Disjunction introduction rule

The disjunction introduction rule in natural deduction permits either inferenceHere $A$ and $B$ are propositions, and $\vee$ denotes logical disjunction. Establishing either disjunct is sufficient to establish the compound proposition.

The truth table explains soundness in classical propositional logic: every assignment making the premise true makes the conclusion true. For example, a proof of $x=0$ also proves $x=0\vee x=1$, without requiring a proof of $x=1$.