Logical biconditional

In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or bidirectional implication or biimplication or bientailment or exclusive nor, is the logical connective used to conjoin two statements

P

{\displaystyle P}

and

Q

{\displaystyle Q}

to form the statement "

P

{\displaystyle P}

if and only if

Q

{\displaystyle Q}

" (often abbreviated as "

P

{\displaystyle P}

iff

Q

{\displaystyle Q}

"), where

P

{\displaystyle P}

is known as the antecedent, and

Q

{\displaystyle Q}

the consequent.

Nowadays, notations to represent equivalence include

↔

,

⇔

,

≡

{\displaystyle \leftrightarrow ,\Leftrightarrow ,\equiv }

.

P

↔

Q

{\displaystyle P\leftrightarrow Q}

is logically equivalent to both

(

P

→

Q

)

∧

(

Q

→

P

)

{\displaystyle (P\rightarrow Q)\land (Q\rightarrow P)}

and

(

P

∧

Q

)

∨

(

¬

P

∧

¬

Q

)

{\displaystyle (P\land Q)\lor (\neg P\land \neg Q)}

, and the XNOR (exclusive NOR) Boolean operator, which means "both or neither".

Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis (antecedent) is false but the conclusion (consequent) is true. In this case, the result is true for the conditional, but false for the biconditional.

In the conceptual interpretation, P = Q means "All P's are Q's and all Q's are P's". In other words, the sets P and Q coincide: they are identical. However, this does not mean that P and Q need to have the same meaning (e.g., P could be "equiangular trilateral" and Q could be "equilateral triangle"). When phrased as a sentence, the antecedent is the subject and the consequent is the predicate of a universal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate).

In the propositional interpretation,

P

↔

Q

{\displaystyle P\leftrightarrow Q}

means that P implies Q and Q implies P; in other words, the propositions are logically equivalent, in the sense that both are either jointly true or jointly false.