Currently following discrete math, and I came across quantifiers.
If there's a predicate that phrases similar to "There is no..." (for example, "There is no human that can fly"), then shouldn't the expression using quantifiers use "$\neg\forall$" instead of "$\neg\exists$"?
It seems to me that both should be correct, although I am clearly wrong.
On
Both statements are very different.
There is no human who can fly means $\neg\exists x [f(x)]$ or if one explicitly refers to humans $\neg\exists x[ h(x)\wedge f(x)]$.
The statement $\neg \forall x[f(x)]$ or more explicitly referring to humans, $\neg \forall x[h(x)\Rightarrow f(x)]$, means that it is not true that all humans can fly.
Consider the difference between these two statements: $$\lnot\forall x\in\mathbf{R}(x^2>0)$$ $$\lnot\exists x\in\mathbf{R}(x^2>0)$$ The first way to understand the difference between $\lnot\forall$ and $\lnot\exists$ is to read directly what these statements mean. The first statement is equivalent to "it is not true that for all real $x$ that $x^2$ is greater than zero. This is clearly true, since $0^2=0$. On the other hand, the second statement means that "it is not true that there exists a real $x$ such that $x^2>0$". But every $x$ other than $0$ fulfills this requirement, so this statement is certainly false.
The second way is to use De Morgan's laws: $$\exists x\in\mathbf{R}(x^2\leq0)$$ $$\forall x\in\mathbf{R}(x^2\leq0)$$
We can easily see now not only that the statements are not equivalent but also which ones are true or false.