Does anything follow from a contradiction?

I suspect that logically, this does not hold, and only works because of a certain definition of implication within modern/symbolic logic is adopted. If we set up the accepted truth table for material implication:

p q p-->q
T T T
T F F
F F T
F T T

(for more on material implication)

we see that in the case of a & ~a --> b, while a & ~a is F, regardless of whether b is T or F, the implication is T.

(This is already covered in Paedo Socrates' post, but I'm typing out the justification for my own reflection and to keep it simple for the reader.)

Now if this statement is taken rather to be a predicator of human behavior or "psychology," perhaps there is some truth to it. That is to say, if someone consciously rejects the principle of non-contradiction and "reasons" accordingly, then it he will accept as true whatever he wants to accept as true, because he is no longer guided by logic. Truth is arbitrarily determined by the subject, and has no relation to reality.

So the next question would then be: is material implication "strong" enough to explain both logical and real necessity? Or is it merely a devised rule of "calculating" or "reasoning" which does not help us think logically but impedes us from doing so?