Common Logical Fallacies

Show your work for each problem.

1.
Argument: If it rains, the ground is wet. The ground is wet. Therefore it rained. Which fallacy is this?
1. a)Circular reasoning
2. b)Denying the antecedent
3. c)Straw man
4. d)Affirming the consequent
Answer: Affirming the consequent
The then part (the consequent, the ground is wet) is given, and the argument concludes the if part. That is affirming the consequent. The ground could be wet because of a sprinkler.
2.
An ad says: 4 out of 5 users improved their grades! What percentage is 4 out of 5?
Answer: 80
$\frac{4}{5} \times 100 = 80$ , so the ad claims $80\%$ . Before trusting it, ask how many users were asked: if only 5, it is a hasty generalization.
3.
Argument: If $x = 4$ , then $x^2 = 16$ . We know that $x^2 = 16$ . Therefore $x = 4$ . Find a counterexample.
Answer: -4
1. What is $4^2$ ? 16
2. Which other number also has a square of 16? -4
3. Is the argument valid? Type 1 for valid or 0 for invalid. 0
4.
Maya shows that $x = 5$ solves $2x + 3 = 13$ . Leo answers: Don't listen to her, she failed the last test. Which fallacy does Leo use?
1. a)Slippery slope
2. b)False dilemma
3. c)Appeal to authority
4. d)Ad hominem
Answer: Ad hominem
Leo attacks Maya instead of her solution: ad hominem. The solution is easy to check: $2 \times 5 + 3 = 13$ , so she is right.
5.
After checking 3, 5 and 7, Tom claims: every odd number greater than 1 is prime. What is the smallest odd number greater than 1 that proves him wrong?
Answer: 9
$9 = 3 \times 3$ is odd but not prime. Three cases were a hasty generalization, and the very next odd number is a counterexample.
6.
Argument: Every integer is either positive or negative. Find the option this false dilemma leaves out.
Answer: 0
1. Which integer is neither positive nor negative? 0
2. How many integers are there from $-3$ to $3$ , including both? 7
3. How many of them are positive or negative? 6
7.
Ana: We should check our answer with a second method. Ben: So you think we can never trust any calculation? That's ridiculous. Which fallacy does Ben use?
1. a)Appeal to authority
2. b)Straw man
3. c)Denying the antecedent
4. d)Hasty generalization
Answer: Straw man
Ana only suggested a second check. Ben replaces her claim with a much more extreme one (never trust any calculation) and attacks that: a straw man.
8.
A school has 500 students and 450 of them use the app StudyFast. 60 students got an A in math, and 54 of them use the app. An ad says: 90% of A students use StudyFast, so the app causes A grades! What percentage of the app users got an A?
Answer: 12
$\frac{54}{450} \times 100 = 12$ . Among the 50 students without the app, $6$ got an A: $\frac{6}{50} \times 100 = 12$ as well. The rate is the same, so the 90% only reflects that 90% of all students use the app. The ad ignores the base rate.
9.
Argument: If a number is a multiple of 10, then it is a multiple of 5. 15 is not a multiple of 10. Therefore 15 is not a multiple of 5. Show that the argument fails.
Answer: 5
1. What is the remainder when 15 is divided by 10? 5
2. What is $15 \div 5$ ? 3
3. How many numbers from 1 to 50 are multiples of 5 but not multiples of 10? 5
10.
Claim: This formula is correct because it always gives the right answers, and we know the answers are right because they come from this formula. Which fallacy is this?
1. a)Correlation vs causation
2. b)Circular reasoning
3. c)Ad hominem
4. d)Slippery slope
Answer: Circular reasoning
The formula is justified by the answers and the answers by the formula. The conclusion is already assumed in the premises: circular reasoning. An independent check, such as testing a known case, would break the circle.
11.
Rule: If a number is divisible by 6, then it is even. Someone affirms the consequent and treats every even number as divisible by 6. How many even numbers from 1 to 20 are counterexamples (even but not divisible by 6)?
Answer: 7
There are 10 even numbers from 1 to 20. Only 6, 12 and 18 are divisible by 6, so $10 - 3 = 7$ even numbers are counterexamples: 2, 4, 8, 10, 14, 16 and 20.
12.
In a class of 40, 25 students play an instrument and 20 of them have a math grade above 80. Of the 15 who don't play, 6 have a grade above 80. Claim: playing an instrument makes you better at math. Compare the two groups.
Answer: 40
1. What percentage of the instrument players have a grade above 80? 80
2. What percentage of the non-players have a grade above 80? 40
3. How many percentage points is the difference between the two groups? 40
13.
A slippery slope: If you skip one homework, you will fall behind, then fail the next test, then fail the whole year. Suppose each of these three steps really happens with probability $0.5$ , independently. What is the chance that all three happen, as a percentage?
Answer: 12.5
1. What is the probability that the first two steps both happen? 0.25
2. What is the probability that all three steps happen? 0.125
3. Write this probability as a percentage. 12.5
14.
Argument: If a number ends in 0, then it is divisible by 5. 35 is divisible by 5. Therefore 35 ends in 0. Check the argument.
Answer: 10
1. What is $35 \div 5$ ? 7
2. What is the last digit of 35? 5
3. How many numbers from 1 to 100 are divisible by 5 but do not end in 0? 10
15.
Argument: If a number is divisible by 9, then it is divisible by 3. 21 is not divisible by 9. Therefore 21 is not divisible by 3. Which statement is correct?
1. a)The argument affirms the consequent, but its conclusion is true
2. b)The argument is valid, because 21 is not divisible by 9
3. c)The argument is valid, because it is modus tollens
4. d)The argument denies the antecedent, and 21 is a counterexample because $21 \div 3 = 7$
Answer: The argument denies the antecedent, and 21 is a counterexample because $21 \div 3 = 7$
The premise negates the if part ( $\neg p$ ) and concludes $\neg q$ : denying the antecedent, an invalid form. 21 itself breaks it: $21 \div 9$ is not a whole number, but $21 \div 3 = 7$ . Modus tollens would need not divisible by 3 as the premise.
16.
A student checks $n^2 - n + 11$ for $n = 1$ to $10$ and gets only primes: 11, 13, 17, 23, 31, 41, 53, 67, 83, 101. She claims the result is always prime. What is the value of $n^2 - n + 11$ for $n = 11$ ?
Answer: 121
$11^2 - 11 + 11 = 121 = 11 \times 11$ , which is not prime. Ten cases in a row were still a hasty generalization: one counterexample disproves the claim.
17.
A headline says: Risk doubles! A side effect rises from 1 in 10000 people to 2 in 10000 people. Find out how big the change really is.
Answer: 0.01
1. How many extra cases are there per 10000 people? 1
2. By what percentage did the risk increase compared with the old risk? 100
3. What is the new risk as a percentage? 0.02
4. By how many percentage points did the risk rise? 0.01
18.
A disease affects 20 of 1000 people. A test is positive for 18 of the 20 sick people, and by mistake also for 42 of the 980 healthy people. Claim: the test finds 90% of sick people, so a positive result means a 90% chance of being sick. What is the real chance, as a percentage?
Answer: 30
1. What percentage of the sick people does the test find? 90
2. How many people test positive in total? 60
3. What percentage of the people who test positive are really sick? 30