How to negate quantified expressions not using DeMorgan's law

Question

I am looking to write an expression from English to a logical expression, then negating that expression.

Give the expression

Some drivers do not obey the speed limit

Let

s(x) = x obeys the speed limit

Let the domain be drivers.

We can express the expression as such

∃x¬S(x)

Then we want to negate the expression.

The way I approached it is

-∃x¬S(x)

But in this document (#3 (a)) it states that the solution is

∀xS(x)

which is making me scratch my head.

The document states

¬∃x¬S(x) ≡ ∀xS(x)

but what law is this? This does not follow DeMorgan's law as that one states

￢∃xQ(x) ≡ ∀x ￢Q(x)

Am I missing something? How do we reach ¬∃x¬S(x) ≡ ∀xS(x)? What law is this?

Answer 1

$$\neg \exists x \ \neg S(x) \overset{Quantifier \ DeMorgan}{=} \forall x \ \neg \neg S(x) \overset{Double \ Negation}{=} \forall x \ S(x)$$

Answer 2

In your last DeMorgan's law quote, let $Q(x)=\lnot S(s).$ then apply the double negation law.

Answer 3

deMorgan's Laws states duality between conjunction and disjunction under negation: $\neg(\phi(\alpha)\land\phi(\beta))=\neg\phi(\alpha)\lor\neg \phi(\beta)$ and $\neg(\phi\lor\psi)=\neg\phi(\alpha)\land\neg \phi(\beta)$ .

The universal quantifier is essentually a conjunction over the domain: it is a statement that the predicate function is true for each from the many entities in the domain. This and that and every other.

The existential quantifier is essentually a disjunction over the domain: it is a statement that the predicate function is true for at least one from many the entities in the domain. This or that or some other.

Then by mathematical induction, deMorgan's Laws are extended to these quantifiers.   $\neg\forall x~\phi(x) = \exists x~\neg\phi(x)$ and $\neg\exists x~\phi(x)=\forall x~\neg\phi(x)$