# Existential quantifier versus Universial quantifier, from a natural language perspective

I am aware about the difference between existential quantifier and universal quantifier and it is always clear for me when to use them,

if the statement starts either with There is/are some ..." or "Everyone/..."

but now I have this statement:

If some people are nicer than any people in area1 , than they must live in area2.

This is the FOL translation:

∀x. (people(x) ^ ∀y.(people(y) ^ in(y,area1) nice(x) > nice(y)) → in(x,area2)

1. Why should I use the universal quantifier when in the statement it talks about some. Should it not be ∃x.
2. Why is it nice(x) > nice(y) and not ^ nice(x) > nice(y)
• Ambiguities of natural language... If we write it as $\exists x [people(x) \land \forall y ( \ldots ) \to in(x,area2)]$ we are forced to assume that there are some people. In natural language, the statement "if some people are ...", it sounds possible that there are none. – Mauro ALLEGRANZA Jun 7 '18 at 13:26
• and regarding question two, if I would use a conjunction then it would mean that all "y" are people, who are not "nice(r)" but the statement is just saying if for all y who live in area2, then those are not nice(r). Is it correct how I distinguish between the two connectives? – xava Jun 7 '18 at 13:35
• Because with the $\land$ you would be saying that everything is a person (including the areas!), and lives in area 1 and is less nice than the $x$ .... (including $x$ itself)!! No, you want to to be saying: if we are dealing with something that is a person and living in area 1, then $x$ is nicer than it – Bram28 Jun 7 '18 at 13:59

The statement can be thought of as:

"If some person is nicer than everyone in area1, then they live in area2"

... which can also be thought of:

"If any one person is nicer than everyone in area1, then they live in area2"

... and now you start to see the connection with the universal: the sentence is really about anyone, i.e. everyone

A little more technical is the following explanation:

At first, the original sentence looks like a conditional, with the antecedent being an existential claim, i.e. something like:

$\exists x [person(x) \land \forall y ((person(y) \land lives(y,area1)) \rightarrow nicer(x,y))] \rightarrow lives(x,area2)$

but the problem with this is that the $x$ in $lives(x,area2)$ is a free variable, and so this is not a sentence. Instead, we realize that the sentence is not a conditional, but a quantified sentence. OK, so should we just extend the scope of the existential to get:

$\exists x \big([person(x) \land \forall y ((person(y) \land lives(y,area1)) \rightarrow nicer(x,y))] \rightarrow lives(x,area2) \big)$

But that isn't right either: now you're basically saying that there is something such that if it is a person nicer than anyone in area 1, then that something lives in area 2 ... well, that would trivially be true if we pick for that something anything that is not a person! (which is why you almost never see an existential of a conditional). Indeed, maybe at this point you realize that this should really be a universal:

$\forall x \big([person(x) \land \forall y ((person(y) \land lives(y,area1)) \rightarrow nicer(x,y))] \rightarrow lives(x,area2) \big)$

That is, we end up saying:

"For any object ... if that object is a person nicer than anyone in area 1, then that person lives in area 2"

... and yes, now we realize that that is just a more convulated way of expressing te original sentence. Or rather: that the original English sentence is a more efficient way of expressing this logic sentence)

Here is another English sentence that makes a similar contraction:

"If a dog wags its tail, then it wants to play"

Now, do you think that this is about some dog? It says "a" dog, and indeed, it is about one dog ... but it is really about any one dog, i.e. any dog, i.e. all dogs. Indeed, it should be clear we should translate this as:

$\forall x ((dog(x) \land WagsTail(x)) \rightarrow WantsToPlay(x))$

If we use the $\land$, i.e.
$\forall x \big([person(x) \land \forall y ((person(y) \land lives(y,area1)) \land nicer(x,y))] \rightarrow lives(x,area2) \big)$
then you would be saying that everything is a person (including the areas!), and lives in area 1 and is less nice than the $x$ .... (including $x$ itself)!!
No, you want to to be saying: If we are dealing with something that is a person and living in area 1, then $x$ is nicer than it .. we want to exclude anything that is not a person or not living in area 1.