Tautology, Contradiction, and Contingency

Definition 12.16 (Tautology). A statement is a tautology, denoted $\mathbb T$, if its truth value is always $T$, no matter the truth values of its component statements.

Definition 12.17 (Contradiction). A statement is a contradiction, denoted $\mathbb F$, if its truth value is always $F$.

Definition 12.18 (Contingency). A statement that is neither a tautology nor a contradiction is a contingency -- its final truth-table column genuinely mixes $T$ and $F$.

Observations.

1. For a tautology, every entry in the formula's final column is $T$.
2. For a contradiction, every entry in the final column is $F$.
3. The negation of a tautology is a contradiction, and the negation of a contradiction is a tautology.
4. The disjunction of a statement with its own negation is always a tautology, and the conjunction of a statement with its own negation is always a contradiction: $p\vee\neg p\equiv\mathbb T$ and $p\wedge\neg p\equiv\mathbb F$.

Worked patterns.

The last column is all $T$, so $p\vee\neg p$ is a tautology (the Law of Excluded Middle).

The last column is all $F$, so $p\wedge\neg p$ is a contradiction. …