I'm trying to rationalize what it physically means to add three spin-1/2 particles. I understand that for a system of two spin-1/2 particles that there are four basis vectors in the new space on account of the tensor product. These are the states, $|1,1⟩, |1,0⟩, |1,-1⟩$, and $|0,0⟩$. I'm not sure I'm understanding where the spin-0 state is coming from.
I rationalize it as having three spins running parallel and one state where they are running anti-parallel. Applying similar logic then to the system with three spin-1/2 particles, I first create a system of two particles with the above states, then add the third particle to the system. For spin-1 with spin-1/2, the parallel state would give me a total spin of 3/2, and the anti-parallel state would give total spin, 1/2.
However, why does the spin-0 state when added to a spin-1/2 particle not give a -1/2 total spin state in addition to a spin-1/2 state (0+1/2). Is this because spin-1/2 added to a zero vector in essence gives 1/2 in an arbitrary direction because spin-0 is directionless?