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Chapter 4: Necessary & Sufficient Conditions

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Necessary & Sufficient Conditions

Necessary and sufficient conditions help us understand and explain the connections between concepts, and how different situations are related to each other. 

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. 
    • 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. When we say kindling requires oxygen, this is equivalent to saying that the presence of oxygen is a necessary condition for firing. 

    • 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. 

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. 
  • 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 coloured. 

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:  
    • This fourfold classification is useful because it provides the starting point for analysing how things are related. 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. 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. 

Necessary and sufficient conditions are also related to the topic of definition. 

When we define a bachelor as a married man, this implies that being an unmarried man is both necessary and sufficient for being a bachelor. 

Using Necessary and Sufficient Conditions to Resolve Disputes 

The concepts of necessary and sufficient conditions are quite simple. 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.

The Write-off Fallacy

    • The concepts of necessary and sufficient conditions are also used in some bad arguments. 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.
    • In principle, a kind 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.

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 different kinds of necessary and sufficient conditions: 

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 the above-mentioned statements. In the first statement, what is being referred to is a logical impossibility. 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. Dissolving gold in water is logically possible but empirically impossible. An empirical possibility is sometimes also known as a 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 travelling from India to France within an hour. 
    • Perhaps such a short trip is possible with some future aeroplanes, 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. 
    • What is currently technologically impossible might well turn out to be technologically possible in the future.

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—a namely legal impossibility. To say that X is not possible in this sense is to say that X is incompatible with the relevant legislation. 

The different senses of possibility apply 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 a legal necessity. 

Exclusive & Exhaustive Possibilities

There are 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 (light rain). 

Second, one possibility can exclude another. If Prabodh is in Japan right now, that excludes the possibility that he is in India. 

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 (inclusive): 

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. 

    • Suppose x is an integer (any of the natural numbers or zero):
    • 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 

References:

Lau, J. Y. (2011). An Introduction to Critical Thinking and Creativity: Think More, Think Better. New Jersey: A JOHN WILEY & SONS, INC., PUBLICATION.



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