CHAPTER 4
NECESSARY AND SUFFICIENT CONDITIONS
Necessary and sufficient conditions help us understand and explain the connections between concepts, and how different situations are related to each other.
4.1 NECESSARY CONDITIONS
To say that X is a necessary condition for Y is to say that the occurrence of X is required for the occurrence of Y (sometimes also called an essential condition). In other words, if there is no X, Y would not exist. Examples:
- Having four sides is necessary for being a square.
- Infection by HIV is necessary for developing AIDS.
- Having the intention to kill someone or to cause grievous bodily harm is necessary for murder.
To show that X is not a necessary condition for Y, we simply find a situation where Y is present but X is not. Examples:
- Eating meat is not necessary for living a healthy life. There are plenty of healthy vegetarians.
- Being a land animal is not necessary for being a mammal. Whales are mammals, but they live in the sea.
In daily life, we often talk about necessary conditions, maybe not explicitly. When we say combustion requires oxygen, this is equivalent to saying that the presence of oxygen is a necessary condition for combustion.
Note that a single situation can have more than one necessary condition. To be a good pianist, it is necessary to have good finger technique. But this is not enough. Another necessary condition is being good at interpreting piano pieces.
4.2 SUFFICIENT CONDITIONS
If X is a sufficient condition for Y, this means the occurrence of X guarantees the occurrence of Y. In other words, it is impossible to have X without Y. If X is present, then Y must also be present. Some examples:
- Being a square is sufficient for having four sides.
- Being a grandfather is sufficient for being a father.
To show that X is not sufficient for Y, we list cases where X occurs but not Y:
- Being infected by HIV is not sufficient for developing AIDS because there are many people who have the virus but have not developed AIDS.
- Loyalty is not sufficient for honesty because one might have to act in a dishonest manner to protect the person one is loyal to.
Note that a single state of affairs can have more than one sufficient condition. Being red and being green are different conditions, but they are both sufficient for something being colored.
4.3 DESCRIBING HOW TWO THINGS ARE CONNECTED
Given any two conditions X and Y, there are four ways in which they might be related to each other:
1. X is both (jointly) necessary and sufficient for Y.
2. X is necessary but not sufficient for Y.
3. X is sufficient but not necessary for Y.
4. X is neither necessary nor sufficient for Y.
Some examples:
1. Being an unmarried man is necessary and sufficient for being a bachelor.
2. Oxygen is necessary but not sufficient for our survival.
3. Having a son is sufficient but not necessary for being a parent.
4. Being rich is neither necessary nor sufficient for a happy life.
This fourfold classification is useful because it provides the starting point for analyzing how things are related. When we think about the relationship between two things X and Y, we can begin by asking whether one is necessary or sufficient for the other. For example, what is the connection between democracy and the rule of law? First, we might say that the rule of law is necessary for democracy. A democracy is impossible if people do not follow legal procedures to elect leaders or resolve disputes. But we might also add that the rule of law is not sufficient for democracy, because the legal rules that people follow might not be fair or democratic. As this example shows, the concepts of necessary and sufficient conditions can be very useful in studying and teaching. When you understand a subject more deeply, you do not just memorize individual pieces of information. You should also be able to understand the connections between the basic concepts, and this includes relationships of necessity and sufficiency.
Necessary and sufficient conditions are also related to the topic of definition. In effect, a definition of X provides conditions that are both necessary and sufficient for X. When we define a bachelor as an married man, this implies that being an unmarried man is both necessary and sufficient for being a bachelor.
4.3.1 Using necessary and sufficient conditions to resolve disputes
The concepts of necessary and sufficient conditions are quite simple, but they are very useful and fundamental concepts. Sometimes when people disagree with each other, especially about some theoretical issue, we can use these concepts to identify more clearly the differences between the parties.
For example, suppose someone claims that computers cannot think because they can never fall in love or be sad. To understand this argument better, we can ask whether this person is assuming that having emotions is necessary for thinking, and if so why? If something is capable of reasoning and deduction, then presumably it can think. But emotions seem to be a different category of mental states. We can imagine a person who is able to think and reason, but who cannot feel any emotion, perhaps due to brain injuries. If this is possible, it shows that emotions are not necessary for thinking.
4.4 THE WRITE-OFF FALLACY
Although the concepts of necessary and sufficient conditions are very important, they are also used in some bad arguments. One such fallacy, which we might call the write-off fallacy, is to argue that something is not important, because it is not necessary or not sufficient for something else that is good or valuable.
For example, some people argue that democracy is not really that important because it is not necessary for having a good government. It is possible for a non-democratic government to work efficiently for the interests of the people, and this might be correct. Furthermore, democracy is not sufficient for good governance either, since citizens can make bad choices and end up electing a bad government. This is also possible. However, it might still be the case that a democratic political system is more likely to produce a good government than other alternatives.1 In principle, a benevolent dictator can be a wise and competent ruler, but the fact is that this is extremely rare, and dictators are more likely to abuse their power. The general lesson here is that a condition C can be an important factor that makes an effect E more likely to happen, even if C is neither necessary nor sufficient for E. It is not enough to write off C as unimportant simply by pointing to isolated cases where there is C but no E, or where there is E but no C.
4.5 DIFFERENT KINDS OF POSSIBILITY
Necessary and sufficient conditions are related to the concept of possibility. To say that X is necessary for Y is to say that it is not possible for Y to occur without X. To say that X is sufficient for Y is to say that it is not possible for X to occur without Y. There are, however, different senses of possibility, and corresponding to these different sense there are different kinds of necessary and sufficient conditions. Let us consider these statements:
- It is impossible to draw a red square without drawing a square.
- It is impossible to dissolve gold in pure water.
- It is impossible to travel from India to France in less than one hour.
- It is impossible to vote in Australia if you are under 18.
The word impossible does not have the same meaning in these statements. In the first statement, what is being referred to is logical impossibility. Something is logically impossible if it is contradictory or against the laws of logic. Thus a round square is a logical impossibility, and it is logically impossible for there to be a red square without there being a square.
But it is not logically impossible to dissolve gold in water. Logic itself does not tell us that this cannot happen. Rather, the impossibility is due to the laws of physics and chemistry that happen to hold in our universe. If our universe had operated differently, then perhaps gold would dissolve in water. Dissolving gold in water is therefore logically possible but empirically impossible. Empirical possibility is sometimes also known as causal or nomological possibility.
The sense in which the third statement is true is again different. The laws of physics probably do not stop us from traveling from India to France within an hour. Perhaps such a short trip is possible with some future airplane, but it is certainly not possible at this point in time. When current technology does not permit a situation to happen, we say that it is technologically impossible, even though it might be both logically and empirically possible. Of course, what is currently technologically impossible might well turn out to be technologically possible in the future.
Finally, voting under the age of 18 is certainly not prohibited by logic, the laws of nature, or current technology. What is meant by impossible in the fourth statement is thus something else—namely legal impossibility. To say that X is not possible in this sense is to say that X is incompatible with the relevant legislation.
Note that what we have just said about the different senses of possibility applies to necessary and must as well. “A square must have four sides” and “it is necessary that a square has four sides” express logical necessity. Whereas “you must be 18 to vote in Australia” is obviously about legal rather than logical necessity.
4.6 EXCLUSIVE AND EXHAUSTIVE POSSIBILITIES
Apart from talking about the ways in which something is or is not possible, there are also some useful terms for talking about the connections between different possibilities.
First, we can speak of a possibility including another. There being rain tomorrow includes the possibility of a heavy rainstorm, and the possibility of just a light drizzle. Second, one possibility can exclude another. If Cinta is in Spain right now, that excludes the possibility that she is in Brazil. Finally, two possibilities can also be independent of each other. Whether it will rain tomorrow presumably does not depend on what you ate for breakfast this morning.
The word exclusive is sometimes used to talk about one possibility excluding another, and it is important not to confuse exclusive with exhaustive.
- A set of possibilities is exhaustive when at least one of them obtains in any logically possible situation (they do not leave out any situation).
- A set of possibilities is exclusive when there is no logically possible situation in which more than one of them obtains (the truth of one excludes the truth of the others).
- In other words, if a set of possibilities is both exhaustive and exclusive, then in any logically possible situation, exactly one of them obtains.
This explanation might be a bit too abstract, so here is an illustration. Suppose x is an integer:
- Two possibilities that are neither exhaustive nor exclusive: x > 3, x > 4. They are not exhaustive because the possibility that x = 2 is not included. They are not exclusive because both of them can be correct, as when x > 5.
- Exhaustive but not exclusive: x > 4, x < 10<
- Exclusive but not exhaustive: x > 4, x = 1
- Exclusive and exhaustive: x > 0, x = 0, x < 0<
EXERCISES
4.1 Suppose we have a definition X = Y. Are the following statements correct about this definition? Why or why not?
a) If the definition is too wide, then X is not necessary for Y.
b) If the definition is too wide, then Y is not necessary for X.
c) If the definition is too narrow, then X is not sufficient for Y.
d) If the definition is too narrow, then Y is not sufficient for X.
e) If X is not necessary for Y, then the definition is too wide.
4.2 Are these statements true or false?
a) If X is logically sufficient for Y, and Y is logically sufficient for Z, then X is logically sufficient for Z.
b) If X is logically necessary for Y, and Y is logically necessary for Z, then X is logically necessary for Z.
c) If X is not necessary for Y, then Y is not necessary for X.
d) Being an intelligent student in the class is necessary for being the most intelligent student in the class.
e) If something is not logically impossible, then it is logically possible.
f) If something is empirically impossible, then it does not actually happen in the world.
g) If something is empirically possible, then it actually happens in the world.
h) If something actually happens in the world, then it is empirically possible.
i) If something is logically possible, then it is empirically possible.
j) If something is empirically possible, then it is logically possible.
k) If something is empirically possible, then it is technologically possible.
4.3 A definition of X provides necessary and sufficient condition for X. See if you can fill in the blanks below correctly with necessary condition or sufficient condition:
a) If the definition is too wide, this means the definition fails to provide the correct_____for X.
b) If the definition is too narrow, this means the definition fails to provide the correct_____for X.
4.4 Determine whether these possibilities are exhaustive and/or exclusive.
a) Inflation goes up. Inflation comes down.
b) P and Q. Neither P nor Q.
c) Sadie and Rita are happy. Sadie and Rita are sad.
4.5 Many management and law school admission tests include questions known as “data sufficiency questions.” There are also similar questions in many recruitment exams. So do try out the following questions. You will be given some information, and then you have to pick the correct answer out of the five choices listed below:
1. Statement 1 alone is sufficient to answer the question, but statement 2 alone is not sufficient.
2. Statement 2 alone is sufficient to answer the question, but statement 1 alone is not sufficient.
3. Both statements are necessary for answering the question, and neither statement alone is sufficient.
4. Either statement by itself is sufficient to answer the question.
5. Statements 1 and 2 together are not sufficient to answer the question.
a) Three stones have a combined weight of 40 kilograms. What is the weight of the heaviest stone?
Statement 1: One stone weighs 10 kilograms.
Statement 2: One stone weighs 20 kilograms.
b) Two students joined the same company at the same time. How much more money per month does trainee X now earn than trainee Y?
Statement 1: Y earns $3000 per month more than when he first started.
Statement 2: X earns $5000 per month more than when she first started.
c) What is the total number of cakes that Elia and Maddalena have eaten?
Statement 1: Elia ate twice as many cakes as Maddalena.
Statement 2: Maddalena ate two cakes fewer than Elia.
4.6 Describe the mistake in this argument in terms of necessary and sufficient conditions:
Students who do not study always fail the exam. Since I have studied, it follows that I will pass the exam.
1 Here is a famous quote from Winston Churchill (1874–1965): “No one pretends that democracy is perfect or all-wise. Indeed it has been said that democracy is the worst form of government except all those other forms that have been tried from time to time.” (Speech in the House on the Parliament Bill, November 11, 1947)