Monday, April 16, 2012

On the limits of formal systems

You can do some very interesting things with formal systems, such as chess, philosophy, or mathematics. You can create true beauty that has the advantage of being eminently useful or making a personal profit. You can make predictions from baseline assumptions and expose inconsistencies. However, there are many things you can’t do, and one of the easiest ones to forget is one of the most important: you can’t prove something using formal logic that exceeds your assumptions. I discuss this further below the fold.

I will start with one of the illustrations that we use for implications, a Venn diagram.

The basic idea of any implication is a subset. The conditions that create the hypothesis (what you assume to be true) are a subset of the conditions that create the conclusion (what you are demonstrating to be true). Since the entire calculus of classical log can be expressed using implications and the word “not”, that means that all of classical logic is basically a verbose form of set theory.

One aspect of this notion is that all implications are downhill, or at best level, in the restrictions imposed upon the model by the statements involved. You reason from the more restrictive, harder to match, less flexible model to the less restrictive, easier to match, more flexible model. This is the only way implications can work, therefore the only way classical logic can work.

Of course, sometimes the implication is just not there. You find yourself with a hypothesis that covers too much ground, and has too much flexibility to support the conclusion.

You can make a couple of choices at this point. One of the most common ones is to increase the number of hypotheses. When you require that all the various hypotheses are true, you are restricting your model even more. You wind up with a much smaller model, as you can see by the blue region.

Now, if we combine the previous two diagrams, we can see that adding additional hypotheses has allowed us to restrict our starting model to the point where we can prove the conclusion.

Of course, most philosophy, and for that matter most mathematics, is done with commonly spoken language as opposed to a formal language. This makes it easy for people to sneak in hypotheses, often without even being aware that is what they are doing. This is even truer when the proof is particularly important to them, such as when it is being used to support a religious position. For example, if someone tells you about a proof that the existence of change (an unrestrictive, easily matched, flexible starting point) can be used to prove the existence of God (a moderately restrictive, not as easily matched, less flexible conclusion), you know that something will be incorrect even before you dig into the proof. Using logic and metaphysics, you can’t prove the existence of God unless you assume such strong hypotheses that, by comparison, God’s existence is less restrictive option. Logic just does not work that way.

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