Can a discrete variable take on multiple values, and thus truth values, simultaneously in the mathematical proof below or elsewhere?

Question

Excerpt from a mathematical proof that serves to prove by contradiction, using the Well Ordered Principle, that predicate P(n) is true for all nonzero integers n.

If n belongs to set C i.e. n >= c, then P(n) is false.

This implies the contrapositive,

If P(n) is true, n has to not be a part of set C i.e n < c.

I know that the above implication is valid but why so? How should I know that n cannot take on 2 values simultaneously -- n being both >= c and < c, thus P(n) being both true and false simultaneously? (Yes I am thinking Dialetheism, maybe even quantum logic but I know nothing other than that quantum logic defies certain aspects of propositional logic.) What is the basis behind a predicate (or any proposition) being either true or false? Or would the simultaneity be possible just that I wouldn't be dealing with a proposition to begin with?

Also, if you don't mind, please do let me know if you feel my understanding of concepts and terminologies (i.e. discrete, predicate, contrapositive, assume) are erroneous. I am extremely new to discrete mathematics and could really use the extra guidance.

Thank you.

Answer 1

Usual mathematics uses Classical logic; thus a statement cannot be both TRUE and FALSE simultaneously.

More specifically, for numbers $n,c$ :

either $n ≥ c$ or $n < c$ (and not both),

because $n ≥ c$ is by definition : $¬ (n < c)$.