THE LOGIC NOTES

Biconditional Glossary

Definition

The material biconditional connective 'IFF' results in a formula A IFF B which is true if and only if A and B have the same truth value.

Connectives with similar truth conditions in natural languages are also taken to express co-implication. The expression 'if and only if' (abbreviated "iff") in English, taken in a truth-functional sense, is the intended analogue of the formal connective.

Comments

The logical biconditional is used to represent the equivalence of different expressions of a proposition. It is especially useful along with the universal quantifier to represent the equality of two sets. So ALLx(Fx IFF Gx) is true iff the set of Fs and the set of Gs are the same set.

In many logical systems, including the one presented here, the biconditional connective is not really one of the primitives. It is present by definition, taking A IFF B to mean

(A IMP B) AND (B IMP A)

or maybe the equivalent

(A AND B) OR (NOTA AND NOTB).

The truth table is:

IFF 0 1
0 1 0
1 0 1

In Logic for Fun, the biconditional connective is just the identity relation on the bool sort, so it can always be written '='.

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