You are writing a proof of "if P then Q," you get stuck halfway, and a tempting thought appears: "what if I just use Q here — it's what I'm trying to show anyway, right?" That instinct feels innocent. It is not. Using the conclusion as a step in its own proof is one of the oldest and most common logical errors in mathematics, and it has a name: begging the question, or in Latin petitio principii.
The short answer: no, you cannot. Not even quietly. Not even as an intermediate step. The whole job of a proof is to derive Q from P — if Q is already allowed in the derivation, you have derived nothing.
The rule in one line
In a direct proof of "if P then Q," the only things you may use are:
- the hypothesis P itself,
- definitions, axioms, and previously proved theorems,
- statements that follow logically from the above via legal moves.
Q is not on this list. Neither is anything logically equivalent to Q, because assuming a logical equivalent is the same as assuming Q wearing a disguise.
Why the rule is strict: a proof is a construction that shows the conclusion cannot fail to hold when the hypothesis does. If you use the conclusion as a brick in the construction, the building stands on itself — remove the imported brick and the whole thing collapses. The proof has no independent support.
A miniature example of the trap
Claim. For every positive integer n, n^2 \ge n.
A student attempt that begs the question:
Since n^2 \ge n, we can divide both sides by n (which is positive) to get n \ge 1, which is true for every positive integer. Hence n^2 \ge n. \blacksquare
Read that carefully. The first line assumes n^2 \ge n — the very thing being proved — and then derives something true from it. The conclusion is reused as a premise. The argument proves nothing about n^2 \ge n; it only shows that if n^2 \ge n were true, then n \ge 1 would also be true. That is a different statement.
The correct direct proof runs the other way:
Let n \ge 1. Multiply both sides by n (which is positive, so the inequality is preserved): n \cdot n \ge n \cdot 1, i.e. n^2 \ge n. \blacksquare
Now the chain starts at a hypothesis (n \ge 1, true for every positive integer) and ends at the conclusion. No cheating.
The circular chain diagram
The sneakier disguises
The obvious version of begging the question — literally writing "Q" inside the proof — is easy to catch. The dangerous cases are the ones where Q shows up in disguise. Three common disguises:
Disguise 1: a logical equivalent of Q. If Q is "\sqrt{2} is irrational," then "\sqrt{2} cannot be written as \dfrac{p}{q} in lowest terms" is the same statement reworded. Using either one in the derivation is the same error.
Disguise 2: a slightly weaker claim whose only proof goes through Q. Suppose you invoke "every even number greater than 2 is the sum of two primes" (Goldbach's conjecture, still unproven) to prove something about primes. If the downstream claim is in fact equivalent to Goldbach, you have begged the question. If it is genuinely weaker and you have an independent proof of it, fine — but the independence must be real.
Disguise 3: a "well-known fact" that is actually just Q. Students sometimes write "it is well known that..." to skip a step, and the well-known fact turns out to be the conclusion itself, restated informally. If you cannot name where the fact is proved (an axiom? a theorem with a reference? a previous step in this proof?), assume it is the thing you still have to show.
Why disguises matter: a proof's legitimacy is preserved under any restatement of the conclusion. A proof that works only because you quietly used Q in a different form is still not a proof. The moment you flag a step as "I am using Q here," the circularity is obvious; the moment you reword Q to hide it, the circularity is camouflaged but no less fatal.
What is allowed, and how to break the loop
If you feel the pull to use Q — if every path forward seems to require something that looks like the conclusion — that is a signal, not a permission slip. The signal is: the direct path may be blocked, and a different proof technique may be needed.
The three escape routes:
- Rework the direct proof. Often the blockage is from using a weaker definition or missing an algebraic manipulation. Go back to the hypothesis and unpack more carefully.
- Switch to proof by contrapositive. Instead of proving "if P then Q," prove "if not Q then not P." The contrapositive is logically equivalent, and the new hypothesis ("not Q") gives you concrete algebraic material to work with.
- Switch to proof by contradiction. Assume P and not Q, and derive an impossibility. This is legal — you are assuming Q's negation, not Q itself — and it often works when direct proof stalls on a negative conclusion.
Each of these techniques is a different legal way to avoid ever writing "Q" as a step.
A three-line diagnostic
Before you submit any proof, run it through three questions:
- Is the hypothesis P actually used somewhere? If not, your proof proves a stronger statement (it holds without P). Usually that means you have smuggled Q in somewhere else.
- Does every step have a justification from the legal list — P, definitions, axioms, prior theorems? If any step is unjustified, that step is probably where Q sneaked in.
- If you cross out the line stating Q at the end, can you still reach the conclusion from the earlier lines? If the only route to Q is a step that says "therefore Q" without derivation, the proof is circular.
Three honest yes / yes / yes answers mean the proof is at least not begging the question. (It may still have other errors, but circularity is ruled out.)
The short summary
- Using the conclusion Q inside its own proof is called begging the question and proves nothing.
- The same error applies to any statement logically equivalent to Q, however reworded.
- If you feel forced to use Q, that is a sign to switch techniques — contrapositive, contradiction, or a cleaner direct proof — not a sign to cheat.
- The legal materials are: the hypothesis, definitions, axioms, and previously proved theorems. Q is never on this list.
Related: Mathematical Proof — Direct Proof · What You Can Assume When You Assume P · Proof by Contradiction · Proof by Contrapositive