Talk:Quantification

message: Should we put in an attention banner?  Yes, I now have pangs of guilt about this article. Please, someone, absolve me of the burden of this guilt. CSTAR 15:29, 19 Nov 2004 (UTC)
replies: text: No guilt feelings are deserved.  What is an attention banner? ---- Charles Stewart 23:29, 20 Nov 2004 (UTC)
replies: text: The following is an attention banner:
CSTAR 23:33, 20 Nov 2004 (UTC)
thread_title:  Attention banner needed? 

message: Let f be a real-valued function on R. 
* A: Pointwise continuity of f on R:
:: \forall x \in \mathbb{R}, \ \forall \epsilon >0, \exists \delta > 0, \forall h \in \mathbb{R}, \quad |h| < \delta  \implies |f(x) - f(x+h)| < \epsilon 
which is the same as 
* A': Pointwise continuity of f on R:
:: \forall \epsilon >0, \ \forall x \in \mathbb{R},  \exists \delta > 0, \  \forall h \in \mathbb{R}, \quad |h| < \delta  \implies |f(x) - f(x+h)| < \epsilon 
This differs from
* B: Uniform continuity of f on R:
:: \forall \epsilon >0, \exists \delta > 0, \forall x \in \mathbb{R},  \forall h \in \mathbb{R}, \quad |h| < \delta\implies |f(x) - f(x+h)| < \epsilon 
by exchange of two quantifiers. Where are the additional variables?--CSTAR 03:20, 6 September 2006 (UTC)
thread_title:  Uniform vs pointwise continuity 

message: There need to be two different pages for "quantification", one which is the formal logic definition, the other which is the process of quantifying information. The latter is a very important topic and has really nothing to do with the former that I can see. There are many books on the history of quantification and the "quantitative spirit" in science (starting around the 18th century), ample material for an entire article. --24.147.86.187 01:48, 4 January 2007 (UTC)
replies: text: The scientific usage of "quantification" might be a synonym of measurement, and we already have a page on that topic. Tim Vickers (talk) 14:52, 24 August 2008 (UTC)

text: Visiting this article almost 6 years after 's suggestion, I wonder why it hasn't been realized yet. Apparently there are no objections to it, are there? - Jochen Burghardt (talk) 08:19, 25 June 2014 (UTC)

text: I'm creating a suggestion about splitting the article in my sandbox, see User:Jochen Burghardt/sandbox/Quantification (extended disambiguation page), User:Jochen Burghardt/sandbox/Quantifier (logic), User:Jochen Burghardt/sandbox/Quantifier (linguistics), and User:Jochen Burghardt/sandbox/Quantification (science). - Jochen Burghardt (talk) 10:25, 26 June 2014 (UTC)
replies: text: Today, I implemented the split suggested above. - Jochen Burghardt (talk) 14:35, 8 August 2014 (UTC)
thread_title: Two meanings of quantification

message: Goldbach's conjecture is that this statement is false, that is, that every natural number greater than 2 is the sum of two primes.
It seemed a bit off-topic to mention Goldbach's conjecture here.  It seems sufficient to mention that the example is related to Goldbach's conjecture, but no more.
--Paul (talk) 00:35, 7 March 2009 (UTC)
replies: text: I think this is fine: first, saying what GC is allows the reader unfamiliar with GC to see that it is the negation of the sentcne above, and secondly, GC is good sort of example, since clicking on the link shows the interesting fact that such a simple sounding sentence is unknown.  If we were to further point out that all such undetermined sentences of arithmetic make use of quantifiers, that might be nice in this article, too. — Charles Stewart (talk) 13:07, 8 March 2009 (UTC)

text: Apart from being relevant or not, Goldbach's conjecture states something different from what is written here. In its present form the statement in the article is just a correct statement. For instance, 11 is not the sum of two primes. —Chris Barista (talk) 15:41, 1 April 2009 (UTC)
thread_title:  is this relevant? 

message: I reviewed this quickly and don't agree with the above complaint. But I see some issues (my apologies if these are in there but my scan missed them):
:1 This article is almost completely about quantification in logic and should be disambiguated as such, with removal of the other possible meanings (but see the problems around "conversational language" below). For instance, re "quantification" as applied to Inductive reasoning, I hunted through my Russell 1912/1977 The problems of philosophy but didn't find anything useful. The world "probability" crops up, as in, "possibility of occurrence given previous observaton", but does the literature support the use of the notion of "quantification" in this context? I don't know. I doubt it: here's a definition from From Encyclopedia Britannica http://www.britannica.com/EBchecked/topic/486102/quantification : 
:: "quantification,  in logic, the attachment of signs of quantity to the predicate or subject of a proposition." 
:2 Include footnotes in the body so a person can figure out which text or reference to go to, to learn more. For instance, the relevant chapters of Reichenbach 1947/1980 is quite sophisticated but doesn't contain info re "paucal and multal", so a person would be clueless as to where to start to find out about these notions.
:3 Infinite versus finite quantification, i.e. "To extend thse operations to an infinite number of terms requires new primitive terms. The correct form of statement therrefore that a conjunction and a disjunction of an infinite number of terms are defined by the operators" (Reichenbach 1980:92):
::∀(x): f(x) ≡ f(x1) & f(x2) &  . . . f(xn), 
::∃(x): f(x) ≡ f(x1) V f(x21) V  . . . f(xn)
:[Something about Wittgenstein goes here] 
:Only one quantifier, usually ∃, is necessary in a formal theory. Hilbert reduced the notion to what he called his " ε-function ", and considered it to be an axiom as opposed to a primitive symbol defined by its usage; note that this operator does play an important role in quantifying conversational language (see Reichenbach below): A(a) --> A(ε(A)). "Here ε(A) stands for an object of which the proposition A(a) certainly holds if it holds of any object; let us call ε the logical ε-function. [The substitution of any given formula for A(a) will be made possible if, instead of  ε(A), we write more precisely εxA(x), εyA(y) . . .]"
:Hilbert uses his ε-function three ways:
: (i) in the common definitions of ∀ and ∃: 
:: ∀a: A(a) <--> A(ε(~A))
:: ∃a: A(a) <--> A(ε(A))
:::Here I've used the modern " ∀a: " for Hilbert's " (a) ", and " ∃a: " for Hilbert's " (Ea) ".
: (ii) in defining the notion of "one and only one" i.e. the object [his words]
: (iii) in the definition of the choice function i.e. "if A(a) holds of several objects, then ε(U) is some one of the objects a of which U(a) holds"
:From Hilbert 1927 The foundations of mathematics, appearing in van Heijenoort 1967:466.
:4 Perhaps add something more re the use of quantifiers in pure logic versus "conversational language". This is covered in Reichenbach; see next entry.
:5 Add something more about negation of quantifiers, i.e. "(For all pigs) It's not the case that (at least one of) pigs fly" is logically equivalent to "There exists no flying pigs". This apparently requires two treatments, one for "logic" (shown in brown), the other for "classes" as they appear in conversation (shown in blue). There's a lot to this and a lot needs to be explained (e.g. what the words "subaltern" mean, for example). And there are subtleties, for example, in the case of conversational classes the domain of S cannot be empty i.e. because the class "sea serpents" is empty, "All sea serpents live in the sea" does not allow us to infer "Some sea serpents live in the sea" (Reichenbach's example). The drawing below combines 2 drawings in Reichenbach that appear on p. 93 and 95, the first one in brown dealing with quantification and negation of functions, the second in blue with the "scholastic" forms that deal with classes (to paraphrase Reichenbach).
:6 Venn diagrams may be useful here. As I review my Reichenbach I see little Venn drawings that I made in the past, and these are still somewhat useful to me. What's important in them is to show the Universe of discourse (as a box surrounding the circles). 
:7 Also, the drawings above may be useful.
If I come up with more ideas I'll add them here. Bill Wvbailey (talk) 15:23, 9 April 2011 (UTC)
thread_title:  Improvement/addition suggestions 

message: There is an error - at least in mathematics the symbols
"\wedge" and "\forall" have two totally different meanings. As the html-code of this page already shows, "\forall" means ALL or FOR ALL, whereas "\wedge" means a logical AND. Also "\exists" means EXIST and "\vee" means a logical OR.  — Preceding unsigned comment added by 178.5.170.2 (talk) 18:53, 31 January 2012 (UTC)
replies: text: I hung flags on these. I looked up the symbols from Arial Unicode MS: ∨ ∧ ∀ ∃, and these are logical OR, logical AND, "for all", "there exists". Also the ∧ with an x under it is odd; it looks like something in Wittgenstein or somewhere (I don't feel like looking it up). Other symbols of course do exist (x) "for all x", and I've seen "E" for "there exists". But I'm not sure wikipedia is in the business of listing an interpretation of every symbol ever used by every mathematician. I know I'm not in that business. I suggest we use the common contemporary symbols. The way to dodge this bullet is to state (in maybe half the words): "Other symbols are in use. The symbolism has varied from era to era, and from author to author. It is a reader's responsibility to learn the symbolism used by the author of the paper or text that they are reading." Bill Wvbailey (talk) 20:00, 31 January 2012 (UTC)
replies: text: Your suggestion sounds good. Actually I'm studying mathematics quite some time now and I had just never seen \wedge and \vee for any other meaning than 'and' or 'or'.  — Preceding unsigned comment added by 178.5.170.2 (talk) 22:55, 31 January 2012 (UTC)
replies: text:  WRONG. see e.g. http://matematyka.pisz.pl/strona/1070.html or http://www.romazur.republika.pl/logika-rachunek_kwantyfikatorow.html - there *are* countries which mostly use these, esp. below college level. also, these AREN'T lowercase \wedge and \vee, they are \bigwedge and \bigvee.Vaxquis (talk) 17:05, 16 February 2012 (UTC)
thread_title:  Notation 

message: It arises only when mixing the metatheory with the theory, for exmple when doing so by using in the metatheory the domains of discourse from the theory. For example, escaping the domain of discourse from second order predicate logic and quantifying the variables over a set, sometimes obscuring it with some shorthand relation and omitting logical connectives from the metatheory (for the universal quantifier, usually the implication is omitted; for the existential quantifier, usually the conjunction is omited).
All these three can be rewritten just by using the correct domain of discourse which is R, and then creating unquantified predicates and connecting them usually with conjunction and implication. In fact, they are informally stated: it's just natural language spelled in formal symbols .
:: \underbrace{\forall x \in \mathbb{R} \, \, \forall \epsilon >0} \, \exists \delta > 0 \, \forall h \in \mathbb{R} \, \left( \, |h| < \delta \, \to \,  |f(x) - f(x+h)| < \epsilon \, \right) 
:: \forall \epsilon >0 \, \underbrace{\forall x \in \mathbb{R} \, \, \exists \delta > 0} \, \forall h \in \mathbb{R} \, \left( \, |h| < \delta  \, \to \, |f(x) - f(x+h)| < \epsilon \, \right)
:: \forall \epsilon >0 \, \underbrace{\exists \delta > 0 \, \, \forall x \in \mathbb{R}} \,  \forall h \in \mathbb{R} \, \left( \, |h| < \delta \, \to \, |f(x) - f(x+h)| < \epsilon \, \right) 
An example that you cannot mix metatheory with theory is Tarski's undefinability theorem
Nesting arises from human tendency to bring mathematical notations closer to natural language.
79.119.14.247 (talk) 12:22, 8 March 2012 (UTC)
replies: text: Perhaps the reason is that your complaint here is not very clear. You seem to be complaining about the use of bounded quantifier notation ((\forall \epsilon > 0)) but the use of bounded quantifiers is beside the point, that section is about nesting of quantifiers ((\forall x)(\exists y)R(x,y)). Tarski's theorem seems to be unrelated as far as I can tell. — Carl (CBM · talk) 12:02, 10 March 2012 (UTC)
thread_title:  Clarify nesting. 

message: In the first two lines, why is the equation display not at the proper height? It is below the normal height of the sentence.Pratyush Sarkar (talk) 02:59, 26 October 2012 (UTC)
thread_title:  About 2.4 Nesting 

message: Is the sigma $\sum$ in the sigma notation of sums really a quantifier? It does bind variables, so according to the definition in this article it apparently is one. 95.90.43.41 (talk) 06:24, 26 June 2013 (UTC)
thread_title:  Sigma notation