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 $0$ $$ \exists x\;(N(x)\to L(x))$$
Best Answer
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.
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.