Implication and Equivalence

Implication and Truth

The implication (conditional) $p \to q$, read as "if $p$ then $q$," is the core connective of mathematical theorems. Here, $p$ is the hypothesis and $q$ is the conclusion.

An implication is false in exactly one case: when $p$ is true and $q$ is false. In all other cases it is true.

When $p$ is false, the implication is vacuously true regardless of $q$. A false premise never makes an implication false; only a true hypothesis paired with a false conclusion does.

Derived Statements and Equivalence

From any implication $p \to q$, we form three related statements:

Common pitfall: Confusing an implication with its converse ($q \to p$) is a major error. $p \to q$ does not guarantee $q \to p$.

The biconditional $p \leftrightarrow q$, meaning "$p$ if and only if $q$" (iff), asserts that both $p$ and $q$ share the exact same truth value. It is true precisely when both $p \to q$ and its converse $q \to p$ hold simultaneously.