Do I Have to Flag the Contradiction Explicitly, or Can the Reader Figure It Out?

You have finished the algebra. The last line reads \gcd(p, q) \geq 2, and three lines earlier you wrote \gcd(p, q) = 1. You can see the collision. The question is: should you say it — stop the proof, point at both lines, and declare "contradiction" — or is it fine to just end the page and trust the reader to put the two together?

The short answer: always flag it. In a formal write-up, the contradiction is the whole point of the method, and pointing at it is not padding — it is the step that converts a wall of algebra into a proof. This article explains why, when, and how.

The rule

Every proof by contradiction should name the contradiction explicitly, in one clean sentence. The standard template is:

"This contradicts [the earlier line / the assumption / a known theorem], which is impossible. Therefore the original claim holds."

That sentence does three jobs at once. It names what just collided, it names what it collided with, and it draws the legal conclusion that the original statement must be true. Skipping any of the three leaves the proof unfinished.

Why this is non-negotiable: the form of proof by contradiction is the reductio ad absurdum rule — "if \lnot P implies false, then P." The "implies false" step has to be on the page, literally written down, for the proof to satisfy the inference rule. Deriving a false statement and stopping is not the same as deriving a false statement and flagging it — in a formal logical system, the flag is the inference step. The reader can often see it, but a proof is not supposed to make the reader do logic in their head.

Three reasons flagging wins

1. It signals "this is why the proof is done." Without the flag, a grader has to re-read the previous two lines and mentally pair them. With the flag, the grader reads one sentence and knows exactly why you think you are finished. This is worth full marks on a borderline answer.

2. It protects against false conclusions. Sometimes what looks like a contradiction is not one — see When Is It Really a Contradiction (vs Just Something Weird)?. Writing the flag forces you to quote the earlier line being denied. If you cannot do that, you do not have a contradiction and you should keep going. The act of writing the flag is itself a check.

3. It records the structure for future reference. Months later, when you or someone else reads the proof, the flag is the anchor. It names the exact pair of statements whose collision drove the conclusion. Unflagged proofs age badly — the logic is harder to reconstruct.

What counts as "flagging"

You do not need a flashing neon sign. Any of these phrasings is a valid flag:

"But we also had \gcd(p, q) = 1, a contradiction."
"This is impossible, since \gcd(p, q) = 1 by assumption."
"Contradiction with the assumption p > 0."
"Hence 2 \mid p and 2 \mid q, contradicting \gcd(p, q) = 1."

Each version names the two colliding statements. The key grammatical ingredients: a linking word (contradicts, contradiction with, but, impossible since) and a quotation of the earlier line. If either is missing, the flag is too weak.

What does not count

Just stopping. Writing \gcd(p, q) \geq 2 and ending the proof there, expecting the reader to recall \gcd(p, q) = 1 from three lines ago, leaves the proof unfinished. Graders typically dock marks.

Flagging without quoting. Writing "contradiction!" by itself names the problem but not the pair. A reader has to search upward to figure out which earlier line is meant. Always quote the earlier statement.

Concluding without the formal step. Flagging the contradiction is only two-thirds of the job. The third piece is the inference: "Therefore the assumption was false, so the original claim is true." If you flag the contradiction and then walk away, a reader might not know whether you are using contradiction or have gone off the rails. The closing sentence is free and should always be written.

A compact model closing

Here is a complete, well-flagged ending you can copy for most proofs by contradiction:

"We have shown X and \lnot X, which is a contradiction. Therefore our assumption [state it] is false, and the original claim [state it] holds. \blacksquare"

Fill in X with the specific colliding pair. Fill in the assumption and the original claim. That is the template. Use it every time.

Flagging the contradiction in a full proof

Claim. There is no positive integer n with n^2 = 2.

Proof (with flag). Suppose, for contradiction, that such an n exists. Then n^2 = 2. Since n \geq 1, we have n^2 \geq 1. Also, since n \geq 2 would give n^2 \geq 4 > 2, we must have n = 1, and then n^2 = 1 \neq 2.

This contradicts n^2 = 2. Therefore no such positive integer n exists. \blacksquare

Notice the final line. It names what collided (n^2 = 1 vs. n^2 = 2), calls it a contradiction, and draws the conclusion. Without that sentence, the reader is left holding two unconnected equations and an implicit request to do logic themselves.

Proof (without flag — weaker). ... so n = 1, and then n^2 = 1. \blacksquare

This version has the contradiction inside it, but a reader has to locate the earlier n^2 = 2 line and pair it up. Most graders will mark this as incomplete.

When flagging can be light

In a textbook or a published paper, authors sometimes abbreviate: "which is impossible," or simply "\Rightarrow\!\Leftarrow" (the double-arrow contradiction symbol). These are fine in context, because the surrounding paragraph makes the collision clear. But they work only when the pair being contradicted is unambiguous — usually when one of them is the assumption itself, stated in the very first line of the proof.

In a student answer or an exam, the safer policy is always: quote the earlier line, name the contradiction, draw the conclusion. Three sentences you can say by rote. They add no cost, and they guarantee full credit.

The habit to internalise

Every proof by contradiction has the same skeleton:

Assume, for contradiction, that [negation of claim].
(derivation)
This contradicts [quoted earlier line], which is impossible.
Therefore [original claim] holds.

Writing lines 1, 3, and 4 is not optional ceremony. They are the proof. The algebra between them is the delivery mechanism, but the delivery mechanism alone is not the proof — the scaffolding around it is. A proof without the flag is like a joke without a punchline: the setup is all there, but nobody laughs.