Overview

In formal logic and mathematics the phrase "if and only if" — often abbreviated as iff — names a biconditional connection between two statements. The biconditional is true exactly when both connected statements share the same truth value: either both true or both false. It is the standard way to express that one assertion is both necessary and sufficient for another.

Definition and notation

Given two propositions A and B, the compound statement "A if and only if B" can be written A if and only if B, A ↔ B, or A ⟺ B. In informal mathematical writing the abbreviation "iff" is widely used. In symbolic accounts the biconditional is equivalent to the conjunction of two conditionals: (A → B) ∧ (B → A). The connective plays the role of an equivalence relation between formulas in many formal systems.

Semantics and truth conditions

The biconditional has a simple truth table: it is true when both A and B are true, and also true when both A and B are false; it is false when exactly one of A or B is true. Because it requires two implications to hold, interpreting "if and only if" asserts both that A is sufficient for B and that A is necessary for B. In propositional logic the biconditional is sometimes presented as the logical XNOR operation from Boolean algebra.

Uses and examples

  • Definition in mathematics: Many mathematical definitions employ "iff" to indicate exact characterizations. For example, a number is even iff it equals twice an integer.
  • Proofs: Biconditionals are commonly established by proving both directions separately: first show A → B, then show B → A.
  • Computer science and engineering: The logical behaviour corresponds to the XNOR gate in digital circuits and to equality checks in Boolean algebra. See XNOR and equivalence for related concepts.

It is important to distinguish the biconditional from a single conditional ("if") and from informal causal language. A single implication A → B asserts only sufficiency (if A then B) but not necessity. Saying "A iff B" asserts both directions. In predicate logic the biconditional extends to formulas with quantifiers, but care is needed: quantifiers interact with implication in ways that sometimes require reformulation of statements before applying an "iff".

Notation, style, and historical notes

Notation varies: English texts use "if and only if" or "iff", while symbols such as ↔ or ⟺ appear in formal contexts. The abbreviation "iff" is conventional in mathematics but may be avoided in more casual exposition in favor of the full phrase. The biconditional concept arose alongside the development of modern symbolic logic and Boolean algebra and remains a central connective for expressing equivalence and precise definitions.

For further background see entries on logic and on mathematics, which cover the broader formal systems where the biconditional operates.