You already hit the important part with "as long as the premises are logical". Since the premises here are flawed, the proof falls apart. It isn't a flaw in formalized proofs. Getting flawless premises is the challenge for trying to use this for philosophical proof - the concepts are typically not as well defined as, say, what you see in geometry class.
Or am I reading too much in the mathematical term 'proof'? Because it sounds more like 'verifiable reasoning' to me.