Before I get into this, I want to explain a mathematical notion called the pigeonhole principle:
Imagine that you have 10 pigeons that you have to fit into 9 pigeonholes. (I know it sounds strange, but just see where I'm going with this.) Notice that no matter how you try to spread them out, you will end up having at least one hole with more than one pigeon. For instance, if you tried to place one pigeon in each hole, you would have 9 holes with one pigeon, and one pigeon left over. That pigeon would have to fit into one of the other holes, so you would have a hole with two pigeons! This would be the best case scenario.. Therefore, you would always have a hole with more than one pigeon... If you wish, try it out with a smaller number. Try drawing five 'pigeons' into four 'holes'. You'll notice that you'll always have a hole with more than one pigeon. This, of course, applies to any number of pigeons and any number of holes - so long as there are more pigeons than holes!
So, what does this have to do with socionics? This principle got me thinking about socionics, and it led me to finding a flaw in the notion that there are exactly 16 relations - and therefore 16 types. Here's what I mean: first, let's make socionics simpler. Let's say there are two quadras, instead of four. Let's also say that everyone fits into one of the two quadras - everyone from quadra A likes each other, everyone from quadra B likes each other, and no one from quadra A or B like each other. Therefore, if you were to poll any group of people under this system, everyone would fit neatly into quadra A or quadra B - there would be no one who likes other people from both quadras! Notice that this issue is similar to the pigeonhole principle, and that socionics has a similar flaw to the two quadra system... Still not convinced? Pretend you are an ILI. You meet someone intellectual you don't like. You call them ILE. Then, you both meet a third intellectual person. None of you like each other. What do you call them then? EII? But then the ILI shouldn't mind them. Oh, they don't? Let's add a fourth intellectual person that none of you like as well then (and so on)... Do you see the problem here? If everyone fits neatly into one of 16 relations, then you get something like this, where the types are 'pigeonholed' (or something like it) into more unlikable relations than are possible...
Of course, there are ways around this: maybe the types are only correlated, or maybe there are exceptions to the rule. On the other hand, maybe socionics is just plain wrong.... Any of these are realistic possibilites. My answer would be that there are more than 16 types - there seems to be so much diversity in the human population on so many levels that this could easily be the case, and there also seems to be some pattern to the quadras and types... Based on this reasoning, more than 16 types seems to be the most intuitive answer...
In any event, what does this prove? It proves that you can't neatly fit a person into one of 16 types. It also proves that in the very least, socionics is not perfect. Remember this when you're arguing with someone that person A has to be (or cannot be) type B...
Jason