Universal Quantification and existential quantification question Im working on this problem but not sure if Im using negation correctly Please express the following statement using the universal quantification (“for all” quantifier) and the existential quantification ("there exists” quantifier): Not all numbers are greater than $2$ $$\lnot \forall x\;(N(x)\land G(x)) $$ There exist some numbers which are less than $0$ $$ \exists x\;(N(x)\to L(x))$$

Universal quantification is restricted by conditional.

'All things $Q$ are $P$', 'All things have property $P$ if they have property $Q$' ... $\forall x~(Q(x)\to P(x))$

Notice how $\forall x~(Q(x)\land P(x))$ claims all things have both properties, which is _not_ what we wish to say.

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Existential quantification is restricted by conjunction.

'Some things $Q$ are $P$', 'Some things with property $Q$ also have property $P$' ... $\exists x~(Q(x)\land P(x))$

Notice how $\exists x~(Q(x)\to P(x))$ may be satisfied if there is nothing with both properties as long as there is something without property $Q$ or with property $P$. (A conditional is true if the _antecedant is false_ or the _consequent true_.) So that is clearly _not_ what we wish to say.