Friday, December 05, 2008

A General Theory

Well, I have been climbing the ladder of infinities for a while now on this blog. Soon I must get back to how this all applies to AI... but not quite yet.

Here is a theory that almost works.

Start with first-order logic. This can be your favorite non-classical variation if you wish; intuitionistic, paraconsistent, relevant, whatever you want. There is only one requirement: it needs to be strong enough to be Turing-complete (more specifically, logical consequence should be completely enumerable but not co-enumerable, thanks to the good old halting problem). Call this the base language.

Add to this your favorite theory of truth. For maximum effectiveness, it should include the infinite hierarchy of truth that I suggested in this post. This is not a difficult requirement: either a revision theory of truth or a fixed-point theory will do, as well as many less-well-known theories, I'm sure... and anyway, the requirement is not totally necessary, as I'll attempt to make clear. Anyway, call this language the language of truth. The theory of truth that you choose will assign actual truth-values to the statements in this language.

Now we have what we consider a solid theory of truth. But, as I pointed out in the message I copied in the previous post, all such theories appear to have referential gaps: some statements will have a status not nameable within the theory, which we can name and reason about as people standing outside of the theory. The type of referential gap will depend on the type of theory of truth that was used. In the most common case, the gap will be sentences that are not assigned either "true" or "false". The theory of truth will be able to state that a sentences is true, or false, but not that it is neither. Defenders of such theories of truth will attempt to claim that we really can't say that; for example, one way of arguing is saying that such sentences have undefined truth values, but we could later add logical conventions that ascribe true or false to them. However, such arguments are self-defeating: the argument needs to refer to the class of sentences that are in this intermediate state, so it generally must invent a label for them (such as "undefined"). This name is precisely the reference gap of the theory, and cannot be stated inside the theory.

So, the next step is to add to the language whatever labels we need in order to fill the gap(s) that exist in the theory of truth. I call this stage a theory of meaning, because I think it is important to point out that the theory of truth is not incomplete just because of the gaps; it may be a complete and valid theory of truth, it just is not a complete theory of logical/mathematical reference. Call the new gap-filling language the first language of meaning. I will generally pretend that there is only one gap, as in the simple case. Call this gap 1-meaningles. (The idea doesn't seem hard to modify to approaches that create multiple gaps.)

Assigning truth values to this language can generally be accomplished by relying on the original theory of truth that we chose. First, label the 1-meaningless sentences as such. Second, modify the theory of truth to act upon 3 truth values rather than the usual 2. This will involve some decisions, but as you can see I am not too concerned with details in this post. Generally, we'll have to decide things like whether it is 1-meaningless or just false to claim that a 1-meaningless sentences is true. Once we've made these decisions, we simply apply the method.

This, of course, creates another gap; having 3 truth values rather than 2 is not so radical as to change the result there. Call the new gap 2-meaningless, and call the language that includes it the second language of meaning. Assign truth-values to this language in the same way, by expanding the method to include 4 truth values.

By now you get the idea. We define 5-meaningless, 6-meaningless, and so on. And if you read the first post I mentioned (this post), then you'll probably also realize that I want to similarly define infinity-meaningless, inf+1-meaningless, inf+inf-meaningless, and so on. More specifically, I want to define a type of meaninglessness corresponding to every ordinal number. As I hinted at the beginning, this rather large hierarchy should smooth out most differences in referential power of the methods involved; so, a really weak initial theory of truth should still do the trick in the end, gaining maximal referential power after unstateably many iterations.

Now for the pleasant surprise. Once I've done this, I can prove that there is no referential gap left. If I had a final gap, it would correspond to an ordinal number larger than all other ordinal numbers (including itself)! This cannot be, so the theory is safe, almost as if it were protected by a magic charm.

For a few weeks now I've been satisfied with this conclusion. But, early on, I realized that I didn't know what the logic should say about a statement like "This sentence is either false or some type of meaningless". A day or so ago I realized what was going on. Each new language can refer to any combination of the truth-states from the levels under it, but obviously there is no top level (since there is no ordinal larger than all others), so we don't have permission to refer to any combination of values from any level we want; we are only allowed to refer to combinations that have some upper bound. The phrase "some type of meaningless" has no upper bound; it attempts to refer to the entire unbounded list.

There is some legitimate mathematical tradition that could be used to justify this limitation of the logic. One is not supposed to tamper with all ordinals at once. So, I could simply say that it's OK to stop here. Of course, I'm using the same sort of self-defeating argument that is so common in this area... in order to say that we can't refer to the list of all ordinals, I am doing exactly that.

Another way out would be to allow the newly-problematic statements to be both true and false, using some paraconsistent logic. This is similarly justified by the motto "one is not supposed to tamper with all ordinals at once". The taboo surrounding the list of all ordinals arises, of course, from the fact that contradictions follow quite quickly from reasoning about it. So, I could somewhat legitimately argue that it is literally an inconsistent yet well-defined mathematical entity.

However, this does not give me maximal expressive power... I will end up wanting to invent new notation to fill reference-gaps in the paraconsistent theory, and on we go.

So, a more direct approach would be to allow unrestricted statements about ordinals, and then do the same thing we've been doing... assign these statements truth values in the obvious ways, then apply an expanded theory of truth to add a truth predicate to that language, then call anything that isn't assigned a value "super-meaningless", expand the theory of truth to give that a predicate, invent 2-super-meaningless, 3-super-meaningless, inf-, and the whole ordinal hierarchy again. Then what? Well, we'll have the same sort of gap for this second ordinal hierarchy. So by doing the whole thing again, we can create super-super-meaningless, super-super-super-meaningless, and infinite versions with any ordinal number of supers . What next? Well, we'll have the same problem again...

But notice what I'm doing...

All of this is quite seriously illegal if we take the notion of ordinal number seriously, because I'm simply constructing a hierarchy of ordinals larger than all ordinals. An ordinal cannot be larger than all ordinals! And even less can there be a whole hierarchy up there...

This demonstrates the force of the two limitative arguments (either "no, you seriously cannot refer to those things" or "sure, go ahead, but you'll derive contradictions"). Even though there really really seems to be a solid next step that can be taken, it is either a brick wall or a gaping ravine...

So, like I said, it seems to be a theory that almost works.

1 comment:

  1. Goso said: "When a buffalo goes out of his enclosure to the edge of the abyss, his horns and his head and his hooves all pass through, but why can't the tail also pass?"