Positively confused about negation

Question

At first, negation was obvious. However, the more I thought about it, the more I got confused on why the answers are what they are. For example,

$$P = \text{The real number } r \text{ is at most } \sqrt{2}$$

whose negation is

$$\neg P = \text{The real number } r \text{ is greater than } \sqrt{2}$$

When I try to think in a precise way about it, I get more confused why it is not "It is not the case...at most $\sqrt 2$. Meaning, it is possible that the following is a possibility : "The real number $r$ is at most $2$".

Can it not be "...at most some $x$" for some $x$ as long as it is not equal to $\sqrt{2}$?

Answer 1

To say :

$r$ is at most $\sqrt{2}$,

means : $r \le \sqrt{2}$.

Here we are using "at most" in a different sense with respect to "some x", that must be translated with the existential quantifier : $\exists$.

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The statement express a relation between two (real) numbers : $r$ and $\sqrt{2}$.

To negate it, we have to express the fact that the two numbers (in that order) do not satisfy the relation.

But the usual translation of "not-(less-or-equal)" is "greater-then".

Thus, the negated statement will be : $r > \sqrt{2}$.

Answer 2

Think of negation as exact opposite of a statement x≤√2 has negated statement of x>√2

• A quatifier here however does make no sense because by stating some sort of x you are violating negation case in this particular example.

Answer 3

Anecdotally, the easiest way to translate is to rephrase the not statement. I.e.

$\urcorner$ ($\forall$) translates to "not for all". In our brains we translate this to, "well there's none". However, if you rephrase it, "not for everything", then it's much easier to understand,namely, there exists some value: $\exists$.