Valid arguments and fallacies
by CHED on June 14, 2017
lesson duration of 7 minutes
under General Mathematics
generated on June 14, 2017 at 08:11 pm
Tags: Logic
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Generated: Jun 15,2017 04:11 AM
Valid arguments and fallacies
( 7 mins )
Written By: CHED on May 28, 2016
Subjects: General Mathematics
Tags: Logic
Resources
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Content Standard
The learner demonstrates understanding of key concepts of propositional logic; syllogisms and fallacies.
Performance Standard
The learner is able to judiciously apply logic in real-life arguments.
Learning Competencies
The learner illustrates different types of tautologies and fallacies.
The learner determines the validity of categorical syllogisms.
The learner establishes the validity and falsity of real-life arguments using logical propositions, syllogisms, and
fallacies.
Introduction 1 mins
Given a set of information, our goal is to logically infer from the given some new information.
For example, we know that only a small portion of the earth's water is freshwater (mostly it's saltwater). We know that if
there there is a limited supply of freshwater, then we should conserve water. Combining these information urges us to
conclude that we should conserve water.
In this lesson, we are going to look at this process of inference formally. We call the set of given information as the
premise, while the derived new information is referred to as the conclusion. The premises and the conclusion form
what we call an argument.
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Define Argument
1 mins
The premises of an argument are intended to act as reasons to establish the validity or acceptability of the conclusion.
We will make this statement more precise later on.
EXAMPLE 1. Explain why the following set of propositions is an argument.
If General Antonio Luna is a national hero, then he died at the hands of the Americans in 1899. General Antonio Luna
is a national hero.
Therefore, General Luna died at the hands of the Americans in 1899.
Solution. The set of propositions is an argument. Its premises are the propositions `If General Antonio Luna is a
national hero, then he died at the hands of the Americans in 1899,' and `General Antonio Luna is a national hero.' The
conclusion, which is flagged by the word `therefore', is the proposition `General Luna died at the hands of the
Americans in 1899.'
Is the argument valid? Is it sound? At the end of the lesson, you will be able to answer these questions.
EXAMPLE 2. Write the following argument presented in the introduction in propositional form and in standard form.
If there is limited freshwater supply, then we should conserve water. There is limited freshwater supply.
Therefore, we should conserve water.
Solution. The premises of this argument are
p1 : If there is limited freshwater supply, then we should conserve water.
p2 : There is limited freshwater supply.
and its conclusion is
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q : We should conserve water.
In symbols, we can write the whole argument in propositional form
(p1 ? p2) ? q,
and in standard form
The following example shows that two arguments may have equivalent logical forms, even if they are different in
content.
EXAMPLE 3. Consider the arguments A and A? given below:
Analyze these arguments separately by answering the following question.
Validity Condition:
Is it logically impossible for the premises to be true and the conclusion false?
If the answer is YES, we say that the argument satisfies the validity condition. The argument is valid.
Solution.
Argument A
Can p ? q and p both be true and q be false?
Suppose that the premises p ? q and p are true. The truth table below shows that both p and p ? q are true only in the
first row. In this row, q is also true.
Hence, YES it is logically impossible for the premises to be true and the conclusion to be false. We say that A satisfies
the validity condition, and so it is valid. In practical terms, this means that if my alarm sounded, then I surely woke up.
Argument A?
Can p ? q and q be true, and p be false?
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Suppose that the premises p ? q and q are true. The truth table below shows that both p ? q and q are true in the first
and third rows. However, in the third row, the conclusion p is false, even if the premises are true.
Argument A? does not satisfy the validity condition. In practical terms, it is possible that I woke up but my alarm did not
sound. I could have woken up due to a bad dream, for example.
In summary, we have the following.
Define Valid Argument 1 mins
Definition.
A valid argument satisfies the validity condition; that is, the conclusion q is true whenever the premises p1,
p2, . . . , pn are all true.
Put another way, for a valid argument, the conditional
(p1 ? p2 ? . . . ? pn) ? q
is a tautology.
EXAMPLE 4. Prove that the argument ((p ? q) ? p) ? q is valid.
This argument is known as Modus Ponens (or Rule of Detachment).
Solution. We only need to show that the ((p ? q) ? p) ? q is a tautology. We can do this using the truth table.
Since ((p ? q) ? p) ? q is a tautology, then the argument is valid.
EXAMPLE 5. Let us consider the arguments in the previous examples.
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Argument A
If my alarm sounds, then I will wake up. My alarm sounded.
Therefore, I woke up.
Argument B
If there is limited freshwater supply, then we should conserve water. There is limited freshwater supply.
Therefore, we should conserve water.
Argument C
If General Antonio Luna is a national hero, then he died at the hands of the Americans in 1899. General Antonio Luna
is a national hero.
Therefore, General Luna died at the hands of the Americans in 1899.
Note that they are all of the form ((p ? q) ? p) ? q, or in standard form
Hence, by Modus Ponens, arguments A, B, and C are all valid. However, this does not mean that the conclusions
are true. Asserting that an argument is valid simply means that the conclusion logically follows from the premises.
In the next table, we display some basic forms of valid arguments in logic. They are also known as rules of inference.
inference.
The associated tautologies of these arguments were already established in Lesson 38, Seatwork 2.
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EXAMPLE 6. Determine whether the following argument is valid.
If Antonio and Jose are friends, then they are Facebook friends. Antonio and Jose are not Facebook friends.
Therefore, they are not friends.
Solution. Let
p : Antonio and Jose are friends.
q : Antonio and Jose are Facebook friends.
Then the given argument is of the form
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Hence, by Modus Tollens, the argument is valid.
EXAMPLE 7. Determine which rule is the basis of each argument below.
(a) Antonio Luna and Jose Rizal like Nelly Boustead.
Therefore, Antonio Luna likes Nelly Boustead.
(b) Antonio Luna is a scientist.
Therefore, either Antonio Luna or Jose Rizal is a scientist.
(c) If the Spaniards imprison Antonio Luna, then he will repent and not join the revolution.
If Antonio Luna regrets not joining the revolution, then he will go to Belgium to study the art of war.
Therefore, if the Spaniards imprison Antonio Luna, then he will go to Belgium to study the art of war.
Solution. (a) Let
p : Antonio Luna likes Nelly Boustead.
q : Jose Rizal likes Nelly Boustead.
Then the given argument is of the form
By the Rule of Simplification, the argument is valid.
(b) Let
p : Antonio Luna is a scientist.
q : Jose Rizal is a scientist.
The argument in standard form is
which is valid by the Rule of Addition.
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(c) Let
p : The Spaniards imprison Antonio Luna.
q : Antonio Luna regrets not joining the revolution.
r : Antonio Luna goes to Belgium to study the art of war.
In standard form, we have
Thus, the argument is valid by the Law of Syllogism.
Define Fallacy 1 mins
Seatwork 1. Use any of the methods in Example 8 to prove that each of the arguments in the Table of Fallacies is not
a valid argument.
Seatwork 2. Determine whether the argument is valid or not. If it valid, identify the rule of inference which justifies its
validity. On the other hand, if it is not valid, construct a counterexample and, if possible, identify what kind of fallacy it
is.
(e.) If Nicanor is a famous author, then he knows how to write. But Nicanor is not a famous author. Hence, Nicanor
does not know how to write. Answer: Invalid (Fallacy of the Inverse)
(f.) If Liwayway is a famous author, then she knows how to write. Moreover, Liwayway knows how to write.
So, she is a famous author. Answer: Invalid (Fallacy of the Converse)
Enrichment 1 mins
Introduce the difference between a valid argument and a sound argument. Recall that earlier we considered the
argument:
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If General Antonio Luna is a national hero, then he died at the hands of the Americans in 1899.
General Antonio Luna is a national hero.
Therefore, General Luna died at the hands of the Americans in 1899.
We already know that the argument is valid by Modus Ponens. The argument satisfies the validity condition. But is the
argument sound?
In this section, we ask the question:
Truth Condition
Are the premises of the argument all generally true?
If the answer is YES, then the argument is said to satisfy the the truth condition.
We know that General Luna was killed by fellow Filipinos at the height of Filipino-American War. Then the premise p ?
q is false because General Luna is a national hero (p is true), but he did not die at the hands of the Americans (q is
false).
Hence, the argument, though it is valid, does not satisfy the truth condition. We say that this argument is not
sound. For an argument to be considered sound it has to satisfy both the validity condition and the truth condition.
Definition.
A sound argument is a valid argument which also satisfies the truth condition.
An argument which does not satisfy either the validity condition or the truth condition is called a bad
argument.
EXAMPLE 11. The following arguments were already shown to be valid.
(a) Antonio Luna and Jose Rizal like Nelly Boustead.
Therefore, Antonio Luna likes Nelly Boustead.
(b) Antonio Luna is a scientist.
Therefore, either Antonio Luna or Jose Rizal is a scientist.
A simple history verification will show that the premises of both arguments are true. Nelly Boustead was the object of
affection of Antonio Luna and Jose Rizal while they were in Spain. They even almost killed each other in a duel
because of her! Moreover, Luna is known as a brilliant general but he is also a scientist. He studied chemistry at the
University of Sto. Tomas, and then he went to Spain where he obtained his license and doctorate in pharmacy.
Hence, these arguments satisfy both the truth condition and the validity condition, and so they are sound arguments.
EXAMPLE 12. Determine whether each of the following arguments is valid, and if each is sound.
If I was born poor, then I cannot serve my country.
(a) I was born poor.
therefore, i cannot serve my mantra
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If I study every day, then I will develop a good work ethic.
(b) I study every day.
Therefore, I cannot serve my country.
Solution. By Modus Ponens, both arguments are valid. Hence, we just have to determine whether they satisfy the
truth condition.
(a) Note that being poor does not prevent one from serving onea?azs country (you can probably think of some
examples). Hence, the given argument is a bad argument.
(b) It is accepted as true that if I study everyday, then I will develop a good work ethic. Now, is the statement `I study
everyday' true? You should know! If you do, then this argument satisfies both the validity and truth conditions, and you
can rightfully assert that you will develop a good work ethic.
Enrichment 1 mins
Introduce the difference between a valid argument and a sound argument. Recall that earlier we considered the
argument:
If General Antonio Luna is a national hero, then he died at the hands of the Americans in 1899.
General Antonio Luna is a national hero.
Therefore, General Luna died at the hands of the Americans in 1899.
We already know that the argument is valid by Modus Ponens. The argument satisfies the validity condition. But is the
argument sound?
In this section, we ask the question:
Truth Condition
Are the premises of the argument all generally true?
If the answer is YES, then the argument is said to satisfy the the truth condition.
We know that General Luna was killed by fellow Filipinos at the height of Filipino-American War. Then the premise p ?
q is false because General Luna is a national hero (p is true), but he did not die at the hands of the Americans (q is
false).
Hence, the argument, though it is valid, does not satisfy the truth condition. We say that this argument is not
sound. For an argument to be considered sound it has to satisfy both the validity condition and the truth condition.
Definition.
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A sound argument is a valid argument which also satisfies the truth condition.
An argument which does not satisfy either the validity condition or the truth condition is called a bad
argument.
EXAMPLE 11. The following arguments were already shown to be valid.
(a) Antonio Luna and Jose Rizal like Nelly Boustead.
Therefore, Antonio Luna likes Nelly Boustead.
(b) Antonio Luna is a scientist.
Therefore, either Antonio Luna or Jose Rizal is a scientist.
A simple history verification will show that the premises of both arguments are true. Nelly Boustead was the object of
affection of Antonio Luna and Jose Rizal while they were in Spain. They even almost killed each other in a duel
because of her! Moreover, Luna is known as a brilliant general but he is also a scientist. He studied chemistry at the
University of Sto. Tomas, and then he went to Spain where he obtained his license and doctorate in pharmacy.
Hence, these arguments satisfy both the truth condition and the validity condition, and so they are sound arguments.
EXAMPLE 12. Determine whether each of the following arguments is valid, and if each is sound.
If I was born poor, then I cannot serve my country.
(a) I was born poor.
therefore, i cannot serve my mantra
If I study every day, then I will develop a good work ethic.
(b) I study every day.
Therefore, I cannot serve my country.
Solution. By Modus Ponens, both arguments are valid. Hence, we just have to determine whether they satisfy the
truth condition.
(a) Note that being poor does not prevent one from serving onea?azs country (you can probably think of some
examples). Hence, the given argument is a bad argument.
(b) It is accepted as true that if I study everyday, then I will develop a good work ethic. Now, is the statement `I study
everyday' true? You should know! If you do, then this argument satisfies both the validity and truth conditions, and you
can rightfully assert that you will develop a good work ethic.
Seatwork 1 mins
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Seatwork 1. Use any of the methods in Example 8 to prove that each of the arguments in the Table of Fallacies is not
a valid argument.
Seatwork 2. Determine whether the argument is valid or not. If it valid, identify the rule of inference which justifies its
validity. On the other hand, if it is not valid, construct a counterexample and, if possible, identify what kind of fallacy it
is.
(e.) If Nicanor is a famous author, then he knows how to write. But Nicanor is not a famous author. Hence, Nicanor
does not know how to write.
Answer: Invalid (Fallacy of the Inverse)
(f.) If Liwayway is a famous author, then she knows how to write. Moreover, Liwayway knows how to write. So, she is a
famous author.
Answer: Invalid (Fallacy of the Converse)
Seatwork 3. Determine whether each of the following arguments is (a) valid, and (b) sound.
(a.) If the solution turns blue litmus paper red, then the solution contains acid. The solution does not contain acid. So,
the solution does not turn blue litmus paper red.
Answer: Valid (Modus Tollens), and Sound
(b.) If the solution turns blue litmus paper red, then the solution contains acid. The solution turns
blue litmus paper red. So, the solution contains acid.
Answer: Valid (Modus Ponens), and Sound
(c.) If you study hard, you refine your communication skills and build up your confidence. If you refine your
communication skills build up your confidence, then your job opportunities increase. Hence, if you study hard, your job
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opportunities increase.
Answer: Valid (Syllogism), and Sound
(d.) Kidnapping is wrong if society disapproves of it. Kidnapping is wrong. So, society disapproves
of kidnapping.
Answer: Invalid (Fallacy of the Converse); thus, not sound
(e.) If overeating causes disease, then it is not healthy. Overeating does not cause disease. So,
overeating is healthy.
Answer: Invalid (Fallacy of the Inverse); thus not sound
(f.) The dinosaurs vanished due to a sudden, extreme drop in temperature. If the dinosaurs vanished
due to a sudden, extreme drop in temperature, then earth must have suffered some sort of
cataclysm millions of years ago. Therefore, the earth must have suffered some sort of cataclysm
millions of years ago.
Answer: Valid (Modus Ponens); Not Sound (there is no generally accepted theory that the first premise is true)
(g.) If ideas are important, then books are important. If ideas change lives, ideas are important.
Hence, if books are important, then ideas change lives.
Answer: Invalid; thus not sound
Seatwork 4. Discuss why each of the following is a bad argument.
(a) If I am not good in math, then I am not intelligent. I am not good in math. Then, I am not intelligent.
Answer: Invalid (Fallacy of the Inverse); thus, bad argument
(b) If I have have dark skin, then I am not beautiful. I have have dark skin. Therefore, I am not beautiful.
Answer: Valid (Modus Ponens), but wrong premise: It is not true that people with dark skin are not beautiful; thus bad
argument
(c) If I drink Gilas power energy drink every day, then I will be good in baskeball. I drink Gilas power energy drink every
day. Hence, I will be good in baskeball.
Answer: Valid (Modus Ponens), but wrong premise: It is not true that people who take a power drink everyday
automatically becomes good in basketball; thus bad argument
Seatwork 5. Construct an argument which
(a) satisfies the validity condition;
(b) does not satisfy the validity condition;
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(c) satisfies the validity condition but not the truth condition;
(d) satisfies both the validity and the truth conditions.
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