Fallacy — Definition, Formula & Examples
A fallacy is a mistake in reasoning: an argument whose conclusion does not actually follow from its premises, even if it looks convincing. In math, fallacies show up as invalid logical steps or as "proofs" that break a rule, such as dividing by zero.
An argument is valid if every assignment of truth values that makes all premises true also makes the conclusion true. A formal fallacy is an argument form that is not valid: at least one interpretation makes every premise true and the conclusion false. Equivalently, the conditional (premises) → (conclusion) is not a tautology. An informal fallacy is an error in content or language, such as ambiguity or an unstated false assumption, rather than in logical form.
Key Formula
Where:
- = The hypothesis (antecedent) of the conditional
- = The conclusion (consequent) of the conditional
- = Implication: "if ... then ..."
- = Conjunction: "and"
How It Works
To test whether an argument is a fallacy, first strip it down to its logical form using letters such as P and Q for the statements involved. Then ask one question: can all the premises be true while the conclusion is false? If you can find even one such case, the form is invalid, and that case serves as your counterexample. In algebraic "proofs," read each line and check that the operation used is legal for every allowed value of the variables. Watch closely for division by an expression that might equal zero, square roots of negative numbers, and steps that quietly assume the result being proved.
Example
Problem: Decide whether this argument is valid: "If an integer n is divisible by 4, then n is even. The number 6 is even. Therefore, 6 is divisible by 4."
Step 1: Translate the argument into symbols. Let P be "n is divisible by 4" and Q be "n is even." The first premise is a conditional and the second asserts Q. The argument then concludes P.
Step 2: Check the premises for n = 6. Every multiple of 4 is even, so the conditional is true. The number 6 is even because it equals 2 times an integer.
Step 3: Check the conclusion. Dividing 6 by 4 leaves a remainder of 2, so 6 is not divisible by 4. The conclusion is false.
Step 4: Both premises are true and the conclusion is false. An argument form that allows this is invalid. This particular pattern, which assumes the converse Q → P, is called affirming the consequent.
Answer: The argument is a fallacy (affirming the consequent). The number 6 is a counterexample: it is even but not divisible by 4.
Another Example
The first example is a formal fallacy, where the logical form itself is invalid. This example is an algebraic fallacy. The logic looks fine, but one step uses an operation that is undefined for the values involved.
Problem: Find the error in this "proof" that 2 = 1. Let a = b. Then a² = ab, so a² − b² = ab − b². Factoring gives (a + b)(a − b) = b(a − b). Dividing both sides by (a − b) gives a + b = b. Since a = b, this becomes 2b = b, so 2 = 1.
Step 1: Check the algebra line by line. Multiplying a = b by a gives a² = ab. Subtracting b² from both sides is legal. Both factorizations are correct identities.
Step 2: Look at the division step. Dividing both sides of an equation by a quantity is only allowed when that quantity is nonzero. Here the assumption a = b forces a − b to be 0.
Step 3: Test with actual numbers to see what goes wrong. Take a = b = 1. Then (a + b)(a − b) and b(a − b) both equal 0, so the equation reads 0 = 0, which is true. But canceling the zero factor turns 0 = 0 into 2 = 1, which is false.
Step 4: From xy = zy you may conclude x = z only when y ≠ 0. The proof cancels a factor equal to zero, so the step does not follow.
Answer: The fallacy is division by zero: the proof divides by a − b, which equals 0 because a = b.
Why It Matters
In an intro-to-proofs or discrete math course, you must justify every step, and graders look specifically for affirming the consequent and circular arguments. Spotting division by zero and misused root rules is essential in algebra and calculus, where these errors produce wrong answers or extraneous solutions. Fallacy detection also underpins work in law, philosophy, and computer science, where program verification checks that code logically guarantees its specification.
Common Mistakes
Mistake: Treating a conditional as if it implied its converse, for example proving Q and then claiming P from P → Q.
Correction: A conditional and its converse are independent statements. To conclude P, you need P itself, Q → P, or the contrapositive form ¬P → ¬Q.
Mistake: Deciding a conclusion is false because the argument for it is fallacious.
Correction: An invalid argument only fails to prove its conclusion. The conclusion might still be true, and a different, valid argument could establish it.
Mistake: Dividing both sides of an equation by a variable expression, such as solving x² = x by dividing by x.
Correction: Dividing by x silently assumes x ≠ 0 and loses the solution x = 0. Factor instead: x² − x = x(x − 1) = 0, which gives x = 0 or x = 1.
Check Your Understanding
Name the fallacy: "If x = 3, then x² = 9. We know x ≠ 3. Therefore x² ≠ 9." Give a counterexample.
Hint: The argument reasons from "not P" to "not Q."
Answer: Denying the antecedent. Take x = −3: then x ≠ 3, but x² = 9.
Is this valid? "Every differentiable function is continuous. f(x) = |x| is continuous at 0. So f is differentiable at 0."
Hint: Compare the form to P → Q, Q, therefore P.
Answer: No. It affirms the consequent. In fact, |x| is continuous at 0 but not differentiable there, because it has a corner.
Find the flawed step: 1 = √1 = √((−1)(−1)) = √(−1)·√(−1) = i·i = −1.
Hint: Ask which property of square roots was used, and under what conditions it holds.
Answer: The step √((−1)(−1)) = √(−1)·√(−1). The rule √(ab) = √a·√b holds for a, b ≥ 0, not for two negative numbers.