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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 $\sqrt{2}$?

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  • $\begingroup$ You've hit a subtle ambiguity of language. $r$ is at most $\sqrt 2$ can be interpreted as either i) $r$ has a specific value and $r \le \sqrt 2$. or as ii) we really don't know what $r$ is but we know it's at most $\sqrt 2$. The negation of i) is $r > \sqrt 2$. The negation of ii) is we dont know that $r$ is most $\sqrt 2$ so $r$ could be anything... well, this is math, not confess our short comings to our shrinks. So $r$ is at most $\sqrt 2$ means i) $r$ is specific and $r\in (-\infty, \sqrt 2]$. That's a statement about $r$; not about what we know about $r$. $\endgroup$
    – fleablood
    Jun 14, 2018 at 7:38
  • $\begingroup$ But that's a clever ambiguity of language and I'm sure we can make some very good paradox sounding puzzles from it. $\endgroup$
    – fleablood
    Jun 14, 2018 at 7:40

3 Answers 3

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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 : $\exists$.


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 : $r > \sqrt{2}$.

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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.
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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: $\exists$.

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