The Principle of Contradiction. In delightful irony, also called ‘ The Principle of Non-Contradiction’.
The Principal Principle of Logico-Mathematical model [in fact of all ‘Analytical Cognition’, to use Immanuel Kant’s expansive phrase] is the Principle of Contradiction.
When your high-school teacher asked you to ‘prove’ something in math-class, he was asking you to show that it all held together nicely. In other words, that you were not contradicting yourself somewhere in the fine-print.
Aristotle’s defense of this pivotal principle is the first formalized application of the Self-Eating Expression in the Western Tradition that I am aware of.
He called it: ‘The First Principle of Rational Knowledge’. It is ‘Aristotle’s Principle’; for it was he who had the courage of conviction to place it on center stage.
This dominant Principle [Virodha in Sanskrit, literally: ‘conflicted, to be countered’] had been known for centuries before Aristotle. But no philosopher before him made as brilliant, forceful and convincing a case for what, in his words: ‘one must have to understand anything whatsoever.’
Two thousand later, Immanuel Kant, who defined the domain of Academic Philosophy for two hundred years, in his Critique of Pure Reason called it the: ‘Principle Sine Qua Non-the universal and fully sufficient principle of all analytic cognition’.
Aristotle’s Principle is the pillar behind the most celebrated claims of High Intellection, of Rationality itself. If you question it you question everything.
Aristotle’s founding of Classical Logic began as an extension of its truth. And Philosophy and Logic, Language and Mathematics, indeed every subject claiming to be rational has had to make peace with its diktats.
In short, this is a pretty important principle.