Logic is a tool for drawing out new
facts from existing ones. A
logical argument can be judged independently of its content--that is, a logical statement can say "If A, then B" where "A" and "B" can be replaced by anything at all. A
logical argument is rather like a
computer program or
mathematical function; it takes
input, operates on it, and produces
output. The input of logic--called an
axiom or
premise--determines the output, called a
conclusion.
Just as you can apply one function to the results of another, you can use the conclusion of one argument as the premise for another. However, this chain of logical arguments has to have a beginning--a starting premise that is not itself the conclusion of any other argument. By definition, this premise is illogical. Thus, we can say not only that everyone is illogical, but that illogic is the necessary foundation for logic.
When engaging in philosophical discussions of the type that are popular among some people on Everything, it is useful to apply logic to each person's arguments. If their logic holds up, you can fairly quickly come back to the illogical judgments that serve as someone's premises. Of course, these premises are often rooted so deeply in someone's nature that they may be uncomfortable about even discussing them, much less having them challenged. However, since these deep premises are the roots of our beliefs and ideas, it is absolutely vital that they be found and examined.
To Yam: Assertions can still be defined as "logical" or "illogical," although that doesn't necessarily say a lot about their truthfulness.
And I agree that assertions should be examined for the evidence in their favor, but what criteria do we apply to determine what constitutes evidence? The variations in rules of evidence are themselves axioms that are usually not based in rationality but in a commitment to a particular world-view.