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?

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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.

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  • I really like that convention, btw- It makes much more sense on a modern keyboard. – Ryder Nov 8 '12 at 0:14
  • It doesn't seem so easy to identify universal and existential wffs after all. Also because, I feel confused as I'm not familiar with above notation at all. Could you please make things a little simple by adding (x) notation? I know I've already accepted this answer but now I'm concerned with how to identify them. Do I need to come up with an another question? – cpx Nov 9 '12 at 8:55
  • @cpx - You can get back to (x) notation by deleting all symbols and all braces, and if you see ∃(...) remove the parentheses around .... I strongly disagree that this makes things "simple", however. You must maintain in your head the groupings I explicitly list with braces and parens, and annotate (x) mentally with "forall", or you will be confused about what the statement is. Also, why are you concerned about how to identify a universal vs. an existential? If this is for a class, ask the professor or TA. If not, note that the two are related by their definition (given above). – Rex Kerr Nov 9 '12 at 9:58
  • Because when you test an invalid argument by checking if all premises true and conclusion is false you need to know if at least one case of existential wff is true for the statement to be true. From what I know if wff has (∃x) quantifier outside of all brackets then it is existential otherwise if it is (x) then it is universal. But if in case quantifier has ~ attached to it then the result is opposite. i.e. We have to eliminate ~ for ~(x)Fx to become (∃x)~Fx for us to infer that it is actually existential. – cpx Nov 9 '12 at 14:00
  • @cpx - Yes, that's right. So eliminate ~ on the outside of quantifier statements. Easy! – Rex Kerr Nov 9 '12 at 16:07

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


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.)

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    Agreed, though not infrequently it's actually ∀(x|Px)Q(x), i.e. an implied subset of the actual universe, so one does have to be careful about assuming existential import. – Rex Kerr Nov 6 '12 at 20:35

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.

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  • What if I had (x)~Fx at the first place? Since we have (x) so, I don't understand how ~(∃x)Fx can be an existential wff. – cpx Nov 6 '12 at 15:25
  • I do see your point, but if we call ~(∃x)Fx universal because it can be re-written as (x)~Fx - then there is nothing that is existential, as any (∃x) can be rewritten as a universal with the addition of a ~. I do like DBK's addition above, that a universal statement also makes no assumption as to the existance of any x in the first place. – Ryno Nov 7 '12 at 9:41
  • I don't agree with then there is nothing that is existential If we had (∃x)Fx which is existential then we cannot change it to be universal with the addition of a ~. Isn't it? – cpx Nov 8 '12 at 10:26
  • ~(x)~Fx - two ~ then! :) – Ryno Nov 8 '12 at 15:37

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