Truth Tables

A truth table lists every possible assignment of true/false to the atomic sentences and computes the formula’s value in each. With $n$ distinct atoms there are $2^n$ rows — two for one atom, four for two, eight for three.

The tool below builds the full table for any formula and classifies it. Try $P \lor \lnot P$, then $P \land \lnot P$, then $(P \rightarrow Q) \leftrightarrow (\lnot P \lor Q)$.

Three kinds of formula

Reading the final column tells you which kind of formula you have:

• A tautology is true in every row — a logical truth, e.g. $P \lor \lnot P$.
• A contradiction is false in every row, e.g. $P \land \lnot P$.
• A contingent formula is true in some rows and false in others — most formulas are contingent.

Logical equivalence

Two formulas are logically equivalent when they have the same truth value in every row. A clean test: $X$ and $Y$ are equivalent exactly when $X \leftrightarrow Y$ is a tautology. The default formula above shows one famous equivalence — $P \rightarrow Q$ is equivalent to $\lnot P \lor Q$, so the biconditional between them comes out true in every row.

Some equivalences worth knowing (each is a tautology when written as a biconditional):

• Double negation: $\lnot\lnot P \leftrightarrow P$
• De Morgan: $\lnot(P \land Q) \leftrightarrow (\lnot P \lor \lnot Q)$ and $\lnot(P \lor Q) \leftrightarrow (\lnot P \land \lnot Q)$
• Contraposition: $(P \rightarrow Q) \leftrightarrow (\lnot Q \rightarrow \lnot P)$

Check any of these by typing it above — a true-in-every-row column confirms it.

Why this matters

Truth tables are a decision procedure: a mechanical, always-terminating method that answers “is this a logical truth?” and “are these equivalent?” for sentence logic. That is exactly the kind of guarantee that makes a logic formal — no cleverness required, just fill in the table.

Takeaways

• A formula over $n$ atoms has a $2^n$-row truth table.
• Tautology (always true), contradiction (always false), contingent (mixed).
• $X$ and $Y$ are logically equivalent iff $X \leftrightarrow Y$ is a tautology.
• The table is a complete decision procedure for sentence logic.

References

• Paul Teller, A Modern Formal Logic Primer, Prentice Hall (1989) — free at tellerprimer.ucdavis.edu. Curriculum follows the primer; explanations and examples here are original.