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April 10, 2009

Conditional Reasoning: Contrapositive, Mistaken Reversal, Mistaken Negation

What is the contrapositive? What do mistaken reversal (converse) and mistaken negation (converse) look like on the LSAT?

While each bite-sized Logical Reasoning argument and each Logic Games rule may seem impossible to understand, they're pretty manageable once you've got a grip on the basics.

That's what this post will teach you.

Original statement:
If I tutor the LSAT in Manhattan, then I tutor the LSAT in New York City.

Symbolized:
Manhattan -> NYC


Mistaken reversal / converse (invalid):
If I tutor the LSAT in New York City, then I tutor the LSAT in Manhattan.

Symbolized:
NYC -> Manhattan

False because this statement implies that I tutor in a different part of NYC (Brooklyn, Queens, Staten Island, or the Bronx).


Mistaken negation / inverse (invalid):
If I do not tutor the LSAT in Manhattan, then I do not tutor the LSAT in New York City.

Symbolized:
Manhattan -> NYC

Again, this statement is false because I could be in another borough of NYC.


Contrapositive (valid):
If I do not tutor the LSAT in New York City, then I do not tutor the LSAT in Manhattan.

Symbolized:
NYC -> Manhattan

This is true. It's impossible for me to tutor the LSAT in Manhattan if don't tutor the LSAT in NYC because Manhattan is in NYC.


It's worth noting the mistaken reversal and mistaken negation are the contrapositives OF EACH OTHER. They are logically equivalent. Why? Because they're flawed for the same reason - they confuse necessary and sufficient conditions.

The sufficient condition:
-appears to the left of the arrow in the "symbolized" sections above
-is often indicated by the words "if" and "when"
-is enough to cause the necessary condition to follow, but it's not necessarily required for the necessary condition to occur
-serves as the evidence

The necessary condition:
-appears to the right of the arrow in the "symbolized" sections above
-is often indicated by the words "then" and "must"
-often appears after a comma
-is required by the sufficient condition
-serves as the conclusion

Why this is important:
Breaking down which parts of the argument are sufficient and necessary allows you to determine the evidence and conclusion. This helps you figure out potential flaws and opportunities to strengthen/weaken the argument.

Further reading:
Wikipedia's article on the contrapositive is solid.



13 comments:

  1. I've been having trouble with this on LR sections lately, especially the Must Be True questions with conditional statements. Though the LR Bible does a good job of telling you where things go, they don't really tell you who, and for me that is the critical difference between being able to understand and apply a concept and pure memorization. Thanks for this!

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  2. Just found this blog and this is what I needed. I've been doing well on the LR section but couldn't figure out why I was having so much trouble on the Logic Games - especially grouping. Finally realized this is why.

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  3. This is VERY helpful thank you!

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  4. I'm still having a bit of trouble with this section.

    You say for the sufficient condition "is enough to cause the necessary condition to follow, but it's not necessarily required for the necessary condition to follow"

    From what I understood, using your example, you tutoring in Manhattan is enough for the necessary condition ( tutoring in NYC) but it is not required for the necessary condition to occur?

    What exactly do you mean by the last part? My understanding is that you tutoring in NYC does not necessarily bring about that you tutor in Manhattan.

    Is that right?

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  5. Yes, that's right. It's possible that I tutor in Queens or Brooklyn instead.

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  6. I'm in the midst of studying for logic games and say, for example, the statement is... (from PrepTest 31 Game 1)...
    If both types of pop are on sale, then all soul is.
    I would symbolize this as: NP + UP --> NS + US
    my question is, I know that for a contrapositive I would switch them and make them negative, but do I ALWAYS change and to or, and, or to and? I seem to be a little confused... for the example I gave, would it be:
    ~NS or ~US --> ~NP or ~NP

    I understand that in some cases it clearly makes sense to switch and and or, but if I had one solid rule to follow, I think I could set myself straight. Thanks for the help!

    -Chelsie

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  7. Chelsie, I was stuck on that game myself. Glad to see I wasn't the only one stuck on a question or two from that game...Great post!

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  8. Chelsie, the rules you're thinking of are called DeMorgan's laws. Just like you said, when you "distribute" a NOT over a series of letters, you also have to switch AND to OR and OR to AND between them.

    ~(A or B) is the same as (~A and ~B)
    ~(C and D and ~F) is the same as (~C or ~D or F)

    So you did it perfectly!

    Note that for DeMorgan's laws, you have to be distributing over only ANDs or only ORs to start. However, you can work from the outside and apply the rule more than once.
    ~(A and (B or C))
    ~A or ~(B or C)
    ~A or (~B and ~C)

    Just make sure the parentheses are in the right place to start, since it affects the result.
    ~((A and B) or C)
    ~(A and B) and ~C
    (~A or ~B) and ~C

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  9. I think the Wikipedia entry really crystallized things for me:

    Let's assume the statement, If A, then B, is true:
    As I understand it, B is the range of all possibilities, which includes A. But A is not the range of all things that cause B—that’s a mistaken reversal.
    If B doesn’t occur at all, A obviously can’t occur since B and all within the range of B doesn’t occur.
    So let’s say B occurs: since there are a whole range (an infinite number?) of possibilities of things that cause B, it's impossible to determine that A caused B.
    But if A occurs, it's impossible to say B does NOT occur (in other words, B MUST occur), since A is in the range of things that cause B to occur.
    My own example: If I bend this pencil, it will break.
    There are a whole range (an infinite number?) of possibilities of what can break the pencil; my bending it is just one in that range of possibilities. So, if I happen upon a broken pencil, it's impossible to determine that I bent it.
    But if I bend it, it will absolutely break, since my bending is in the range of things that causes it to be broken. If the pencil isn’t broken at all, then obviously I didn’t bend it.
    One more: If A is on the committee, B is on the committee.
    There are an infinite range of things that can cause B to be on the committee, none of which can possibly be determined if the only information given is that B is on the committee. But since A is one in an infinite range of possibilities that cause B to be on the committee, it follows that if A is on the committee, B must be on the committee. There is no situation where A is on the committee and B is NOT on the committee.
    If B is not on the committee, neither A or anyone else can be on the committee (unless there’s another rule in the game..)

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  10. Steve, can you elaborate more on the contrapositive?

    What would be the contrapositive of the following?

    if A happens, B or C or both happen.
    --> is it B+C do not happen, A does not happen?

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  11. Can someone tell me if its possible to have contrapositives for "most" or "some" statements. I don't think they should but I'm not a 100% confident. Steve your input would be really appreciated.

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  12. "This is true. It's impossible for me to tutor the LSAT in Manhattan if don't tutor the LSAT in NYC because Manhattan is in NYC."

    Unless you were tutoring someone in Manhattan, Kansas...........

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  13. I am a little confused about this one.
    "If Sarah were a concert pianist for a major Orchestra, she would be famous. She is not a concert pianist since she is not famous."
    Is this right or wrong?
    I think it is wrong because it didn't state the "major Orchestra" part in the second sentence.
    So, the contrapositive statement would be right if the question stem becomes "she is not a concert pianist for a major Orchestra since she is not famous. Am I right?
    Or the reason it is wrong because the author does not condsider that Sarah could be a concert pianist with a minor Orchestra, which means didn't consider all the possible conditions in making the conditional reasoning.
    Hope you could answer this question for me. Thank you so much

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