Negation can Create Confusion

This distinction becomes complicated when conditions get complex. Our definitions and tests hold for all features, whether positive or negative (such as not having hair) and whether simple or conjunctive (such as having both a beard and a mustache) or disjunctive (such as having either a beard or a mustache). Thus, not having any hair (anywhere) on your head is a suffi- cient condition of not having a beard, so not having a beard is a necessary condition of not having any hair on your head. But not having any hair on your head is not necessary for not having a beard, because you can have some hair on the top of your head without having a beard. Negation can create confusion, so we need to think carefully about what is being claimed to be necessary or sufficient for what.

Even in simple cases without negation, conjunction, or disjunction, there is a widespread tendency to confuse necessary conditions with sufficient conditions. It is important to keep these concepts straight, for, as we will see, the tests concerning them are fundamentally different.

Which of the following claims are true? Which are false?

1. Being a car is a sufficient condition for being a vehicle.

2. Being a car is a necessary condition for being a vehicle.

3. Being a vehicle is a sufficient condition for being a car.

4. Being a vehicle is a necessary condition for being a car.

5. Being an integer is a sufficient condition for being an even number.

6. Being an integer is a necessary condition for being an even number.

7. Being an integer is a sufficient condition for being either an even number or an odd number.

8. Being an integer is a necessary condition for being either an even number or an odd number.

9. Not being an integer is a sufficient condition for not being an odd number.

10. Not being an integer is a sufficient condition for not being an even number.

11. Being both an integer and divisible by 2 without remainder is a sufficient condition for being an even number.