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I need to negate the following statement : "Some integers are not odd"

I have the below, where O(x) is "odd" $$ \exists x (\neg O(x)) $$

Would the negation be $$ \forall x (O(x)) $$

I'm confused, since if so, this statement does not make sense since there can be even numbers, does the negation have to be real?

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    $\begingroup$ Why doesn't it make sense? The original statement is true (some integers are not odd) so it makes sense that the negation of that original statement becomes false. $\endgroup$ – JMoravitz Jul 21 '16 at 3:12
  • $\begingroup$ I guess I misunderstood the concept of the negation, is the negated statement correct? "for all x, x is odd" $\endgroup$ – splinks Jul 21 '16 at 3:17
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    $\begingroup$ Sensible statements can be false. "The sky is orange" makes sense but is false. "The sky is not orange" makes sense and is true. "The sky is Tuesday" doesn't make sense. The negation of a sensible true statement needs to be a sensible false statement, and vice versa. $\endgroup$ – Graham Kemp Jul 21 '16 at 3:31
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As alluded to in Graham Kemp's comment, you seem to be mixing up true statements with meaningful statements. Meaningful statements are sometimes called syntactically correct statements, and can be either true or false.

The negation of a true statement is false, and the negation of a false statement is true. So it stands to reason that, when you negated $\color{blue}{\exists x \; \lnot O(x)}$ ("some integers are not odd"), a true statement, you got $\color{red}{\forall x O(x)}$ ("all integers are odd"), a false statement.

On the other hand, the negation of a meaningful statement is meaningful. In this case, "some integers are not odd" and "all integers are odd" are both meaningful things to say (they make sense), even if one of them is completely wrong.

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Don't worry about it.   Sensible statements can be either true or false.   "The sky is orange" makes sense but is false.   "The sky is not orange" makes sense and is true.   "The sky is Tuesday" doesn't make sense.


In the domain of integers, $\exists x~\neg O(x)$ means "There exists an integer which is not odd."   This statement is true.

The negation of that statement is indeed, $\forall x~O(x)$ , which means "Every integer is odd."   This statement is false, as it should be.

We expect the negation of a true statement to be a false statement.   That is what negation means.


$$\neg \exists x~{\neg O(x)}~ \iff ~\forall x~{O(x)}$$

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