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April 7, 2011

Negating Conditional Statements in Logical Reasoning

A common technique for solving LSAT Logical Reasoning questions (particularly, Necessary Assumption questions) is to negate each of the answer choices. The correct answer choice, when negated, destroys the argument by preventing the conclusion from logically following from the evidence.

Sometimes, answer choices contain conditional statements, rather than simply containing a single clause.

The proper negation of a conditional statement can often be trickier than the negation of a single clause.

When negating a conditional statement, keep in mind that your goal is NOT to negate the variables themselves.

For example, if we have the statement X → Y, we can do 4 different types of modifications that involve negating the variables included, but none of them is truly a negation of the statement as a whole.

We could negate the sufficient condition, resulting in NOT X -→ Y.

We could negate the necessary condition, resulting in X -→ NOT Y.

We could negate both the sufficient condition and the necessary condition, resulting in NOT X → NOT Y.

We could take the contrapositive of the statement, resulting in NOT Y → NOT X.

However, this isn't what you should be doing when your goal is to negate a conditional statement.

A conditional statement is composed of a sufficient condition and a necessary condition.

It's claiming that one thing is sufficient to guarantee, to require, another thing to occur.

The negation of this concept would be that the thing previously claimed to be sufficient to guarantee another is no longer sufficient.

So, if we had been originally told that X requires Y, the negation of that statement would be that X does not require Y -- that X is no longer sufficient to guarantee Y.

Take the following statement:

If I have pizza, then I will be happy.

P → H

Suppose someone then claims this statement is not true. (For example, they say that if I had pizza, but was repeatedly punched in the face, I wouldn't be happy despite my possession of pizza).

As such, pizza is not truly sufficient to guarantee my happiness, because I also need to not be repeatedly punched in the face in order to be happy.

We can diagram this information as P --/--> H

It's simply P, followed by an arrow with a slash through it, followed by H.

Just as a conditional statement is valid information and useful information, knowing that two particular variables do NOT have a sufficient-necessary relationship is also useful information.

It tells us that one thing alone is NOT enough to require another.

Please see, for example, the diagramming technique used in my recent blog post on Logical Reasoning: Inference Questions and the Contrapositive.

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Examples of diagramming in this way:

Parallel Flaw question:

Take, for example, a Parallel Flaw question - PrepTest 36 (December 2001), Section 3, Question 19 (page 275 in Next 10).

The evidence tells us that liking turnips is not sufficient to guarantee that one likes potatoes.

We can diagram this as T --/--> P.

(Liking turnips doesn't require that you like potatoes.)

The (flawed) conclusion tells us that liking potatoes is not sufficient to guarantee that one likes turnips.

We can diagram this as P --/--> T.

(Liking potatoes doesn't require that you like turnips.)

Even though this is telling us that we DON'T know something (just because someone likes turnips, this doesn't guarantee that they like potatoes), this doesn't mean that it's not worth writing down or knowing.


Necessary Assumption question:

Additionally, let's look at the choices in a Necessary Assumption question - PrepTest 33 (December 2000), Section 1, Question 19 (page 157 in Next 10). Negating answer choices in Necessary Assumption questions is a useful technique, as I described in my blog post titled, Necessary Assumption Questions, Negation Test, and Must Be True Qs.

Choice A can be rewritten to state, "If a demagogue can enlist the necessary public support to topple an existing regime, then a comprehensive general education system must have been in place" or DNPS → CGES

To negate this, we can say, "a demagogue can enlist the necessary public support to topple an existing regime EVEN IF a comprehensive general education system is not in place" or DNPS ---/--> CGES


Choice B can be rewritten to state, "General awareness of injustice in a society requires literacy" or GAI → L

To negate this, we can say, "general awareness of injustice in society DOES NOT require literacy" or GAI --/--> L


Choice C can be rewritten to state, "If you have a comprehensive system of general education, then you will tend to preserve benign regimes' authority" or CSGE → TPBRA

To negate this, we can say, "Even if you have a comprehensive system of general education, it may not tend to preserve benign regimes' authority" or CSGE --/--> TPBRA


Choice D can be rewritten to state, "If you have a lack of general education, then your ability to differentiate between legitimate and illegitimate calls for reform will be affected" or LGE → ADA

To negate this, we can say, "Even if you lack general education, your ability to differentiate between legitimate and illegitimate calls for reform won't necessarily be affected" or LGE --/--> ADA


Choice E can be rewritten to state, "If a benign regime doesn't provide comprehensive general education, then it'll be toppled by a clever demagogue" or NOT PCGE → TCD

To negate this, we can say, "Even if a benign regime doesn't provide comprehensive general education, it won't necessarily be toppled by a clever demagogue" or NOT PCGE --/--> TCD


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Please leave your questions for each other about properly negating conditional statements (or negating any kind of statements at all) in the comments and help each other out!




10 comments:

  1. Finally! After having this explained by other test preps, your explanation makes perfect sense and is easiest to digest.

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  2. "If A, then B" is a conditional relationship. Attack the arrow and thus negate its conditionality. This means that A is not sufficient to require B. So:
    A->B becomes A-/->B, which one might write in words like this: Even if A, then not necessarily B. (Even if is not conditional BTW.) By destroying the conditionality, you can apply this negated answer to the argument. If it the argument falls apart, then you have found the necessary assumption.

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  3. Hi Steve,

    I am a little confused after reading this. For example, when working on logic games, the conditional A --/--> B means that if A occurs, B WILL NOT OCCUR; but using your example, you would refer to the negated conditional as: if A occurs, B is not required (IT MAY OR MAY NOT OCCUR).

    To say that B will not occur is not the same as saying that B may or may not occur. Can you please explain when it's appropriate to interpret the negated conditional as a certain WILL NOT versus using an uncertain MAY OR MAY NOT?

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    Replies
    1. the way I understand this diagramming off the conditional if a then B simply refers to a necessary condition but it doesn't guarantee that B will follow
      the reason for this is because the difference between necessary and sufficient is based on what is required – the necessary and what guarantees an outcome the sufficient condition. I hope this helps

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  4. Well you just said it. When you are in LG then you know that it will not occur.

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  5. Hi Steve,
    Does A--/-->B mean there is no relationship between A and B?
    So a) A is not sufficient to make B happen, as well as b) B is not necessary if A occurs?

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  6. This is way late but to the post above, it could mean dogs are not necessarily border collies, but dogs are always mammals. IE Whenever A occurs, it is not a given that B follows or is true.

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  7. I'm learning logic in Math class, and I just don't understand the difference between a contrapositive and a negation. Or maybe they are the same thing?

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  8. a necessary condition is a required condition
    a sufficient condition is a guaranteed condition

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  9. For choice A, why do you write 'even if' and 'not'? When I tested the theory I just wrote "a comprehensive general education system does not need to be in place". Is that valid or did I err somewhere?

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