why is negation before the quantifier and not Property?

Question

I am studying square of opposition. Some dogs have fleas is shown in the text as (see bottom left sentence in image)

$\neg \forall x [D_{x} \supset \neg F_{x}]$

Why is it not (notice the negation changed) ?

$\forall x [\neg D_{x} \supset \neg F_{x}]$

Reason 1

Reason I say so is because the second option translates into English pretty well: "Some dogs have fleas" is equivalent to "Not all dogs have not fleas" which is equivalent to "For all X, if x is not dog then x has not fleas" which is what the 2nd option says

As a quick reference, See Tidman mentions english translation in 7.3 that $\forall x [D_{x} \supset F_{x}]$ translates "For all x , if x is a dog then it has fleas"

Reason 2

Secondly $\neg \forall x$ translates into something funny like "For all not x" and messes with the idea of "universe of discourse"

Tidman Page 180:

Answer 1

Because "Some dogs have fleas" is the same as "Not all dogs have no fleas".

The last one is symbolized with : $\lnot \forall x (Dog(x) \to \lnot Fleas(x))$.

This in turn is equivalent to : $\exists x (Dog(x) \land Fleas(x))$, which is the equivalent symbolization of "Some dogs have fleas".

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Your proposed symbolization : $\forall x (\lnot Dog(x) \to \lnot Fleas(x))$ means that : "Every object in the universe, if it is not a dog, it has no fleas", which means : "Every object in the universe, if it has fleas, then it is a dog", and this is not the same as "Some dogs have fleas".

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$¬∀x$ does not translate into something funny like "For all not x" but in "Not for all x", i.e. into "Some x not".

Answer 2

Think about it like this: \begin{align*} &\textsf{Some dogs have fleas.} \\ &\iff \textsf{It is not the case that (all dogs do not have fleas).} \\ &\iff \textsf{It is not the case that (for all $x$, if $x$ is a dog, then $x$ does not have fleas).} \\ &\iff \textsf{It is not the case that ($\forall x~ [D(x) \supset {\sim} F(x)]$).} \\ &\iff {\sim} (\forall x ~ [D(x) \supset {\sim} F(x)]) \\ \end{align*}