Biconditional
A biconditional statement "P if and only if Q" is true when both P and Q have the same truth value, meaning each implies the other.
Formula
P \leftrightarrow Q \text{ (P if and only if Q)}
Definition
A biconditional says "P if and only if Q," meaning both P and Q are always either both true or both false: one happens exactly when the other does. Formally, $P \leftrightarrow Q$ is true when $P$ and $Q$ have the same truth value, equivalent to $(P \to Q) \wedge (Q \to P)$; mathematical definitions and characterization theorems are typically stated as biconditionals. In model theory, $P \leftrightarrow Q$ being a theorem means $P$ and $Q$ are true in exactly the same models, and in category theory, isomorphism ($A \cong B$) is the categorical analogue of biconditional equivalence.
Example
"A shape is a square if and only if it has four equal sides and four right angles": both conditions always go together. "$n$ is even $\leftrightarrow$ $n^2$ is even" requires proving both directions: "even $n$ implies $n^2$ even" and "$n^2$ even implies $n$ even" (often via contrapositive). The completeness of a metric space can be characterized by multiple equivalent properties: Cauchy sequences converge iff every closed ball is complete iff the Baire category theorem holds, an "iff" chain revealing a web of equivalent formulations.
Key Insight
"If and only if" (often abbreviated "iff") means the relationship goes both ways, the strongest kind of "if-then" connection, and every mathematical definition is implicitly a biconditional, understanding this makes definitions more useful. Discovering that two seemingly different properties are logically equivalent is among the deepest results in mathematics, often revealing hidden unity in apparently different areas of the subject.