You suffer from the entrenched notion in Western society that dualism is correct. Your analysis of true and false above is correct, but only in a logical forum where there are two values of truth. The Yin and Yang is a great example how dualisms are in fact one thing.
Dualism is an archaic notion which needs to be removed from our consciousness.
It has done much more harm than good. Learning to see the ontological existence of the whole as opposed to only seeing a part of the picture will go far in correcting the errors that we make on a regular basis due to our misunderstanding of the nature of reality. There is no reason wht we can not extend this type of thinking to arrive at a closer understanding of what is really going on around and within us. Let me know. Your analysis of true and false above is correct, but only in a logical forum where there are two values of truth.