Friday, August 22, 2008

Other's Inventions

My goal mentioned in the previous post was creating a logic that can reason about its own semantics (somehow sidestepping Tarski's proof of the impossibility of this). Well. This is a very ambitious goal. Luckily, I am not the first to desire such a thing, and there is a vast body of work on the subject. I find not one solution, but three:

Fixed Point Theories
Supervaluation Theory
Revision Theory

There are other theories than these, but these are the ones I saw mentioned the most.

It is comforting to see that "grounding" is often mentioned in connection with these theories, although the word is not being used in quite the same way I am currently using it. The word comes up particularly often in association with fixed-point theories.

Some of the differences between the three theories are superficial. For example, fixed-point theories are often stated in terms of three truth values: true, false, and neither-true-nor-false. Revision theorists reject this idea, and instead simply fail to assign a truth value to some sentences. This makes no difference for the math, however.

So, in a sense, the three theories are very similar: each describes some method of assigning truth values to some, but not all, sentences. Each of the methods sound overly complicated at first, compared to typical paradox-free mathematical settings, in which truth-assignment is very straightforward. But, once one realizes that the complexity is needed to avoid assigning truth values to meaningless statements such as paradoxes, each of the methods seems fairly intuitive.

Unfortunately, each method yields different results. So which one is right?

I've got to do a lot more research, to find out more about how they differ, other alternative theories, and so on. I suspect the fact that I am looking for concrete results rather than merely a philosophically satisfying logic will help guide me somewhat, but I am not sure.

Also, it appears that most of these logics are developed as modifications of arithmetic, rather than set theory. But, I am looking for a logic that can serve as a foundation for mathematics. It appears that a "direct attack" on the self-reference problem does not automatically get me what I want in terms of the general reference problem. So, I still need a good theory on how mathematical facts about uncountable sets can be grounded...

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