### 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**

**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**

**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**

**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.

Next LSAT: November 25th