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Math in Society: Mathematics for liberal arts majors

Section 1.4 Logical Fallacies

Subsection 1.4.1 Logical Fallacies

In the last section we saw that logical arguments are invalid when the premises are not sufficient to guarantee the conclusion, and that even if an argument is valid it may be unsound if the premises are not true. There are other ways that a logical argument my by invalid or unsound. One of the more common ways this can occur is if the argument is a fallacy.
A fallacy is a type of argument that appears valid but uses a logical error to persuade or deceive. Fallacious arguments are especially common in advertising and politics, so it is important as informed citizens to recognize when we are being presented with a fallacious argument and to not be persuaded by it.

Subsection 1.4.2 Common Logical Fallacies

There are many logical fallacies, and some go by more than one name. Below we introduce a few of the more common fallacies that you will be asked to recognize by name, but there are many others.

Subsection 1.4.3 Personal Attack (Ad hominem)

A personal attack argument attacks the person making the argument while ignoring the argument itself. A personal attack is not the same as an insult. Rather, a personal attack claims that there is something wrong with the person or group in order to cast doubt on their character and discredit their argument.

Example 1.4.1.

“Jane says that whales are fish, but she’s only in the second grade so she can’t be right.”
Here the argument is attacking Jane, not the validity of her claim, so this is a personal attack.

Example 1.4.2.

“Jane says that whales are fish, but everyone knows that they’re really mammals. She’s so stupid.”
This certainly isn’t very nice, but it is not a personal attack since a valid counterargument is made (“they really are mammals”) along with a personal insult.

Example 1.4.3.

“Mr. Smith is a college dropout, so his stance on education reform cannot be trusted.”
Here the argument uses the fact that Mr. Smith did not complete their college degree to discredit their ideas on education reform, so it is a personal attack.

Subsection 1.4.4 Appeal to Ignorance

An appeal to ignorance argument assumes something is true because it hasn’t been proven false.

Example 1.4.4.

“Nobody has proven that photo isn’t of Bigfoot, so it must be Bigfoot.”
This is an example of an appeal to ignorance since the fact that no one has been able to prove the picture of Bigfoot is false is being used as evidence that it is Bigfoot.

Subsection 1.4.5 Appeal to Authority

An appeal to authority argument attempts to use the authority of a person to prove a claim. An authority could be an expert such as a doctor or scholar, or someone who is admired like a celebrity or sports figure. While an authority can provide strength to an argument, problems can occur when the person’s opinion is not shared by other experts, or when the authority is irrelevant to the claim.

Example 1.4.5.

“A diet high in bacon can be healthy; Doctor Atkins said so.”
Here, an appeal to a doctor’s authority is used for the argument. This generally would provide strength to the argument, except that the opinion that eating a diet high in saturated fat runs counter to general medical opinion. More supporting evidence would be needed to justify this claim.

Example 1.4.6.

“Jennifer Hudson and Oprah lost weight with Weight Watchers, so their program must work.”
In this example there is an appeal to the authority of celebrities. While their experience does provide evidence, it provides no more than any other person’s experience would.

Subsection 1.4.6 False Dilemma

A false dilemma argument falsely frames an argument as an “either or” choice without allowing for additional options.

Example 1.4.7.

“Either those lights in the sky were an airplane or aliens. There are no airplanes scheduled for tonight, so it must be aliens.”
This argument is a false dilemma since it ignores the possibility that the lights could be something other than an airplane or aliens.

Subsection 1.4.7 Straw Man (or Straw Person)

A straw person argument involves misrepresenting the argument in an oversimplified, distorted and less favorable way to make it easier to attack.

Example 1.4.8.

“Senator Khouri has proposed reducing military spending by 10%. Apparently, she wants to leave us defenseless against attacks by terrorists.”
Here the arguer has represented a 10% funding cut as equivalent to leaving us defenseless, making it easier to attack Senator Khouri’s position.

Subsection 1.4.8 Post Hoc

A post hoc argument claims that because two things happened sequentially, then the first must have caused the second.

Example 1.4.9.

“Every morning the rooster crows just before dawn. It must be his crow that makes the sun rise.”
Here the arguer is saying the rooster caused the sun to rise, but it is more likely that the sun rising caused the rooster to crow.

Example 1.4.10.

“Today I wore a red shirt and my football team won! I need to wear a red shirt every time they play to make sure they keep winning.”
This person is saying their team won because they wore a red shirt. This type of superstition is quite common in sports even though we really know they are unrelated.
Sometimes there may be more than one fallacy that seems reasonable. Consider this argument: “Emma Watson says she’s a feminist, but she posed for these racy pictures. I’m a feminist and no self-respecting feminist would do that.” Could this be ad hominem, saying that Emma Watson has no self-respect? Could it be appealing to authority because the person making the argument claims to be a feminist? Could it be a false dilemma because the argument assumes that a woman is either a feminist or not, with no gray area in between?
We have described just six of the many types of logical fallacies. Once you learn to recognize these you will also likely become aware of many others. There are many lists of logical fallacies online.

Exercises 1.4.9 Exercises

Exercise Group.

Determine which type of fallacy each argument represents.
1.
John Bardeen’s work at the Advanced Institute for Physics has progressed so slowly that even his colleagues call him a plodder. Hence, it is prudent at present not to take seriously his current theory relating how strings constitute the smallest of subatomic particles.
2.
You will tell the general manager that I made the right choice in dealing with that customer. After all, I’m the shift manager, so my decisions are always right.
3.
It was his fault, Officer. You can tell by the kind of car I’m driving and by my clothes that I am a good citizen and would not lie. Look at the rattletrap he is driving and look at how he is dressed. You can’t believe anything a dirty, longhaired hippie like that might tell you. Search his car; he probably has pot in it.
4.
We can go to the amusement park or the library. The amusement park is too expensive, so we must go to the library.
5.
Nearly all heroin addicts used marijuana before trying heroin. Clearly marijuana use leads to heroin addiction.
6.
The oven was working fine until you started using it, so you must have broken it.
7.
Old man Brown claims that he saw a flying saucer in his farm, but he never got beyond the fourth grade in school and can hardly read or write. He is completely ignorant of what scientists have written on the subject, so his report cannot possibly be true.
8.
You should use Sparkle brand toothpaste since four out of five dentists recommend it.
9.
She didn’t say that I couldn’t borrow her car, so I figured it was fine if I borrowed it for the weekend.
10.
If you think that teens should be taught about contraceptive measures then you want to give kids license to have sex with no consequences.
11.
You’re either part of the solution or part of the problem.
12.
No one can prove that God exists, therefore God does not exist.
13.
You’re clearly just too young to understand.
14.
You should start off every morning with Champions brand cereal. It is what Michael Jordan eats, so you know that it must be very good for you.
15.
No one on the council objected to the idea that he proposed, so everyone must think it is a good idea.
16.
Why should we believe your testimony? You haven’t had a steady job since 2003.
17.
A huge percentage of diagnosed cases of autism came very soon after vaccinations for the measles. These vaccinations must be causing autism.
18.
If you are against this war then you must hate America.
19.
Just look at her face. How could anyone vote for that?

20.

There are a number of fallacies that were not discussed in this section. Do an internet search for the following fallacies. Provide both a definition and at least one example.
  1. Slippery Slope
  2. Circular Reasoning
  3. Appeal to Emotion
  4. Red Herring
  5. Whataboutism