I’m not yet good enough at writing posts to actually properly post something but I hoped that if I wrote something here people might be able to help me improve. So obviously people can comment however they normally would but it would be great if people would be willing to give me the sort of advice that would help me to write a better post next time. I know that normal comments do this to some extent but I’m also just looking for the basics – is this a good enough topic to write a post on but not well enough executed (therefore, I should work on my writing). Is it not a good enough topic? Why not? Is it not in depth enough? And so on.

Is your graph complete?

The red gnomes are known to be the best arguers in the world. If you asked them whether the only creature that lived in the Graph Mountains was a Dwongle, they would say, “No, because Dwongles never live in mountains.”

And this is true, Dwongles never live in mountains.

But if you want to know the truth, you don’t talk to the red gnomes, you talk to the green gnomes who are the second best arguers in the world.

And they would say. “No, because Dwongles never live in mountains.”

But then they would say, “Both we and the red gnomes are so good at arguing that we can convince people that false things are true. Even worse though, we’re so good that we can convince ourselves that false things are true. So we always ask if we can argue for the opposite side just as convincingly.”

And then, after thinking, they would say, “We were wrong, they must be Dwongles, for only Dwongles ever live in places where no other creatures live. So we have a paradox and paradoxes can never be resolved by giving counter examples to one or the other claim. Instead of countering, you must invalidate one of the arguments.”

Eventually, they would say, “Ah. My magical fairy mushroom has informed me that Graph Mountain is in fact a hill, ironically named, and Dwongles often live in hills. So yes, the creature is a Dwongle.”

The point of all of that is best discussed after introducing a method of diagramming the reasoning made by the green gnomes. The following series of diagrams should be reasonably self explanatory. A is a proposition that we want to know the truth of (the creature in the Graph Mountains a Dwongle) and not-A is its negation (the creature in the Graph Mountains is not a Dwongle). If a path is drawn between a proposition and the “Truth” box, then the proposition is true. Paths are not direct but go through a proof (in this case P1 stands in for “Dwongles never live in mountains” and P2 stands in for “Only Dwongles live in a place where no other creatures live). The diagrams connect to the argument made above by the green gnome. First, we have the argument that it mustn’t be a Dwongle because of P1. The second diagram shows the green gnome realising that they have an argument that it must be a Dwongle too due to P2. This middle type of diagram could be called a “Paradox Diagram.”

Figure 1. The green gnomes process of argument.

In his book, Good and Real, Gary Drescher notes that paradoxes can’t be resolved by making more counterarguments (which would relate to the approach shown in figure 2 before, which when considered graphically is obviously not helpful, we still have both propositions being shown to be true) but rather, by invalidating one of the arguments. That’s what the green gnomes did when they realised that Graph Mountain was actually a hill and that’s what the final diagram in figure 1 shows the result of (when you remove a vertex, like P1, you remove all the lines connected to it as well).

Figure 2. Attempting to resolve a paradox via counter arguments rather than invalidation.

The interesting thing in all of this is that the first and third diagrams in figure 1 look very similar. In fact, they’re the same but simply with different propositions proven. And this raises something: It can be very difficult to tell the difference between an incomplete paradox diagram and a completed proof diagram. The difference between the two is whether you’ve tried to find an argument for the opposite of the proposition proven and, if you do find one, whether you’ve managed to invalidate that argument.

What this means is, if you’re not confident that your proof for a proposition is true, you can’t be sure that you’ve taken all of the appropriate steps to establish its truth until you’ve asked: Is my graph complete?