What is the difference between $\neg\exists x$ and $\neg\forall x$ ? Is $\neg\exists x$ ever used?

Question

I am sure that $\exists x$ means "there exists some $x$" so $\neg\exists x$ should mean "there is no $x$" but is that not exactly what $\neg\forall x$ is defined as?

Answer 1

No. $\neg \forall x$ means 'not every $x$'. So, for example, not every number is even. But that does not mean that there are no even numbers at all.

Other examples:

There are no unicorns .. so we write $\neg \exists x \ Unicorn(x)$ ... as you say: "there is no $x$ that is a unicorn'

Not everything is an apple ... so we write $\neg \forall x \ Apple(x)$

This also helps:

$\neg \exists x \ P(x)$ is equivalent to $\forall x \ \neg P(x)$ ... if there is not any $x$ that is a $P$, then nothing is a $P$, i.e. everything is a 'non-$P$' ... and vice versa

$\neg \forall x \ P(x)$ is equivalent to $\exists x \ \neg P(x)$ ... if it is not true that everything is a $P$, then there must be some things that are a 'non-$P$' ... and vice versa

Answer 2

This is a good question relating to domain of quantification. The existential and universal quantifiers have implicit or explicit domains. If the domain has less than two elements then the two quantifiers are equivalent. They effectively have identical meaning. However, if two or more elements are in the domain then they are different. The statement that there exists at least one element with a property is not the same as the statement that all of the elements have the property.

Thus $\neg\exists x$ and $\neg\forall x$ are not equivalent. For example, the statement "There oes not exist an even prime" is false because $2$ is an even prime. The statement "Not all primes are even" is true because $3$ is not an even prime. Thus the two statements have different truth values.