Truth Tables

Evaluate compound propositions; spot tautologies and contradictions.

The idea

A truth table records the complete behaviour of a compound proposition: its truth value under every possible assignment of truth values to its variables.

A proposition is a statement that is either true or false, written $T$ or $F$. Compound propositions are assembled from simpler ones by the connectives $\lnot p$ (not), $p \land q$ (and), $p \lor q$ (or), $p \to q$ (if–then) and $p \leftrightarrow q$ (if and only if). The truth value of a compound depends only on the truth values of its parts, never on what the parts are about, so a table can capture it in full.

The table has one row for each assignment of $T$ and $F$ to the variables and one column for each piece of the expression; we fill the columns from the inside outward until the last holds the value of the whole. With $n$ variables there are $2^{n}$ rows, so the work is finite and purely mechanical. The figure carries this out for the compound $\lnot p \lor q$.

The finished column classifies the proposition: true in every row, a tautology; false in every row, a contradiction; true in some rows and false in others, contingent.

Ways to work on it

• Walkthrough. Connectives, the conditional's one false row, and classification.
• Practice. Evaluate a compound at a given assignment.
• Hardest. Classify a statement as tautology, contradiction, or contingent.

Not sure where to start? Take the ten-question placement test.