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

StatementSymbolsValid/InvalidDescription
1. In order to be irreplaceable one must always be different.irreplaceable → differentGivenGiven
2. If you are different, then you are irreplaceable.different → irreplaceableInvalidConverse
3. If you are not different, then you are not irreplaceable.~different → ~irreplaceableValidContrapositive
4. If you are not irreplaceable, then you are not different.~irreplaceable → ~differentInvalidInverse

Next LSAT: September 16

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.

Answer

(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.

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

~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.

Answer

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.

Answer

~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.

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 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.

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.

Invalid Inferences

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Next LSAT: September 16