"For all" or "for each"

Question

In mathematical context, or in the context of mathematical logic, is there a difference between:

This is valid for each x.

and

This is valid for all x.

?

If both have the same meaning, which is preferable?

This is off-topic: it is not about English language and usage, but about the special language used in maths. — Colin Fine, Commented Dec 23, 2013 at 18:04
@ColinFine: So the use of English is scientific texts is off-topic? — Yiorgos S. Smyrlis, Commented Dec 23, 2013 at 18:06
You were asking whether in the context of mathematics there was a difference between the phrases. This is not part of English in a technical context, this is part of the special variety of English used in a technical context, which is different. — Colin Fine, Commented Dec 23, 2013 at 18:44
It is certainly about language, since these are some of the same quantifiers that occur in all natural languages. Specifically, for each X, for every X, for all X are standard ways to express the universal (∀) quantifier, while some X, (at least) one X, and there exist(s) (an) X are standard ways to espress the existential (∃) quantifier. These two are what McCawley calls "the logicians' favorite quantifiers", though there's lots more quantifiers than that in every natural language. — John Lawler, Commented Dec 23, 2013 at 22:05

Rory Alsop · Accepted Answer · 2013-12-23 18:20:01Z

2

Both are valid here, as is 'every' as GEdgar mentions.

I would suggest that 'every' and 'all' are slightly more grammatical in this usage.

answered Dec 23, 2013 at 18:20

Rory Alsop

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