Talk:Existential quantification

overkill

Someone may just want to find a simple description of existential quantifier. It may be a overkill to direct the reader to the article "existential quantification". -- User:Wshun

How do you learn about a symbol without knowing what it means? The existential quantifier (what you call the symbol) is not a mathematical concept; existential quantification is. This isn't MathWorld, thank goodness; we're not writing a dictionary, and we don't have a separate article on every variation in terminology.

OTOH, you've started material related to uniqueness quantification, so I'd better start that article!

-- Toby Bartels 02:52 25 Jul 2003 (UTC)

Feel free to redirect it to existential quantification, but please explain what an existential quantifier is in that article in a way that the explanation is easy to locate.

I did not start uniqueness quantification. You can read the history of this article. wshun 03:18 25 Jul 2003 (UTC)

You're right, I have Poor Yorick to thank for stimulating me into action.

Looking over the quantification articles just now, I think that they're too unwieldy, and in even worse ways than not being able to find these variant terms easily. I need to go to bed now and will be gone for a couple of days, but Sunday I plan to rearrange all of these articles to make everything easier to find. Then I'll want your opinion on how well I did! ^_^ In the meantime, I'm not moving anything anymore. -- Toby Bartels 03:55 25 Jul 2003 (UTC)

OK, done. Please look and give your opinion. -- Toby Bartels 21:09, 2 Aug 2003 (UTC)

Skolemization

Um, what's wrong with my definition of skolemization? —Ashley Y 11:24, 2004 Aug 20 (UTC)

Skolemization is a method of reordering quantifiers to move existential quantifiers to the left, like this:

$\forall {a}{\in }\mathbf {A} .\exists {b}{\in }\mathbf {B} .P(a,b)$

is equivalent to

$\exists {f}{\in }\mathbf {A} \rightarrow \mathbf {B} .\forall {a}{\in }\mathbf {A} .P(a,f(a))$

---

Your understanding is correct, but

• A->B is not a standard notation for the set of functions from A->B
• It moves from first-order logic (where Skolemization is normaly used) to second order logic (quanitfication over functions)

Normally, the existance of f is postulated implicitely as a concrete function, not a higher-order variable. In that case, a formula and its Skolemized form are not equivalent (because Interpretations and hence Models change), but equisatisfiable.

--Stephan Schulz 09:45, 14 Oct 2004 (UTC)

---

Right and right. Besides that, a section on skolemization doesn't seem fitting in an article on existential quantification. At most it should be linked to under See also or something.

Nortexoid 10:51, 5 Apr 2005 (UTC)

Consider the following proposition:

$\exists {n}{\in }\mathbf {S} \,P(n)$

If S is the empty set, is this statement true or false?

-- David 00:43, 18 Oct 2005 (EDT)

False. Why could it possibly be otherwise? If this relates to Skolemization, remember that we always require a non-empty universe. --Stephan Schulz 07:21, 18 October 2005 (UTC)

Odd numbers and natural numbers

In the "Basics" sections, it is stated that

"For some odd number n, n·n = 25" is logically equivalent to "For some natural number n, n is odd and n·n = 25".

But isn't that false, because the second statement introduces an unnecessary restriction on n, namely that it be positive? An "odd number" can be negative, but a "natural number" which is also odd cannot.

I'm going to change the first statement to specify that the n in question should be positive. I would change the second, but it'd be best to leave the language of that one just as it is, as it's used later in the article (or something very close to it is, rather).

Brad Gibbons (talk) 08:16, 30 October 2009 (UTC)