Implication and Equivalence

If, Then, and Exactly When

The implication (conditional) $p \to q$ — "if $p$ then $q$" — is the connective at the heart of every theorem. Its truth table has one subtle feature: $p \to q$ is false in exactly one case, when $p$ is true but $q$ is false. In every other case it is true — including, crucially, when $p$ is false, where the implication is vacuously true regardless of $q$. The premise $p$ is called the hypothesis and $q$ the conclusion.

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

• Converse $q \to p$ — swap hypothesis and conclusion.
• Inverse $\neg p \to \neg q$ — negate both.
• Contrapositive $\neg q \to \neg p$ — negate and swap.

The essential fact: an implication is logically equivalent to its contrapositive, but not to its converse or inverse. "If it is raining, the ground is wet" and "if the ground is not wet, it is not raining" say the same thing; but "if the ground is wet, it is raining" (converse) does not follow. Confusing an implication with its converse is one of the most common reasoning errors.

The biconditional $p \leftrightarrow q$ — "$p$ if and only if $q$" — asserts that $p$ and $q$ have the same truth value; it is true exactly when both $p \to q$ and its converse $q \to p$ hold. This is the "iff" of definitions and equivalences: $p \leftrightarrow q$ means each implies the other, so they stand or fall together.

Common pitfall: treating an implication as equivalent to its converse (affirming the converse). $p \to q$ does not give you $q \to p$. It is equivalent to its contrapositive $\neg q \to \neg p$. Also, when $p$ is false, $p \to q$ is vacuously true — a false hypothesis never makes an implication false; only a true hypothesis with a false conclusion does.