**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*.