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Every once in a while I encounter phrases like

and let $f_i$ be increasing $\forall 1 \leq i \leq n.$

Although the meaning is clear, I find it a horrible quantifier abuse. The quantifiers, as symbols, have a rather precise meaning and definitely are not just shorthands for expressions "for all" and "exists" (the more, it TeX it takes the same amount of keystrokes, and even in hand-writing 6 letters instead of one symbol is not that high a cost).

How to deal with them? Do you know any on-line text that explains it that I could link? Or maybe I am just overreacting and this is how "nowadays" math should be written?

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    $\begingroup$ With my students I am considering using a bat... $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 25, 2013 at 18:30
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    $\begingroup$ $\forall$ such questions, leave a comment explaining $\exists$ a time and place for such symbols, etc. If all $\in$ work together, we can make a $\Delta$. $\endgroup$
    – GeoffDS
    Commented Aug 25, 2013 at 18:54
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    $\begingroup$ Why is this on meta? $\endgroup$
    – mrf
    Commented Aug 25, 2013 at 20:48
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    $\begingroup$ @mrf I'm asking how to handle a specific behavior, not "Why I shouldn't abuse quantifiers like this?", does this answer your question? $\endgroup$
    – dtldarek
    Commented Aug 25, 2013 at 20:57
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    $\begingroup$ There is a question on using quantifiers. Just add your answer and link it. $\endgroup$
    – Michael Greinecker
    Commented Aug 26, 2013 at 5:00
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    $\begingroup$ @Mariano: I believe that studies showed that Batman is about 63% more effective than just a bat. $\endgroup$
    – Asaf Karagila Mod
    Commented Aug 26, 2013 at 11:28
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    $\begingroup$ @AsafKaragila Imagine those headlines: "Batman teaches students how to use quantifiers!"... $\endgroup$
    – dtldarek
    Commented Aug 26, 2013 at 11:43
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    $\begingroup$ ... and shortly afterwards "Do we need math in school?" $\endgroup$
    – dtldarek
    Commented Aug 26, 2013 at 11:44
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    $\begingroup$ @dtldarek: Well, if Batman teaches intro to logic then I'm sure no one will argue that it's needed in schools. See also SMBC(a) and SMBC(b). $\endgroup$
    – Asaf Karagila Mod
    Commented Aug 26, 2013 at 11:46
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    $\begingroup$ @MichaelGreinecker: The link in your comment refers back to this post? $\endgroup$
    – user53259
    Commented Aug 26, 2013 at 18:28
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    $\begingroup$ @Adriano: Yes, but the quantifier is still unnecessary: just write for $i=1,\ldots,n$. $\endgroup$
    – Brian M. Scott
    Commented Aug 27, 2013 at 9:47
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    $\begingroup$ More distressing than quantifier abuse are quantifier manipulations that lead to nonsense that the student would be unlikely to write if ordinary mathematical English were used. $\endgroup$
    – André Nicolas
    Commented Aug 28, 2013 at 2:28
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    $\begingroup$ @Brian M. Scott: I would go even further and say just write $f_{1},$ $f_{2}, \; \dots, \; f_{n},$ unless there is a reason to introduce the dummy variable $i$ at this point in the exposition. $\endgroup$
    – Dave L. Renfro
    Commented Aug 28, 2013 at 17:55
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    $\begingroup$ Is 'for all $1\le i\le n$ much better? $1$ is normally less than or equal to $i\le n$ for all values of $1$, but I suppose it depends what $i$ is. $\endgroup$
    – John Gowers
    Commented Aug 28, 2013 at 18:09
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    $\begingroup$ How do you quantify the abuse? $\endgroup$
    – copper.hat
    Commented Aug 28, 2013 at 22:42

2 Answers 2

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In my opinion the above example isn't bad because of the formal aspects (according to the rules one should write $\forall_{x\in X} \varphi(x)$) but it is just ugly from the linguistic point of view (as perfectly pointed out by Graphth). Simply (as written in the answers to the linked non-meta question) in plain text one should avoid quantifiers. $\def\N{\mathbb N} \def\R{\mathbb R}$

However. It is often the formula ($\varphi(x)$) that is important and the quantifier $\forall_{x\in X}$ is obvious from the context. Then we usually write like in the example "function $f_n$ is increasing for all $n$" rather than "for all $n$ function $f_n$ is increasing". And if we have a complex formula written with symbols in a separate line ($$), then in my opinion it may be a good notation to add $\forall n\in \N$ at the end of the line rather than struggling with adding text-style "for all natural $n$" to the symbol-style formula (it's firstly mixing styles and secondly unhandy in TeX).

Similarly, if we have a long pointed list of formulas with different quantifiers (written at the end), it may also be more clear (easier to compare or look back to when reading further paragraphs) if written with symbols "$\forall_{n\in \N}\ \forall_{r\in \R}\ r>n$" rather than words "for all natural $n$ and real $r$ bigger than $n$". When I just want to check some trivial conditions ("was it $<$ or $\leq$?") reading symbols is faster than reading words.

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    $\begingroup$ I am curious about the use of the quantifiers you have here. I don't think I have seen them used before with the arguments as subscripts. Where is this commonly done? $\endgroup$
    – Tobias Kildetoft
    Commented Aug 30, 2013 at 18:31
  • $\begingroup$ @TobiasKildetoft Oh, it's just $\TeX$. Normally in handwriting I place the variable below the quantifier and in non-inline $\TeX$ it works perfectly by using _, but the inline $\TeX$ (invoked by a single dollar) saves vertical space by putting the variable in the subscript. $\endgroup$
    – savick01
    Commented Aug 30, 2013 at 22:39
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    $\begingroup$ @savick01, that only happens because you put the variable as a subscript: you wrote $\forall_{x\in X}\phi(x)$ instead of $\forall {x\in X},\phi(x)$ and therefore you got $\forall_{x\in X}\phi(x)$ instead of $\forall{x\in X},\phi(x)$. That is neither standard nor done automatically by TeX. In a display you could also say $\mathop\forall_{x\in X}\phi(x)$ to get $$\mathop\forall_{x\in X}\phi(x)$$ but that is very nonstandard (and pretty ugly IMHO! (That \forall is not a «growing» operator being not the least important of the reasons)) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 1, 2013 at 4:47
  • $\begingroup$ @MarianoSuárez-Alvarez, the last notation is what is commonly used in Warsaw and what I intended (we also write $\leqslant $ instead of $\leq$ just like in Russia, so it all is probably a matter of history). But I was wrong thinking that what I wrote renders by standard to $$\mathop\forall_{x\in X}\varphi(x).$$ In my opinion $\forall$ is not that far from the "growing/limiting" operators: $$p\in \bigcap_{i\in I} U_i \iff \mathop\forall_{i\in I} p\in U_i \\ p\in \bigcup_{i\in I} U_i \iff \mathop\exists_{i\in I} p\in U_i.$$ $\endgroup$
    – savick01
    Commented Sep 1, 2013 at 9:37
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    $\begingroup$ I meant \forall is not growing in the technical sense that it isn't in TeX! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 1, 2013 at 9:38
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The problem that I see in the given example is distinct from what savick01 wrote about. The quantifier symbol $\forall$ ought to be followed immediately by the variable that it quantifies. What is written in the example would be read as "for all one $\dots$", which is not what was intended. This problem is easily corrected by writing $\forall i\in[1,n]$.

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  • $\begingroup$ Another solution is the following: put "$1 \leq i \leq n$" below the quantifier so that $i$ lies below the pick of $\forall$. One may argue that it's just formatting, but I've seen such notation many times and find it clear and elegant. $\endgroup$
    – savick01
    Commented Aug 30, 2013 at 23:04

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