Necessary Conditions and Sufficient Conditions - concepts from LSAT official

-- 作者：yws

-- 发布时间：2004-7-26 15:31:00

-- Necessary Conditions and Sufficient Conditions - concepts from LSAT official

I was very confused about necessary and sufficient conditions, i think, may be that\'s why i am weak in those logic questions.

Luckily, i just got an official LSAT book, it helps me alot to understand the differences and what we can infer from the statements with these conditions.

The following is from the book, i hope it can help anyone of you chinese friends too :P

Necessary Conditions and Sufficient Conditions

“You don’t deserve praise for something unless you did it deliberately.”

“Tom deliberately left the door unlocked.”

The first statement expresses a necessary condition. Doing something deliberately is a necessary condition for deserving praise for doing it.

The first statement says that you have to do something deliberately in order to deserve praise for doing it. It doesn’t say that any time you do something deliberately you thereby deserve praise for doing it. So the mere fact that Tom did something deliberately is not enough to bring us to the conclusion that Tom deserves praise for doing it.

However, if the first statement had said “If you do something deliberately then you deserve praise for doing it,” it would be saying that doing something deliberately is a sufficient condition for deserving praise for doing it.

The necessary condition above could have been stated in several different ways:

“You deserve praise for something only if you did it deliberately.”

“You don’t deserve praise for something if you didn’t do it deliberately.”

“To deserve praise for something, you must have done it deliberately.”

All the three statements mean the same thing, and none of them says that doing something deliberately is a sufficient condition for deserving praise.

Sufficient conditions can also be expressed in several different ways:

“If it rains, the sidewalks get wet.”

“Rain is all it takes to get the sidewalks wet.”

“The sidewalks get wet whenever it rains.”

These statements each tell us that rain is a sufficient condition for the sidewalks getting wet. It is sufficient, because rain is all that it takes to make the sidewalks wet. But notice that these statements do not say that rain is the only thing that makes the sidewalks wet. There are other possibilities, such as melting snow. So these statements do not express necessary conditions for wet sidewalks, only sufficient conditions.

How Necessary Conditions work in Inferences?

N: “You deserve praise for something only if you did it deliberately.”

If we are also given a case that satisfies the necessary condition, such as

“Tom deliberately left the door unlocked.”

We cannot legitimately draw an inference. Since the mere fact that Tom did something deliberately is not enough to bring us to the conclusion that Tom deserves praise for doing it.

Statements that express necessary conditions can play a part in legitimate inferences, of course, but only in combination with the right sort of information.

1) Suppose that in addition to statement N we are told, the result the necessary condition for occurs, such that:

“Tom deserves praise for leaving the door unlocked.”

This allows us to conclude that Tom deliberately left the door unlocked.

Since statement N says that you have to do something deliberately in order to deserve praise for doing it, conclusively, Tom must have deliberately left the door unlocked if he deserves praise for what he did.

2) Or, suppose that in addition to statement N we are told, the necessary condition is not met, such that:

“Tom did not leave the door unlocked deliberately.”

This allows us to conclude that Tom does not deserve praise for leaving the door unlocked.

So, in general when you have a statement that expresses a necessary condition, it allows you to infer something in just two cases: (1) you can infer that the necessary condition is met from knowing that the thing it is the necessary condition for occurs; (2) you can infer from knowing that the necessary condition is not met that the thing it is the necessary condition for does not occur;

For a necessary condition statement "B if only A", we can write as ( B --> A), i.e. B can prove A.

(1) the thing the necessary condition for occurs à the necessary condition is met

(B -> A)

(2) the necessary condition is not met à the thing the necessary condition for does not occur

(~A --> ~B)

How Sufficient Conditions work in Inferences?

Statements that express sufficient conditions can also serve as a basis for inferences.

S: “If it rains, the sidewalks get wet.”

(1) If we are told that the sufficient condition is satisfied (i.e. told that it is raining), then we can legitimately draw inference that the sidewalks are getting wet.

(2) Suppose that in addition to statement S we are told that the sidewalks did not get wet. Since the sidewalks get wet whenever it rains, we can conclude with complete confidence that it didn’t rains.

So in general, when you have a statement that expresses a sufficient condition, it allow you to infer something in just two cases: (1) if you know that the sufficient condition is met, then you can infer that the thing it is the sufficient condition for occurs; (2) you can infer that the sufficient condition is not met from knowing that the thing it is the sufficient condition for does not occur.

For a sufficient condition statement "if A, B", we can write as (A --> B), i.e. A can prove B.

(1) the sufficient condition is met à the thing the sufficient condition for occurs (A --> B)

(2) the thing the sufficient condition for does not occur à the sufficient condition is not met (~B --> ~A)

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-- 作者：yws

-- 发布时间：2004-7-26 15:43:00

The following are what i interpret about "both necessary and sufficient conditions", i.e. "if and only if", derived from the above concepts:

For a necessary condition statement "B if only A", we can write as ( B --> A), i.e. B can prove A.

(1) the thing the necessary condition for occurs à the necessary condition is met

(B -> A)

(2) the necessary condition is not met à the thing the necessary condition for does not occur

(~A --> ~B)

For a sufficient condition statement "if A, B", we can write as (A --> B), i.e. A can prove B.

(1) the sufficient condition is met à the thing the sufficient condition for occurs (A --> B)

(2) the thing the sufficient condition for does not occur à the sufficient condition is not met (~B --> ~A)

Thus,

for a necessary and sufficient conditions statement "B if and only A",

we can write as ( B <--> A), i.e. B can prove A and A can prove B too.

I hope what i interpret is correct and helpful, as i just kind of understanding the concepts.

Please feel free to give me any comments or suggestion. Thanks.

-- 作者：lawyer_1

-- 发布时间：2005-1-28 23:25:00

上面的文章摘之“The Official LSAT SuperPrep”，但有一处小错误，文章中"B if only A", 是笔误，应该为"B only if A"。如果是"B if only A", 则应该是A---〉B。

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https://forum.chasedream.com/dispbbs.asp?boardid=24&id=61430

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