This is a computer science heavy post, so if you don't have the stomach for a lot of abstract, mathematical, theoretical ideas, you'd better skip this one.
I hope you would read it though.
Right now you can say that there are two big camps in artificial intelligence: those whose methods are based on logic, and those whose methods are based on mathematics.
One logic based system would be Cyc, a knowledge base which can reason. So if you put in "All men are mortals; Socrates is a man;" the program can tell you that "Socrates is a mortal." Well, not in plain English (although professor Chris Riesbeck and others are working on that), but it has an understanding of what those words mean. It connects, say, George Bush with USA, presidency, war on Iraq, and so on. Basically, it's the "common sense" that a lot of people feel computers don't have.
On the other hand are stochastic or statistical systems, generally labeled "machine learning" systems. They operate mostly on numbers, although some use strings and other data structures as well. Most of the cool-sounding AI techniques are in this category, including neural networks and genetic algorithms. These programs are essentially giant calculators, which take past events into account and try to mathematically create the optimal solution. For example, a photograph with a sun in it would have a much brighter spot somewhere on the surface, and the program could analyze the distribution of light over the surface to categorize whether a photo has the sun in it or not. It doesn't really know what a sun is (a star that gives off heat and light), but it does the job.
Obviously, both sides have their advantages and weaknesses, or there wouldn't be two camps. The logicians require large amounts of data beforehand - where the computer stores all that common knowledge - and it could take a very long time to logically derive anything. Using the Socrates example again, it also knows that men have two legs and dogs don't have four legs, so it might conclude that Socrates is not a dog, which is not very useful at all. The main problem here is how to sort through all this junk. For stochastic methods, the problem is the opposite. It is relatively faster than logic, but it doesn't really know much about the thing it's doing. You can say that there is no creativity involved, no way to do something unexpected. They are in a sense limited to what they were programmed to do, within a very specific domain.
So the idea I had, and have held it for a while now, is to introduce statistics into a knowledge base. This is how it will work. Cyc is essentially built on connections between ideas, a hierarchy of different relationships. Buying might be a subset of all possession transfers, which in turn might be a subset of all human interactions.
The problem is that each idea, object, and relationship has so much other stuff connected to it, you don't know where to begin. Example: a story about animals making false cries of danger might induce thoughts of Boy who Cried Wolf. There is however a lot of junk in both those stories, when the central concept is very abstract, that of deceiving others as entertainment and a trade off between humor and protection. Humans can find connections without a problem, but machines have to explore millions of concepts to hit on the right now.
So what you can do (or try do to; this is a theoretical discussion) is put a weight on each relationship. Wolves might be more connected to animals than to dogs, and birds more to doves than to penguins. Humans have a tendency to use shortcuts, cognitive heuristics so we don't just stand and think the whole day. Statistical information about the strength of each link would be similar, allowing the machine to know which links are more commonly encountered, and therefore should be explored first.
Taking knowledge bases as relationships between objects, the entire KB can be seen as a mathematical graph. And with statistical linkage associated with each link/edge, guess what? We now have a undirected weighted graph on our hands. The theorem proofer can now rely on the large amount of graph algorithms (Dijkstra's algorithm, A* search) to either find the shortest path to proof a statement, or to generate new statements. There is still the matter of the machine not knowing what results are important, but that's a different problem.
One last thing about the statistics: one way to get the different weights between different (but related) objects would to be search another knowledge base: Wikipedia. Although Wikipedia is written in English, the text is still somewhat machine readable. For example, the machine will know to link to other pages if the user encloses something in double brackets ([[ ]]). So one easy way to find the links is take the inverse of the number of links it takes to link two wikipedia pages. This is in fact another graph traversal problem, but much more easily solved. So objects which are link directly to each other (say dog and animal) would have a much higher chance of being evaluated than objects far apart (say dog and black holes).
I think that's one way to take advantage of both worlds. I don't know if this has been done before, and I'm not really in a mood to find out.