P and ~P are inherently associated by a logical difference and neither of the two can be dismissed without dismissing the other one.
Law of Excluded Middle (LoEM) states that for any proposition P either that proposition P is true, or its negation ~P is true.
In other words, LoEM holds that either (1) P is true which means that ~P is dismissible, or (2) ~P is true and P is dismissible.
As such, LoEM postulates an absolute difference between a proposition “P” and its negation “not P”.
But what is “not P”? It’s simply that which is “different from P”: “not to be P” implies “to be different from P”, it does not imply an absolute difference from P so that either “P” or “not P” should be dismissible. Not at all. An operation of negation does not bring a right to dismiss either the negation or that which is being negated.
“P” and “not P” are simply different. Neither can be dismissed.
Moreover, “not P” is part of “P” itself. What makes “P” P is associated to the content of “not P”.
As we know, “not P” is the negation of “P”; therefore, to dismiss “not P” is to dismiss “P” itself.
To dismiss the one is to dismiss the other. To dismiss the other is to dismiss the one.
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