Negating a Quantified Statement

Denying a for-all claim

The negation of a statement is true exactly when the statement is false. "For all n, P(n)" is false as soon as one n fails, so its negation is "there exists n with not P(n)". The quantifier swaps from for all to there exists, and the inside is denied.

The negation does not say that every case fails. Denying that every case works leaves room for many cases that do work. It insists only on one case that does not.

Denying a there-exists claim

"There exists n with P(n)" is false only when no n works, that is, when every n fails. So its negation is "for all n, not P(n)". The swap runs the other way: there exists becomes for all.

The negation of "some triangle has two right angles" is "no triangle has two right angles", which says that every triangle has fewer than two right angles. The negation of "there exists a whole number n with $n^2=2$n² = 2" is "for all whole numbers n, $n^2\ne 2$n² ≠ 2".

The claims at the top are for all x, $x\le 6$x ≤ 6, and there exists x with x > 6. Each is the negation of the other: "not every x is at most 6" says the same as "some x is more than 6". Lift $x_5$x₅ past the line and both change at once.

Not all primes are odd

To negate a statement, put "not" in front of it and move the "not" inward one step at a time. Each time it passes a quantifier, the quantifier swaps. When it reaches the inside, the inside is denied.

Start from "it is not true that every prime is odd". Moving the "not" past "every" turns it into "some", and the inside becomes "is not odd": some prime is not odd. That prime is 2, so the negation is true and "every prime is odd" is false.

With two quantifiers, the "not" passes both. The negation of "for all n, there exists m with m > n" is "there exists n such that for all m, $m\le n$m ≤ n": some whole number n is at least as large as every whole number m. That is false, because m = n + 1 is larger, so the original is true.

The first six primes. Every one is odd except 2, so "some prime is not odd" is true and "every prime is odd" is false. One column is all the negation needs.

Denying too much

Negating only the inside, and keeping the quantifier, gives "every prime is not odd", that is, "no prime is odd". That claims far more than the negation does, and it is false: 3 is an odd prime.

A statement and its negation are never both false. "Every prime is odd" is false, so its negation must be true. "Some prime is not odd" is true; "no prime is odd" is false, so it cannot be the negation.

A conditional inside

Many for-all statements have a conditional inside. "Every multiple of 4 is even" is "for all n, if n is a multiple of 4, then n is even". A conditional is false only when its hypothesis holds and its conclusion fails, so the negation is "there exists n that is a multiple of 4 and is not even". No such n exists, since 4k = 2(2k) is even, so the original is true.

The usual mistakes

Keeping the quantifier. The negation of "for all n, $n^2>n$n² > n" is "there exists n with $n^2\le n$n² ≤ n", which n = 1 makes true. "For all n, $n^2\le n$n² ≤ n" claims the failure happens everywhere, and n = 2 breaks it.

Negating the inside wrongly. The negation of $n^2>n$n² > n is $n^2\le n$n² ≤ n, not $n^2<n$n² < n, because equality was never ruled out.

Negating a conditional as a conditional. The negation of "every train that stops at Airport stops at Central" is not "every train that stops at Airport does not stop at Central". It is "some train stops at Airport and does not stop at Central".

A train timetable

In the application below, six trains stop at three stations. The negation of "every train that stops at Airport also stops at Central" asks for one train that stops at Airport and not at Central, and the timetable has two.