Aristotle’s definition of the Principle, which he extolled as: ‘The Surest Principle..Itself not an assumption…but a beginning for all other axioms‘. The one Principle: ‘Which one must have to understand anything whatsoever’, is as follows:
‘It is impossible for the same thing at the same time to belong and not to belong to the same thing and in the same respect.’
Here is Immanuel Kant, 2000 years after Aristotle:
‘The proposition: ‘No subject can have a predicate that contradicts it’ is called the Principle of Contradiction..and we must hold [this Principle] to be the universal and fully sufficient principle of all analytic cognition..and the fact that no cognition can be at variance with this Principle without nullifying itself, constitutes the Principle Sine Qua Non..‘
Aristotle’s founding of Classical Logic began as an extension of its truth. And Philosophy and Logic and Mathematics since his day have had to make peace with its diktats.
Aristotle’s Principle is the pillar behind the most celebrated claims of High Intellection, of Rationality itself. If you question it you question everything.