Reply to Comment on " on the Original Proof by Reductio Ad Absurdum of the Hohenberg-kohn Theorem for Many-electron Coulomb Systems "

Any mathematical proof is a game. As a game, it is based on a definite set of rules of logic reasoning which altogether constitutes the subject of logic. One of the simplest rules of the theory of logic is a denial of the truth of a given proposition that is expressed as a sentence. The truth of a proposition has to be denied by asserting its negation. Assuming, for example, that the proposition p := {Everyone is wise} (see Ref. [1]) is false, I assert instead s := {Everyone is unwise}. s seems to be a negation of p. However, s is definitively not the logical negation of p that in logic is defined as ‘not p’ or p := {the proposition that is true when p is false and false when p is true} [1,2]. Therefore, s is false if p is true, but not certainly true if p is false, and hence, s 6= p. The standard method of asserting the negation of a simple sentence consists in attaching the word ‘not’ to the main verb of the sentence. However, the assertion of the negation of a compound proposition (sentence) is not that trivial. Consider the set of negations within the context of the Hohenberg-Kohn theorem [3] (see also Ref. [4] and Ref. [5] for the notations):