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For Aristotle, the fundamental law of truth is A = A, that is, things are what they are, and they aren't anything else. This is also known as the law of identity. On this basis, he claims that the world is knowable, because things in the world have an identity. Yet there is another tradition of thought which has also been very influential, and that is that there is no essence to things. Instead, everything is in flux. A = A and A = not-A. One person who realized this possibility was Heraclitus. To reveal the contradictions in reality, he formulated many riddles.
Even though Heraclitus believed that there was a reason for everything, a logos, he also said that people would not know the logos if they saw it, and they would mistake it for something else.
Other thinkers have embraced contradiction as a fundamental fact, and completely abandoned any simple logos based on a law of identity. In the poem Song of Myself, Walt Whitman suggests a person cannot be defined at all. He says “Do I contradict myself? / Very well then I contradict myself. / (I am large, I contain multitudes)” (Whitman 1891 : 78). Joss Wheadon conjures Whitman in this 2013 commencement (graduation) address to the students of Wesleyan College.
Hegelian and Marxian dialectic is an attempt to show that contradictions lead to changes, and that reality itself is an ongoing process of working through contradictions. See for instance Dialectics for Kids.
It is worth noting that most of the things one hears in university lectures are presented as if they were true because of the law of identity. That is to say, lecturers usually present knowledge as a set of propositions as absolute facts. Most of the things that are actually worth knowing about the world–whether that is the structure of the universe, the evolution of life, the formation of human societies, or any contemporary social problem–can only be understood if one embraces the fact that they are defined by contradictions and cannot be captured in a simple statement of fact.
Whitman, Walt. 1891 . “Song of Myself.” In Leaves of Grass, 1891-92 edition. Philadelphia: David McKay, Publisher. http://whitmanarchive.org/published/LG/1891/.
Horn, Laurence R. 2014. “Contradiction.” In The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), edited by Edward N. Zalta. http://plato.stanford.edu/archives/spr2014/entries/contradiction/.