Definition 3 (Tautology). A tautology is a proposition that is always true for any of its variables.
Definition 4 (Contradiction). A contradiction is a proposition that is always false for any value of its variables.
Theorem 4.4. If proposition P is a tautology then ~P is a contradiction, and conversely.
Proof. If P is a tautology, then all elements of its truth table are false (by Definition 1), so all elements of the truth table for ~P are false, therefore ~P is a contradiction (by Definition 2). ⬜
Example 2. "It is raining or it is not raining" is a tautology, but "it is not raining and it is raining" is a contradiction.
Remark 2. Example 1 used De Morgan's Law ~(p ∨ q) ≡ ~p ∧ ~q.