// laws and rules · Logical equivalence
Definition of the biconditional_
The biconditional p ⇔ q (“p if and only if q”) is defined as the conjunction of the two implications p ⇒ q and q ⇒ p. The formula (p ⇔ q) ⇔ ((p ⇒ q) ∧ (q ⇒ p)) is a tautology and explains why “if and only if” proofs always have two parts.
Example
(p ⇔ q) ⇔ ((p ⇒ q) ∧ (q ⇒ p))
What the variables mean
- ▸ p: “A number is divisible by 2”
- ▸ q: “A number is even”
In plain words
“A number is divisible by 2 if and only if it is even” means: if it is divisible by 2 it is even, and if it is even it is divisible by 2.
Truth table
| p | q | p ⇔ q | p ⇒ q | q ⇒ p | (p ⇒ q) ∧ (q ⇒ p) | (p ⇔ q) ⇔ ((p ⇒ q) ∧ (q ⇒ p))★ |
|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T |
| T | F | F | F | T | F | T |
| F | T | F | T | F | F | T |
| F | F | T | T | T | T | T |
Classification: Tautology · 4 rows
Statement
(p ⇔ q) ⇔ ((p ⇒ q) ∧ (q ⇒ p)). The biconditional is true when p and q share the same truth value; the double implication demands that neither direction is broken.
That is why it is read “if and only if” (abbreviated “iff”): the “if” covers q ⇒ p and the “only if” covers p ⇒ q.
Why it is a tautology: reading the table
Rows p = T, q = T and p = F, q = F: p ⇔ q is T. Both implications are T (antecedent and consequent equal), so the conjunction is T. They agree.
Row p = T, q = F: p ⇔ q is F; p ⇒ q is F, so the conjunction is F. Row p = F, q = T: p ⇔ q is F; q ⇒ p is F and the conjunction is F. The columns are identical in all four rows.
How it is used in proofs
Every theorem of the form “A if and only if B” is proved in two steps: (⇒) assume A and derive B, and (⇐) assume B and derive A. This law is the justification for that scheme.
In systems where ⇔ is not primitive, this equivalence is literally its definition.
Examples
Mathematics: “A triangle is equilateral if and only if its three angles are equal” requires proving both directions separately.
Programming: boolean equality `a == b` is the biconditional. It is true when `a` and `b` have the same value, i.e. when `(!a || b) && (!b || a)`.
Relation to other laws
Replacing each implication by material implication gives p ⇔ q equivalent to (¬p ∨ q) ∧ (¬q ∨ p), and by distribution to (p ∧ q) ∨ (¬p ∧ ¬q).
Its negation is exclusive or: ¬(p ⇔ q) is equivalent to p ⊕ q, which is true when the values differ.
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related operators
Frequently asked questions
Why do “if and only if” proofs have two parts? ▼
Because the biconditional is the conjunction of two implications, and each must be proved.
What is the difference between ⇒ and ⇔? ▼
⇒ only requires q to be true when p is; ⇔ additionally requires p to be true when q is. The biconditional is stronger.
Is the biconditional the same as equality? ▼
For truth values, yes: p ⇔ q is T exactly when p and q coincide.