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In the last lesson you learned that the rule of contraposition (or, shortly, contraposition) is a valid inference. However, beware of invalid inferences that may be confused with contraposition:
The converse of a conditional is another
conditional which has the same two
clauses, but in the reverse order.
The converse of
A → B
is:
B → A
The Fallacy of the Converse:
A fallacious inference from a conditional
to its converse. The two conditionals are
not equivalent and generally cannot be
inferred one from another.
The inverse of a conditional is another
conditional which has the same two
clauses, but negated.
The inverse of
A → B
is:
~A → ~B
The Fallacy of the Inverse:
A fallacious inference from a conditional
to its inverse. The two conditionals are
not equivalent and generally cannot be
inferred one from another
Let’s use this example to teach the inverse and the converse:
If I press the stop button (A), then the motor will turn off (B).
Invalid Inference: reverse A → B into B → A
The motor is off (B); therefore I pressed the stop button (A).
(There are other ways it could have shut off.)
Nice guys finish last: NG → FL
Then you take the converse of that:
If you finish last, you are a nice guy: FL → NG
To reason that you must be a nice guy because you finished last is the Fallacy of the Converse.
Invalid Inference: negate A → B into ~A → ~B
I didn’t press stop (~A); therefore the motor is on (~B).
(There are other ways it could have shut off.)
George: “Every instinct I have is always wrong.”
George’s principle: I (instinct) → W (wrong).
Jerry Seinfeld: “If every instinct you have is wrong, then the opposite would have to be right.”
Seinfeld uses the Fallacy of the Inverse to negate this: ~I → ~W.
So, if George just does the opposite of his instincts, he’ll always be correct (the opposite of wrong).
“In order to be irreplaceable one must always be different.” (Coco Chanel)
irreplaceable → different
1. If you are different, then you are irreplaceable.
different → irreplaceable
This is an invalid inference because this is the converse of the original statement. A statement and its converse are not logically equivalent.
2. If you are not different, then you are not irreplaceable.
~different → ~irreplaceable
This is a valid inference because this is the contrapositive of the original statement. A statement and its contrapositive are logically equivalent.
3. If you are not irreplaceable, then you are not different.
~irreplaceable → ~different
This is an invalid inference because this is the inverse of the original statement. A statement and its inverse are not logically equivalent.
Statement | Symbols | Valid/Invalid | Description |
1. In order to be irreplaceable one must always be different. | irreplaceable → different | Given | Given |
2. If you are different, then you are irreplaceable. | different → irreplaceable | Invalid | Converse |
3. If you are not different, then you are not irreplaceable. | ~different → ~irreplaceable | Valid | Contrapositive |
4. If you are not irreplaceable, then you are not different. | ~irreplaceable → ~different | Invalid | Inverse |
Next LSAT: October 09
The Fallacy of the Converse and the Fallacy of the Inverse end up as the same thing (see video).
Note: the above video mentions ‘truth tables’, which we do not cover here. We will do truth tables at At Least One.
Identify whether the following inferences are valid or invalid.
Whenever a siren is heard, my dog becomes scared. My dog did not become scared.
If the above statements are true, which one of the following is also true?
(A) A siren is not heard.
(B) A siren is heard.
(A) A siren is not heard.
Statement |
Symbols |
Valid/Invalid |
Description |
1. Whenever a siren is heard, my dog becomes scared. |
siren → dog scared |
Given |
Given |
2. If my dog does not become scared, then a siren is not heard. |
~dog scared → ~siren |
Valid |
Contrapositive |
3. If a siren is not heard, then my dog does not become scared. |
~siren → ~dog scared |
Invalid |
Inverse |
4. If my dog becomes scared, then a siren is heard. |
dog scared → siren |
Invalid |
Converse |
“In order to write about life first you must live it.” (Ernest Hemingway)
write → live
1. If you did not live life, then you cannot write about it.
~live → ~write
This is a valid inference because this is the contrapositive of the original statement. A statement and its contrapositive are logically equivalent.
2. If you lived life, then you can write about it.
~write → ~live
This is an invalid inference because this is the inverse of the original statement. A statement and its inverse are not logically equivalent.
3. If you did not write about life, then you cannot live it.
live → write
This is an invalid inference because this is the converse of the original statement. A statement and its converse are not logically equivalent.
“If you’re changing the world, you’re working on important things.” (Larry Page)
change → important things
1. If you are not changing the world, then you are not working on important things.
~change → ~important things
This is an invalid inference because this is the inverse of the original statement. A statement and its inverse are not logically equivalent.
2. If you are not working on important things, then you are not changing the world.
~important things → ~change
This is a valid inference because this is the contrapositive of the original statement. A statement and its contrapositive are logically equivalent.
3. If you are working on important things, then you are changing the world.
important things → change
This is an invalid inference because this is the converse of the original statement. A statement and its converse are not logically equivalent.
“Those who cannot remember the past are condemned to repeat it.” (George Santayana)
~remember → condemned
1. If you remembered the past, then you are not condemned to repeat it.
remember → ~condemned
This is an invalid inference because this is the inverse of the original statement. A statement and its inverse are not logically equivalent.
2. If you are condemned to repeat the past, then you did not repeat it.
condemned → ~remember
This is an invalid inference because this is the converse of the original statement. A statement and its converse are not logically equivalent.
3. If you are not condemned to repeat the past, then it means you remembered it.
~condemned → remember
This is a valid inference because this is the contrapositive of the original statement. A statement and its contrapositive are logically equivalent.
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 / 11 correct
Next LSAT: October 09