Only If, and If and Only If

Necessary and Sufficient showed that “only if” names something needed. This lesson draws the arrows. “You get a refund only if you keep the receipt” means: refund, so receipt. The arrow runs from refund to receipt and not back. Keeping the receipt does not promise a refund.

“If and only if” runs both ways. “The gate opens if and only if the card is valid” means two things: valid card, so gate open; gate open, so valid card. Mathematicians shorten it to iff. Each direction is a separate promise, and each can fail on its own.

Everyday speech blurs the two. “I will come if it is sunny” often means if and only if. A rule written with “if” is often read as “only if.” When a rule matters, ask the person who wrote it which arrows they meant, and write both down.

The bonus letter

Esther’s contract says, “A bonus is paid only if the team meets its target.” The team meets it, and no bonus arrives. Esther is angry until her colleague Wen draws the arrow: bonus, so target. The sentence never promised the reverse.

The next year she asks for different words: “A bonus is paid if and only if the team meets its target.” Now the arrow runs both ways. When the target is met, the bonus is owed, and when it is missed, nobody can expect one.

Try this today Rewrite one rule you follow as an arrow: which way does it point, and does the person who wrote it think it points both ways?