The Contrapositive

Every if-then sentence has a twin that says the same thing. “If the shop is open, the light is on” means “if the light is off, the shop is closed.” Swap the parts and deny both, and you have the contrapositive, true whenever the original is.

Two other turns look natural and are not safe. “If the light is on, the shop is open” swaps the parts without denying them. That is the converse, and a cleaner working late makes it false. “If the shop is closed, the light is off” denies both parts without swapping. That is the inverse, and the same cleaner breaks it.

Arguments built on those turns have names. Affirming the consequent runs from the then-part back to the if-part. Denying the antecedent runs from the missing if-part to the missing then-part. Both feel like logic, and both fail.

The allergy card

Nadia’s allergy card says, “If a dish contains peanuts, the menu marks it with a star.” She sees a dish with no star and orders it with confidence: no star, so no peanuts. That is the contrapositive, and it holds if the card is right.

Her friend Baptiste sees a starred dish and says, “That one has peanuts.” Nadia points out that the star might mark another allergen. Baptiste has read the converse, which the card never promised.

Try this today Take one if-then rule you live by and write its contrapositive; then write its converse and ask whether you believe that too.