Logical biconditional

In logic and mathematics, the logical biconditional (sometimes known as the material biconditional) is the logical connective of two statements asserting "p if and only if q", where q is an antecedent and p is a consequent. This is often abbreviated p iff q. The operator is denoted using a doubleheaded arrow (↔), a prefixed E (Epq), an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to (p → q) ∧ (q → p), or the XNOR (exclusive nor) boolean operator. It is equivalent to "(not p or q) and (not q or p)". It is also logically equivalent to "(p and q) or (not p and not q)", meaning "both or neither".

The only difference from material conditional is the case when the hypothesis is false but the conclusion is true. In that case, in the conditional, the result is true, yet in the biconditional the result is false.

In the conceptual interpretation, a = b means "All a 's are b 's and all b 's are a 's"; in other words, the sets a and b coincide: they are identical. This does not mean that the concepts have the same meaning. Examples: "triangle" and "trilateral", "equiangular trilateral" and "equilateral triangle". The antecedent is the subject and the consequent is the predicate of a universal affirmative proposition.