Talk:Universal quantification

Separate article

I disagree to make a seprate article about universal quantifier while there is a good univeral quantification article. Sure they are different, but wikipedia is not a dictionary, meaning we don't need a separate article for each different topics but closely related with one another. -- Taku 03:47 11 Jun 2003 (UTC)

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$\forall {({\textrm {(mensen)-(enkelen)}})}:{\textrm {kaka}}$

Consider the following proposition:

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

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

-- David 00:23 18 Oct 2005 (EDT)

If $S=\varnothing$, the statement is meaningless, because the quantification presumes S contains elements. Neocapitalist 14:24, 10 December 2005 (UTC)

No, the statement is vacuously true. – Smyth\^talk 19:32, 3 February 2006 (UTC)

Added Things to Properties

I've added some stuff on negation and rules of inference; if somebody could double-check me, and add the algebraic properties, I'd be much obliged. Neocapitalist 14:24, 10 December 2005 (UTC)

ASCII or Unicode

Is there a ASCII or Unicode value for this symbol I couldn't find it and thought it would be a good thing to have on the page the web address is %E2%88%80

The unicode for "∀" is U+8704. I'll go insert it right now. Aeron (talk) 07:15, 5 March 2008 (UTC)

Your unicode does not work with my XP SP3 IE8 even though it works with Firefox. Instead it is a little square. I don't want to 'force' everyone in the visible universe to upgrade simply because there is a little square so I politely suggest that your immediate insertion be one that works for all platforms. jtlenaghan@hotmail.co.uk. 2012-2-1 about 19:42. — Preceding unsigned comment added by 94.196.119.234 (talk) 19:42, 1 February 2012 (UTC) Thank you. Once I have finishd editing I should preview/review the page before saving. jtlenaghan@hotmail.co.uk 2012-2-1 about 20:18 GMT (UTZ). — Preceding unsigned comment added by 94.196.119.234 (talk) 20:19, 1 February 2012 (UTC)

Integer numbers instead of natural ones

I'm quoting from the example:

For all natural numbers n, 2·n = n + n.

Shouldn't that word be better integer numbers, instead of natural ones? I mean, that formula works too with negative values! I'm not totally sure, so this is why I don't want to directly modify it. But if I'm true, following references may be edited as well. Regards, Néstor —Preceding unsigned comment added by 88.24.185.92 (talk) 14:23, 23 September 2008 (UTC)

I agree; it even works with real numbers ( since 2x = x+x whatever x is), WHO CAME UP WITH THIS CRAZY EXAMPLE??? — Preceding unsigned comment added by 85.211.139.192 (talk) 19:42, 5 September 2011 (UTC)

I see it as even worse, the "true" example isn't even true. 2 is a composite number, but 2 · 2 is not greater than 2 + 2. Mehnadnerd (talk) 07:56, 8 January 2017 (UTC)mehnadnerd

No, 2 isn't composite. Jochen Burghardt (talk) 19:23, 8 January 2017 (UTC)

Attempts

I removed 'attempts to' from the lead; if universal quantification is merely an 'attempt' then if it fails somewhere, there should be a discussion of the (alleged) failure; to be sure, the subject of quantification isn't trivial, but I didn't see anywhere in the article that supported 'attempts'.Zero sharp (talk) 13:48, 4 November 2008 (UTC)

Not equivalent

$\exists {x}{\in }\mathbf {X} \,\lnot P(x)$

Generally, then, the negation of a propositional function's universal quantification is an existential quantification of that propositional function's negation; symbolically,

$\lnot \ \forall {x}{\in }\mathbf {X} \,P(x)\equiv \ \exists {x}{\in }\mathbf {X} \,\lnot P(x)$

These are equivalent only if X is non-empty set. It must be given that at least one person exists, because if no person exists, then it does not follow from "It is not the case that, given any living person x, that person is married" that "There exists a living person x such that he is not married", so it's not a good way to introduce that symbol.--90.191.147.186 (talk) 08:49, 20 July 2010 (UTC)

If X is empty then $\forall {x}{\in }\mathbf {X} \,P(x)$ is true and $\exists {x}{\in }\mathbf {X} \,\lnot P(x)$ is false, regardless of what P is. So the negation of the first still has the same truth value as the second. — Carl (CBM · talk) 03:55, 21 July 2010 (UTC)

Basics Section Unclear

I am having a very difficult time understanding what the Basics section is trying to convey. If "2·n" is defined as a notation for functions of the form "n+n", then what is the point of mentioning natural numbers and that the notation is true? Isn't it true by definition? It also seems unusual to use the number 2; if it's an arbitrary choice, why not use a letter so as to not be as confusing?

The next problem comes in when the same notation gets used for a new function: "2·n > 2 + n". Was this a typo? Neither the ">" character nor the change in form were explained. Is it implicit that the old notation should be substituted and yield "n + n > 2 + n"? It is said that this is false when n is replaced by 1, "1 + 1 > 2 + 1", but it is not made clear how this violates the function definition or is even relevant. — Preceding unsigned comment added by 128.237.245.218 (talk) 17:15, 29 August 2012 (UTC)

It has been explained to me that the "·" is a common expression for multiplication and that the use of "2" was intentional rather than arbitrary. In addition, ">" is meant to signify "is greater than" for comparison. Perhaps the article should include a section that makes these conventions more clear. In any case, I undid my earlier mistaken edit and hope that eventually this article makes more sense regarding how function definitions are in/valid. — Preceding unsigned comment added by 128.237.245.218 (talk) 19:19, 29 August 2012 (UTC)

Intro like reading cement

All wikipedia articles and (particularly ones on symbols, math, and stats) should start with a 'good enough' layman definition on a single line that is easy reading and gives the close enough meaning. That's all 95% of visitors need. Something like "In the context of math and logic '∀' is a symbol representing 'for all' and refers to universal quantification" — Preceding unsigned comment added by 172.250.254.17 (talk) 01:58, 24 July 2018 (UTC)

Notation problem in "As Adjoint" section

IMO the following text is wrong and needs to be changed to correctly communicate that what it refers to as ${\mathcal {P}}(!)$ is the inverse image of $!$.

The more familiar form of the quantifiers as used in first-order logic is obtained by taking the function f to be the unique function $!:X\to 1$ so that $P(1)=\{T,F\}$ is the two-element set holding the values true and false, a subset S is that subset for which the predicate $S(x)$ holds, and

${\begin{array}{rl}{\mathcal {P}}(!)\colon {\mathcal {P}}(1)&\to {\mathcal {P}}(X)\\T&\mapsto X\\F&\mapsto \{\}\end{array}}$

I'm not sure how best to do this. Theoh (talk) 22:29, 13 September 2021 (UTC)