Common Logical Fallacies

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.

Affirming the Consequent

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.

Denying the Antecedent

Argument: If a number is divisible by 4, then it is even. 10 is not divisible by 4. Therefore 10 is not even. Find the error and fix the argument.

Write the form: $p$: divisible by 4. $q$: even. Given $\neg p$, the argument concludes $\neg q$. = Denying the antecedent

Test the conclusion: $10 \div 4 = 2.5$, but $10 \div 2 = 5$ = 10 is even, the conclusion is false

Find the valid form: Modus tollens uses $\neg q$: if a number is not even, then it is not divisible by 4. = $\neg q \to \neg p$ is valid

Fix the argument: 15 is not even. Therefore 15 is not divisible by 4. = A valid argument

Answer: The argument denies the antecedent. The number 10 shows the conclusion can be false. A valid version starts from not even, for example: 15 is not even, so 15 is not divisible by 4.

A False Dilemma with Zero

Argument: Every number is either positive or negative. 0 is not positive. Therefore 0 is negative. What is wrong?

Find the either-or premise: The first premise allows only two options: positive or negative. = Two options are offered

Look for a missing option: $0$ is neither positive nor negative. = A third option exists

Name and fix the fallacy: The premise is a false dilemma. Correct version: every number is positive, negative or zero. = False dilemma

Answer: The first premise is a false dilemma, because zero is a third option. With the corrected premise, the conclusion 0 is negative no longer follows.

Correlation Is Not Causation

In a school of 200 students, 120 eat breakfast and 90 of them pass the math test. Of the 80 who skip breakfast, 40 pass. Claim: eating breakfast causes better math results. Check the numbers and the reasoning.

Pass rate with breakfast: $\frac{90}{120} \times 100 = 75$ = $75\%$

Pass rate without breakfast: $\frac{40}{80} \times 100 = 50$ = $50\%$

Describe what the data shows: The two things happen together: a correlation of 25 percentage points. = Correlation, not proof

Look for a hidden third factor: Students who sleep enough or have a calm morning routine may both eat breakfast and study better. = Possible confounding variable

Answer: The pass rates are $75\%$ and $50\%$, so there is a correlation. It does not prove causation: a third factor, like sleep, could explain both. A fair test would compare similar groups in a controlled experiment.

It Worked for 4 Cases

A student notices that $2^n - 1$ is prime for the primes $n = 2, 3, 5, 7$ and concludes: if $n$ is prime, then $2^n - 1$ is always prime. Test the claim with $n = 11$.

Check the first cases: $2^2 - 1 = 3$, $2^3 - 1 = 7$, $2^5 - 1 = 31$, $2^7 - 1 = 127$. = All four are prime

Name the type of reasoning: Four cases are used to claim a rule for all primes. = Hasty generalization

Test $n = 11$: $2^{11} - 1 = 2048 - 1 = 2047$ = $2047$

Factor the result: $2047 = 23 \times 89$ = Not prime: a counterexample

Answer: The claim is a hasty generalization. The first four primes work, but for $n = 11$ the result is $2047 = 23 \times 89$. Many examples never prove a rule, one counterexample disproves it.

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.

Mistake: Thinking that a fallacy proves the conclusion is false

Why: 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 $x = -2$.

Mistake: Confusing affirming the consequent with modus ponens

Why: 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).

Mistake: Treating many examples as a proof

Why: 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.

Mistake: Calling every mention of an expert an appeal to authority

Why: 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?

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.

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.

News Headlines and Totals

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

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.

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.

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.

A fallacy is an error in reasoning: the conclusion does not follow from the premises

Affirming the consequent ($p \to q$, $q$, so $p$) and denying the antecedent ($p \to q$, $\neg p$, so $\neg q$) are invalid forms

Modus ponens ($p \to q$, $p$, so $q$) and modus tollens ($p \to q$, $\neg q$, so $\neg p$) are the valid forms

Informal fallacies include ad hominem, straw man, false dilemma, hasty generalization, circular reasoning, slippery slope and appeal to authority

Correlation is not causation: look for a third factor and compare rates, not totals

Examples never prove a general rule, but one counterexample disproves it

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

A: 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.

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

A: 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.

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

A: 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.

Q: How can I tell correlation from causation?

A: 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.

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Common Logical Fallacies

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Logic & Sets / Logical Reasoning

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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