# Translation of "for some" as existential quantifier

In English one can write the existential statement ∃ xA : P(x) as follows:

P(x) for some xA.

Is there a similar construction in German? My best guess is:

P(x) für irgendein xA.

1. Is this use of irgendein correct? Does it correctly reflect the meaning of existential quantifier?
2. Does irgendein need to be declined in this case and, if so, how? I don't know what grammatical gender x should take. Is it das Element x, die Variable x or der Buchstabe x ?
3. How would it be for more than one variable, e.g. ∃ x,yA : P(x) ?

EDIT

To clarify: It's important to distinguish here between a universal statement and an existential statement. I'll be more specific about why I would like a translation of the "for some" construction by providing a concrete example:

Let B = {(n,2n) : nN} and let bN. Then b = 2m for some mN.

Here m is not an arbitrary element, since it depends on b. I want to avoid constructions such as "Es existiert m ..." because I want to introduce b before m in a concise way.

• This use of "for some" is ubiquitous in mathematics written in English. (I am a mathematician and native English speaker.) My trouble with the word "beliebig" in this context is that it suggests that x is arbitrary, which it is not. Nov 25 '21 at 19:55
• @dwolfeu That's correct. Don't use "beliebig" in this context.
– Uwe
Nov 25 '21 at 20:13
• @ShegitBrahm "for some" is unambiguously existential. The problematic phrase is "for any", which can indeed be interpreted as existential or universal (and should therefore be avoided).
– Uwe
Nov 25 '21 at 20:17
• I'm thinking the best way to answer this kind of thing to to go on German Wikipedia and read some math articles. For example this article uses es geben and existieren about equally often. Mathematical jargon isn't a constant though, and a lot depends on the field and the time period. Nov 26 '21 at 0:02
• The last time I did math it was Es gibt ein or Es gibt ein oder mehrere - Do not use irgendein or beliebig as it may imply any which implies all
– TaW
Nov 26 '21 at 9:10

Your unease is justified, “irgendein” would be unclear, as it could be read as “any” and hence universal quantification. At least it would be so unusual that you should not use it. That aside, you need to decline, but luckily all letters are neuter, so “für irgendein x” is grammatically correct.

Unfortunately, it has been a while since I have written any mathematics in German, and I am not sure what the best option is. You could just write “für ein x“, but that is not as clear as “for some x”. A more precise version is

für (ein) geeignetes x

or in plural

für geeignete x,y.

It sounds more formal than “for some”, so I believe it is less common. It is possible that it sounds a bit old-fashioned, but you can definitely use it. I think I would mostly go with the longer “es existiert x, so dass” construction.

The formal statement ∃ xA : P(x) using the existential quantifier reads as

Es existiert ein xA, so dass P(x) gilt

or

Es gibt ein xA, so dass P(x) gilt.

It is unnecessary to say mindestens ein or something like that. You can also say

P(x) für ein xA.

or

Es gilt P(x) für ein xA.

Saying für irgendein would be highly unusual. If you really want, you can say für mindestens ein, but it is completely unnecessary.

• "It is unnecessary to say mindestens ein or something like that" But it is not uncommon to clearly distinguish from "Es gibt genau ein" (There is exactly one). Nov 26 '21 at 11:24
• @rexkogitans Correct. To express uniqueness quantification one often uses the quantifier ∃ ! . That is, one writes ∃ ! x P(x) . It includes more than ∃, therefore it needs the additional "genau". See en.wikipedia.org/wiki/Uniqueness_quantification . Nov 26 '21 at 11:58
• Typo: existiert Nov 26 '21 at 17:06
• @rexkogitans When I teach my students first-order logic I have to emphasize "at least one" a few times. But after that is should be understood without further clarification that "∃ x ∈ A" means "there exists at least one x in A". So it is only necessary when teaching beginners. Nov 27 '21 at 18:29
• @MichaelKay Not in a mathematical context. And the phrase "P(x) für ein x ∈ A" is not used in everyday language. Nov 28 '21 at 23:51

From my days with formal math at school and university I remember wordings like

Es existiert mindestens ein x ∈ A, für das gilt: (...)

1. Yes, it is correct, and it is in my opinion the clearest way to state the fact if you want to avoid the lengthier "Es existiert ein x, für das P(x) gilt". Other possibilities are "für ein x ∈ A" or "für ein geeignetes x ∈ A" (as suggested by Carsten S). Do not use "für x ∈ A" or "für ein beliebiges x ∈ A"; these are more likely to be interpreted as universal.

2. It must be declined, but since you need neuter gender for a variable without further specification, "irgendein" is correct. If you specify the type, say "für irgendein Element x", "für irgendeine Matrix A", or "für irgendeine natürliche Zahl n", it depends on the gender of "Element", "Matrix", or "Zahl".

3. "P(x,y,z) für irgendwelche x, y, z" or "P(x,y,z) für geeignete x, y, z" is fine.

• Wouldn't "irgendein" imply that P(x) holds true for any x ∈ A one could choose, meaning for all x ∈ A? Then, it would express ∀ x ∈ A : P(x) rather than ∃ x ∈ A : P(x), wouldn't it? In addition to the wording I mentioned in my answer, maybe "einige" would fit better than "irgendein". Nov 25 '21 at 17:34
• @dwolfeu "irgendein" is declined in the same way as "ein": "für ein x -> für irgendein x".
– Uwe
Nov 25 '21 at 20:10
• @dwolfeu "irgendein" is the correct spelling; "irgend ein" is non-standard.
– Uwe
Nov 25 '21 at 20:11
• @Uwe But "irgendein" (especially in contrast to just "ein") means "any member of a group, regardless which member", correct? Think for example of an illusionist's trick, "Nehmen Sie eine Karte, irgendeine" - "Take a card, any card". So, if you say that "irgendein" element of the set fulfills some condition, you basically say "regardless which element of the set you chose, it fulfills the condition". That will only work if all elements fulfill the condition. Also, the question speaks of "for some x", so there's already a plural there. Nov 25 '21 at 20:35
• @HenningKockerbeck The some in for some isn't necessarily plural. It's the same use of some as in I just met some guy. (see en.wiktionary.org/wiki/some > Determiner > Definition 4). Nov 26 '21 at 4:56

I can't really agree with any of the suggestions so far.

• “Es existiert ein x...” suggests there's exactly one, which the ∃ statement doesn't imply. Better would be “es existieren xA...”. Both are clunky, nowhere near as natural as “some” in English.

• “Es gibt ein x...” has the same uniqueness problem. This isn't as awkward as “existiert”, in fact, if anything it's a bit too colloquial.

• P(x) für irgendein x...” is even more colloquial, and very unclear what's actually meant logically.

• “Für geeignete x” is nice language-wise, but it doesn't really express existance, it's rather something like ∀ xA : P(x) ⇒ Q(x), where Q is whatever property is discussed afterwards in the text.

• P(x) für ein beliebiges x aus A” is even worse, this suggests ∃ xA ∧ ∀ xA : P(x), which is much a much stronger proposition.

What I would write depends on the context.

• In the initial statement of a theorem, be precise. “Für A-quibilotschige P existiert immer xA sodass P(x) erfüllt ist.”

• If this is the final conclusion, then I would just write “...daraus folgt die Existenz von x”, where the actual domain and properties were already mentionen earlier.

• If you write this to bring x in scope for doing further work on it, I would say something like “Sei xA sodass P(x). (N.b.: Existenz eines solchen x folgt aus Mustermann et al. 1999)”

• If this is just a side remark regarding something the reader is probably already familiar with, just hand-wave something simple like “es gibt xA mit P(x)” or “P(x) gilt für gewisse xA”.

• I do not agree that "ein" would be read as "genau ein". Nov 28 '21 at 19:53
• I totally agree with @CarstenS. Es gibt ein x or Es existiert ein x just states existence of at least one x. If it's exactly one, it would be genau ein x. Nov 29 '21 at 3:48
• I didn't say it states ∃!, only that it suggests it. Yes, if you want to express precisely ∃! then you need to say “genau ein”. If you want to express precisely ∃ then you should say neither “genau” nor “ein”. Again, this doesn't always matter; if it can be looked up elsewhere what some theorem states exactly then “ein / eines” is alright if it improves the sentence structure. But for results of your own, it's no good leaving any possible ambiguity. Nov 29 '21 at 9:18
• You're wrong. There is no ambiguity.at all. Neither ∃ nor ∀ have anything to do with quantity. Nov 30 '21 at 2:46
• @Olafant ∀ does not have to do with quantity, but ∃ has to do with quantity. It's exactly what it's all about: that the quantity is 1 or larger. Nov 30 '21 at 6:08