I should admit here and now that i haven't really taken a full course on logic, but this seems to fall under the same catagorizes as everything else does.

In the traditional logic they teach in undergraduate CS, member of sets are described using quantifiers(the universal and existential quantifier namely). The meaning of the expression changes based on what order the quantifers are put in. Existential quantifers "depend" on the universal quantifer they come after. This concept is a bit difficult for new students no thanks to the fact that human language is often very abigous about the meaning of sentences. Scope ambitguity is a big problem in language.

What always seemed weired to me was that it was never proven to me that all in my classes that all of the different possible relations between universal and existential quantifers could be expressed by lining them up in a row with different permutations. Turns out this was wrong like I thought. Some Swedish dude came up with "Independance Friendly Logic", which is a not too shabby notation for describing all the possible relations. You can use skolemization, but thats lame. I personally diagram the relations using circles and dots. Universal quantification are circles, existential quantification is dots. If a dot is in a circle then it depends on that universally quantified variable. In this system FOL is just all the diagrams which have concentric circles.

but ill nitpick on all this a little more later