Universal Quantifiers

Question

What is the difference between ∀x∈U : ~p(x) and ~∀x∈U : p(x) ?? Could anybody give any English sentences explaining both of them?

Answer 1

The quantifiers are duals of each other: $\neg\forall\neg\equiv\exists$ and $\neg\exists\neg\equiv\forall$. Thus, $\neg\forall\equiv\exists\neg$.

So $\neg(\forall x\in U) p(x) \equiv (\exists x\in U) \neg p(x)$, which means "there is some $x$ in $U$ such that $\neg p(x)$".

The first sentence says "every $x$ in $U$ is such that $\neg p(x)$" -- quite different.

Answer 2

Suppose you live in a world called U.

For the first one suppose every person in U loves Taylor Swift and let p(x) be the statement : x doesn't love Taylor Swift. For every x in U, p(x) is false so ~p(x) is true.

For the second one suppose U is the real world where some people just don't love Taylor Swift and let p(x) be the statement : x loves Taylor Swift. So p(x) is false for some x in U. So the statement "everybody loves Taylor Swift" which can be expressed as $\forall$ x $\in$ U : p(x), is false.

Answer 3

$\forall$ is a universal quantifier, and we have some predicate $p(x)$, an "English" sentence would be something along the lines of "Every x that is a U is not a p(x)", i.e. if $p(x)$ denotes something like x has property $P$, then every $x$ in $U$ does not have property $P$, like is not blue or is not yellow or is not a prime number.

The next logical statement is negating the universal quantifier. By the property of universal quantification, its negation is the existential. So in the second statement you are saying exists some $x$ in $U$ such that $p(x)$ is true, like for example, if $U$ is a set that contains all integers and $p(x)$ denotes $x$ is a prime, then the statement is "There exists some prime in the integers 1 to 100".

Answer 4

The first ($\forall x\in U : \neg p(x)$) says: "The property $p$ does not hold for any object in $U$."

The second ($\neg \forall x \in U : p(x)$) says: "It is not the case that the property $p$ holds for every object in $U$."

Examples, if $U$ is all fish and the property is "x swims", the first says that no fish swim and the second says that there is a fish that doesn't swim.