If and only if

Example of A is a proper subset of B

Example of C is no proper subset of B

If and only if (iff) is a biconditional logical connective between statements. In logic and related fields such as mathematics and philosophy, "if and only if" signifies that either both statements are true or both are false. The expression is used to state that two statements are both necessary and sufficient for each other. The notation p ↔ q represents a biconditional statement; it can be read as "p if and only if q," where p and q are propositions.

Definition

In formal logic, a biconditional statement is true whenever both parts have the same truth value. The truth table for p ↔ q (where p and q are propositions) is as follows:

p	q	p ↔ q
True	True	True
True	False	False
False	True	False
False	False	True

This shows that p ↔ q is true if p and q are both true or if p and q are both false.

Usage

In mathematics, "if and only if" is used to convey that two statements are equivalent. For example, in the context of set theory, one might say, "A set is empty if and only if its cardinality is zero." This means that a set being empty is necessary and sufficient for its cardinality to be zero.

In philosophy, especially in logical positivism and analytic philosophy, "if and only if" statements are crucial in formulating definitions and constructing precise arguments.

Notation

The symbol used to represent "if and only if" in formal logic is "↔" or "⇔". In mathematical texts, it is common to see the abbreviation "iff" for "if and only if" to save space and reduce repetition.

Properties

The biconditional operator has several important properties:

• Commutativity: p ↔ q is equivalent to q ↔ p.
• Associativity: Given any three propositions p, q, and r, the statement (p ↔ q) ↔ r is equivalent to p ↔ (q ↔ r).
• Distributivity: The biconditional operator does not distribute over other logical operators in the same way as conjunction and disjunction do.

Examples

1. "A square is a rectangle if and only if it has four right angles." This statement asserts that having four right angles is both necessary and sufficient for a shape to be a square. 2. In number theory, "A number is even if and only if it is divisible by 2." This means that being divisible by 2 is a necessary and sufficient condition for a number to be even.

See also

• Logical connective
• Necessary and sufficient condition
• Propositional calculus

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