Free LSAT Course > Logical Reasoning > Formal Logic > Some Invalid Inferences

#### 2. Invalid inferences

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:

### Converse and Inverse

#### Converse

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.

#### Inverse

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:

#### 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.

### Table Guide

 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: Jun 10/Jun 11

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.

### The Converse and Inverse of a Conditional Statement Video Summary

• 0:04 – The converse of a conditional statement is when we reverse the order of the sufficient and necessary conditions. (p → q; converse: q → p)
• 0:17 – Note: A conditional statement and its converse have different meanings.
• 0:50 – For example: If it’s a dog, then it’s a mammal. Diagram this as: D → M.
• 1:19 – Its inverse is: If it’s a mammal, then it’s a dog. Diagram this as: M → D.
• 1:37 – 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 – But its converse is false because a mammal does not necessarily have to be a dog. There are other kinds of mammals that are not dogs.
• 2:08 – A conditional statement and its converse are not logically equivalent.
• 2:09 – The inverse of a conditional statement is 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 – The converse and the inverse (which is the contrapositive of the converse) of a conditional statement are logically equivalent.
• 3:51 – But they are not logically equivalent to the original conditional and the original contrapositive.
• 3:59 – 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 animals like cats that are not dogs, but are still mammals.

#### Valid or Invalid?

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.