How can negation change the truth value of a proposition?

In formal logic, specifically within the propositional calculus, negation is a truth-functional operator that acts upon a proposition to invert its truth value.

Every proposition is assumed to be either true or false, but never both. The negation operator, symbolized as $~$, functions as follows:1

• If a proposition $p$ is true, its negation $~p$ ("not $p$") is false.1
• If a proposition $p$ is false, its negation $~p$ is true.1

Because negation is a truth-functional operator, the truth value of the resulting compound proposition is determined entirely by the truth value of the original argument. This systematic inversion ensures that the negation of a statement always holds the opposite truth value of the original, providing a foundational mechanism for logical reasoning and the construction of truth tables.12