Analyzing Edmund's speech "Why 'bastard'? Wherefore 'base'?" show how this is a distingushing mark of Edmund in King Lear.
Edmund's speech shows that anger at being stigmatized for being illegitimate and denied an inheritance—the "land" that will come to Edgar—is a distinguishing mark of Edmund's. Another distinguishing mark of Edmund's that emerges from the speech is his sense of agency: Edmund is not about to take his illegitimacy lying down. He sees it as unfair and arbitrary, a manmade law that he rejects. As he says, he had no say in the accident of his parents not being married when he was conceived and born, and he doesn't see why he should suffer for it.
Throughout this speech, Shakespeare has Edmund playing on words that mean illegitimate—when he says "thou, nature, art my goddess," he is playing on the words "nature/natural" to describe an illegitimate child. He says he worships this nature because it is not manmade and arbitrary like the "rule" that says only a legitimate child can inherit. Second, he plays on related words such as base, baseness, and bastard that are used to describe illegitimate children and asks what meaning they have in reality: how is he is different, after all, from his legitimate brother when he has a "mind as generous, a shape as true" as Edgar's? This shows yet another distinguishing mark of Edmund's: he deceives himself, for he is not at all as "generous" or "true" as his brother. Yet as so often in Shakespeare, the case is complicated: the implication is that it may well be the manmade artifice of "illegitimacy" that has warped Edmund so that he comes to resemble his "base" label.
In the opening scene, Edmond is present when Gloucester, talking to Kent, makes crude references to his conception and birth. He publicly reacts as expected.
In scene 2, in a soliloquy, Edmond tells us how he really feels. He appeals to Nature as his goddess. Exactly what does he mean? When most people think of nature, they think of positive nature, flowers, trees, furry little animals, etc. Edmond is not appealing to those forces of nature. Rather he is talking to the negative power of nature.
In his world, he is marginalized due to circumstances beyond his control. That his mother was not married to Gloucester is not his fault yet society labels him a bastard.
When he compares himself to his brother, he discovers that physically and mentally they are alike. It is not as if he was born with a B branded on his forehead, yet in his world it exits none the less.
He takes comfort in the fact that at least there was fun in his illegitimate conception as compared to the duty performed in legitimate conception.
He justifies his betrayal of his brother and his father.
"Now gods, stand up for bastards!"
Your question refers to Edmund's opening speech in act1 sc.2 of King Lear. This solo speech holds the crucial key to Edmund's character, his advocacy of 'Nature' by virtue of being a natural child, a bastard, of Gloucester. Because of being born out of wedlock, he is illegitimate and marginalised as per social convention. Gloucester's legitimate son, Edgar, is due to enjoy all property rights, and Edmund is destined to be neglected and deprived.
In this speech, Edmund raises questions against social stgmatisation of the bastards, for no faults of theirs. He is as well-shaped in body and mind as all other legitimates in the world, and yet he is set aside as base:
Why bastard? Wherefore base?
When my dimensions are as well compact,
My mind as generous, and my shape as true,
As honest madam's issue? Why brand they us
With base? With baseness? Bastardy? Base? Base?
Edmund banters at Edgar's 'legitimacy', and ventilates his grudging anguish for discriminations against the so-called bastards. Since he is a 'natural' child, he vows to go by 'nature', to go by the principle of the 'survival of the fittest', not bothering for others, but only bothering to promote self-interest.