2. Contrapositive

We just studied if A, then B. The contrapositive is if you reverse and negate that.

If momma bear crosses the river, so does baby bear. Contrapositive: If the baby didn’t cross, then momma didn’t, either.

Those who contrapose will do well on the LSAT (contrapose: the act of making a contrapositive). Why? You can take conditional rules and double them by contraposing the original conditional.

Original Conditional

Step 1:
Reverse Values

Step 2:
Negate Values to make Contrapositive

Two Valid Conditionals

And, poof! It becomes two valid conditionals after you contrapose the original (reverse the statements and negate them).

Original:

Contrapositive:

Contrapositive Video Summary

  • 0:19 – Let us consider the statement: If Hilda is in law school, then Hilda took the LSAT. We can diagram this as: law school → LSAT.
  • 0:49 – From this statement, we can infer the following ideas:
    • Taking the LSAT is a requirement for going to law school. (necessary condition)
    • If you don’t meet the requirement for something then you don’t get that thing.
    • “can’t have the first without the second”
  • 1:17 – We can then infer from these that since the LSAT is a requirement, this means that if Hilda didn’t take the LSAT, then she’s not in law school. We can notice that this is also a conditional statement: If Hilda didn’t take the LSAT, then Hilda isn’t in law school.
  • 2:02 – This shows us that whenever we’re given a conditional relationship, we can automatically infer another conditional relationship from it.
  • 2:10 – We can diagram the second statement by taking the necessary condition of the first statement, making it sufficient then negating it; and taking the sufficient condition of the first statement, making it necessary then negating it. We can diagram the second statement as: ~LSAT → ~law school.
  • 2:40 – We can see that the second statement is the exact reverse of the first one. This second statement that we inferred from the first one is called the “contrapositive”.
  • 2:56 – Note: Whenever you are given a conditional statement, you can always infer its contrapositive.
  • 3:09 – Let us consider another example: If the switch is flipped, then the light turns on. Its diagram is: light switch → light.
  • 3:21 – Note: The contrapositive does not depend on the content of the statement. It’s only concerned with the form of the statement.
  • 3:29 – We can then infer from this statement its contrapositive using the procedure before. It then becomes: If the lights don’t turn on, then the switch isn’t flipped. Its diagram is: ~light → ~light switch.
  • 3:58 – As another example, consider the statement: If you love me, then you’ll take a shower. Its diagram is: love me → shower.
    Its contrapositive is: If you don’t take a shower, then you don’t love me. Its diagram is: ~shower → ~love me.
  • 4:18 – Recap: The “contrapositive” is a valid inference from the conditional statement. To diagram it, simply switch and negate the conditions.

Next LSAT: October 28th87

Example of a conditional statement: It don’t mean a thing if it ain’t got that swing. We can diagram this as follows: ~has swing → ~mean anything. Hence, we can infer from this its contrapositive which is: mean anything → has swing.

Another example: You must take the A Train, if you want to go to Harlem. We can diagram this as: go to Harlem → take the A train. We can then infer its contrapositive: ~take the A train → ~go to Harlem.

Interactive Learning Games

Contrapositive Drills

If I don’t get into law school, then I plan on going to business school.

Contrapositive

  1. Locate the sufficient indicator: If
  2. Locate the conditional phrase right after the sufficient indicator: “. . . I don’t get into law school . . .” (This phrase is the sufficient condition.)
  3. Abbreviate the sufficient condition: ~LS (“. . . not get into law school . . .“)
  4. Locate the necessary condition: “. . . I plan on going to business school.” (The phrase other than the sufficient condition is the necessary condition.)
  5. Abbreviate the necessary condition: GBS
  6. Combine the two conditions in the form of: sufficient → condition
    Diagram: ~LS → GBS (don’t get into law school → plan on going to business school)
  7. Then, reverse the positions of the conditional phrases: GBS → ~LS
  8. Finally, negate their values to form the contrapositive:
    ~GBS → LS (did not go to business school → got into law school)

If you want to become a doctor then you shouldn’t smoke.

Contrapositive

  1. Locate the sufficient indicator: If
  2. Locate the conditional phrase right after the sufficient indicator: “. . . you want to become a doctor . . .” (This phrase is the sufficient condition.)
  3. Abbreviate the sufficient condition: BD
  4. Locate the necessary condition: “. . . you shouldn’t smoke.” (The phrase other than the sufficient condition is the necessary condition.)
  5. Abbreviate the necessary condition: ~S (“. . . should not smoke . . .“)
  6. Combine the two conditions in the form of: sufficient → condition
    Diagram: BD → ~S (want to become a doctor → not smoke)
  7. Then, reverse the positions of the conditional phrases: ~S → BD
  8. Finally, negate their values to form the contrapositive:
    S → ~BD (smoke → do not want to become a doctor)

All non-professionals in the program use software to improve their driving.

Contrapositive

  1. Locate the sufficient indicator: All
  2. Locate the conditional phrase right after the sufficient indicator: “. . . non-professionals in the program . . .” (This phrase is the sufficient condition.)
  3. Abbreviate the sufficient condition: ~P (“. . . not professionals . . .“)
  4. Locate the necessary condition: “. . . use software to improve their driving.” (The phrase other than the sufficient condition is the necessary condition.)
  5. Abbreviate the necessary condition: USD
  6. Combine the two conditions in the form of: sufficient → condition
    Diagram: ~P → USD (non-professionals → use software to improve their driving)
  7. Then, reverse the positions of the conditional phrases: USD → ~P
  8. Finally, negate their values to form the contrapositive:
    ~USD → P (not use software to improve driving → professional)

“No one can make you feel inferior without your consent.”
-Eleanor Roosevelt

Contrapositive of Quote

  1. Locate the sufficient indicator: No
  2. Locate the conditional phrase right after the sufficient indicator: “. . . one can make you feel inferior . . .” (This phrase is the sufficient condition.)
  3. Abbreviate the sufficient condition: MFI
  4. Locate the necessary condition: “. . . without your consent.” (The phrase other than the sufficient condition is the necessary condition.)
  5. Abbreviate the necessary condition: ~C
  6. Using the sufficient indicator “no” means putting a tilde (~) on the necessary condition for negation: ~(~C) = C
  7. Combine the two conditions in the form of: sufficient → condition
    Diagram: MFI → C (make you feel inferior → with consent)
  8. Then, reverse the positions of the conditional phrases: C → MFI
  9. Finally, negate their values to form the contrapositive:
    ~C → ~MFI (without consent → cannot make you feel inferior)

“No one can be at peace unless he has his freedom.”
-Muhammed Ali

Contrapositive of Quote

  1. Locate the necessary indicator: Unless
  2. Locate the conditional phrase right after the necessary indicator: “. . . he has his freedom.” (This phrase is the necessary condition.)
  3. Abbreviate the necessary condition: F
  4. Locate the sufficient condition: “No one can be at peace . . .” (The phrase other than the necessary condition is the sufficient condition.) *Although this phrase uses the indicator “no”, the idea of the phrase also means “one cannot be at peace”.
  5. Abbreviate the sufficient condition: ~P (“. . . one cannot be at peace . . .“)
  6. Using the necessary indicator “unless” means putting a tilde (~) in the sufficient condition for negation: ~(~P) = P
  7. Combine the two conditions in the form of: sufficient → condition
    Diagram: P → F (be at peace → has freedom)
  8. Then, reverse the positions of the conditional phrases: F → P
  9. Finally, negate their values to form the contrapositive:
    ~F → ~P (has no freedom → not at peace)
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