Truth Tables

Precise definition

A truth table lists the truth value of a compound proposition for every assignment of its atomic propositions. With $n$ distinct atomic propositions there are $2^n$ rows. A tautology is true on every row, a contradiction false on every row, and a contingency true on some rows and false on others.

Notation and mathematical language

Standard precedence is often $\neg$, then $\land$, then $\lor$, then $\to$, then $\leftrightarrow$, but parentheses are safest. The implication $P\to Q$ is false only on the row $P=T,Q=F$. Two formulas are equivalent when their final columns agree on all rows.

Conceptual picture

Intermediate columns turn a complicated sentence into small, checkable operations. A consistent row order—such as blocks of $T$ and $F$ whose lengths halve from left to right—ensures no assignment is omitted or duplicated.

Fully worked example

Interpretation and application

Truth tables verify small digital circuits, access rules, and program conditions. Their row count grows exponentially, so algebraic laws, proof, or automated methods become preferable for many variables; the table remains an exact reference for the connectives.

Common mistakes