Suppose I want to 'negate' the following expression:
$\exists m$ s.t. $\forall n > 0$, …
Would the negation be of the form:
(a) $\not \exists m$ s.t. …
or
(b) $\forall m \exists$ …
or
(c) Does it depend on your definition of negation
And if the answer (as I suspect) is (b), can you explain why it is (b) rather than (a)?
Best Answer
It depends on how far you take negation into the interior of the statement.
(a) is fine, provided "everything after s.t." is identical to the initial statement's "everything after s.t.".
(b) is fine, provided whatever follows $\forall m \exists n ...$ is the negation of what follows after the "$\exists m \forall n ... $" in the initial statement.
Without knowing your quantified statement in it's entirety, we can't really say which is the correct negation. But negation is negation, and the first and second forms are equivalent, provided pushing the negation inward proceeds correctly in the case of (b).
For example
Suppose we have for $m, n \in \mathbb Z$, $$\exists m \forall n (m+n = n)\tag{1} $$
It's negation can be written: $$\lnot \exists m \forall n (m+n= n)\tag{$\lnot 1_1$}$$
The negation can also be written, and is equivalent to the negation immediately above, as follows: $$\forall m \lnot\big(\forall n (m+n=n)\big)\tag{$\lnot 1_2$}$$
Which is equivalent to: $$\forall m \exists n \Big(\lnot(m+n = n)\Big)\tag{$\lnot 1_3$}$$
Which is equivalent to: $$\forall m\exists n\big(m+n\neq n\big) \tag{$\lnot 1_4$}$$