The idea of ‘paradox' first emerged with the Eleatic philosopher Zeno of Elea, following in the steps of his master the philosopher Parmenides, and it caused an unprecedented revolution in thought, for -as Zeno himself claimes- it lead to the birth of the art of dialectic. Even though this dialectical method and binary logic -later adopted by Aristotle- became a widespread standard in philosophical methodology, it still lacks firm logical foundations, and recently many began realizing this, even after the prominence of Aristotlean logic continued for 2000 years.
Aristotlean logic allows for paradoxicality because it is binary and dualistic, that is; it views different categories as irreconcilable, and pairs or opposites (such as: good and evil, night and day, life and death) not as emerging from one source, but as contradictory polar opposites or thesis and antithesis. Heraclitus, as many philosophers, however, was able to see the oneness beyond the duality, which is the origin of all things and the harmony that holistically unifies everything. We can thus say that Heraclitus perspective was universal while Aristotle's perspective was partitive, and particular.
In this essay I argue that that only in a particular system of philosophy (such as Aristotle's) can the idea of paradox exist, for the famous law of non-contradiction is not a law associated with pure logic, but rather a law constrained within the domain of Aristotelean or binary logic. This is evidenced by the fact that many Pre-Socratic philosophers, such as Heraclitus, used a logic which didn't regard opposites as contradictory, but rather saw them as harmonious pairs which complete each other and which in the end are emerging from a oneness. Some sources even claim that Heraclitus outright denied the law of non-contradiction.
Heraclitus states in his Fragments that: “This principle (logos, the hidden harmony behind all change) bound opposites together in a unified tension, which is like that of a lyre, where a stable harmonious sound emerges from the tension of the opposing forces that arise from the bow bound together by the string." 
So now we are faced two ‘opposite' positions. Firstly, that of Heraclitus, who claims that opposites are reconcilable, and if this position is true this consequently renders the concept of ‘paradox' futile. Secondly, that of Aristotle, who claims that opposites are irreconcilable, and if his position is true then ‘paradox' is possible.
In this essay I will use a purely logical analysis to elucidate the differences between the Heracletean position and the Aristotelean, and how they're associated with two different modes of thinking, the first being oneness and the second duality, and I will show how oneness has prominence over duality. I will also investigate whether paradoxicality is possible. Or will I -by virtue of this attempt of mine to prove the superiority of Heraclitus's position over Aristotle's, or one position over the other- be falling into another paradox?