= Logical implication
{title2=$\to$}
Logical implication forms the <proposition> $A\to B$, read “if $A$, then $B$”. In <classical propositional logic>, it is false exactly when $A$ is true and $B$ is false. Equivalently, it has the same <truth table> as $\neg A\vee B$, using <logical negation> and <logical disjunction>. A false hypothesis makes the <logical implication> true regardless of its conclusion; thus it asserts neither the hypothesis nor a causal connection.
For example, for every <natural number> $n$, <divisibility> by four implies that $n$ is an <even number>: $n=4k=2(2k)$. At $n=8$, both statements are true. At $n=6$, the hypothesis is false and the conclusion true; at $n=3$, both are false. The only forbidden case is a multiple of four that is not an <even number>.
In <Intuitionistic propositional logic>, a proof of $A\to B$ consists of a construction transforming any proof of $A$ into a proof of $B$. The <implication introduction rule> proves $B$ under a temporary assumption $A$; the <modus ponens rule> applies the result when $A$ is available. The classical identification with $\neg A\vee B$ is not generally intuitionistically valid.
Back to article page
