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

This is valid for each x.


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
    Dec 23 '13 at 18:04
  • 1
    ... and what about This is valid for every x ?
    – GEdgar
    Dec 23 '13 at 18:04
  • @ColinFine: So the use of English is scientific texts is off-topic? Dec 23 '13 at 18:06
  • 1
    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
    Dec 23 '13 at 18:44
  • 4
    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. Dec 23 '13 at 22:05

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

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

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