Teacher Guide: Common Logical Fallacies

Learn to spot and fix the most common logical fallacies, from affirming the consequent to correlation vs causation, with math examples.

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10 questions on Logical Reasoning. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Distinguish valid argument forms from affirming the consequent and denying the antecedent
Identify common informal fallacies in written arguments and explain the error
Use a counterexample to show that a conclusion does not follow
Recognize misused statistics: totals instead of rates, small samples and correlation presented as causation
Rewrite a fallacious argument so that it becomes valid or honest

Prerequisites

• If-Then Statements: hypothesis, conclusion and the truth table of $p \to q$
• Converse and Inverse: why a true statement can have a false converse
• Contrapositive: the equivalent form $\neg q \to \neg p$
• Percentages and rates, such as $\frac{90}{120} \times 100$

Discussion Starters

1. If it rains, the street is wet. The street is wet. Did it rain? What else could have happened?
2. Can an argument with a fallacy still have a true conclusion? Give an example.
3. Find a headline that compares totals. What number would you need to make a fair comparison?
4. When is it reasonable to trust an expert, and when does it become an appeal to authority?

Common Misconceptions

If the conclusion is true, the argument must be valid

Remediation: Show an invalid argument with a true conclusion: if a number is divisible by 6, then it is even. 12 is even, so 12 is divisible by 6. The conclusion is true, but the same form gives a false result for 14.

A strong correlation proves that one thing causes the other

Remediation: Use ice cream sales and sunburns: both rise in summer. Ask students to name the hidden third factor (sunny weather) before discussing controlled experiments.

If something worked many times, it will always work

Remediation: Let students compute $2^n - 1$ with a calculator for $n = 2, 3, 5, 7$ , ask for a conjecture, then reveal $n = 11$ and the factors $23 \times 89$ . The surprise makes the point stick.

Differentiation Ideas

For Struggling Students:

• Give a table with the four forms and let students match each argument to a row
• Start with everyday arguments before moving to numbers
• Provide the counterexample and ask students to explain why it breaks the argument

For On-Level Students:

• Name the fallacy in ten short arguments and write a corrected version of each
• Compute rates from news data and decide whether a headline is fair
• Use the truth table of $p \to q$ to explain why affirming the consequent fails

For Advanced Students:

• Research a real statistical claim and identify the possible confounding variables
• Use a calculator to find the next prime $n$ after 11 for which $2^n - 1$ is not prime
• Write a short debate speech that avoids every fallacy in the lesson, then let classmates check it

Standards Alignment

MP3 (CCSS.MATH.PRACTICE.MP3)
Construct viable arguments and critique the reasoning of others
HSS.ID.C.9 (CCSS.MATH.CONTENT.HSS.ID.C.9)
Distinguish between correlation and causation
HSS.IC.B.6 (CCSS.MATH.CONTENT.HSS.IC.B.6)
Evaluate reports based on data

Lesson Resources

visualInteractive Truth Table
Students find the row of $p \to q$ where both premises are true but $p$ is false
activityFallacy Detectives
Groups collect ads and headlines, name the fallacy and rewrite the claim honestly
worksheetValid or Fallacy?
Sort 12 arguments into modus ponens, modus tollens, affirming the consequent and denying the antecedent

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A fallacy is an error in reasoning: the conclusion does not follow from the premises, even if the argument sounds convincing. There are two families.

1. Formal fallacies are mistakes in the form of an argument. Two of them come from if-then statements

p \to q

. Compare them with the two valid forms:

\begin{array}{l|c|c|c} \text{Form} & \text{Premises} & \text{Conclusion} & \\ \hline \text{Modus ponens} & p \to q,\ p & q & \text{valid} \\ \text{Modus tollens} & p \to q,\ \neg q & \neg p & \text{valid} \\ \text{Affirming the consequent} & p \to q,\ q & p & \text{invalid} \\ \text{Denying the antecedent} & p \to q,\ \neg p & \neg q & \text{invalid} \end{array}

Affirming the consequent is the converse error (

q \to p

), and denying the antecedent is the inverse error (

\neg p \to \neg q

). The truth table shows why they fail: in the row where

p

is F and

q

is T, the statement

p \to q

is T and

q

is T, so both premises are true, but

p

is false. (The interactive table checks the converse error: its last column is F in exactly this row.)

2. Informal fallacies are mistakes in the content of an argument:

Ad hominem: attacking the person instead of the argument.
Straw man: changing the other person's claim into a weaker one, then attacking that.
False dilemma: offering only two options when there are more.
Hasty generalization: a general rule from too few cases.
Circular reasoning: the conclusion is hidden in the premises.
Slippery slope: claiming one small step must lead to a disaster.
Appeal to authority: true because a famous or powerful person said so.
Correlation vs causation: two things happen together, so one must cause the other.

Important: a fallacy shows that the argument does not prove its conclusion. The conclusion itself may still be true or false.

Worked Examples

Argument: If a number is divisible by 6, then it is even. 14 is even. Therefore 14 is divisible by 6. Is the argument valid?

Name the parts

$p$ : the number is divisible by 6. $q$ : the number is even. → Rule: $p \to q$

Find which part is given

We are told that 14 is even, so the conclusion $q$ is given, and the argument concludes $p$ . → Form: $p \to q,\ q$ , therefore $p$

Compare with the valid forms

Modus ponens needs $p$ as a premise, not $q$ . This is affirming the consequent. → Invalid form

Check with a calculation

$14 \div 6 = 2$ remainder $2$ → 14 is not divisible by 6

Answer: The argument is invalid: it affirms the consequent. 14 is even but not divisible by 6, so the conclusion is even false here.

Common Mistakes

Thinking that a fallacy proves the conclusion is false

Why it's wrong: A fallacy only shows that the argument does not prove its conclusion. Take the rule if $x = 2$ , then $x^2 = 4$ . Concluding $x = 2$ from $x^2 = 4$ is invalid, yet $x$ might still be $2$ .

Correct: Say the conclusion does not follow. To show it is false you need a counterexample, here $x = -2$ .

Confusing affirming the consequent with modus ponens

Why it's wrong: Both use $p \to q$ and look alike. The difference is which part is given as a premise.

Correct: Given $p$ , you may conclude $q$ (valid). Given $q$ , you may not conclude $p$ (affirming the consequent).

Treating many examples as a proof

Why it's wrong: Checking 3 or even 100 cases is still a hasty generalization if the claim is about all numbers.

Correct: Use examples to make a conjecture, then prove it with an argument that covers every case, or find a counterexample.

Calling every mention of an expert an appeal to authority

Why it's wrong: Citing a relevant expert together with evidence is normal. The fallacy is using a name instead of evidence, especially outside the person's field.

Correct: Ask: is this person an expert on this exact topic, and what evidence supports the claim?

Why It Matters

Fallacies are everywhere, and many of them hide behind numbers:

Mathematics: Checking a formula for 4 cases is not a proof. For the primes $n = 2, 3, 5, 7$ the number $2^n - 1$ is prime, but for $n = 11$ it gives $2047 = 23 \times 89$ .
News and statistics: Headlines often turn a correlation into a cause, or report big totals without dividing by the population.
Advertising: 8 out of 10 users liked it! sounds strong until you learn that only 10 people were asked.
Debates and social media: Ad hominem attacks and straw men replace real arguments, and false dilemmas hide better options.
Science: Researchers use control groups precisely because correlation is not causation.

Once you can name a fallacy, you can explain exactly what is missing and how to fix the argument.

Real World Applications

Advertising Claims

Ads love percentages and famous faces. A small sample and a celebrity are a hasty generalization and an appeal to authority working together.

Example:

An ad says: 80% of users got better grades, says a famous football player! The player is not an expert on learning, and the 80% may come from very few people.

1Try It Yourself

A study app has 5000 users. The ad claim of 80% comes from a survey of 10 users, and 8 of them answered yes.

What percentage of all users were actually asked?

Step 1: Write the mathematical expression

Write the share of users who were asked, as a percentage:

News Headlines and Totals

Headlines often compare totals without dividing by the size of the group. Big cities have big totals of almost everything.

Example:

City A has 4000 deaths a year and City B only 300, so a headline says City A is far more dangerous. But City A has 2000000 people and City B has 100000.

2Try It Yourself

City A: 2000000 people, 4000 deaths a year. City B: 100000 people, 300 deaths a year. Compare the rates per 1000 people.

Which city has the higher death rate per 1000 people?

Step 1: Write the mathematical expression

Write the rate per 1000 people for City A:

Road Safety Statistics

A famous claim says most accidents happen close to home, so driving near home is dangerous. It ignores the base rate: how much driving happens near home.

Example:

If 85% of all kilometers are driven near home, we would expect about 85% of accidents there even if the risk were the same everywhere.

3Try It Yourself

In a region, 700 of 1000 accidents happen within 10 km of home, and 85% of all driving happens within 10 km of home.

Does the data show that driving near home is more dangerous?

Step 1: Write the mathematical expression

Write the number of accidents expected near home if the risk were the same everywhere:

Key Takeaways

1A fallacy is an error in reasoning: the conclusion does not follow from the premises
2Affirming the consequent ( $p \to q$ , $q$ , so $p$ ) and denying the antecedent ( $p \to q$ , $\neg p$ , so $\neg q$ ) are invalid forms
3Modus ponens ( $p \to q$ , $p$ , so $q$ ) and modus tollens ( $p \to q$ , $\neg q$ , so $\neg p$ ) are the valid forms
4Informal fallacies include ad hominem, straw man, false dilemma, hasty generalization, circular reasoning, slippery slope and appeal to authority
5Correlation is not causation: look for a third factor and compare rates, not totals
6Examples never prove a general rule, but one counterexample disproves it

Frequently Asked Questions

If an argument has a fallacy, is its conclusion wrong?

Not necessarily. The fallacy shows that the argument does not prove the conclusion. The conclusion may still be true for other reasons, so you need a counterexample to show that it is false.

What is the difference between a formal and an informal fallacy?

A formal fallacy is wrong because of its structure, whatever the topic: affirming the consequent is invalid for numbers, weather or anything else. An informal fallacy is wrong because of its content, such as a missing option or an attack on a person.

How many cases do I need to check to prove a rule?

No number of cases is enough for a rule about infinitely many numbers.

2^n - 1

is prime for the primes

n = 2, 3, 5, 7

and still fails for

n = 11

. Examples help you guess, and a proof makes it certain.

How can I tell correlation from causation?

Ask three questions: could a third factor cause both, could the cause go the other way, and was there a controlled experiment? Without a fair comparison, a correlation only shows that two things happen together.

Glossary

Fallacy: An error in reasoning that makes an argument fail to prove its conclusion
Formal fallacy: An error in the form of an argument, such as affirming the consequent
Informal fallacy: An error in the content of an argument, such as a straw man or a false dilemma
Affirming the consequent: The invalid form $p \to q$ , $q$ , therefore $p$
Denying the antecedent: The invalid form $p \to q$ , $\neg p$ , therefore $\neg q$
Confounding variable: A hidden third factor that affects two things and makes them look connected

Formula Card

Modus ponens (valid)

p \to q,\ p \ \therefore\ q

The hypothesis is given, so the conclusion follows

Modus tollens (valid)

p \to q,\ \neg q \ \therefore\ \neg p

The conclusion is false, so the hypothesis is false

Affirming the consequent (invalid)

p \to q,\ q \ \therefore\ p

The converse error: $q$ can be true while $p$ is false

Denying the antecedent (invalid)

p \to q,\ \neg p \ \therefore\ \neg q

The inverse error: $q$ can be true while $p$ is false

Teacher Guide: Common Logical Fallacies

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Advertising Claims

News Headlines and Totals

Road Safety Statistics

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Modus ponens (valid)

Modus tollens (valid)

Affirming the consequent (invalid)

Denying the antecedent (invalid)

More in This Topic

Introduction to Logical Reasoning

Deductive Reasoning

If-Then Statements

Converse and Inverse