Difference of "for all" and "for each" in a certain context

Question

I was thinking about if there is any difference, depending of the context, by saying "for each" and "for all" and I think I may found one context for this.

Considering the definition of Cauchy sequences:

It states that for every $\epsilon>0 \exists $ an $N$ st.. blablabla... Of course there's no unique $N$ which works for all $\epsilon>0$, because if there was it should be the maximum $N$ of all $\epsilon>0$ we considered. But then we could choose an smaller value of $\epsilon$ (let's say $\epsilon_1$), therefore $N(\epsilon_1)$ would be greater than the others, so there's no such thing as an unique $N$ that works for all $\epsilon>0$.

So, "for all" in this context means the same as "for each", because we cant have an $N$ which works for all $\epsilon>0$ at the same time, so we have for each $\epsilon>0$ an N st blbabla...

But suppose there exists an unique $N$ which works for all $\epsilon>0$. So, in this assumption no matter what $\epsilon$ you choose, the distance beetwen the elements of the sequence whose indices are greater or equal than $N$ would be always less or equal than $\epsilon$. So for all $\epsilon>0$ there exists an unique $N$ st blablabla...

Now, if I say for each $\epsilon>0$ there exists an unique $N$ st blablabla... It just states that exists an unique $N$ that works for each $\epsilon>0$ but it doesnt ensures that $N$ works for all $\epsilon>0$, so it has a different meaning by saying "for each" and "for all" in this context. What do you think about that??

Of course my assumption is false, but there could be an similar context which is true, and by using the same reasoning, "for each" and "for all" would have different meaning.

Answer 1

"For each" and "for all" are synonymous. However, the order in which you interleave "for each" (or "for all") and "for some" makes a difference and you can make life very hard for yourself if you make the mistake of putting one at the beginning and one at the end:

1. for each $\epsilon > 0$, for some $N> 0$, $1/N < \epsilon$
2. for some $N > 0$, for each $\epsilon > 0$, $1/N < \epsilon$
3. for each $\epsilon > 0$, $1/N < \epsilon$, for some $N > 0$

1 is true, 2 is false and 3 is ambiguous. (I think you were thinking of a sentence along the lines of 3 when you began to think that "for each" and "for all" were not synonymous.)

Answer 2

So, if I understand you, you are suggesting that the keywords "each" and "every" should map to the order in which we write mixed universal and existential quantifiers, regardless of the spoken order.

• "There's some $y$, that makes it true, for every $x$" should be interpreted as: $~\exists y~\forall x~\operatorname{It}(x,y)$ .
• "There's some $y$, that makes it true, for each $x$" should be interpreted as: $~~\forall x~\exists y~\operatorname{It}(x,y)$ .

Well, it would be useful if naïve speech was that consistent in the use of "for each" and "for every", and if everybody always used "for all" for one rather than the other.   Unfortunately that is counterfactual; it can't be relied on.   There are enough exceptions to always require double checks on the speaker's intent.

It would be far less confusing and error prone to encourage students to learn to use the same order in both speech and symbols.

• "There's some $y$, that makes it true, for all $x$" should be interpreted as: $\boxed ?$ .

Answer 3

There is one context in which "all" and "each" cannot be interchanged, and that is when describing an algorithm of some sort.

Put half of each of the fruits into the red box and the other half into the blue box.

Put half of all the fruits into the red box and the other half into the blue box.

Following the first results in each fruit being split up into two halves that go into different boxes.

Following the second results in the fruits being divided into two groups that go into different boxes.

Similarly consider:

There is an upper bound for each function in the sequence.

There is an upper bound for all functions in the sequence.

The first states that the each function in the sequences has an upper bound, while the second is usually taken to state that there is a common upper bound for all the functions in the sequence. (Here it is irrelevant what exactly "upper bound" means.)

Since Rob Arthan considers the second sentence in the above example as ambiguous, here is a much clearer pair:

There is a unique cardinality for each set.

There is a unique cardinality for all sets.

And for people who already know too much about cardinality and hence their interpretation is biased, here is an even clearer example:

There is one teacher for each student.

There is one teacher for all students.

The moral of this is to use "each" whenever referring to each one individually and to use "all" whenever referring to all of them as a whole collection (IF one insists on writing in natural language with quantifiers on both sides of the statement). In many cases they are interpreted to mean the same thing, but not always, as I showed.

If there are two or more quantifiers, I personally recommend putting all the quantifiers in front. Sometimes that sounds a bit unnatural in English, in which case one should use commas and add words like "common" or "same" and choose carefully between "each" or "all" to ensure that misinterpretation is impossible.

Answer 4

The key point in mathematical English is that the order of the quantifiers in English should match the intended formal order. So, if we write out the quantifiers very explicitly:

• There is an $x$ such that, for all $y$, $P(x,Y)$ is $$(\exists x)(\forall y)P(x,y)$$

• There is an $x$ such that, for each $y$, $P(x,y)$ is also $$(\exists x)(\forall y)P(x,y)$$

• For all $y$ there is an $x$ such that $P(x,y)$ is $$(\forall y)(\exists x)P(x,y)$$

• For each $y$ there is an $x$ such that $P(x,y)$ is also $$(\forall y)(\exists x)P(x,y)$$

As you can see, "all" and "each" are synonymous in that sense. The order is all that matters, if things are written out in enough detail (the "such that" phrasing).

In more general English phrasing, without the "such that", it can be very hard to tell exactly what is meant, as the answer by user21820 shows. In some cases, you need to have enough context to already know what is intended (or be able to figure it out).

There is one more issue: many authors will write the innermost quantifier after the body of the formula, like this:

• There is an $x$ such that $P(x,y)$ for all $y$ is $$(\exists x)(\forall y)P(x,y)$$ for many authors.

This convention is not universal, and it can be confusing if you don't already know what is intended, but it is common enough to point out (especially in analysis with the definitions of limits and continuity).