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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.
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 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
Next LSAT: January 16/1787
“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
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.
(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
“Those who don’t believe in magic will never find it.”
-Roald Dahl
“No man’s knowledge here can go beyond his experience.”
-John Locke
“It always seems impossible until it’s done.”
-Nelson Mandela
“If you expect nothing from somebody you are never disappointed.”
-Sylvia Plath
“If you judge people, you have no time to love them.”
-Mother Theresa
“No one can make you feel inferior without your consent.”
-Eleanor Roosevelt
“No one can be at peace unless he has his freedom.”
-Muhammed Ali
If I don’t get into law school, then I plan on going to business school.
If you want to become a doctor then you shouldn’t smoke.
All non-professionals in the program use software to improve their driving.
This is an adaptive drill: The questions will get harder or easier depending on your performance. You can't go backwards or change prior answers.
Complete: 0 / 5 correct
In this lesson you learned how to make a valid inference: the contrapositive. The next lesson will review invalid inferences.
Next LSAT: January 16/17