Talk:Quantification

Old material

This contains material from Universal quantification and Existential quantification. When I finish transferring items (tomorrow?), then I'll remove stuff from those articles that applies to quantification in general and appears here. Then those articles can focus on aspects unique to their particular sort of quantification. For now, those articles remain complete, without relying on this one. -- Toby Bartels 10:30, 31 Jul 2003 (UTC)

These changes have now been made. -- Toby Bartels 21:29, 3 Aug 2003 (UTC)

This is a important article; it was well written but I think some clarification on the rule of formation would be helpful. However, I believe in the scientific method. TEAR IT APART. After all, I'm just a vacuum cleaner.CSTAR 03:56, 12 May 2004 (UTC)

I have continued to make some changes to discuss natural language quantification, informal mathematical quantification and the formal semantics of quantification (at least what is objective quantifcation, as opposed to substitute quantification). Again I think a lot can be removed. Please have a look at this and modify. CSTAR 05:42, 15 May 2004 (UTC)

I will add some stuff on multal and paucal quantification ( quantification such as for almost every x blah...) and then I'll call it quits on this article.

Could we merge the other articles on existential and universal quantification into this one?CSTAR 23:12, 15 May 2004 (UTC)

In Mathematics section there is following phrase:

A similar analysis applies to the disjunction, 1 is prime, or 2 is prime, or 3 is prime, etc.

1 is NOT prime number, but it does not break the disjunction. I'm not sure if it should be fixed as English is not my mothertongue, but this phrase may cause some confustion. Mathemonkey (talk) 21:28, 1 May 2008 (UTC)

New intro

The new intro is innaccurate. Quantification is not limited to size. Unless there is any objection soon I will revert it.CSTAR 14:03, 19 Nov 2004 (UTC)

Your change is sound, in that what you have written is better than what you replaced, but I've got some reservations about the way the relation between quantifiers and quantities is being treated. I'm not a linguist, and am not entirely comfortable with the following material, but it seems to me that in natural language, mass noun (eg. two pints of beer) and count noun constructions (eg. four cats) are in many respects the paradigmatic instances of quantifiers, with some, no and all being more in the line of special instances. It seems backwards to me to relegate the quantity-related aspects of quanitifers to the back seat. ---- Charles Stewart 14:18, 19 Nov 2004 (UTC)

Sigh, yes, I have to agree with your assertion:

I've got some reservations about the way the relation between quantifiers and quantities is being treated.

In fact the article's treatment of quantification is almost entirely restricted to quantification in formal languages (Mea culpa). Unfortunately, I don't know much about this, I admit. When I first stumbled on this page it only dealt with formal quantification and moreover even with that restricted focus, I thought the material was not very informative and in fact a little condescending to the reader.

Of course feel free to revert back to any previous state or some suitable middle.

Sooner or later I will tackle this problem. I can't think of a 30 second edit that will resolve the issue, and I have little time for WP right now, so I am OK with leaving it as is. ---- Charles Stewart 23:29, 20 Nov 2004 (UTC)

Attention banner needed?

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)

No guilt feelings are deserved. What is an attention banner? ---- Charles Stewart 23:29, 20 Nov 2004 (UTC)

The following is an attention banner:

CSTAR 23:33, 20 Nov 2004 (UTC)

Uniform vs pointwise continuity

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)

Two meanings of quantification

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) Insert non-formatted text here

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)

I don't think 'measurement' is an adequate synonym for quantification even though their meanings do overlap. I propose we create a new page called 'quantification (science)' and remove from this article all the empirical science stuff. Then it might prove ?desirable?? to rename this page as 'quantification (logic)' to reduce any remaining confusion. Any views on this idea? thanks Peter morrell 06:41, 4 September 2008 (UTC)

Maybe this is stupid, but I was actually looking for what ist in German called `Quantor'. This is the mathematical (international) symbol language of mathematics. The symbols are called `Quantoren' (plural). Maybe this is also part of the confusion here. Quantors themselves do not only quantify things in mathematics. They are more or less an extremly short and precise way to write down relations between structures or - more generally - almost any mathematical statement can be written with them. — Preceding unsigned comment added by 178.5.170.2 (talk) 18:43, 31 January 2012 (UTC)

Visiting this article almost 6 years after Peter morrell'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)

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)

Today, I implemented the split suggested above. - Jochen Burghardt (talk) 14:35, 8 August 2014 (UTC)

Changes made to article undone

I recently changed an erroneous statement in the article which said, "1 is prime" to "1 is not prime." The entry has since been changed back to "1 is prime."

It is widely accepted by mathematicians that 1 is not a prime number. If 1 were to be prime then the fundamental theorem of arithmetic would have to be changed drastically. As it stands, the fundamental theorem of arithmetic holds that every natural number is either prime or can be expressed as a unique, finite product of prime numbers. If 1 were to be prime then every natural number would have an infinite number of prime factors. I am going to change this back. Please, leave the statement "1 is not prime" alone. Eyes down, human. (talk) 01:19, 25 August 2008 (UTC)

The article is not saying 1 is prime. It is using an example:

1 is prime, or 2 is prime, or 3 is prime, ...

as an example of an OR statement. Since it is apparently causing confusion to have an or statement in which one disjunct is "1 is prime", I'll change it to something else. — Carl (CBM · talk) 01:47, 25 August 2008 (UTC)

is this relevant?

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)

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)

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)

Someone gets mugged in New York every 10 minutes

Is that true? I'm guessing it's intended as a joke, although slightly inappropriate. —Preceding unsigned comment added by Money is tight (talk • contribs) 00:31, 2 December 2009 (UTC)

It's just an example of an ambiguous sentence. — Carl (CBM · talk)

Example matrices

These are the 8 different formulas, using the loving-relation Lxy and the quantifiers ∀ and ∃. You may concider adding them to this or an other article.

No column/row is empty: $\forall x\exists yLyx$: Everyone is loved by someone. $\forall x\exists yLxy$: Everyone loves someone.	The diagonal is nonempty/full: $\exists xLxx$: Someone loves himself. $\forall xLxx$: Everyone loves himself.	The matrix is nonempty/full: $\exists x\exists yLxy$: Someone loves someone. $\exists x\exists yLyx$: Someone is loved by someone. $\forall x\forall yLxy$: Everyone loves everyone. $\forall x\forall yLyx$: Everyone is loved by everyone.	Hasse diagram of the implications
One row/column is full: $\exists x\forall yLxy$: Someone loves everyone. $\exists x\forall yLyx$: Someone is loved by everyone.

Watchduck (talk) 23:31, 7 December 2010 (UTC)

Copied from Watchduck's talk page:

I'm angling for some in-line (i.e. footnoted) sources for the Quantification article. Your drawings are interesting and engaging. Did you dream these up, or is there a source for them? And the number "16" for the possible drawings is also interesting . . . again is there a source, or did you derive this? I'd say the drawings are worthy of inclusion in the article if you can source their origin.

If you created them yourself, and as these vaguely resemble Venn diagrams, is a way that you could (re-)generate them so they mimic the familiar Venn diagrams? (Is that even possible? Haven't thought it thru . . ..) Personally I wouldn't regard such drawings as O.R., but rather as "examples". I know you've put a lot of work into your drawings as they exist now; they are just a bit foreign even to someone like myself who knows Karnaugh maps pretty much backwards and forwards . . .

If you choose to respond I suggest that that you do so at Talk:Quantification, but if you'd rather, respond at my talk page. Bill Wvbailey (talk) 02:37, 9 April 2011 (UTC)

It is common to represent a binary relation by its logical matrix, and as everyone else we may do so as well. As a binary relation on a set A is a subset of the Cartesian product A² = A × A, this is the most natural representation one could imagine. A link to Logical matrix should be given in the description. If you want to add something else, you may google for "matrix of a relation" etc.

Maybe the ten examples themselves should be sourced, but that should be found in a random textbook about the subject. I've got them from the german article de:Prädikatenlogik, as well as the implications shown in the Hasse diagram.

I think this can be used without hesitation, because the facts are easily checked, and they are represented in a common way.

(16 possible drawings? These are 8, but indeed there are 8 negations too. But this is not about powers of two. The next number after the 8 is a 46: Wikiversity: Formulas in predicate logic)

Watchduck (talk) 11:37, 9 April 2011 (UTC)

This is just silly

By the time this article gets to Natural Language it has probably lost anyone who actually wanted to understand the subject. Anybody who understands the article doesn't need it for an explanation. Who is it written for (apart from the authors)?Cannonmc (talk) 14:47, 8 April 2011 (UTC)

Improvement/addition suggestions

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(x_n),

∃(x): f(x) ≡ f(x₁) V f(x₂₁) V . . . f(x_n)

[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 ε_xA(x), ε_yA(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)

Reichenbach § 19. Negation of Operators

The rules in a "square of opposition" per Reichenbach 1980:91-96

Connonmc's complaint reiterated

I am sorry but the "improvement" is as incomprehensible (to the layman) as the original article. There is no point in a Wikipedia article that can only be understood by people who know what the subject is about in the first place Cannonmc (talk) 12:06, 11 April 2011 (UTC)

No one's made any changes to the article, that I know of. The above are only suggestions in the process of mouldering away, probably.

In certain areas of human knowledge, there may be no way of, to use the pejorative "dumbing it down", i.e. reducing a discussion/teaching below a certain level of background-knowledge that is required for the discussion. Problem is, what is that level? What is the necessary background knowledge necessary? I'd say these notions are difficult for almost anyone under the age of about 15-16 years.

As a starting point: For an eye-opening discussion in very elegant simple prose see Bertrand Russell 1912/1997 The problems of philosophy, Oxford University Press, NY, ISBN 0-19-511552-N; Russell wrote this during his work on his and Whitehead's Principia Mathematica. You can buy this book at the e.g. Borders, for not very much money ($10.95).

To avoid clutter on this talk page I've cc'd to here the contents that was here as of 04 April 2011. There I can work on it without disturbing other editors. If you want to see what I'm working on, please feel free.

The problem of "induction" as it relates to "Quantification" : The article does not develop the philosophy of this. But as I was working on what is now here I realized that (and Russell treats this as well) the notion of "Induction" needs development. Induction is what our minds undergo when we repeatedly experience an association of sensations (e.g. { this_particular_dogshow, blue_dogs } to the point that we expect the association to happen every time the first element appears ( e.g. this_particular_dogshow ). Some folks call this "animal logic". There's a great little parable in Russell about the farmer that feeds the chicken every morning until the chicken comes to expect it as a "rule" or "law" of its existence: just as surely as the day breaks every morning the farmer brings the grain in a pail. Then one day instead of a pail of grain the farmer arrives with a hatchet. . .. From this, it would seem that we naturally by a priori or "built-in" mechanisms, "generalize". Some possible associations for the dog-show example: "At this particular dog show, all the dogs are blue ", or perhaps "This dog show always has blue dogs". This needs further development. Bill Wvbailey (talk) 18:49, 12 April 2011 (UTC)

Notation

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)

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)

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)

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)

Clarify nesting.

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)

I removed an entire section, because of bias. It's correct just because "falsehood implies truth". The entire section I removed: Nesting. No one reacted, so I reverted back.

Observation: Formal logic makes the base for "Foundations of mathematics" and formal logic can be described and studied within Mathematics.

The concept of "Nesting" is not incorrect. It is superfluous. One may add it to a theory but, with respect to Occam's razor, it should be added with a parpose. This purpose may be to help understanding of the mathematical formal language by the human Intuition (psychology).

The natural language examples should need the mention that, when taken out of context, they may have more than one meaning. A possible reason for such examples is that "we don't like to overcomplicate" what we say, so we use expressions with missing information, but which was presented in the past. 79.119.14.247 (talk) 18:17, 8 March 2012 (UTC) 79.119.7.169 (talk) 07:30, 10 March 2012 (UTC) 79.119.7.169 (talk) 10:41, 10 March 2012 (UTC)

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)

About 2.4 Nesting

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)

Sigma notation

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)