Is there a better alternative to the phrase, 'it holds that'?

Question

The following phrases abound in my writing:

There exists [whatever] such that [whatever].
For all [whatever] it holds that [whatever].

Lately, I've been feeling that the phrase 'it holds that' is overly long-winded. The only substitute I can think of is 'we have that' which is just as bad. I've solved the problem in my personal writing by using the abbreviation 'iht = it holds that' (along with sth = such that), but this isn't appropriate for more formal pieces.

Is there a better phrase?

Quite often you need do nothing more than reverse the sentence, e.g., ‘$f(x)\in A$ for each $x\in B$’, or use a comma (‘For each $x\in B$, $f(x)\in A$’). — Brian M. Scott, Jun 29 '13 at 02:00
@user46080, as in: for all $x,y; x+y=y+x$? It would be better to have something separating the $x,y$ and the $x+y=y+x$. — goblin GONE, Jun 29 '13 at 02:00
You have something: a comma. If you feel that it has too little visual weight (and I sometimes do), reverse the sentence: $x+y=y+x$ for all $x,y\in A$. — Brian M. Scott, Jun 29 '13 at 02:01
@BrianM.Scott, thank you for your comment. I used to do the reversal thing, but lately I've been trying to make more writing more 'linear' and predictable. I don't know if this is good practice or bad practice (still experimenting!). — goblin GONE, Jun 29 '13 at 02:02
If a comma is insufficient visual space, and to avoid scope ambiguity due to reversing the sentence, then I too marginally prefer "we have that" to "to holds that". — ryang, May 28 '22 at 23:53

score 12 · Answer 1 · answered Sep 21 '14 at 10:24

12

Personally, I would recommend not saying things like "It holds that" which few, if any, native English speakers would say. Instead, say "we have that" or just "we have", or perhaps replace "It holds that x > y" with "The inequality x > y holds".

Similarly, I would recommend not saying things like "We denote by x the distance between y and z". Instead, say "We let x denote the distance between y and z".

answered Sep 21 '14 at 10:24

Jeffrey Shallit

1,518

I'm a native speaker, I can promise you that. – goblin GONE Jul 23 '16 at 07:01
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Many native English-speaking mathematicians have absorbed unusual phrasings such as "it holds that" and "We denote by x y" and "associated to" by reading mathematical papers written by non-native speakers. Once absorbed, they seem natural. But ask someone not involved in the field, and they'll say, "Wut?!" – Jeffrey Shallit Jul 23 '16 at 10:32
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For example, compare this google ngram search for "associated to" versus "associated with". Although most native English speakers are extremely unlikely to say "associated to", I'd guess the probability goes up greatly if you are a mathematician. https://books.google.com/ngrams/graph?content=associated+to%2C+associated+with&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cassociated%20to%3B%2Cc0%3B.t1%3B%2Cassociated%20with%3B%2Cc0 – Jeffrey Shallit Jul 25 '16 at 09:34
Interesting... $;!$ – goblin GONE Jul 25 '16 at 09:35
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This google ngram search shows that "it holds that" is extremely rare in English prose, but has been slowly increasing in popularity since 1820: https://books.google.com/ngrams/graph?content=it+holds+that&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cit%20holds%20that%3B%2Cc0 – Jeffrey Shallit Nov 11 '17 at 11:13
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Curiously, it seems to be booming in British English, while it's stagnating in AmE: https://books.google.com/ngrams/graph?content=it+holds+that&year_start=1800&year_end=2008&corpus=6&smoothing=3 – Michaël Cadilhac Mar 19 '19 at 18:49

score 5 · Accepted Answer · answered Jun 29 '13 at 03:13

I don't see anything wrong with such phrases. I think they're perfectly idiomatic, and I don't see them as long-winded. I'd much rather occupy another couple centimeters of space on the page than confuse the reader with an ambiguous statement.

In the second case, I personally prefer "we have that" or "we have". I don't care as much for "it holds that", because my brain briefly searches for the antecedent of "it" and there's a moment of grammatical dissonance.

amWhy · Answer 3 · 2013-06-29T16:34:21.720

Technically speaking, whatever the property/predicate/implication following

"for all [(such and such) $\in$ (domain)], ..."

...logically speaking, it needs NO connecting phrase or words; a comma will often do.

But admittedly, when writing mathematical exposition, I think it helps with parsing to use an "intermediate" phrase: "for all [such and such], we have that...". Whether, or what to use, depends on the context, the number of quantified variables, and the complexity of the predicates which follow. But my own preference is ", ...we have that"... And although it does strike me as "repetitive" in my own writing (in that I use it so often), I wouldn't worry about it being "long-winded."

I also often write "$\exists x$ such that..." or "$\exists x$ for which...", even though it may not, technically, be necessary.

Using just a comma can result in an expression that's hard to parse. "For all $x \in X$, $y \in Y_x$, $z \in Z_{x,y}$." Which part is an assumption and which a conclusion? — Nate Eldredge, Jun 29 '13 at 03:04
@Nate Absolutely, and nothing I said suggests otherwise. Your comment has my upvote! — amWhy, Jun 29 '13 at 16:37

Is there a better alternative to the phrase, 'it holds that'?

3 Answers3