Difference between "for any" and "for all"?

Question

Though searching for previous questions returns thousands of results for the query "for any" "for all", none specifically address the following query:

I'm reading a textbook in which one definition requires that some condition holds for any $x,\ x' \in X,$ and right afterwards another definition requires that some other condition holds for all $x,\ x' \in X.$

Is there a difference between 'for any' and 'for all'?

Answer 1

This seems to depend on the context: "For all $x \in X \ P(x)$" is the same as "For any $x∈X \ P(x)$" On the other hand "If for any $x∈X \ P(x)$, then $Q$" means that the existence of at least one $x\in X$ with $P(x)$ implies $Q$, so $P(x)$ doesn't need to hold for all $x \in X$ to imply $Q$.

Answer 2

"Any" is ambiguous and it depends on the context. It can refer to "there exists", "for all", or to a third case which I will talk about in the end.

https://en.oxforddictionaries.com/definition/any

Oxford Dictionary:

1. [usually with negative or in questions] Used to refer to one or some of a thing or number of things, no matter how much or how many.
   [as determiner] ‘I don't have any choice’
   which means there does not exist a choice
2. Whichever of a specified class might be chosen.
   [as determiner] ‘these constellations are visible at any hour of the night’
   which means for all hours

Also in Oracle database I remember a query to return the employees where employee.salary > any(10,20,30,40) which means the salary of the returned employee must be bigger than 10 or 20 or 30 or 40 which means "there exist" one salary in that tuple such that the employee.salary is bigger than it.

Same ambiguity is in math, since math did not come from nothing, rather it is a notation system for the language.

However:

Some times "any" is used for the meaning of "any" and not "exist" or "all". For example, in the definition of the little o asymptotic notation we have:

$o(g(n))$ = { $f(n)$: for any positive constant c>0, there exists a constant $n_0>0$ such that $0 \leq f(n) < c g(n)$ for all $n \geq n_0$ }

Here "any" means "any" which is two things "there exists" and "for all" how??

1. If you take any as "for all c" then the meaning is wrong because $n_0$>0 is attached to some choice of c and for each c there may be a different $n_0$>0. And you can not find a fixed $n_0$ for all c that satisfies the remaining because c can go very close to zero like c=0.000....00001

2. If you take any as "there exist c" then the meaning is wrong also because for some c the remaining may apply but for another c the remaining may not apply.
   Example: let $f(n)=n$ and $g(n)=2n$:
   If $c=1$ then $n < 1 \times (2n)$ for $n \geq n0>0$
   But if $c=0.1$ then $n > 0.1 \times (2n)$ for all $n\geq n0>0$

So here "any" means "any" which is for all but one at a time , so in the little o asymptotic notation "any" means for all c>0 pick one at a time and the remaining should be satisfied.

Conclusion: Either do not use "any" in Math or explain to the reader what it means in your context.

Answer 3

1. Here's a comparison of the various translations of $$∀x{∈}F\; P(x).$$

   • “For all elements $x$ in $F,\,P(x)$ holds” sometimes sounds like the property $P$ might belong to $F$ as a whole rather than to its individual members: “for all members of the family, they have a house” (1 house in total? or 5?). Contrast with “for each member of the family, they have a house” (definitely 5 houses in total). (“For all $x$” is Mathlish.)

   • “For every element $x$ in $F,\,P(x)$ holds”, despite ‘every’ too having a collective sense, says that the property $P$ is common to the members of $F.$

   • “For each element $x$ in $F,\,P(x)$ holds” directly attributes the property $P$ to individual members of $F.$

   • “For any element $x$ in $F,\,P(x)$ holds” doesn't strongly communicate that the property $P$ belongs to each and every member of $F;$ nevertheless, “for any $x$” typically means “given an arbitrary $x$”, which, logically, is synonymous with “for every $x$” and “for each $x$”.

2. The phrase ‘not any’ actually means ‘not some’ instead of ‘not every’.

   For example, observe that the sentences within each group below are equivalent to one another:

   • She does $\color\red{\textbf{not}}$ have $\color\red{\textbf{any}}$ disease.

     She does $\color\red{\textbf{not}}$ have $\color\red{\textbf{some}}$ disease.

     $\color\red{\boldsymbol{\lnot \exists}} x\, Dx$

     $\forall x\, \lnot Dx$

     For every disease, she does not have it.

   • She does not have every disease.

     $\lnot \forall x\, Dx$

     $\exists x\, \lnot Dx$

     For some disease, she does not have it.

3. Consider this logical relationship: \begin{align} &&\forall x\;\big(Px\to Q\big)\tag{$\color{brown}{1a}$}\\ &\equiv{}&\big(\exists x\,Px\big)\to Q\tag{$\color{brown}{1e}$}\\&\large\not\equiv{}&\big(\forall x\,Px\big)\to Q\tag{$\color{blue}{2a}$}\\&\equiv{}&\exists x\;\big(Px\to Q\big).\tag{$\color{blue}{2e}$}\end{align}

   For each/every/any $x,$ if $x$ has property $P,$ then $Q$ is true.	$\color{brown}{(1a)}$
   If some $x$ has property $P,$ then $Q$ is true. $\boxed{\textbf{Any }x}$ has property $P$ implies that $Q$ is true. $\quad\quad(\Large❔)$ $\boxed{\textbf{If for any }x},\,x$ has property $P,$ then $Q$ is true.$\quad\quad(\Large❔)$ $\boxed{\textbf{If any }x}$ has property $P,$ then $Q$ is true. $\quad\quad\quad\quad(\Large❔)$	$\color{brown}{(1e)}$
   If each/every $x$ has property $P,$ then $Q$ is true.	$\color{blue}{(2a)}$
   For some $x,$ if $x$ has property $P,$ then $Q$ is true.	$\color{blue}{(2e)}$

   When the word ‘any’ is embedded in a conditional's antecedent, the idiomatic meaning is typically—though not always—sentence $\color{brown}{(1e)}$ instead of sentence $\color{blue}{(2a)};$ that is, ‘if any’ frequently means ‘if some’ instead of ‘if every’; sometimes, its intended meaning is ambiguous.

   For example, Wikipedia's definition of set disjointedness “A collection of two or more sets is disjoint $\boxed{\text{if any}} \,$two distinct sets of the collection are disjoint” requires external clarification whether three sets are disjoint

   • if some pair of distinct sets is disjoint

   or

   • if every pair of distinct sets is disjoint;

   it turns out from reading between the lines two sections down that Wikipedia intends the ‘if every’ reading, even as the ‘if some’ reading is probably more immediate. No wonder authors disagree between these two contrasting definitions!

   Furthermore, compare

   • $\boxed{\text{If}}\,$the dog understands $\boxed{\text{any}}\,$command, then it is a genius.

   with

   • $\boxed{\text{If}}\,$the dog understands $\boxed{\text{any}}\,$command, then it has been trained.

Summary: ‘for each’ and ‘for every’ are excellent translations of the universal quantifier; the word ‘any’ sometimes corresponds to ‘∃’ instead of ‘∀’, and in technical writing should be used judiciously.

Answer 4

Despite its score, ryang's post is totally wrong.

He incorrectly claims that "the phrase ‘not any’ actually means ‘not some’ instead of ‘not every’.", and says that the following are equivalent and examples of this claim:

(1) She does not have any disease.
(2) She does not have some disease.

This is bogus, as is obvious to any native English speaker. Consider another pair:

(3) He does not understand any complicated sentence.
(4) He does not understand some complicated sentence.

He also incorrectly attempts to connect his incorrect claim with the FOL equivalence between "∀x ( P(x) ⇒ Q )" and "∃x ( P(x) ) ⇒ Q". This is quite obvious once you consider the following:

(5) If any student fails the course, he/she must retake the course.

Obviously, the pronoun "he/she" refers to "any student" that "fails the course". So one is NOT permitted to express this in anything close to the form of "∃x ( P(x) ) ⇒ Q"!

~ ~ ~

The true answer is that ryang is NOT describing actual English, and hence is not at all an answer to Bernd's question (and one wonders how many upvoters actually know proper English).

In proper English, "any" and related determiners "anyone" and "anything" function by inducing a ∀-quantifier at the global level, but not passing any indirect-statement boundary (including modal boundary) unless forced to. The determiners "some" and "every" and "each", on the other hand, stay as local as possible. And "some" means "any" when forced to extend beyond a local region.

The correct analysis of the above key examples are as follows:

(3) He does not understand any complicated sentence.
(3') ∀X∈ComplicatedSentences ( he does not understand X )

(4) He does not understand some complicated sentence.
(4') ∃X∈ComplicatedSentences ( he does not understand X )

(5) If any student fails the course, he/she must retake the course.
(5') ∀X∈Students ( X fails the course ⇒ X must retake the course )

Here are more examples:

(6) If some student fails the course, he/she must retake the course.
(6') ∀X∈Students ( X fails the course ⇒ X must retake the course )

(7) If every student of Paul fails some course, he will be fired.
(7') ( ∀X∈StudentsOf(Paul) ∃C∈Courses ( X fails C ) ) ⇒ ( Paul will be fired )

(8) If any student of Paul fails some course, he will be fired.
(8') ∀X∈StudentsOf(Paul) ( ∃C∈Courses ( X fails C ) ⇒ ( Paul will be fired ) )

(9) If everyone agrees on any additional rule, we will adopt it.
(9') ∀R∈AdditionalRules ( ∀X∈People ( X agrees on R ) ⇒ We will adopt R )

(10) If this job can be done by anyone, we would not need to hire any expert.
(10') ∀E∈Experts ( ( ∀X∈People ( this job can be done by X ) ) ⇒ we would not need to hire E ) )

In particular take a good look at (9) and (10), which demonstrates that my explanation is correct. In (9), "any additional rule" is quantified at the global level. In contrast, in (10), "done by anyone" is quantified only within the "can ..." boundary, but "any expert" is quantified at the global level.

Finally, the use of "if" instead of "iff" (if and only if) in a definition in conventional mathematics is an abuse of terminology and once you fix that then it becomes obvious that the quantifier represented by "any" does not pass the "iff". This fact about "iff" being a hard boundary for quantification is a convention but also partially logically motivated, but it will take too long to explain why...