3.2: Necessary and Sufficient Conditions

One sentence is a sufficient condition for another sentence if the truth of the first would guarantee the truth of the second. The truth of the first is enough – all you need, sufficient – to ensure the truth of the second. Having your head cut off is a sufficient condition for being dead. There are many other ways to die, but decapitation is enough to ensure it.

And one sentence is a necessary condition for a second if the truth of the second sentence is required—is needed, is necessary—for the truth of the first sentence. For example, paying your tuition fees is a necessary condition for graduating. If you are to graduate, you must pay your fees.

In the previous example, paying your fees is necessary, but not sufficient, for graduating. It is necessary, because no one graduates without paying their fees, but it is not sufficient because there are other things you must also do (like pass the required number of credit hours).

Requirements and prerequisites are usually necessary conditions. They are things that you must do to achieve a certain goal, but by themselves they do not guarantee success. For example, studying is a necessary condition for Sufficient condition: enough, a guarantee Necessary condition: a condition that must be met Requirements and prerequisites are necessary conditions 3.2 Necessary and Sufficient Conditions 69 passing this course (you must study to pass it), and practicing is a necessary condition for becoming a good basketball player (you must practice). But neither, alone, is sufficient.

How does all this relate to conditionals?

1. The antecedent of a conditional is a sufficient condition for the consequent.
2. The consequent of a conditional is a necessary condition for the antecedent.

We have treated sentences as necessary and sufficient conditions, but it is also useful to think of properties in this way. For example, the property, or characteristic, of being a dog is sufficient for that of being an animal. Having the first property is enough to insure having the second. And the property or characteristic of being an animal is necessary for that of being a dog. Nothing is a dog unless it is an animal.

Only sentences can serve as antecedents or consequents of conditionals, but we can often turn talk about properties into talk about sentences. For example, instead of saying that the property of being a dog is sufficient for the property of being an animal, we could say that if something is a dog then it is an animal. This involves subtleties you will learn about if you take a course in formal logic, but they aren’t relevant here. When we do this, antecedents are still sufficient conditions for their consequents, and consequents are still necessary conditions for their antecedents. The best way to understand these concepts is to work through the following exercises on necessary and sufficient conditions.

Exercises on Necessary and Sufficient Conditions

1. “If you get a cholera shot, then you’ll be safe in the villages.”

Here the claim is that getting a cholera shot is a __________condition for being safe in the villages.

2. “You must get a cholera shot, if you’re going to get a visa to India.”

Here the claim is that getting a cholera shot is a __________ condition for getting a visa to India.

3. “Lots of people with rich friends do not succeed in national politics. But no one succeeds in national politics without rich friends.”

Here the claim is that having rich friends is a __________ condition (but not a__________ condition) for succeeding in national politics.

4. “Show me a good loser, and I’ll show you a loser.”

Here the claim is that being a good loser is a __________ condition for being a loser.

5. “Someone is a bachelor if, and only if, they are an unmarried male.”

Here the claim is that being an unmarried male is a __________condition for being a bachelor.

6. “OU will go to a Bowl game only if they have a winning season.”

Here the claim is that having a winning season is a __________ condition for going to a Bowl game.

7. “OU won’t go to a Bowl game unless they have a winning season.”

Here the claim is that having a winning season is a __________ condition for going to a Bowl game.

8. “. . . liberal democracy has arisen only in nations that are market-oriented, not in all of them, but only in them” [Charles E. Lindblom, Politics and Markets].

Here the claim is that a society’s being market oriented is a __________ (but not a __________ condition) for liberal democracy.

9. “If wishes were horses, then beggars would ride.”

Here the claim is that wishes being horses is a __________ condition for beggars to ride.

10. “Hard work guarantees success.”

Here the claim is that working hard is a __________ condition for being successful.

11. “The defendant is only guilty of first-degree murder if she planned the crime out beforehand.”

Here the claim is that planning the crime out beforehand is a __________ condition for being guilty of first-degree murder.

12. “I’m not about to run in this heat.”

Here the claim is that the heat is a __________ condition for not running.

13. “Show me someone who likes your cooking, and I’ll show you someone who needs a tongue transplant.”

Here the claim is that liking your cooking is a __________ condition for needing to get their tongue replaced.

14. “A contract is binding only when there is no fraud.”

Here the claim is that the absence of fraud is a __________ condition for a contract to be binding.

15. “Nothing ventured, nothing gained.”

Here the claim is that taking a chance is a __________ condition for gaining something.

16. “You must pass the final to pass this course.”

Here the claim is that passing the final is a__________ condition for passing the course.

17. “You will pass this course only if you pass the final.”

Here the claim is that passing the final is a __________ condition for passing the course.

18. “You will pass this course if only you pass the final.”

Here the claim is that passing the final is a __________ condition for passing the course. (This sentence is tricky; compare this sentence to the one right above it.)

19. “You will not pass this course unless you pass the final.”

Here the claim is that passing the final is a __________ condition for passing the course.

20. “John will marry Sue only if she agrees to have three children.”

Here the claim is that agreeing to have three children is a __________ condition for his agreeing to marry her.

21. “You won’t be happy, if you buy it at Sturdley’s.”

Here the claim is that buying it at Sturdley’s is a __________ for being unhappy.

22. “A person is a brother just in case he is a sibling who identifies as male.”

Here the claim is that being a sibling who identifies as male is a __________ condition for being a brother.

Answer

1. Here the claim is that getting a cholera shot is a sufficient condition for being safe in the villages. The claim here is that if you get a shot, you’ll be safe. So, the sentence says that getting a shot is enough – it’s sufficient for being safe.
2. Here the claim is that getting a cholera shot is a necessary condition for getting a visa to India. Requirements and prerequisites are usually necessary conditions. Here the claim is that you must get the shot if you are to get the visa. But there are other necessary conditions as well, e.g., not being a convicted felon. So, getting the shot is not sufficient.
3. Here the claim is that having rich friends is a necessary (but not a sufficient condition) for succeeding in national politics. This sentence says that a requirement, a necessary condition, for success in national politics is having rich friends. But since lots of people with rich friends do not succeed, being rich is not sufficient.
4. Here the claim is that being a good loser is a sufficient condition for being a loser. Hint: it will help here to rephrase the claim in a more explicitly conditional form, as “If you show me a good loser, I’ll show you a loser.”
5. Here the claim is that being an unmarried male is a both a necessary and sufficient condition for being a bachelor.
6. Here the claim is that having a winning season is a necessary condition for going to a Bowl game.
7. Here the claim is that having a winning season is a necessary condition for going to a Bowl game.
8. Here the claim is that a society’s being market oriented is a necessary (but not a sufficient condition) for liberal democracy.
9. Here the claim is that wishes being horses is a sufficient condition for beggars to ride.
10. Here the claim is that working hard is a sufficient condition for being successful.
11. Here the claim is that planning the crime out beforehand is a necessary condition for being guilty of first-degree murder.
12. Here the claim is that the heat is a sufficient condition for not running.
13. Here the claim is that liking your cooking is a sufficient condition for needing to get their tongue replaced.
14. Here the claim is that the absence of fraud is a necessary condition for a contract to be binding.
15. Here the claim is that taking a chance is a necessary condition for gaining something.
16. Here the claim is that passing the final is necessary condition for passing the course.
17. Here the claim is that passing the final is a necessary condition for passing the course.
18. Here the claim is that passing the final is a sufficient condition for passing the course.