Converse and Inverse

In the last lesson you learned that the contrapositive is a valid inference. But there are invalid inferences, like the converse (flipping a conditional) and the inverse (taking the negative of a conditional).

A → B

If I press the stop button (A), then the motor will turn off (B).

Fallacy of the Converse

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.

Fallacy of the Inverse

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.

Next LSAT: November 25th

Valid or Invalid?

Identfiy whether the following inferences are valid or invalid.

“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.

Answer

~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.

Answer

live → write

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.

Answer

~write → ~live

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.

Answer

~change → ~important things

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 working on important things, then you are not changing the world.

Answer

~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.

Answer

important things → change

This is an invalid inference because this is the inverse of the original statement. A statement and its inverse 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.

Answer

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.

Answer

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.

Answer

~condemned → remember

This is a valid inference because this is the contrapositive of the original statement. A statement and its contrapositive are logically equivalent.

The Fallacy of the Converse and the Fallacy of the Inverse end up as the same thing (see video).

The Converse and Inverse of a Conditional Statement Video Summary

  • 0:04 – We get the converse of a conditional statement when we reverse the order of the sufficient and necessary conditions. (p → q; converse: q → p)
  • 0:17 – Note: These are not logically equivalent statements. A conditional statement and its converse have different meanings.
  • 0:50 – For example: If it’s a dog, then it’s a mammal. We can diagram this as: D → M.
  • 1:19 – Its inverse then is: If it’s a mammal, then it’s a dog. We can then diagram this as: M → D.
  • 1:37 – We can reasonably say that the original statement is true. If it’s a dog, then it is a mammal because one of the properties of a dog is being a mammal.
  • 1:51 – However, we see that its converse is false because a mammal does not necessarily have to be dog. There are other kinds of mammals that are not dogs.
  • 2:08 – The point here is simple. A conditional statement and its converse are not logically equivalent.
  • 2:09 – We get the inverse of a conditional statement when we negate both its sufficient and necessary conditions. (p → q; inverse: ~p → ~q)
  • 2:25 – This is like taking the contrapositive of the converse of a conditional. *p → q; converse: q → p; inverse: ~p → ~q (contrapositive of the converse)
  • 3:27 – We can then infer that the converse and the inverse (which is the contrapositive of the converse) of a conditional statement are logically equivalent.
  • 3:51 – However, they are not logically equivalent to the original conditional and the original contrapositive.
  • 3:59 – Let us return to our previous example: If it’s a dog, then it’s a mammal. We diagram this as: D → M.
  • 4:12 – Its inverse is: If it’s not a dog, then it’s not a mammal. We can then diagram this as: ~D → ~M. This is false because there are other non-dog animals like cats that are not dogs, but are still mammals.
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Next LSAT: November 25th