# Iff

Iff is a short­hand for “if and only if”. Its log­i­cal sym­bol is $$\leftrightarrow$$.

“A iff B” ($$A \leftrightarrow B$$) is quite dis­tinct from “if A then B” ($$A \rightarrow B$$). Con­sider the stipu­la­tion “If the dog barks, then it will soon bite”. This would not obli­gate the dog to bark a warn­ing be­fore bit­ing. The “if” re­la­tion isn’t sym­met­ri­cal. As such, the dog might some­times bite spon­ta­neously, with no bark­ing at all.

If we wanted to en­sure that bit­ing is always fore­warned by bark­ing, we would in­stead stipu­late “Iff dog barks, then it will soon bite”. This is equiv­a­lent to

• “the dog barks if and only if it will soon bite”

• “If the dog barks then it will soon bite, and if the dog bites it will have barked be­fore­hand”.

• “The dog barks only when it will soon bite”

With “iff”, the im­pli­ca­tion runs in both di­rec­tions.

Parents:

• Mathematics

Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.