two inputs x and y. 

A "full adder" does the full job: Most of the time, you need to process two inputs, plus a carry from another (half or full adder). A "full adder" adds three inputs x, y and z.