# Conditional Statements and Contrapositives (Chapter 13)

• July 27, 2016 at 9:48 pm #2314
Participant

Dear Mr. Kim,

Thank you for creating such a great resource for LSAT prep. With your book I’ve already come a long way from where I started in understanding various concepts on the test. Today I got stuck on a certain part of Chapter 13.

So far, I’ve been able to grasp the concept of creating the contrapositive for any conditional statement. Also, the logic behind the statement and contrapositive makes sense with context (rain and park example). Without context, I can figure out how compound conditional statements and their contrapositives work logically. However, I am really struggling to identify the logic behind the contrapositive of non-compound statements (although I can easily write out the contrapositives). I’ve tried creating my own contexts for the problems, but this does not always seem to work.

In other words, I can write out the contrapositives for non-compound statements, but I don’t understand the logic behind them.

1. Original: A ——> B Contrapositive: /B ——> /A

I think I can see the logic in this one. A triggers B; if no B, then no A because A would have triggered B. This is not to say B can’t exist without A. Just that if no B it’s guaranteed no A.

2. Original: /X ——> Y Contrapositive: /Y ——> X

I do not see how the logic works in this example. This contrapositive does not seem to be an equal opposite to the original statement even though it must be.

3. Original: L ——> /M Contrapositive: M ——> /L

I think I understand this one. L triggers no M; if M then no L because M can’t be included if L is.

4. Original: /G ——> /H Contrapositive: H ——> G

If no G then no H, how can H guarantee G? I don’t see how these are equal opposites.

I’m sure I’m missing something very basic, but I can’t figure it out. Thanks so much for any help, and thanks again for writing such a great study resource.

• July 27, 2016 at 10:10 pm #2315
LSAT Dan
Participant

With respect to #4, an example might clear it up:
/C –> /V. (If you’re not a citizen, you cannot vote)
V –>. C. (If you can vote, you’re a citizen).

It works the same way it does in the example where nothing is negated. V must guarantee C, because (original statement) if we didn’t have C, we couldn’t have V.

For #2, consider a game show where the winner can choose one (but only one) of two prizes:

/C –> V. (If you don’t take the car, you can have the vacation).
/V –> C. (If you don’t take the vacation, you can have the car).

/V guarantees C, because if we didn’t have C, we could have V.*

^ In this example, C and V would be “eligibility for the car/vacation”.

Hope this helps

-Dan

• July 28, 2016 at 11:39 am #2317
Participant

Thank you so much for your reply, LSAT DAN! Yes, the examples definitely do help to clear up some of the confusion with numbers 2 and 4. Independent of the examples though, I’m still having trouble reasoning why the contrapositives of these statements do, in fact, create equal and opposite logical statements. By themselves, the contrapositives of numbers 2 and 4 seem to say something totally different than their original statements.

For number 2, /C –> V; having no C triggers V. So far so good. But to me its equal and opposite /V –> C seems to stand by itself as a different statement altogether. Even though no C ensures V, why does no V then guarantee C? I can use number 4, scenario 2, on page 180 as an example of my confusion: “O will be performed if L is not.” (/L –> O) Not singing L triggers singing O; how can it then be guaranteed that not singing O will guarantee L? In the vacation and car example, you had to choose one, and one would prevent the other. In the singing example though, there are other songs available, and not all songs will even be used. Why is one guaranteed to exist with the other’s absence? If that is a true logical conclusion, is it then also logical to conclude with this conditional statement that either C or V will definitely be included in an in/out ordering game and either C or V will go in each group of a grouping game (ingredient and car service examples)?

Number 4 now makes sense! V must guarantee C, because if we didn’t have C, we couldn’t have V cleared it up for me. Thank you! Like with the first example, it seems to follow that C does not necessarily you’ll have V.

Maybe I’m getting hung up on something that doesn’t much matter (or maybe I’m missing something totally obvious). I’m taking Mr. Kim’s advice seriously though, and want to make sure I fully understand all underlying logic behind a contrapositive before I use the technical “reverse and negate” method. Thanks again for your help!

• July 28, 2016 at 1:25 pm #2318
LSAT Dan
Participant

Here’s another way to look at #2, and one that I think you’ll find helpful. Let’s go back to your example; I’ll cut and paste the way you phrased it:

“O will be performed if L is not.” (/L –> O) Not singing L triggers singing O; how can it then be guaranteed that not singing O will guarantee L?

So, our initial rule is “If not L, then O.” To analyze the contrapositive, take “not O” as your starting point:

YES :
NO: O

There are two possibilities for L; it will be sung, or it won’t. Let’s look at them separately. We’ll call the two possibilities “Variation 1” and “Variation 2,” respectively:

YES: L
NO: O

YES: (neither)
NO: LO

Notice that variation 2 violates our original rule, “If not L, then O”! So variation 2, just given the initial rule, is not a possibility. Which means that once O is in the “No” column, L must be in the “Yes” column.

In other words… /O –> L – the contrapositive.

Basically, putting both L and O out violates the original rule, so as soon as EITHER one is out, the other must be in. Hope this helps you get your head around it; these definitely take some getting used to.

-Dan

• July 28, 2016 at 3:01 pm #2320