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Answers

**Example 1**

If all dogs are mammals, then all dogs are reptiles.

All dogs are mammals.

Therefore, all dogs are reptiles.

This argument is valid. The fact that the first premise and conclusion are false doesn’t mean the argument form is logically invalid. This same argument form can be used to make good arguments. The argument form is “If A, then B. A. Therefore, B.” A good argument with this argument form is the following:

- If all dogs are mammals, then all dogs are animals.
- All dogs are mammals.
- Therefore, all dogs are animals.

How can we prove the argument is valid? We can show that it’s impossible to form a formal counterexample. We can assume the argument is invalid and prove that such an assumption is impossible because it will lead to self-contradiction.

The easiest way to realize that this argument form is valid is to realize what it means to say “If A, then B.” This statement means “If A is true, then B is true” or “B is true whenever A is true). That also implies that if B is false, then A must be false.

We can prove the argument form is valid using the following reasoning:

- The counterexample must have true premises, and a false conclusion.
- In that case we assume that “If A, then B” is true because it’s a premise, A is true because it’s a premise, and B is false because it’s our conclusion.
- In that case ‘A’ must be false because “if A, then B” is assumed to be true, and ‘B’ is assumed to be false. (Consider the statement, “If dogs are mammals, then dogs are animals.” If we find out that dogs aren’t animals, then they can’t be mammals. If the second part of a conditional statement is false, then the first part must be false.)
- Therefore, ‘A’ is true and false. That’s a contradiction.
- The assumption that the argument is has true premises and a false conclusion leads to a contradiction.
- Therefore, the argument form can’t be invalid.
- Therefore, the argument must be valid.

**Example 2**

If all dogs are mammals, then all dogs are animals.

All dogs are animals.

Therefore, all dogs are mammals.

Although the premises and conclusion are true, the argument form is invalid. The argument form is the following:

- If A, then B.
- B.
- Therefore, A.

We can then replace the variables to create a counterexample that uses this argument form with true premises and a false conclusion. The variables will be replaced with the following statements:

A: All dogs are reptiles.

B: All dogs are mammals.

This leads to the following counterexample:

- If all dogs are reptiles, then all dogs are animals.
- All dogs are animals.
- Therefore, all dogs are reptiles.

Both premises are true, but the conclusion is false. Therefore, the argument form must be invalid.

**Example 3**

Either it’s wrong to indiscriminately kill people, or it’s not wrong to kill someone just because she has red hair.

It’s wrong to kill someone just because she has red hair.

Therefore, it’s wrong to indiscriminately kill people.

This time the premises are true, the conclusion is true, and the argument form is valid. The argument form is the following:

- Either A or not-B. (“Not-B” means “B is false.”)
- B (is true).
- Therefore A.

An example of a good argument with this argument form is the following:

- Either dogs are warm-blooded or dogs aren’t mammals.
- Dogs are mammals.
- Therefore, dogs are warm-blooded.

Let’s try to prove this argument is valid by proving it’s impossible to provide a counterexample. We can assume it’s invalid only to find out that such an assumption will lead to a contradiction.

- We assume the premises are true and the conclusion is false.
- We assume ‘A’ is false because it’s the conclusion.
- We assume ‘B’ is true because it’s a premise.
- We assume “Either A or not-B” is true.
- “Either A or not-B” requires that either A is true or not-B is true.
- We know A is false, so not-B must be true.
- Therefore, B and not-B are both true. That’s a contradiction.
- Therefore, the the assumption that the premises are true and conclusion is false leads to a contradiction.
- Therefore, the argument form can’t be invalid.
- Therefore, the argument form is valid.

**Example 4**

Either disciplining people is always wrong or it’s not always wrong to discipline people for committing crimes.

Disciplining people hurts them.

Therefore, disciplining people is always wrong.

This argument is invalid, and it’s already a counterexample because the premises are true and the conclusion is false. The argument form is the following:

- Either A or not-B.
- C
- Therefore, B.

Another counterexample is the following:

- Either murder is always appropriate or it’s not always appropriate to murder people for making you angry.
- Murdering people hurts them.
- Therefore, murder is always appropriate.

**Example 5**

It’s often good to give to charity.

If it’s often good to give to charity, then the Earth is round.

Therefore, the Earth is round.

This argument is logically valid, even though the premises seem to lack relevance. Logical validity doesn’t guarantee relevance.

The argument form is the following:

- A.
- If A, then B.
- Therefore, B.

This is basically the same argument form as the first example, so no further proof of validity is required.

**Example 6**

The death penalty sometimes leads leads to the death of innocent people.

Therefore, the death penalty sometimes leads to the death of innocent people.

This argument is circular, but it’s still logically valid. The argument structure is the following:

- A.
- Therefore, A.

We can prove the argument is valid by proving that it’s impossible to have a counterexample. Such an argument looks like the following:

- We must assume the premise is true and the conclusion is false.
- We assume ‘A’ is false because it’s the conclusion.
- We assume ‘A’ is true because it’s the premise.
- Therefore, ‘A’ is true and false.
- The assumption that the premise is true and conclusion is false leads to a contradiction.
- Therefore, the argument form can’t be invalid.
- Therefore, the argument form must be vaild.

**Example 7**

Murder is always wrong.

Sometimes murder isn’t wrong.

Therefore, the death penalty should be illegal.

The premises contradict each other, but the argument is still valid because it’s impossible for the premises to be true and the conclusion to be false at the same time. We can tell that both premises can’t be true at the same time, so it’s impossible to make a counterexample because that would require both premises to be true. The argument form looks like the following:

- A.
- Not-A
- Therefore, B.

We can prove this argument to be valid by showing why a counterexample can’t be given:

- We assume the premises are true and the conclusion is false.
- ‘A’ is assumed to be true.
- Not-A is assumed to be true.
- ‘B’ is assumed to be false.
- Therefore, ‘A’ is true and false.
- Therefore, the assumption that the premises are true and conclusion is false leads to a contradiction.
- Therefore, the argument form can’t be invalid.
- Therefore, the argument form must be valid.

**Example 8**

It’s wrong to refuse to hire the most qualified applicant due to irrelevant criteria.

Therefore, it’s wrong to refuse to hire the most qualified applicant due to the color of her skin.

This argument might sound like it’s valid, but it’s technically invalid with the following argument form:

- A.
- Therefore, B.

A counterexample would be the following:

- It’s good to help people.
- Therefore, it’s good to help prisoners escape from prison.

The reason why the argument might sound valid is because we have an assumption that the color of an applicant’s skin is irrelevant criteria. We could then make the argument valid using the following reasoning:

- It’s wrong to refuse to hire the most qualified applicant due to irrelevant criteria.
- If it’s wrong to refuse to hire the most qualified applicant due to irrelevant criteria, then it’s wrong to refuse to hire the most qualified applicant due to the color of her skin (because skin color is irrelevant criteria).
- Therefore, it’s wrong to refuse to hire the most qualified applicant due to the color of her skin.

The argument form is now:

- A.
- If A, then B.
- Therefore, B.

This argument form is the same as was used in example 1 and has already been proven to be valid.

**Example 9**

We should try to keep an open mind.

Therefore, either rocks exist or rocks don’t exist.

This argument has a premise that seems irrelevant to the conclusion, but it’s still logically valid because the conclusion will be true no matter what. It can’t be invalid because a counterexample requires the conclusion to be false. The argument form looks like the following:

- A.
- Therefore, B or not-B.

We can prove this argument is valid by proving that we can’t have a counterexample using the following reasoning:

- Let’s assume that we can develop a counterexample, so the premise is assumed to be true and the conclusion is assumed to be false.
- We assume ‘A’ is true because it’s a premise.
- We assume “B or not-B” to be false because it’s a conclusion.
- B or not-B is true. (If ‘B’ is false, then “B or not-B” is true. If ‘B’ is true, then “B or not-B is true.)
- Therefore ‘B’ is true and false.
- The assumption that the premise is true and conclusion is false leads to a contraction.
- Therefor, the argument form can’t be invalid.
- Therefore, the argument form is valid.

**Example 10**

All cats are mammals.

Therefore, some cats are mammals.

This argument is invalid despite the fact that it might look valid. The statement “All cats are mammals” is equivalent to “if something is a cat, then it’s a mammal” and the statement “some cats are mammals” is equivalent to “there is at least one cat and it’s a mammal.” We can then reveal the logical structure as the following:

- If X exists then it’s a Y.
- Therefore, an X exists and it’s a Y.

The problem here is that it’s the existential fallacy—we can’t assume that something exists in a conclusion when no premise claims something to exist. In this case we can’t assume a cat exists just because all cats are mammals. A counterexample could be the following:

- If you are found guilty for killing everyone on Earth in a court of law, then you will go to prison.
- Therefore, someone was found guilty for killing everyone on Earth in a court of law, and that person went to prison.

The main difference between these two arguments is that we know that cats exist. That’s the hidden premise that can be used to fix the argument:

- If something is a cat, then it’s a mammal.
- A cat exists right now.
- Therefore, a cat exists and it’s a mammal.

(Updated 11/24/2013)