A particularly good quote...

...from Nietzsche:

Not being able to contradict is proof of an incapacity, not of "truth."
-The Will to Power, #515, March-June 1888

In other words, whatever is non-contradictory merely is lacking the will or the strength to overcome its simplicity (a Hegelian term meaning its identity with itself or self-equivalence) and encounter or reside in contradiction. As soon as it overcomes this, it will be powerful, for it will have to possess not only the level of strength necessary for itself to exist (as it does in merely being non-contradictory) but also the level of strength necessary for itself to hold itself together despite its contradiction. Now, this contradictory thing is not true by virtue of this strength either, but it certainly can assert itself and control the merely non-contradictory due to it. This is why--as Freud, a Nietzschian, notes--the most domineering and efficient forces throughout history seem to articulate themselves contradictorily, "unreasonably:" think of Fascism ("in order to be a people, we must become solely a state!"), colonialism ("in order to live better, you must allow us to destroy your way of life!"), sometimes democracy ("in order to have freedom, you must always comport yourself as a free person!").
If we understand all this, Nietzsche's critiques of classical logic, like this late fragment, become clearer and much more poignant:

If, according to Aristotle, the law of contradiction is the most certain of all principles, if it is the ultimate and most basic, upon which every demonstrative proof rests, if the principle of every axiom lies in it; then one should consider all the more rigorously what presuppositions already lie at bottom of it. Either it asserts something about actuality, about being, as if one already knew this from another source; that is, as if opposite attributes [of a thing under consideration as to whether it is contradictory] could not be ascribed to it [because contradiction everywhere is, for some reason, impossible]. Or the proposition means: opposite attributes should not be ascribed to it. In that case, logic would be an imperative, not to know the true, but to posit and arrange a world that shall be be called true by us.
-The Will to Power, #516, Spring-Fall 1888

Nietzsche here basically outlines the superego: logic, rationality, the non-contradictory--all this is merely the result of an imperative to-be-logical, -rational, etc.... or else! Freud saw it the same way. In The Ego and the Id the quintessential statement of the superego to the ego is specified by Freud as the following: "You ought to be like this (like your father)" (Pg. 30).