Differences between “for each” and “for all” according to Pugh's book

Question

On page 6 of C. C. Pugh's Real Mathematical Analysis (2017)：

Avoid reading $\forall$ as "for all," which in English has a more inclusive connotation.

Instead, one should read $\forall$ always as for each, as the writer suggests.

Could anyone please point out for me what the more inclusive connotation Pugh is referring to?

Related. Obviously, in English there is a difference between "all" and "each", like there is in other languages across Europe, but I don't think that a specific difference between "for all" and "for each" is clear to native speakers either. Unsurprisingly so. However, in some languages only one of the two analogous is in use as a translation of $\forall$. — , Nov 30 '20 at 07:49

score 4 · Accepted Answer · answered Nov 30 '20 at 07:56

I think he wants to make sure the meaning is not read as "for all at once" instead of the intended "for each separately (whichever one you care to choose)".

For those who have used the notation for years the meaning is intuitive. Perhaps there is a point of confusion at the early learning stage. Most would not be dogmatic about it - "For all ..." is after all a colloquial rendering of a precise mathematical statement (which is why we have the notation in the first place).

score 3 · Answer 2 · answered Nov 30 '20 at 07:44

I'm not really sure what Pugh's getting at here. My best guess is that he's worried that one might have trouble distinguishing

"For all $x$ there is some $y$ such that [stuff]"

from

"There is some $y$ such that for all $x$ [stuff]"

on the grounds that "for all" sort of sounds like it's describing a single condition which must be met all at once.

OK this doesn't quite match up with Pugh's "more inclusive connotation," but it really is the best guess I have.

But to me this is a huge reach: quantifier alternation is a genuinely difficult topic for many people, language be darned, and I don't expect "For each" to do much better than "For all" in this respect. Certainly nobody will look at you askance if you read "$\forall$" as "for all."

Conversely, nobody will look at you askance if you read it as "for each" either. If that choice of language makes quantifier structure easier to parse, you should absolutely do it. But there's nothing wrong with "for all" here.

score 3 · Answer 3 · answered Nov 30 '20 at 08:13

I'll take a crack at it.

In English, to a non-mathematician, the statement "All green crows pay taxes" might be regarded as false because it suggests that there is at least one green crow that is paying taxes.

Alternatively, even to a non-mathematician, the statement "For each crow that is green, the crow pays taxes" might be regarded as true, because the wording suggests that in order to dispute the statement, the non-mathematician must be concerned about finding a tax-dodging green crow.

So the alternative wording seems to emphasize that in order to dispute the statement, you must find a tax-dodging green crow. Then the non-mathematician will ask himself: How can I do that if there are no green crows in the first place?

Differences between “for each” and “for all” according to Pugh's book

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