What ever the object is, so too is consciousness. Consciousness is not the self, because simply put, there is no self. Is is not a relation, for there is not self for it to stand in relation to. Consciousness is the object. As to a fact.
Facts are neither true nor false.
Only statements about facts can be true or false. For instance, it is a fact that my pet is a dog, and there is a tree in my front yard. These are facts, they are not changed in the slightest no matter what I think or say about them. My propositions, however, might be true or false, about them. For instance, if I said that my pet is a cat, then according to the way the world is, my proposition is false.