Contrapositive

We just studied conditionals, sentences that either have the form “If A, then B”, or could be rephrased in terms of such a form. Now we introduce the notion of a contrapositive of a conditional. The contrapositive of “If A, then B” is “If not-B, then not-A”. Or, in symbols:

The contrapositive of
A → B
is:
~B → ~A

The tilde sign “~” stands for negation. “~P” is read “It is not the case that P”, or shortly “Not-P”. Here is an example of a conditional and its contrapositive, first in ordinary language, then in symbols.

If mamma bear crossed the river, so did baby bear. Contrapositive: If the baby didn’t cross, then mamma didn’t, either.

M → B
~B → ~M

This lesson focuses on the first one, the contrapositive. 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 into a new inferences. So, you can turn one conditional statement into a second one.

Original Conditional

Step 1:
Reverse the clauses.

Step 2:
Negate the clause.

Two Conditionals

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

Contraposing twice brings us back where we started:

The contrapositive of
~B → ~A
is:
A → B

Why? Let us apply the steps suggested above to ~B → ~A. The result is:

~~A → ~~B

According to the rule of double negation, ~~P is equivalent to P, so ~~P and P can be replaced by each other. In the case above ~~B and ~~A are replaced by B and A respectively, and the result is:

A → B

With the aid of Steps 1 and 2 and the rule of double negation, we can see also that:

The contrapositive of
A → ~B
is:
B → ~A

The contrapositive of
~A → B
is:
~B → A

The rule of contraposition:

The rule of contraposition is a rule of inference. It tells us that from a conditional we can validly infer its contrapositive. In other words, if we have A → B as a premise, we can infer ~B → ~A as a conclusion. Remember that the contrapositive of ~B → ~A is A → B. So the rule tells us that the two conditionals are equivalent. This means that they can be inferred one from another, and also that they can replace each other. So the rule tells us:

The rule of contraposition:
A → B is equivalent to ~B → ~A
A conditional and its contrapositive are equivalent

Original:

Contrapositive:

Contrapositive Video Summary

  • 0:19 – Consider the statement: If Hilda is in law school, then Hilda took the LSAT. We can diagram this as: law school → LSAT.
  • 0:49 – 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.
  • 1:17 – Since the LSAT is a requirement, this means that if Hilda didn’t take the LSAT, then she’s not in law school. This is also a conditional statement: If Hilda didn’t take the LSAT, then Hilda isn’t in law school.
  • 2:02 – Whenever we’re given a conditional relationship, we can automatically infer another conditional from it.
  • 2:10 – 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. Diagram the second statement as: ~LSAT → ~law school.
  • 2:40 – 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: Given a conditional statement, you can always infer its contrapositive.
  • 3:09 – 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 but only the form of the statement.
  • 3:29 – Infer from this statement its contrapositive using the procedure before. It becomes: If the lights don’t turn on, then the switch isn’t flipped. Its diagram is: ~light → ~light switch.
  • 3:58 – Another example: 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: September 1687

Video Summary

“It don’t mean a thing if it ain’t got that swing.”

Diagram this as:
~has swing → ~mean anything.

Its contrapositive is:
mean anything → has swing.

“You must take the A Train, if you want to go to Harlem.”

Diagram this as:
go to Harlem → take the A train.

Infer its contrapositive:
~take the A train → ~go to Harlem

Contrapositive Drills

There are conditional claims within the following quantified sentences. Find them, put them into symbols, contrapose them, translate back into English and quantify them. The result should be a sentence that has the same meaning as the original sentence.

If it rains, then the event will be canceled. It rained.

If the above statements are true, which one of the following must also be true?
(A) If the event was canceled, then it rained.
(B) The event would have to be canceled.

Answer

(B) The event would have to be canceled.

Statement

Symbols

Valid/Invalid

Description

1. If it rains, then the event will be canceled.

rain → canceled

Given

Given

2. If the event is not canceled, then it did not rain.

~ canceled → ~rain

Valid

Contrapositive

3. If the event is canceled, then it rained.

canceled → rain

Invalid

Converse

4. If it does not rain, then the event will not be canceled.

~rain → ~ canceled

Invalid

Inverse

“Every problem is absurdly simple when it is explained to you.”

-Sherlock Holmes

Contrapositive

  1. Locate the keyword indicator: when (sufficient indicator)
  2. Locate the conditional phrase right after the keyword indicator: “. . . when it is explained to you.” (This is the sufficient condition.)
  3. Abbreviate the sufficient condition: ETY
  4. Locate the phrase other than the sufficient condition: “Every problem is absurdly simple . . .” (This phrase is the necessary condition.)
  5. Abbreviate the necessary condition: PAS
  6. Combine the two phrases in the form of: sufficient → condition
    Diagram: ETY → PAS
  7. Reverse the positions of the conditional phrases: PAS → ETY
  8. Finally, negate them both to form the contrapositive:
    Contrapositive: ~PAS → ~ETY

“Those who don’t believe in magic will never find it.”

-Roald Dahl

Contrapositive

  1. Locate the keyword indicator: those who (sufficient indicator; means the same as “people who”)
  2. Locate the conditional phrase right after the keyword indicator: “. . . don’t believe in magic . . .” (This is the sufficient condition.)
  3. Abbreviate the sufficient condition: ~BM (We use a negation because of the word ‘don’t’.)
  4. Locate the phrase other than the sufficient condition: “. . . will never find it.” (This phrase is the necessary condition.)
  5. Abbreviate the necessary condition: ~F (We use a negation because of the word ‘never’.)
  6. Combine the two phrases in the form of: sufficient → condition
    Diagram: ~BM → ~F
  7. Reverse the positions of the conditional phrases: ~F → ~BM
  8. Finally, negate them both to form the contrapositive:
    Contrapositive: ~(~F) → ~(~BM) = F → BM

“No man’s knowledge here can go beyond his experience.”

-John Locke

Contrapositive

  1. Locate the keyword indicator: No (sufficient indicator)
  2. Locate the conditional phrase right after the keyword indicator: “. . . man’s knowledge here . . .” (This is the sufficient condition.)
  3. Abbreviate the sufficient condition: K
  4. Locate the phrase other than the sufficient condition: “. . . can go beyond his experience.” (This phrase is the necessary condition.)
  5. Abbreviate the necessary condition: BE
  6. Using the sufficient indicator “no” means we have to negate the necessary condition using (~): ~(BE) = ~BE
  7. Combine the two phrases in the form: sufficient → necessary
    Diagram: K → ~BE
  8. Reverse the positions of the conditional phrases: ~BE → K
  9. Finally, negate them both to form the contrapositive:
    Contrapositive: ~(~BE) → ~K = BE → ~K

“It always seems impossible until it’s done.”

-Nelson Mandela

Contrapositive

  1. Locate the keyword indicator: until (necessary indicator)
  2. Locate the conditional phrase right after the necessary indicator: “. . . it’s done.” (This is the necessary condition.)
  3. Abbreviate the necessary condition: D
  4. Locate the phrase other than the necessary condition: “It always seems impossible . . .” (This phrase is the sufficient condition.)
  5. Abbreviate the sufficient condition: SI
  6. Using the necessary indicator “until” means we have to negate the sufficient condition (~): ~SI
  7. Combine the two phrases in the form: sufficient → necessary
    Diagram: ~SI → D
  8. Reverse the positions of the conditional phrases: D → ~SI
  9. Finally, negate them both to form the contrapositive:
    Contrapositive: ~D → ~(~SI) = ~D → SI

“If you expect nothing from somebody you are never disappointed.”

-Sylvia Plath

Contrapositive

  1. Locate the keyword indicator: If (sufficient indicator)
  2. Locate the conditional phrase right after the keyword indicator: “. . . you expect nothing from somebody . . .” (This is the sufficient condition.)
  3. Abbreviate the sufficient condition: ENS
  4. Locate the phrase other than the sufficient condition: “. . . you are never disappointed.” (This phrase is the necessary condition.)
  5. Abbreviate the necessary condition: ~D (We use a negation because of the word “never”.)
  6. Combine the two phrases in the form: sufficient → necessary
    Diagram: ENS → ~D
  7. Reverse the positions of the conditional phrases: ~D → ENS
  8. Finally, negate them both to form the contrapositive:
    Contrapositive: ~(~D) → ~ENS = D → ~ENS

“If you judge people, you have no time to love them.”

-Mother Theresa

Contrapositive

  1. Locate the keyword indicator: If (sufficient indicator)
  2. Locate the conditional phrase right after the sufficient indicator: “. . . you judge people . . .” (This is the sufficient condition.)
  3. Abbreviate the sufficient condition: JP
  4. Locate the phrase other than the sufficient condition: “. . . you have no time to love them.” (This phrase is the necessary condition.)
  5. Abbreviate the necessary condition: ~LT (We use negation because of the word “no”.)
  6. Combine the two phrases in the form: sufficient → necessary
    Diagram: JP → ~LT
  7. Reverse the positions of the conditional phrases: ~LT → JP
  8. Finally, negate them both to form the contrapositive:
    Contrapositive: ~(~LR) → ~JP = LT → ~JP

“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)

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)

Diagram Conditional Statements

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Question 5
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In this lesson you learned how to make a valid inference: the contrapositive. The next lesson will review invalid inferences.

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Next LSAT: September 16