What do “universal” and “existential” mean in logic?

Question

What's the difference between "universal" and "existential" when used in the context of wff (well-formed formulas)?

We have a universal quantifier, which can be written as (x), and an existential quantifier, which can be written as (∃x).

Now, let's take ~(∃x)Fx as an example. Would you call it a universal or existential wff?

What do "universal" and "existential" really mean when we are talking about logic in general?

Answer 1

Personally, I greatly dislike the (x) notation, as it invites confusion--when you place something other than x inside parentheses, is it universal quantification, or is it to show grouping? Therefore, I will use ∀(x){...} and ∃(x){...} to mean "for all" and "there exists" respectively so as to be perfectly clear what the variable is and what the predicate is.

With this done, we have either ~∀(∃x){Fx}, which makes no sense because ∃x is not a variable, or ~(∃(x){Fx}) which makes sense and means "it is not true that there exists an x such that Fx". In fact, in predicate calculus you usually define

∃(x){Fx} iff ~(∀(x){~Fx})

So basically, an existential is just a not-universal. Now,

~(∃(x){Fx}) iff ~(~(∀(x){~Fx})) iff ∀(x){~Fx}

which is a universal, not a not-universal. So I would call it universal.

Answer 2

In addition to Rex Kerr's answer, let me answer the following:

What do Universal and Existential really mean in general when we are talking about logic?

Universally quantified statements are usually interpreted as saying

For all x, P(x) holds

or

For every x, P(x) holds

These statements are "not existential" in the sense that they do not make an existence claim over x. It may be helpful to think of ∀(x)P(x) as meaning

If anything is an x, then for all x, P(x) holds

(Note, however, that we usually assume the universe of discourse to be non-empty. In this case ∀(x)P(x) has an existential import.)

Answer 3

Existential - There exists x such that... (says something about some x) Universal - All x are... (says something about every instance of x)

Yes, as with so many things in logic, adding a "Not" can change that (idiomatically) as ~(∃x)Fx is the same as (x)~Fx, and does specify something about all x, not just some of them, but as far as I'm aware this is still an "Existential" statement. Willing to be corrected on that last point, but as far as I'm aware, as long as you are talking about (∃x), it's Existential.

Answer 4

There is universal axioms of logic. I will call this . @ Universal is X @ Axiom is A @ Logic is L. Hence XA=L. Existentialism = E @ Unrealistic = U @ Realality = R Hence UR=E. Basicly if universal logics are accepted natural reality's. Existensialism is mind created unnatural postulated reality's and should be cancelled out by universal axioms of logic. HENCE XA= L+ UR=E- HENCE L+ CANCELS OUT E- Scientology the mystical route to freedom and happyness; via clearing hangups and engrams.Jargon and claimed benifits. Existentialism is irrationality that promotes stupidness. Existentialism releaves the mind of responsibility to identify and integrate reality.Universal axiomatic logics are actions not based on opinions of author or anyone else. The same moral actions exsist for all locations cultures and ages @ universal.