Logical Fallacies — Mathematics Knowledge System

Overview

A logical fallacy is an argument whose conclusion does not actually follow from its premises, even though it may look persuasive. Formal fallacies are errors in the argument's logical structure itself — the argument would be invalid no matter what statements are substituted into its variables — and are typically named by the invalid inference pattern they mimic. The two most common formal fallacies are affirming the consequent (from P → Q and Q, wrongly concluding P) and denying the antecedent (from P → Q and ¬P, wrongly concluding ¬Q); both are easily confused with the genuinely valid rules modus ponens (P → Q, P ⊢ Q) and modus tollens (P → Q, ¬Q ⊢ ¬P). This is distinct from informal fallacies (such as ad hominem or appeal to authority), which are errors in content or context rather than structure — a formal fallacy is invalid for every substitution of its propositional variables, which is exactly why truth tables and formal proof can catch it mechanically.

Intuition

Picture the implication 'if it rains (P), the ground gets wet (Q).' Modus ponens is sound: told it is raining, you correctly conclude the ground is wet. But affirming the consequent reasons backwards: told the ground is wet, it is tempting but wrong to conclude it rained — a sprinkler could have wet the ground instead, so Q being true doesn't pin down P. Denying the antecedent makes a similar mistake in the other direction: told it did NOT rain, it is tempting but wrong to conclude the ground is dry — again, the sprinkler could still be running. Both fallacies feel intuitively compelling because they resemble the two genuinely valid rules (modus ponens and modus tollens), but a single counterexample scenario (the sprinkler) is enough to show the implication P → Q alone never forces the reverse or negated conclusion.

Formal Definition

Definition

The two classic formal fallacies, contrasted with the valid inference rules they resemble:

P \to Q,\ P \;\vdash\; Q \quad \text{(modus ponens — valid)}

Modus ponens

P \to Q,\ Q \;\nvdash\; P \quad \text{(affirming the consequent — invalid)}

Affirming the consequent

P \to Q,\ \neg Q \;\vdash\; \neg P \quad \text{(modus tollens — valid)}

Modus tollens

P \to Q,\ \neg P \;\nvdash\; \neg Q \quad \text{(denying the antecedent — invalid)}

Denying the antecedent

Worked Examples

Let P = 'divisible by 4' and Q = 'divisible by 2'. The argument has the form P → Q, Q, therefore P.
$P \to Q,\ Q \;\therefore\; P$
This is exactly the pattern of affirming the consequent, which is invalid — and indeed the conclusion is false, since 6 is divisible by 2 but not by 4.
$6 \text{ is divisible by } 2 \text{ but } 6 \nmid 4\text{-divisibility}$

Answer: This commits the fallacy of affirming the consequent; the conclusion (6 is divisible by 4) is in fact false, which confirms the argument form is unsound.

Practice Problems

Difficulty 2/10

Classify: 'If n is prime, then n is odd or n = 2. Suppose n is not prime. Therefore n is neither odd nor 2.' Is this valid, and if not, which fallacy does it commit?

Difficulty 3/10

Classify: 'If a quadrilateral is a square, then it has four right angles. This quadrilateral has four right angles. Therefore, it is a square.' Which fallacy, and give a counterexample.

Difficulty 4/10

Explain why modus tollens (P → Q, ¬Q ⊢ ¬P) is valid while denying the antecedent (P → Q, ¬P ⊢ ¬Q) is not, using truth tables.

Quiz

Summary

A logical fallacy is an argument whose conclusion does not follow from its premises; formal fallacies fail due to invalid structure alone.
Affirming the consequent (P → Q, Q ⊢ P) and denying the antecedent (P → Q, ¬P ⊢ ¬Q) are the two classic invalid patterns, easily confused with the valid modus ponens and modus tollens.
A single counterexample scenario, or a truth-table check across all rows, is enough to expose a formal fallacy.

References

WebsiteWikipedia — Formal fallacy