logical foundations of logic

Tonight I am wondering what logic is made of? In other words, if logic is a system of relatedness (through inference or deduction or whatchawannacallit), what does it relate? "Propositions" seems to be a popular answer to this question, but I don't find it particularly helpful, since it is not clear what a proposition is. Is it a kind of statement? Is it whatever is left of a statement when intentionality is subtracted out? Is it something that is done to statements?

I've got Aristotle laying it out two different ways, in the first chapter of Prior Analytics. He says a proposition is a "statement," sure enough, "affirming or denying something of something," but when he gets down to dividing propositions up into kinds, he says a demonstrative proposition is an "assumption of one of two contradictories," rather than saying that it simply is "one of two contradictories" and that a dialectical proposition is a "choice." I think either of these lines could be assimilated to other, just by pointing out that anything we do as logical practicioners we do through statements. I.e., nothing is assumed unless someone makes a statement, to himself at least, to the effect that this one of the two contradictories is the case. Or on the other hand, a proposition can be called a statement even if it isn't one, because what it does it does in a statement.