Assume the Negation — But Still Check Your Moves Preserve Meaning

The misconception: "In a proof by contradiction, I assume the opposite and then just push symbols around until something breaks. The 'assumption' does all the work, so I don't have to think carefully about each step — the mechanism handles it."

This is wrong in a specific, costly way. The assumption sets up the hypothesis, yes — but every step after it is a normal deduction that has to be logically valid. If you make an invalid move — dividing by something that might be zero, squaring an inequality, losing a case — you will produce a false statement that looks like a contradiction but was actually generated by your own bad algebra. The proof will be invalid even though the claim it was defending might be true.

This article shows the trap with concrete examples, gives the two moves to audit, and explains why "just mechanically push symbols" feels fine but fails.

Where the trap comes from

A proof by contradiction has a seductive structure: one assumption at the top, one contradiction at the bottom. Between them, a chain of symbolic steps. Because the whole argument is wrapped in "suppose, for contradiction, that...", students get the impression that the wrapping absolves them of needing to be careful — as if the word "suppose" let them take algebraic shortcuts they would not dare take in a direct proof.

It does not. The assumption only sets the starting point. Every arrow from one line to the next is a regular inference and has to be valid on its own terms.

Why the assumption does not license careless steps: "suppose P" means "take P as a premise and derive consequences." It does not mean "take P as a premise and derive any symbolic string via any manipulation." The inference rules you use — algebra, substitution, factoring, case splits — are the same ones as in a direct proof, with the same validity requirements. The logical weight of "suppose" is at the very first step only; every following step bears its own burden.

Example: a fake contradiction that feels real

Claim: There is no real number x satisfying x^2 + 1 = 2x.

Fake proof (don't do this). Suppose there is such an x. Then x^2 + 1 = 2x, so x^2 - 2x + 1 = 0, so (x - 1)^2 = 0, so x = 1. Now divide both sides of the original equation by x: x + 1/x = 2. "But" — the student says — "x + 1/x \geq 2 for x > 0 with equality only at x = 1, and the strict inequality breaks the supposition."

Where is the error? The student concluded their proof at a place that is not a contradiction. Setting x = 1 gives (1)^2 + 1 = 2(1) = 2, which is true — there is no contradiction. The original claim is simply false: x = 1 really does solve x^2 + 1 = 2x. The student was so committed to "we assumed, so there must be a contradiction somewhere" that they invented one by misreading the equality case of an inequality as the "contradictory" branch. The form of the reasoning wore the look of a contradiction proof without any of its substance.

Lesson: "suppose" does not mean "a contradiction exists somewhere." It means "let us see if one does." If you cannot honestly produce the red line (see the Highlight the Contradiction Sentence article), the original claim might be true — in which case your "contradiction proof" was attempting to prove something false.

The two audits every step needs

When you write a step like "from line 4, therefore line 5," check two things:

Audit 1 — Is the inference rule valid in general? Algebra has side conditions. You cannot divide by zero. You cannot take the square root without acknowledging both branches. You cannot square an inequality without considering signs. Each algebraic move has a contract; break it, and your "consequence" was not actually implied by the previous line.

Audit 2 — Is the inference valid under the assumption you are operating in? This is subtler. When you assumed "x is even," all steps that follow must be consistent with x being even. If you then divide both sides of an equation by x, you need to note that x \neq 0 — which you may or may not be granted, depending on the setup. If the assumption constrains x to be non-zero automatically, fine; if not, you just introduced a case split you did not handle.

Why Audit 2 is harder than Audit 1: the assumption adds information to every subsequent line. Sometimes that information licenses moves you could not make in general ("we know q \neq 0 because p/q is a fraction, so dividing by q is safe"). Sometimes it forbids moves you could make in general. Staying aware of what the assumption grants and forbids is part of the craft of a contradiction proof; forgetting it is how fake contradictions are manufactured.

The "n is odd; assume n is even" scenario

Here is the exact scenario the misconception describes. You want to prove a claim of the form "n is odd." You assume, for contradiction, that n is even. You start doing algebra: n = 2k for some integer k, and so on. What could go wrong?

Several things:

You forget that every substitution must respect the domain. If the original statement was about positive integers, "n = 2k" requires k \geq 1, not just k \geq 0. Forgetting this lets you accidentally include n = 0 in your argument and derive bogus conclusions.
You do a manipulation that only works for odd numbers, forgetting you are now in the "even" world. For instance, if you cite a lemma that only applies when n is odd, you have falsely applied a non-premise. The lemma does not hold under your current assumption.
You divide by something that the assumption makes zero or undefined. For example, if the assumption "n is even" forces some expression n - \text{(odd)} to be odd, and you then try a manipulation requiring that expression to be zero, you are dividing by non-zero — fine — but if the roles are reversed, you might be dividing by zero and creating a false equation.

Audit 1 and Audit 2, applied to every line, catch all three mistakes. "Mechanical" pushing does not.

The right way to run a contradiction proof

Here is the discipline to develop:

Write the assumption clearly as a line at the top, tagged as "suppose, for contradiction."
At every step, annotate which rule justifies the move (algebra, definition, a named theorem, a substitution). Do not skip these annotations even in rough work.
When you invoke a theorem or definition, pause for three seconds and ask whether its hypotheses are satisfied under your current assumption.
When you hit what looks like a contradiction, run the two-part check from When Is It Really a Contradiction before stopping.

This sounds heavy, but it is not. After a few proofs done this way, the audits happen automatically in your head and leave only the ones that genuinely need work.

Find the invalid step

Claim: No positive integer n satisfies n^2 = n.

"Proof":

(a) Suppose, for contradiction, that n is a positive integer with n^2 = n.

(b) Divide both sides by n: n = 1.

Find the error.

Line (b) is valid — n is a positive integer, so n \neq 0, and dividing is allowed. Line (c) is where the proof collapses: the student noticed that n = 1 satisfies the equation and treated "satisfies the equation" as "contradicts the supposition." But the supposition was that some positive integer satisfies the equation — finding one does not contradict that, it confirms it.

Result: The claim is false; n = 1 is a counterexample. The "contradiction proof" was attempting to prove something untrue, and the attempt failed at Audit 1 on line (c): "therefore, contradiction" was not justified by what came before.

The lesson is the same as before: the word "suppose" at the top does not guarantee a contradiction at the bottom. Every step — including the final "so, contradiction" — must earn its justification.

Treat every line of a contradiction proof the way you would treat a line of a direct proof: as an inference that needs to be valid on its own merit. The "suppose" at the top is just one more premise; the rest of the work is ordinary careful reasoning. Students who skip the audits do not save time — they produce proofs that cannot be trusted, and lose marks when a grader tries to find the red line and discovers there isn't one.