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 my Logical Reasoning course.

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.