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(Some report that my question is similar to another post. However, that post is talking about writing the "proof", rather than "stating" the theorem. "Proving" a theorem is NOT of the same structure and situation as "stating" a theorem. So this question is not duplicated to the other! Do not let it to be closed! And by the way, I'm also the OP of that question...)

In writing a textbook, when we need to state a theorem that is a universal quantification, we can use the word

"for all ..."(or equivalently "for every", "for any", "for arbitrary", "for each")

or

"let ...",

Which of these ways is more ideal? Why?

Although I think writing as "for all" is the more natural way to reflect the logical structure, that is a universal quantifier $\forall$, the popular style I have seen tends to use "let".

Any theoretical aspect or experience is welcome.

Example set 1.

  1. For all natural numbers $n$, if $n$ is even, then $n$ squared is even.

  2. Let $n$ be a natural number. If $n$ is even, then $n$ squared is even.

Example set 2.

  1. Let $A,B$ be two sets. If for all $x\in A$, $x\in B$, then we say $A$ is a subset of $B$.

  2. For all pairs $A,B$ of sets, if for all $x\in A$, $x\in B$, then we say $A$ is a subset of $B$.

Example set 3.

  1. Let $Y$ be a subspace of $X$. Then $Y$ is compact if and only if every covering of $Y$ by sets open in $X$ contains a finite subcollection covering $Y$. (Munkres Topology Lemma 26.1)

  2. For all subspaces $Y$ of $X$, $Y$ is compact if and only if every covering of $Y$ by sets open in $X$ contains a finite subcollection covering $Y$.

Example set 4.

  1. For every $f:X\to Y$ being a bijective continuous function, if $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (adapted by me, maybe ill-grammared?)
  2. For every bijective continuous function $f:X\to Y$, if $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (adapted by me.)
  3. Let $f:X\to Y$ be a bijective continuous function. If $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (Munkres Topology Theorem 26.6)

New added example set 5 (I skipped the quantification on $E,f:E\to\mathbb{R},L,c$, just focus on the key part here.)

  1. If "$\forall\varepsilon>0,\exists\delta>0,\forall x\in E,0<|x-c|<\delta\rightarrow |f(x)-L|<\varepsilon$", then we say $f(x)$ converges to $L$ when $x$ approaches $c$.
  2. If, for all $\varepsilon>0$, there exists $\delta>0$ such that for all $x\in E$, if $0<|x-c|<\delta$ then $|f(x)-L|<\varepsilon$", then we say $f(x)$ converges to $L$ when $x$ approaches $c$. (Using "for all")
  3. If let $\varepsilon>0$, there is $\delta>0$, such that let $x\in E$, if $0<|x-c|<\delta$ then $|f(x)-L|<\varepsilon$", then we call $f(x)$ converges to $L$ when $x$ approaches $c$. (Using "let". I think this type is not natural. But I can't tell why.)
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Eric
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    If the conditions on the object are large like "equicontinuous sqeuence of $L^1$ functions", then I would go for "Let", but if the conditions on the object are very small, like "natural number" or "compact set" or "Noetherian module" etc. then I would go for "for all". So for the first and third example, I would go for "For all", but for the second one I would go for let because we are asking for pairs of subspaces, which seems odd. It's all opinion though, and this is mine. I'm not even sure I'm consistent on this though! – Sarvesh Ravichandran Iyer Oct 10 '16 at 08:58
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    Related to while-proving-a-universal-quantification-we-begin-by-"Let ... in"-or-“Arbitrary choose …” ? – Mauro ALLEGRANZA Oct 10 '16 at 09:08
  • @MauroALLEGRANZA But in "proving a theorem" and "stating a theorem", I'm not sure it is the same situation. – Eric Oct 10 '16 at 09:16
  • I note that you reorganized the first phrase in Examples 1.1 and 4.2. The same less awkward reorganizations in Examples 2 and 3 are "For all pairs of sets, $A, B$, ...", "For all subspaces, $Y \subset X$, ...". Why the extra commas? Appositives. Also, in several places, you have written "called" where "say" would be more appropriate. Also, I know the standard is to italicize, but everyone would benefit from correct punctuation here: 'we say "$A$ is a subset of $B$".' – Eric Towers Oct 10 '16 at 12:44
  • @EricTowers Thanks for mentioning that. – Eric Oct 10 '16 at 13:01
  • You can't say "...such that let $x \in E$..." (it's not grammatical). Better to say "...such that for $x \in E$...". Also note that you can't have a verb after "we call X ..." - it has to be a noun or an adjective. So we can say "we call X a continuous function" or "we call X convergent", but not "*we call X converges" - but you can say "we say X converges" :-) – psmears Oct 10 '16 at 14:17
  • @psmears I see. Another question: do you think it is good if we all write in such "$\forall$-$\exists$" style? i.e. If $\forall\varepsilon>0,\exists\delta>0,\forall x\in E,0<|x-c|<\delta\rightarrow |f(x)-L|<\varepsilon$, then we say $f(x)$ converges to $L$ when $x$ approaches $c$. – Eric Oct 10 '16 at 14:23
  • Or "$\forall A,B$, ($\forall x\in A,x\in B)\underset{\text{def}}{\Longleftrightarrow} A$ is a subset of $B$ " – Eric Oct 10 '16 at 14:31
  • A nice thing about "let" vs. "for all" is that it reduces the temptation to replace words with symbols. – asmeurer Oct 10 '16 at 17:54
  • @asmeurer Can you explain further more? – Eric Oct 10 '16 at 17:56
  • I mean for students (particularly those new to writing proofs). They are tempted to use $\forall$ instead of writing out "for all", and it makes their mathematics harder to read. There is no symbol for "let", so the temptation to do that is gone if you use let. – asmeurer Oct 10 '16 at 18:05
  • An additional consideration is how the proof would start. On the one hand, if my theorem states "For all widgets $w$, ...", then I invariably start the proof with "Let $w$ be a widget". On the other hand, if my theorem states "Let $w$ be a widget. Then ...", I immediately start reasoning about $w$ in the proof. (I go even further than this. If my theorem says "Let $w$ be a widget. If $w$ is green, then ...", then my proof tends to start with "Assume $w$ is green.") – Magdiragdag Oct 10 '16 at 21:26
  • (reformatting my above comment) That is more a matter of style - personally I'd say that the "$\forall \exists$" style is too abbreviated and confusing, and the more wordy style is easier to read−and in general it is preferred in books I've read−but sometimes the "$\forall \exists$" can make a statement more precise (where putting it in words can leave it ambiguous). – psmears Oct 11 '16 at 09:14
  • Just to add a small comment: I think "For all natural numbers n, if n is even, then n squared is even." sounds weird. First you say "for all" and then you are like "not really for all, only for these". Thus I would either write "For all even numbers n ..." or "Let n be a natural number. If n is even...". –  Oct 11 '16 at 09:16
  • Your question begins with an up arrow pointing at noting, and a mention of "that other question" with no question linked or named. If there's a banner there suggesting a duplicate, only you can see it. – TRiG Oct 11 '16 at 14:14
  • @TRiG Got it! Thanks for telling me. – Eric Oct 11 '16 at 15:30

6 Answers6

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If you're doing informal mathematics, there's really no difference. I guess from a type-theoretic perspective, it's kind of the difference between $$x:\mathbb{R} \vdash P(x) \qquad \mbox{and} \qquad \vdash (\forall x:\mathbb{R})\,P(x).$$

The former is arguably better, because it doesn't presuppose that we're working in a category that interprets universal quantification. So "let" is preferable to "for all" for this reason. But, again, unless you're doing highly formal mathematics, it's not really worth worrying about. (I say that, but a part of me finds the question fascinating, and I've just gone and favourited it.)

goblin GONE
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    I haven't studied type theory or category yet, but I feel what you mentioned is exactly what I want. Can you explain a little bit more to help me understand the two formula you gave? – Eric Oct 10 '16 at 09:21
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    @Eric, sure. Basically, $x:\mathbb{R} \vdash P(x)$ means "in the context $x\in \mathbb{R}$, it holds that $P(x)$." Whereas $\vdash (\forall x:\mathbb{R}),P(x)$ "in the empty context, it holds that for all $x \in \mathbb{R}$, we have $P(x)$." Type theorists tend to prefer the symbol $:$ to $\in$, because their account of the meaning of this symbol is completely different to the materialistic account familiar to most mathematicians. – goblin GONE Oct 10 '16 at 11:19
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    I don't see how this comes anywhere near answering the question. This question isn't asking about type theory, category theory, or formal systems; it's asking about the English language and proof writing. – Tanner Swett Oct 11 '16 at 01:55
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    @TannerSwett, the answer to all such questions is: "If you're doing informal mathematics, there's really no difference." What else really needs to be said here? By the way, type theory is basically entirely about proofs; look up the phrase 'propositions as types, proofs as programs' if you're interested. – goblin GONE Oct 11 '16 at 02:09
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    @TannerSwett, yes, I indeed asked for the formal mathematics, not English language. In particular, the whole question is concerned about the correspondence with human language and the formal logic structure behind these. My original post have a tag on logic, but it had been removed by someone else. – Eric Oct 11 '16 at 04:21
  • @Eric Ah, it sounds like I misunderstood your question. My apologies. – Tanner Swett Oct 11 '16 at 12:15
  • @TannerSwett Don't worry, that's fine! When I post the question, I originally want to ask the formal reason and aspect of using "for all" versus "let". I guess "let" is not formalized in logic, so it is not evident to me that we should use "let", rather than "for-*" family. But given that many textbooks use "let", I'm also curious about if there is any critical reason that the word "let" prevails over the others. So I also welcome any suggestions. – Eric Oct 11 '16 at 15:56
  • @goblin I want to ask: is the vocabulary "for all"(or the notation $\forall$) dispensable in formal mathematics(e.g. the type theory you mentioned)? Or the another question, does it mean that theoretically, all the definitions and theorems in undergraduate mathematics have no need to (or, have better not) use for all ($\forall$), just using let? – Eric Oct 18 '16 at 15:16
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    @Eric, if you're doing abstract algebra, 95% of uses of "for all" are "non-essential" uses that don't actually require the universal quantifier. For example, an integral domain is a commutative ring $R$ satisfying $x:R ,y:R \vdash xy=0 \rightarrow x = 0 \vee y=0$. So there's no need for universal quantifiers. But sometimes, you really do need them. For instance, the principle of function extensionality says that if I'm given sets $X$ and $Y$ and functions $f : X \rightarrow Y$, then: $$\forall x \in X(f(x)=g(x)) \vdash f=g$$ I don't think it's possible to get rid of the $\forall$'s here. – goblin GONE Oct 19 '16 at 00:37
  • @goblin Thanks! By the way, can you recommend some type theory (or related) books? I want to sit on my chair and self-study them! :) – Eric Oct 19 '16 at 04:07
  • @Eric, I'd love to be able to say "yes", but everything I know about type theory, I've learned by thinking about category theory, operads, Lawvere theories, universal algebra, and by thinking very carefully about propositions as types. That last link will take you to nLab, which is a good place to get a feel for what's out there. – goblin GONE Oct 19 '16 at 04:30
  • @Eric, here's an extremely useful nLab page. – goblin GONE Oct 19 '16 at 04:31
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In your first example, either is fine. In your second example, 2 is ungrammatical - you cannot just replace "Let" with "For all", the "be" has to be deleted (or replaced with "that are"). Otherwise, either is fine.

In all three examples, you'll notice that your "For all" example results in a long, slightly awkwardly-phrased sentence with at least three clauses. The "Let" version divides the sentence into two simpler sentences, so the reader can process it one step at a time.

In general, "For all" is okay as long as the thing you're quantifying over is small and doesn't really require any work to understand. If the reader's going to have to think about it even a little - e.g. "system of linear equations in five variables" - I'd use "Let".

Also - and I'm pretty sure this is just a personal preference - I try to avoid having more than two parts to a sentence in a mathematical theorem, if I can. "If $X$ then $Y$" is fine, but "If $X$ then if $Y$ then $Z$" is ugly. Similarly, "For all $x$, if $Y$ then $Z$" is complicated, and it gets worse the more complicated $x$, $Y$, and $Z$ are.

Reese Johnston
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  • I'd added the fifth example set. What do you think about that case? – Eric Oct 10 '16 at 09:18
  • Both 1 and 2 are unwieldy and hard to understand. 3 isn't grammatical - you can't use "be" like that. For this case, I wouldn't use either - I'd use "suppose". That is: "Suppose that for every $\epsilon > 0$ there is a $\delta > 0$ such that for every $x \in E$, if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$. Then we say $f(x)$ converges to $L$ as $x$ approaches $c$." – Reese Johnston Oct 10 '16 at 09:40
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As a personal point of view I would first "fix" what I'm working with with a let, and then state my property with for all if needed.

So my canonical form would be:

Let a be something, b be something, and c be something. If for all d such that something on a,b,c and d then we have something great on a,b,c

I think it is the more readable way to state a theorem.

So to go through your examples:

Example set 1.

Let $n$ be a natural number. If $n$ is even, then $n$ squared is even.

Example set 2.

Let $A,B$ be two sets. If for all $x\in A$, we have $x\in B$, then we say that $A$ is a subset of $B$.

Example set 3.

Let $Y$ be a subspace of $X$. $Y$ is compact if and only if every covering of $Y$ by sets open in $X$ contains a finite subcollection covering $Y$. (Munkres Topology Lemma 26.1)

Example set 4.

Let $f:X\to Y$ be a bijective continuous function. If $X$ is compact and $Y$ is Hausdorff, then $f$ is a homeomorphism. (Munkres Topology Theorem 26.6)

** example set 5**

Let $E,f:E\to\mathbb{R},L,c$ be things. If for all $\varepsilon>0$, there exists $\delta>0$, such that for all $x\in E$, $0<|x-c|<\delta\rightarrow |f(x)-L|<\varepsilon$", then we say $f(x)$ converges to $L$ when $x$ approaches $c$.

wece
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    In #2: I consider "If for all $x\in A$, $x\in B$ ..." quite poor style. (I know the OP did the same thing.) Commas often have mathematical meanings, so one should essentially never separate two mathematical statements with a mere comma. Here, it looks like the $x\in B$ is still part of the hypothesis's quantifier. "If $x\in B$ for all $x\in A$" is better, or "If $x\in A$ implies $x\in B$" perhaps. (And all the math quantifiers in a row in #5 is basically unreadable; the OP's options with more words are better.) I hope @Eric read this too. – Greg Martin Oct 11 '16 at 07:13
  • @GregMartin I edited to take into account your comment. Is it better now? Thanks. – wece Oct 11 '16 at 07:38
  • I don't think it is appropriate for $d$s to smother on $a$s, $b$s or $c$s in a mathematical textbook. Otherwise, I agree. :) – tomasz Oct 11 '16 at 08:13
  • @tomasz :D I didn't even know this word before you comment ... Thanks for pointing out the typo ... (spell checking works in mysterious ways ...) – wece Oct 11 '16 at 08:17
  • I liked the smothering version! As for your edits, yes, #2 is fine now (we've rediscovered the whole purpose of "we have" in math writing), and #5 is much better. In definitions, I prefer to put the term being defined before the definition itself. And I'd still rather avoid the use of the $\to$ in the final implication (as well as the comma separating $x\in E$ and that implication). – Greg Martin Oct 11 '16 at 08:19
  • Well, a small caveat: the "for all $x$ (...)" form can work well for very short and self-contained statements. At nontrivial complexity, they are prone to become (unnecessarily) more confusing. – tomasz Oct 11 '16 at 08:19
  • @tomasz I personally prefer the $\forall x$ form but it can be confusing if you work with logics using $\forall$. Anyway having a row of variable depending of each others is hard to digest in any form. (ps: Thanks for the new word "caveat" :) – wece Oct 11 '16 at 08:25
  • I don't see why the $\forall$-$\exists$ style seems not be well accepted by others. It is pretty clear when seeing, and it is not hard to loud it. Furthermore it actually reflects the logic structure behinds. (see the limit definition case) – Eric Oct 11 '16 at 08:34
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Some of this is a matter of style - which isn't very important. But there are a few "technical" problems with some of your examples.

Examples 2.1 and 2.2:

If for all $x \in A$, $x \in B$, then ...

If you parse this as "If ( for all $x \in A$ and $x \in B$ ) ..." it doesn't make sense. The intended meaning is "For all $x$, if $x \in A$ implies that $x \in B$, we say ...".

Since you are considering the $x$'s "one at a time" here, I would prefer "for each/any/every $x$" to "for all $x$".

Example 3.2:

I would prefer "for each/any/every subspace" to "for all subspaces", for the same reason as above.

Examples 4.1, 4.2

The grammar seems a bit convoluted here. The main statement is of the form "If (the premises) then (the conclusion)" but you have a subsidiary clause (stating the properties of $X$ and $Y$) after the premises. I would re-order 4.1 as something like

If $X$ is compact and $Y$ is Hausdorff, then every bijective continuous function $f:X \to Y$ is a homeomorphism.

and similarly for 4.2.

Example 5.3

"If let $\epsilon > 0$ ..." is not correct English grammar. You need a clause after the "If", but "let $\epsilon > 0$ ... then $|f(x)−L|< \epsilon$" is a complete sentence, not a clause within the bigger sentence starting with "If".

alephzero
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  • For your first paragraph, doesn't anybody call the $\forall$ "for all"? (which is equivalent to "for *") – Eric Oct 11 '16 at 01:43
  • In your re-order Examples 4.1, if it just started by "If $X$ is compact and $Y$ is Hausdorff", it relatively lacks the feel of "universal quantification". The quantification became implicit. – Eric Oct 11 '16 at 01:47
  • I disagree with the sentiment that style is not very important. I think all mathematicians come across papers that are nigh on impossible to read precisely because of their terrible style. – tomasz Oct 11 '16 at 08:16
  • @alephzero why you said "If you parse this as "If ( for all x∈ ) ..." it doesn't make sense. "? Due to grammartical reason or logical reason? – Eric Oct 18 '16 at 05:17
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One instance where this can matter is when working with formal logic, particularly quantifier logic. In that case, $\forall$ has a very specific meaning, and it is very important to clearly distinguish the formal logic itself and the meta-logic of proving things about the formal logic. For that reason, it may be clearer to avoid the phrase "for all" entirely from the meta-logic, to avoid any confusion with the $\forall$ symbol.

asmeurer
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  • Wow! I'm the first time to heard about this! But it is too common to see a sentence like "for all $\epsilon>0$, there exists $\delta>0$, ...". Isn't it proper? – Eric Oct 10 '16 at 18:15
  • Isn't this only true when reasoning about logic? When reasoning about math, the point is to use logic. – djechlin Oct 11 '16 at 03:23
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I might misunderstand your question, and I'm not a logician, but I can't resist to give an answer here. In my point of view it is more important to be clear and easy to read than to write a logically 100% correct statement.

I think that in some of your examples, you should use neither "Let" or "For all", but rather use words, to make the statements easier to digest (I'm well aware, and respect that others think different). Suggestions:

Example 1

The square of an even number is even.

If it is not clear enough that this holds for all even numbers, then maybe:

The square of every even number is again even.

Example 2

We say that $A$ is a subset of $B$ if every element of $A$ also belongs to $B$.

Example 4

Every bijective continuous mapping from a compact space to a Hausdorff space is a homeomorphism.

mickep
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    Here's my interpretation of your contribution: the statements must still be 100% logically correct—but that can often be accomplished through only (or mostly) words, as these examples show. Math symbols $\ne$ rigor! – Greg Martin Oct 11 '16 at 07:15
  • Yes, you are correct. Thanks for clarifying. – mickep Oct 11 '16 at 09:10