--  作者：valarie

--  发布时间：2004-7-21 23:17:00

--  Please come in - For those confused by Sufficient and Necessary conditions

If you still are not clear about Sufficient and Necessary conditions, please read the following article, which I found helpful.

The Concepts of

Necessary and Sufficient Conditions 

Contents

Introduction

Definition of "necessary condition"

Definition of "sufficient condition"

Necessary conditions that are not jointly sufficient

Sufficient conditions that are not necessary

The concept of converse relations

"Is a necessary condition for" and "is a sufficient condition for" are converse relations

Four possible combinations

Practice exercise #1

Practice exercise #2

Different kinds (or modes) of necessary condition

1. Introduction

Everyone is familiar with the concept of a necessary condition. For example, we all know that air is necessary for (human) life. Without air, there is no (human) life. Similarly, a microscope (or some other instrument) is necessary for human beings to see viruses. (Viruses are too small to be seen by the naked eye.)

Similarly, everyone is familiar with the concept of a sufficient condition. For example, it suffices (i.e. it is sufficient for) an object\'s having four sides that it is a square. Or, again, it is sufficient for your having something to drink that you have a glass of Coca-Cola®. 

2. Definition of "necessary condition"

Definition: A condition A is said to be necessary for a condition B, if (and only if) the falsity (/nonexistence /non-occurrence) [as the case may be] of A guarantees (or brings about) the falsity (/nonexistence /non-occurrence) of B. 

So common is this notion of necessary condition that there are, not surprisingly, a great many ways to express that something is a necessary condition. Here are a number of examples, all - more or less - saying the same thing:

"Air is necessary for human life."

"Human beings must have air to live."

"Without air, human beings die (i.e. do not live)."

"If a human being is alive, then that human being has air (to breathe)."

In an `if-then\' statement (such as the last example immediately above), the clause that follows the "then" (i.e. the so-called `consequent\') states the necessary condition for the antecedent (i.e. the clause immediately following the "if"). Thus that some human being has air (to breathe) is a necessary condition for that human being\'s being alive.

Let\'s look at some further examples (again, all saying pretty much the same thing). Notice the use of "if-then" (in the sixth [highlighted] example):

"Having a microscope (or some other instrument) is a necessary condition for (our) seeing viruses."

"A microscope (or a similar instrument) is needed to see viruses."

"Human beings must use (have) a microscope to see viruses."

"Human beings cannot see viruses without a microscope."

"Anyone who sees viruses has (uses) a microscope."

"If someone sees viruses, then that person uses a microscope." NOTE

"Without a microscope, a person cannot see viruses."

"If a person does not have (the use of) a microscope, then that person does not see viruses."

"Whoever lacks a microscope does not see viruses."

"One must have a microscope to see viruses."

But of course, as we well know, in general a necessary condition is not a sufficient condition. All sorts of conditions may be necessary for others, but do not - by themselves - suffice for, or guarantee, those others. 

3. Definition of "sufficient condition"

Definition: A condition A is said to be sufficient for a condition B, if (and only if) the truth (/existence /occurrence) [as the case may be] of A guarantees (or brings about) the truth (/existence /occurrence) of B. 

For example, while air is a necessary condition for human life, it is by no means a sufficient condition, i.e. it does not, by itself, i.e. alone, suffice for human life. While someone may have air to breathe, that person will still die if s/he lacks water (for a number of days), has taken poison, is exposed to extremes of cold or heat, etc. There are, in fact, a very great many conditions that are necessary for human life, and no one - or even just a few of them - will suffice for [or guarantee] human life. Or, further, consider the property of having four sides. While having four sides is a necessary condition for something\'s being a square, that single condition is not, by itself, sufficient (to guarantee) something\'s being a square, i.e. some four-sided things (e.g. trapezoids) are not squares. There are several necessary conditions for something\'s being a square, and all of these must be satisfied for something\'s being a square:

x has (exactly) four sides

each of x\'s sides is straight

x is a closed figure

x lies in a plane

each of x\'s sides is equal in length to each of the others

each of x\'s interior angles is equal to the others (they are each right [i.e. 90o] angles)

the sides of x are joined at their ends

The foregoing is a complete set of necessary conditions, i.e. the set comprises a set of sufficient condition for x\'s being square.

Frequently the terminology of "individually necessary" and "jointly sufficient" is used. One might say, for example, "each of the members of the foregoing set is individually necessary and, taken all together, they are jointly sufficient for x\'s being a square."

Caution: In this example, we have been able, with ease, to list a set of individually necessary conditions that is also sufficient for something\'s being a square. However, we must not generalize from this simple example and believe that it is usually, or often, a straightforward task to specify sets of conditions that are individually necessary and jointly sufficient. Sometimes it is much easier to specify (some, or many, of the) necessary conditions even though we are unable to specify a set that is jointly sufficient. Other times, the converse is true: for some cases it will be easier to specify sufficient conditions without our being able to specify individually necessary ones. 

4. Necessary conditions that are not sufficient

Example 4.1 - A set of conditions that are individually necessary without being jointly sufficient.