biconditional

A biconditional is a logical connective that combines two statements P and Q into a statement of the form “P if and only if Q.” It expresses both directions: if P is true, then Q must be true, and if Q is true, then P must be true. A biconditional is true exactly when P and Q have the same truth value, either both true or both false.

In formal logic, the biconditional is written as P ↔ Q. It is logically equivalent to the conjunction of the two conditionals, (P → Q) ∧ (Q → P). It is also equivalent to (P ∧ Q) ∨ (¬P ∧ ¬Q), which makes the notion of truth-value equality explicit.

Examples help illustrate the idea. A number is even if and only if it is divisible by 2. Here, “even” and “divisible by 2” share the same truth value for any integer. Another example: a quadrilateral is a rectangle if and only if it is a parallelogram with four right angles. In each case, the biconditional asserts a precise, two-way dependence between the two conditions.

Use in proofs and definitions often hinges on the biconditional’s requirement to establish both directions. Proving a biconditional typically involves proving P → Q and Q → P separately. In natural language, biconditionals can be clearer but also risk ambiguity, so precise formulation is important in mathematics and logic.