I encountered this in database subject which I am not able to understand even after lot of efforts.
In relational calculus what do you mean by ¬(∀x)(¬P(x)) ?
As far as I know, (∀x)(P(x)) means P(x) is true for all x.
(∀x)(¬P(x)) means P(x) is not true for any x but all other tuples belonging to other tables are TRUE.
(¬∀x)(¬P(x)) means some of the tuples belonging to other tables are TRUE.
Kindly elaborate. If an example can be given, it will be really nice.
$\neg(\forall x)(\neg P(x))$ and $(\neg\forall x)(\neg P(x))$ are equivalent expressions.
Both are read as: "Not for all $x$, $P(x)$ is false".
This is also equivalent to: "For some $x$, $P(x)$ is true"
$$\neg \color{silver}(\forall x\color{silver}{)(}\neg P(x)\color{silver}) \qquad\iff\qquad \color{silver}(\exists x\color{silver}{)(}P(x)\color{silver})$$