How does the law of contradiction work?

The law of contradiction, often referred to as the law of noncontradiction, is a fundamental principle of logic stating that it is impossible for a proposition and its negation to both be true at the same time and in the same sense.12

Symbolically, for any proposition p, the law is expressed as:
$$~(p·~p)$$
In this formula, the symbol $~$ represents "not" and $·$ represents "and," meaning it is impossible for both p and not-p to be true simultaneously.2

Historically, Aristotle identified this principle as an axiom of thought. It dictates that a thing cannot both be and not be of a specified kind or in a specified manner at the same time. While traditional logicians viewed this as a foundational law, some modern systems, such as many-valued logics or those developed by Jan Łukasiewicz, have explored frameworks where this law does not hold, often to account for propositions that are neither strictly true nor false.23