Tautology — Definition, Formula & Examples

A tautology is a compound logical statement that is true in every possible case, no matter whether its individual parts are true or false. For example, "It will rain today or it will not rain today" is always true.

In propositional logic, a tautology is a compound proposition whose truth value is TRUE under every possible assignment of truth values to its component propositional variables. A tautology is denoted by the symbol $\top$ and is logically valid by form alone.

How It Works

To determine whether a statement is a tautology, you build a truth table listing every combination of truth values for its variables. If the final column shows TRUE in every row, the statement is a tautology. A statement with

n

variables requires

2^n

rows. If even one row evaluates to FALSE, the statement is not a tautology.

Worked Example

Problem: Use a truth table to prove that the statement p ∨ ¬p ("p or not p") is a tautology.

Step 1: List all possible truth values for p. Since there is one variable, there are two rows.

p \in \{T, F\}

Step 2: Compute ¬p for each row. When p = T, ¬p = F. When p = F, ¬p = T.

\neg T = F, \quad \neg F = T

Step 3: Evaluate p ∨ ¬p for each row using the disjunction rule (true when at least one operand is true).

T \lor F = T, \quad F \lor T = T

Answer: Every row yields TRUE, so p ∨ ¬p is a tautology.

Why It Matters

Recognizing tautologies lets you identify steps in a proof that are logically guaranteed, which is essential in geometry proofs and discrete math. In computer science, tautology checking underlies circuit simplification and automated theorem proving.

Common Mistakes

Mistake: Confusing a tautology with a statement that happens to be true. For instance, calling "2 + 3 = 5" a tautology.

Correction: A tautology is true because of its logical structure, not because of specific facts. "2 + 3 = 5" is a true arithmetic fact, but it is not a compound logical form that is true under all truth-value assignments.