An exploration of the foundations of logic and philosophy
First imagine a black line on a white piece of paper. It separates one half from the other and we say the line has 0 width.
Replace the line with a black paper half-way over the white paper. We still have a sharp line separating one side from the other but it is as if just half of the logic that the previous line if 0 width is represented. We are invited to imagine a variation of Venn diagrams which have different logical implications than we are used to, implying a logical context which isn't classical. Perhaps we can use this clue to reason clearly about things which are otherwise paradoxical?
Then see if this experiment is repeatable. Instead of a black paper we place another white paper over the first. Now we have constructed a line again and we can also tell a little more about what happened in the previous step. We know there is a line there because we constructed it but the contrast has disappeared so we can't see clearly where the line is. This is some kind of logic of camouflage.
Let's imagine we have an object which we compare with itself it as it is represented on each side of our line. In the 0-width example the object is clearly either part of one half of the paper or the other. In the last example with our invisible line the object's position is unknown. But, just as we know there is a boundary by the act of constructing the invisible line we can insert the information that the position of the object has changed, even repeatedly. What can we do with the black-on-white paper construction to extract information from the system? What type of information do we need to insert to produce meaningful observations?
Since the parallel postulate states that there is exactly one line through a point parallel to another line, what do our modified logics reveal about unity and equality? If our 0 width line is a number line and the parallel postulate is a statement about Cauchy completeness, what do our extended logics construct as number-like objects?