If and only if symbol (↔)

How to read it

OpenStax’s Contemporary Mathematics calls the biconditional a two-way contract and shows that p ↔ q is the same as (p → q) ∧ (q → p): an implication in each direction. Oscar Levin’s Discrete Mathematics puts it as two parts, an implication and its converse, the “if” and the “only if”. The words are often shortened to “iff”: forall x: Calgary calls it the snappier word, and NIST’s Digital Library of Mathematical Functions lists “iff” for “if and only if” beside ⟺ for “is equivalent to”.

Also searched as: ↔, ⇔, biconditional symbol, double arrow in logic, equivalence symbol, iff.

↔, ⇔ or ≡

forall x: Calgary notes that books which write the conditional with an arrow use ↔ for the biconditional, and books which use the horseshoe ⊃ use the triple bar ≡. Mathematics texts also write ⇔ or the long ⟺. The triple bar has other jobs too: NIST’s Digital Library of Mathematical Functions lists ≡ both for “equals by definition” and for equivalence in modular arithmetic, so the context decides.