Examples of Valid & Invalid Logical Reasoning
‘I have described formal
logic, said a little about why it’s important for proper reasoning, and
described how we can prove arguments to be logically invalid through
counterexamples. I will now give examples of valid and invalid logical arguments
to help illustrate the difference and help us learn how to identify the
difference in everyday life. I will give 10 examples of arguments that could be
either valid or invalid, but I won’t give the answers to the examples right
away to give you a chance to think about the answers on your own. Then I will
give the answers in another section below and justify my answer using
counterexamples when possible. I will also prove that various argument forms
are valid. It’s possible that the arguments below make use of poor reasoning
that is unrelated to logical validity, but logical validity is my only concern
here’.
Problem solving
Example 1
1. If all dogs are mammals, then all
dogs are reptiles.
2. All dogs are mammals.
3. Therefore, all dogs are reptiles.
Reptiles:
a vertebrate animal of a class that includes snakes, lizards, crocodiles,
turtles, and tortoises. They are distinguished by having a dry scaly skin and
typically laying soft-shelled eggs on land.
Example 2
1.
If all dogs are
mammals, then all dogs are animals.
2.
All dogs are animals.
3.
Therefore, all dogs
are mammals.
Example 3
1.
Either it’s wrong to
indiscriminately kill people, or it’s not wrong to kill someone just because
she has red hair.
2.
It’s wrong to kill
someone just because she has red hair.
3.
Therefore, it’s wrong
to indiscriminately kill people.
Example 4
1.
Either disciplining
people is always wrong or it’s not always wrong to discipline people for
committing crimes.
2.
Disciplining people
hurts them.
3.
Therefore,
disciplining people is always wrong.
Example 5
1.
It’s often good to
give to charity.
2.
If it’s often good to
give to charity, then the Earth is round.
3.
Therefore, the Earth
is round.
Example 6
1.
The death penalty
sometimes leads to the death of innocent people.
2.
Therefore, the death
penalty sometimes leads to the death of innocent people.
Example 7
1.
Murder is always
wrong.
2.
Sometimes murder isn’t
wrong.
3.
Therefore, the death
penalty should be illegal.
Example 8
1.
It’s wrong to refuse
to hire the most qualified applicant due to irrelevant criteria.
2.
Therefore, it’s wrong to
refuse to hire the most qualified applicant due to the color of her skin.
. Example 9
1.
We should try to keep
an open mind.
2.
Therefore, either
rocks exist or rocks don’t exist.
Example 10
1.
All cats are mammals.
2.
Therefore, some cats
are mammals
Answers
Example 1
1.
If all dogs are
mammals, then all dogs are reptiles.
2.
All dogs are mammals.
3.
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:
1.
If all dogs are
mammals, then all dogs are animals.
2.
All dogs are mammals.
3.
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:
1.
The counterexample
must have true premises, and a false conclusion.
2.
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.
3.
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.)
4.
Therefore, ‘A’ is true
and false. That’s a contradiction.
5.
The assumption that
the argument is has true premises and a false conclusion leads to a
contradiction.
6.
Therefore, the
argument form can’t be invalid.
7.
Therefore, the
argument must be valid.
Example 2
1.
If all dogs are
mammals, then all dogs are animals.
2.
All dogs are animals.
3.
Therefore, all dogs
are mammals.
Although the premises and conclusion are true, the argument
form is invalid. The argument form is the following:
1.
If A, then B.
2.
B.
3.
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.
B: All dogs are mammals.
This leads to the following counterexample:
1.
If all dogs are
reptiles, then all dogs are animals.
2.
All dogs are animals.
3.
Therefore, all dogs
are reptiles.
Both premises are true, but the conclusion is false.
Therefore, the argument form must be invalid.
Example 3
1.
Either it’s wrong to
indiscriminately kill people, or it’s not wrong to kill someone just because
she has red hair.
2.
It’s wrong to kill
someone just because she has red hair.
3.
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:
1.
Either A or not-B.
(“Not-B” means “B is false.”)
2.
B (is true).
3.
Therefore A.
An example of a good argument with this argument form is
the following:
1.
Either dogs are
warm-blooded or dogs aren’t mammals.
2.
Dogs are mammals.
3.
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.
1.
We assume the premises
are true and the conclusion is false.
2.
We assume ‘A’ is false
because it’s the conclusion.
3.
We assume ‘B’ is true
because it’s a premise.
4.
We assume “Either A or
not-B” is true.
5.
“Either A or not-B”
requires that either A is true or not-B is true.
6.
We know A is false, so
not-B must be true.
7.
Therefore, B and not-B
are both true. That’s a contradiction.
8.
Therefore, the the
assumption that the premises are true and conclusion is false leads to a
contradiction.
9.
Therefore, the
argument form can’t be invalid.
10.
Therefore, the
argument form is valid.
Example 4
1.
Either disciplining
people is always wrong or it’s not always wrong to discipline people for
committing crimes.
2.
Disciplining people
hurts them.
3.
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:
1.
Either A or not-B.
2.
C
3.
Therefore, B.
Another counterexample is the following:
1.
Either murder is
always appropriate or it’s not always appropriate to murder people for making
you angry.
2.
Murdering people hurts
them.
3.
Therefore, murder is
always appropriate.
Example 5
1.
It’s often good to
give to charity.
2.
If it’s often good to
give to charity, then the Earth is round.
3.
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:
1.
A.
2.
If A, then B.
3.
Therefore, B.
This is basically the same argument form as the first
example, so no further proof of validity is required.
Example 6
1.
The death penalty sometimes
leads leads to the death of innocent people.
2.
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:
1.
A.
2.
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:
1.
We must assume the
premise is true and the conclusion is false.
2.
We assume ‘A’ is false
because it’s the conclusion.
3.
We assume ‘A’ is true
because it’s the premise.
4.
Therefore, ‘A’ is true
and false.
5.
The assumption that
the premise is true and conclusion is false leads to a contradiction.
6.
Therefore, the
argument form can’t be invalid.
7.
Therefore, the
argument form must be vaild.
Example 7
1.
Murder is always
wrong.
2.
Sometimes murder isn’t
wrong.
3.
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:
1.
A.
2.
Not-A
3.
Therefore, B.
We can prove this argument to be valid by showing why a
counterexample can’t be given:
1.
We assume the premises
are true and the conclusion is false.
2.
‘A’ is assumed to be
true.
3.
Not-A is assumed to be
true.
4.
‘B’ is assumed to be
false.
5.
Therefore, ‘A’ is true
and false.
6.
Therefore, the
assumption that the premises are true and conclusion is false leads to a
contradiction.
7.
Therefore, the
argument form can’t be invalid.
8.
Therefore, the
argument form must be valid.
Example 8
1.
It’s wrong to refuse
to hire the most qualified applicant due to irrelevant criteria.
2.
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:
1.
A.
2.
Therefore, B.
A counterexample would be the following:
1.
It’s good to help
people.
2.
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:
1.
It’s wrong to refuse
to hire the most qualified applicant due to irrelevant criteria.
2.
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).
3.
Therefore, it’s wrong
to refuse to hire the most qualified applicant due to the color of her skin.
The argument form is now:
1.
A.
2.
If A, then B.
3.
Therefore, B.
This argument form is the same as was used in example 1 and
has already been proven to be valid.
Example 9
1.
We should try to keep
an open mind.
2.
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:
1.
A.
2.
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:
1.
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.
2.
We assume ‘A’ is true
because it’s a premise.
3.
We assume “B or not-B”
to be false because it’s a conclusion.
4.
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.)
5.
Therefore ‘B’ is true
and false.
6.
The assumption that
the premise is true and conclusion is false leads to a contraction.
7.
Therefor, the argument
form can’t be invalid.
8.
Therefore, the
argument form is valid.
Example 10
1.
All cats are mammals.
2.
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:
1.
If X exists then it’s
a Y.
2.
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:
1.
If you are found
guilty for killing everyone on Earth in a court of law, then you will go to
prison.
2.
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:
1.
If something is a cat,
then it’s a mammal.
2.
A cat exists right
now.
3. Therefore, a cat exists and it’s a
mammal.
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