Tautologies, Contradictions and Contingent Statements

A full truth table (Building a Truth Table) shows every row a formula can be in. Looking at the main column, three things can happen, and each has a name.

Key idea

A tautology is true in every row. A self-contradiction is false in every row. A contingent statement is true in at least one row and false in at least one row.

Why it works

Tautologies are true because of their form alone. $p \lor \neg p$ is true whatever $p$ says, since one of the two disjuncts must be true. Self-contradictions are false by form alone. Contingent statements depend on the facts: whether $p \to q$ is true depends on what $p$ and $q$ say about the world. Logic cannot settle a contingent statement; it can only show which rows make it true.

A tautology: the main column is T in every row.

Worked examples

Worked example: A tautology

Classify $(p \land q) \to p$.

The only way to make a conditional F is a T antecedent with an F consequent.
If $p \land q$ is T, then $p$ is T, so the consequent cannot be F.
So no row is F.

Answer: Tautology

Worked example: A self-contradiction

Classify $(p \to q) \land (p \land \neg q)$.

$p \land \neg q$ is T only in the row $p$ T, $q$ F.
In that row $p \to q$ is F, so the conjunction is F.
In every other row $p \land \neg q$ is F, so the conjunction is F.

Answer: Self-contradiction

Worked example: Contingent

Classify $(p \lor q) \to p$.

Row $p$ T, $q$ T: T → T is T.
Row $p$ F, $q$ T: T → F is F.
It has a T and an F.

Answer: Contingent

Common mistake

Calling a formula a tautology after checking a few rows. One F anywhere means it is not a tautology, and that row might be the last one you check. Fill the whole column. A shortcut that is safe: to show a formula is not a tautology, one F row is enough; to show it is one, every row must be checked (or you need an argument that no row can be F, as in the first example).

Tautologies matter beyond this lesson. An argument is valid exactly when the conditional "if all the premises, then the conclusion" is a tautology, so every valid argument hides a tautology inside it. And a formula is a self-contradiction exactly when its negation is a tautology, so checking one kind is the same work as checking the other.

Worked example: Tautology from an argument

Modus ponens has premises $p \to q$ and $p$ and conclusion $q$. Which formula is a tautology because of that?

Join the premises with $\land$ and make them the antecedent.
The conclusion is the consequent.

Answer: $((p \to q) \land p) \to q$, shown in the table above.

All T: tautology.
All F: self-contradiction.
Some of each: contingent.
The negation of a tautology is a self-contradiction, and the reverse.

Tautologies, Contradictions and Contingent Statements

Worked examples

Practice

Builds on

Leads to

Sources and further reading