My next post was *going* to be an essay showing how the halting problem, Godel's theorem, Russel's paradox, and several other things can all be demonstrated using one method, called the diagonal method. Perhaps this post is still coming, but I have not felt inspired to write the whole thing out yet. For the moment:
Some Important/Crazy Questions
1. OK, so there's classical logic. In classical logic, everything is either true or false: not neither, not both. Classical logic has some oddities, which have led people to propose alternative logics. Two such logics are: intuitionistic logic, and paraconsistent logics. In some sense, these are opposite solutions to the same problem. (See here.) I like to think of it this way: both logics admit that there are true cases, false cases, and border cases.
They differ in how they deal with the border cases. Intuitionistic logic calls them "neither true nor false". Paraconsistant logic calls them "both true and false". This choice results in very different treatments. This notion of a "border case" has surprisingly been formalized using topology, and that is where I got the metaphor. (See here.) But If there is such a nice relationship between the two, shouldn't they be usefully equivalent, or perhaps part of some larger system? A movie I recently watched made a random reference to the idea of a four-valued logic, one with "true", "false", "both", and "neither". How would this work? (Googling the idea seems to bring up random unrelated stuff.)
2. Another alternative logic I have heard of is called "quantum logic". This is literally a logical system to work with the oddness of quantum mechanics. Strange! So my second question is: how does this logic work? How does it relate to paraconsistant and intuitionistic logics?
3. For applications in artificial intelligence, it is very useful to attempt a seamless integration of probability and logic, rather than simply using logic to reason about probabilities. (This is of course the goal of the Alchemy system I recently mentioned, as well as many other AI systems.) The natural question is how these alternative logics generalize to alternative theories of probability. For intuitionistic logic and paraconsistant logic, possible probabilitic versions readily present themselves: intuitionistic probability theory would let probabilities sum to less than one, and paraconsistant probability theory would let them sum to more than one. I was amazed to hear that the quantum logic has a simple probabilitic version: allow probabilities to have both real-number and imaginary-number components. (I heard this on the AGIRI mailing list. See here, second post in the thread.) How should all these things be interpreted??
Googling all of this stuff comes up with many results, so perhaps I will have a post that answers some of these questions soon. Meanwhile, I am still convinced that some form of nonmonotonic logic that can reason about incomputable models of the world is the important direction to go.