Individuals perceive things differently and each is entitled to their own view.
Someone who is mentally ill may perceive something totally different than anyone around them (I am not crazy! Everyone IS out to get me!) A mentally ill person may perceive things differently on different days depending on their medications, physical well being (lack of sleep, nutrition, etc.), and other factors. Whatever their perception is at that time, it is their belief and no amount of arguing will change their mind at that point.
Accept that the issue will not be resolved at the time. If you can put yourself in the other person's shoes, it may help you understand why they feel the way they do even if you don't agree with it. Have they suffered a loss (of job, spouse, social status, property, etc.) that makes them fearful of their future? Is there something threatening their independence (health, blindness, memory loss, financial difficulties?) How would you feel in a similar situation?
In my experience, if the medications are not in balance, the mentally ill person may perceive things to be completely different from reality, but to them IT IS their reality at the moment and may be pretty frightening. I have observed people who are seeing someone who is not there and having a conversation with them; someone hearing voices; and persons in a paranoia experience that truly are convinced people are out there after them. It is terrifying even as an observer. If you are ever faced with this, do not try to convince the person these things are not realities, but keep them safe from harming themselves or others and get them to trained mental health professionals that can assess their condition and assess their immediate needs.