According to the YouTube video What Is Consciousness, somebody can know that they are conscious but doesn't know that other people are conscious. I think it might not even be possible for one to prove about oneself that they are conscious. You might be thinking, "How can I think I'm conscious and not be conscious? If I think I'm conscious, that means I am conscious." That just shows that if you make the assumption that you think you are conscious, then you are conscious. How do you prove the assumption that you think you are conscious? You might be thinking you can prove it as follows; "I think I think I'm conscious. Therefore, I think I'm conscious." The problem is that you have not yet proven the assumption that you think you think you're conscious.
Not everybody knows that they're conscious. I once read on a Quora answer about somebody who convinced their younger brother that the latter was dead and then the younger brother started crying while saying "I'm dead." How does anyone then know that they're conscious? Is it by defining knowledge in such a way that you don't have to prove something first to know that it's true? Did we invent a totally new mathematical system and define knowledge in such a way that you know something when you're proven it in that system, where the system allows for the rule of inference that once you see something, you can say that you saw it and anything that mathematically follows from 2 statements you previously said using another rule of inference, you can also say?