Philosophy Index

Biconditional

The biconditional or material equivalence operator is used to symbolize “if and only if” statements in symbolic logic. The biconditional is a truth-functional operator in logic.

The symbol is used to indicate a biconditional relationship.

“If and only if P, then Q” may be formalized as P ↔ Q.

The phrase “if and only if” is sometimes abbreviated as iff, and so statements like “iff P, then Q” should be taken to mean P ↔ Q, while “if P, then Q” simply means P → Q.

Note: Some logicians use the symbol ≡ to serve as the biconditional operator. However, this site reserves that symbol to indicate the relationship of equivalence.

Truth Values

The following truth table illustrates the possible truth values for a biconditional expression P ↔ Q, given all possible valuations for propositions P and Q. Specifically, P ↔ Q is only true if P and Q share the same truth value, either both true, or both false.

P Q P ↔ Q
T T T
T F F
F T F
F F T

Expressed as conditionals or a disjunction

The operator ↔ can also be expressed as two conditional statements. For example, the phrase “if and only if P, then Q” is equivalent to saying “If P, then Q, and if Q, then P”. Formally:

(P → Q) ∧ (Q → P)

We may also express a biconditional as a disjunction of two conjunctions, specifically saying “P and Q, or neither”:

(P ∧ Q) ∨ (¬P ∧ ¬Q)

Material and Logical Equivalence

The function of the biconditional operator is also known as material equivalence. This should not be confused with logical equivalence (≡).

Material equivalence, the biconditional ↔, refers to an operation between two propositions or formulae, which states that if one is true, then the other is true, and if one is false, then the other is also false. Logical equivalence, ≡, on the other hand, refers to a relationship between two formulas, specifically that they imply each other. These are similar concepts but are used in different ways.

The symbol ≡ is sometimes used for material equivalence, causing even more confusion. On this site, the symbol ↔ will always be used to mean material equivalence, or the biconditional operator, and ≡ will be reserved for logical equivalence.