Common Mistakes in Arguments

Introduction

In this lesson, we’ll learn about common mistakes in writing a proof, also called fallacies. Mostly, these fallacies are due to misinterpretation or misconception about the implication operation. These fallacies create wrong arguments or faulty proofs. One should be careful when using inference rules and avoid these fallacies while building arguments and making proof.

Fallacy of affirming the conclusion

Suppose we have the following two hypotheses:

• $H_1$: $p\Rightarrow q$.
• $H_2$: $q$.

From these hypotheses, if we conclude $p$, this is called the fallacy of affirming the conclusion. We can not conclude $p$ from $H_1$ and $H_2$ because $p$ can be true or it can be false.

Further, note that the following proposition,

$$\left(p\Rightarrow q\right)\land q \Rightarrow p,$$

is not a tautology. It is rather a contingency. So, be cautious that we can not conclude $p$, given $H_1$ and $H_2$.

Examples

As our first example, let’s see what is wrong with the following argument.

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If Sam buys a burger, then he gets a free drink. Sam got a free drink. Hence, he bought a burger.

Let’s look at the structure of the argument to find the problem it has.

• $s_1$: Sam buys a burger.
• $s_2$: Sam gets a free drink.

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The given information is as follows:

• $s_1 \Rightarrow s_2$: If Sam buys a burger, then he gets a free drink.

• $s_2$: Sam gets a free drink.

From this information, we can not conclude $s_1$, because it will be a fallacy to affirm the conclusion. When $s_1 \Rightarrow s_2$ and $s_2$ are true, $s_1$ can be true, or it can be false.

In our second example, we will look at a scenario first, then we will analyze it.

Scenario

An engineering firm announces to its employees that if they perform their job really well, (then) they will get a salary raise. At the year-end, everybody got a salary raise. Charlie and Smith talked to each other and came to know that both got a salary raise. Both are concluding about the other person that they perform their job really well. In reality, Charlie performed her job really well while Smith did not perform his job really well. Smith got the salary raise with a warning in a personal meeting with his boss.

Analysis

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• $c_1$: Charlie performed her job really well.
• $c_2$: Charlie got a salary raise.
• $m_1$: Smith performed his job really well.
• $m_2$: Smith got a salary raise.

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Now the facts are as follows.

• $c_1\Rightarrow c_2.$

• $c_2.$

• $m_1 \Rightarrow m_2.$

• $m_2.$

Smith is concluding $c_1$ about Charlie from $\left(c_1\Rightarrow c_2\right)$ and $c_2$, which is the fallacy of affirming the conclusion.

Likewise, Charlie is concluding $m_1$ about Smith from $\left(m_1 \Rightarrow m_2\right)$ and $m_2$, which is the fallacy of affirming the conclusion.

Fallacy of denying the hypothesis

If we have the following two hypotheses:

• $H_1$: $p\Rightarrow q.$
• $H_2$: $\neg p.$

To conclude $\neg q$, from these hypotheses is called the fallacy of denying the hypothesis. We can not conclude $\neg q$ from $H_1$ and $H_2$ because $\neg q$ can be true, or it can be false.

Further, note the following proposition:

$$\left(p\Rightarrow q\right)\land \neg p \Rightarrow \neg q.$$

It is not a tautology. It is rather a contingency. So, be cautious that we can not conclude $\neg q$ if we are given $H_1$ and $H_2$.

Examples

For the first example, let’s see what is wrong with the following argument:

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If Willy is a fish then she can swim. But Willy is not a fish. Hence, Willy can not swim.

• $w_1$: Willy is a fish.
• $w_2$: Willy can swim.

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The given information is as follows:

• $w_1 \Rightarrow w_2$: If Willy is a fish, then she can swim.

• $\neg w_1$: Willy is not a fish.

From here we can not conclude $\neg w_2$, that is the fallacy of denying the hypothesis. Given that, $\left(w_1 \Rightarrow w_2\right)$ and $\neg w_1$ are true, $\neg w_2$ can be true or it can be false.

In the second example, first, we’ll describe a scenario and then a solution, showing the problem due to the fallacy of denying the hypothesis.

Scenario

Alvin, a calculus teacher, announced the following policy in class.

“Any student who scores ‘A’ grade in the final examination will get an ‘A’ grade in the course.”

After the result was declared, three students approached Alvin to contest their grades. Marva got an “A” grade in the final examination, while Zane and Harry got a “B” grade. Marva and Zane got an “A” grade in the course, while Harry got a “B” grade.

Name	Grade in Final	Grade in Course
Marva	A	A
Zane	B	A
Harry	B	B

The students are confused that if the teacher has kept his words or not. The questions in their minds are as follows.

1. Zane has not secured an A in the final examination. How can he get an A in the course?

2. If Zane got an A in the course, why has Harry not gotten an A? Both have the same grade in the final examination.

These students asked Alvin to justify that he is consistent with his announced policy.

Analysis

Alvin reiterated the policy that students who scored an “A” grade in the final exam got an “A” grade in the course. Then they discussed each case separately.

The cases of Marva and Harry have no confusion, and everybody is okay with that.

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For the case of Zane, Alvin did the following analysis:

• $z_1$: Zane scored an “A” grade in the final examination.

• $\neg z_1$: Zane did not score an “A” grade in the final examination.

• $z_2$: Zane got an “A” grade in the course.

• $\neg z_2$: Zane did not get an “A” grade in the course.

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The policy which Alvin announced was as follows:

• $z_1\Rightarrow z_2$: If Zane scores an “A” grade in the final exam (then), he will get an “A” grade in the course.

As far as this policy remains true, Alvin is consistent. In the result, $z_1$ is false and $z_2$ is true, making $z_1\Rightarrow z_2$ true. So, in the case of Zane, Alvin is consistent with his policy.

The confusion in the minds of students was as follows:

$$\left(z_1\Rightarrow z_2\right)\land \neg z_1\Rightarrow \neg z_2.$$

The students were wrongly concluding $\neg z_2$ from $\left(z_1\Rightarrow z_2\right)$ and $\neg z_1$. They were confused due to the fallacy of denying the hypothesis.