### Arguments and Contrapositives | Necessary and Sufficient Assumptions I spend a great deal of time talking about the difference between Necessary Assumption and Sufficient Assumption questions in the LSAT's Logical Reasoning section.

Arguments assume a link between the evidence and conclusion presented - this link can often easily be framed as a conditional statement.

Because the contrapositive of this statement is simply a rewording of the argument itself, the contrapositive of that statement is both necessary and sufficient for that argument to work.

As such, it can serve as the correct answer to both Necessary Assumption and Sufficient Assumption questions.

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Necessary Assumption
Let's start with the fact that the contrapositive of an argument's evidence-conclusion link can serve as a necessary assumption.

I mean that if X -> Y is an argument, then NOT Y -> NOT X is a necessary assumption (an assumption required) for that argument to be valid.

After all, if the contrapositive were negated, then the original statement would not be valid either, and the argument wouldn't be valid. As such, the original statement requires the contrapositive to be true as well.

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Sufficient Assumption

Additionally, the contrapositive of an argument's evidence-conclusion link can serve as a sufficient assumption.

I mean that if X -> Y is an argument, then NOT Y -> NOT X is an assumption that is sufficient for the argument to be valid. What I mean is that if we're told, as new information, that NOT Y -> NOT X is valid, it must be the case that the argument itself (X -> Y) is also valid. This is because if the contrapositive of a statement is valid, then the original must also be valid, since they're logically equivalent.

This is all a bit abstract, but let's look at it with a couple of examples from real LSAT questions:

Necessary Assumption example:

(Please see PrepTest 36 (December 2001 LSAT), Section 3, Question 16 - page 275 in Next 10)

In this argument, the stimulus tells us (paraphrased):

Because reptiles can't make big behavioral changes when the environmental changes a lot, reptiles aren't capable of engaging in advanced thought

In shorthand, the argument is saying:

NOT capable of big behavior changes with environmental changes -> Not capable of complex thought

The contrapositive of this statement would be something like:

Capable of complex thought -> capable of big behavioral changes with environmental changes

In other words:

If an animal is capable of complex thought, then it must be capable of making big behavioral changes as the environment goes through big changes.

Choice D of this question pretty much says just that.

Again, if an original conditional statement that forms the core of an argument is considered to be true, then it is required that its contrapositive also be true in order for that argument to work.

Sufficient Assumption example:

(Please see PrepTest 36 (December 2001 LSAT), Section 1, Question 26 - page 261 in Next 10)

In this argument, the stimulus tells us (paraphrased):

Because Vermeer used expensive props, it must not be due to a scarcity of props that he kept using the same props over and over.

In shorthand, the argument is saying:

\$ props -> NOT due to small # of props that V kept reusing them

The contrapositive of this statement would be something like:

If it were due to a small # of props that V kept reusing them, then NOT \$ props.

In other words:

If it were due to a small number of props that Vermeer kept reusing the same ones, then he wouldn't have been using expensive props in the first place.

Choice E of this question pretty much says just that.

Again, if we're told, as new information in an answer choice, that the contrapositive of the argument is guaranteed to be true (or is "assumed"), then the original version of that conditional statement (the one in the argument) must also be true, and the argument is valid.

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1. 2. The Vermeer SA example begs one question: in the stim. how would one see that it is conditional given the lack of any SA or NA signal word?

3. "it was clearly"

4. The stem clearly says "follows logically if which one of the following is assumed", i.e. NA.

5. Anonymous (2/6/12): The indication that the conclusion is a conditional is the presence of 'thus it was clearly'. This should tip you off that what follows (not for lack of props that the recurrent items were used) is a Necessary Condition. The question does a tricky job adding a filler statement in there between the sufficient and necessary conditions (while we might speculate...). The Sufficient condition is the props used are expensive.

We can symbolize this as: Expensive Props --> ~Lack of props that recurrent items were used.

Also: Your statement "follows logically if which one of the following is assumed" is incorrect. This stem is typically associated with 'Sufficient' Assumption questions. (Look for words like 'requires', 'demands', 'necessary', etc for "Necessary Assumption" questions.

Hope this helps

6. I'm not really getting this section. I don't really understand the difference between sufficient assumption and necessary assumption questions. I also don't understand the importance of knowing the difference between these two question types. Can you help explain this a little more? I read the other blog entries on this but it just isn't clicking.

1. Was this clarified for you? If so, can you help me understand? I am also confused even after reading the other blogs.

7. the word "dearth" threw me off when I originally did this question, as I didn't know what it meant (dearth: A lack or scarcity of something). Considering its a key term in locating the right answer, its what made the question difficult for me.

I don't think the vocabulary is as arcane on newer tests.

8. Thanks to Clay for the explanation :). But I am still having a difficult time understanding how this is a conditional argument? I know what the indicator words are for necessary conditions (must, then, required, unless, until, except, without, etc.). I just don't understand how "it was clearly" indicates a necessary condition. Some help would be greatly appreciated!! :)

9. Thanks to Clay for the explanation :). But I am still having a difficult time understanding how this is a conditional argument? I know what the indicator words are for necessary conditions (must, then, required, unless, until, except, without, etc.). I just don't understand how "it was clearly" indicates a necessary condition. Some help would be greatly appreciated!! :)

10. So...after some thought I think I understand why #26 is conditional. I also think that I may have a better understanding of the difference between necessary and sufficient assumptions. Someone please correct me if I am wrong (or right).

First, I will address why #26 is conditional. Start with the question stem-which indicates that we are dealing with a sufficient assumption. All sufficient assumptions assume P-->C (that a premise leads to the conclusion). For #26, the premise is "the props V used were expensive" and the conclusion is "it was clearly not for a lack of props that the recurrent items were used". Answer choice E is the contrapositive of this.

As for the difference between necessary and sufficient assumptions, the formula for a necessary assumption is premise + answer choice (which provides the missing link between the premise and conclusion)= the conclusion. And for sufficient assumptions the formula is premise + conclusion while the correct answer will provide the contrapositive of that relationship...

But...now that I think about it...my formula for the necessary assumption is for general necessary assumptions, that formula will not work for necessary assumptions that are conditional. When a necessary assumption is conditional, look for an answer choice that addresses the contrapositive-just like you would for a sufficient assumption.

11. So...after some thought I think I understand why #26 is conditional. I also think that I may have a better understanding of the difference between necessary and sufficient assumptions. Someone please correct me if I am wrong (or right).

First, I will address why #26 is conditional. Start with the question stem-which indicates that we are dealing with a sufficient assumption. All sufficient assumptions assume P-->C (that a premise leads to the conclusion). For #26, the premise is "the props V used were expensive" and the conclusion is "it was clearly not for a lack of props that the recurrent items were used". Answer choice E is the contrapositive of this.

As for the difference between necessary and sufficient assumptions, the formula for a necessary assumption is premise + answer choice (which provides the missing link between the premise and conclusion)= the conclusion. And for sufficient assumptions the formula is premise + conclusion while the correct answer will provide the contrapositive of that relationship...

But...now that I think about it...my formula for the necessary assumption is for general necessary assumptions, that formula will not work for necessary assumptions that are conditional. When a necessary assumption is conditional, look for an answer choice that addresses the contrapositive-just like you would for a sufficient assumption.

12. THANK YOU!