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As an instructor for discrete math, I would like to know what the rules are for scoping variables in natural language, so that I can explain the them to my students.

To which of the following statements would it make sense to append, "Then $Q(n)$ is true."?

  1. Suppose that for all $n \in \mathbb{N}$, $P(n)$ is true.
  2. Suppose that $P(n)$ is true for all $n \in \mathbb{N}$.
  3. For all $n \in \mathbb{N}$, suppose that $P(n)$ is true.
  4. Suppose that for some arbitrary $n \in \mathbb{N}$, $P(n)$ is true.
  5. Suppose that $P(n)$ is true for some arbitrary $n \in \mathbb{N}$.
  6. For some arbitrary $n \in \mathbb{N}$, suppose that $P(n)$ is true.
  7. Let $n \in \mathbb{N}$. Suppose $P(n)$ is true.

To me, for 1. and 2., the scope of $n$ is closed at the end of the sentence, so it doesn't make sense to refer to $Q(n)$ afterwards. But I can't articulate why.

I think that 3.-7. are all okay, with my preference in my own writing being 3. $<$ 4. $=$ 5. $=$ 6. $<$ 7.

If you do not agree with this answer, let me know as well.

Arbuja
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Glenn Sun
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    I agree with you. 1 and 2 are clearly introducing $(\forall n.P(n))$ as an assumption, so $n$ is not in scope. 3 is confusing. 7 is most explicit, but 4-6 are fine. Other options to consider are simply "Suppose $P(n)$" or "Suppose $P(n)$, where $n$ is a natural number". You might not need to emphasize "arbitrary". – Karl Jan 21 '25 at 18:39
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    I agree with @Karl that the forms with "arbitrary" are probably best avoided. Another thing that must be avoided is writing the quantification phrases at both ends of the sentence: people often write sentences of the form "for all $x$, $P(x, y)$ holds, for some $y$", but that's ambiguous: it could mean either $\forall x\exists y(P(x, y))$ or $\exists y \forall x(P(x, y))$ and these are not equivalent in general. – Rob Arthan Jan 22 '25 at 01:57
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    ... also if "all" were changed to "some" in 1 and 2, then they would probably by OK-ish: it is common to read an existential assumption as introducing the bound variable for later use. Probably not best practice, but just about acceptable I think. – Rob Arthan Jan 22 '25 at 02:02
  • People also often write "for some" to also mean "for some particular", which translates to "there exists". So there is merit to being specific and saying "for some arbitrary", though of course the clarity is debatable. Generally agreeing with Rob. – Glenn Sun Jan 22 '25 at 03:46
  • I think your intuitions are spot on. Not sure if your intuitions translate to any precise 'rules', but given your attention to detail and again given your intuitions I think you'll do a fine job relating this to your students. In fact, why don't you give your students these exact 7 sentences and have them reflect on which ones they think make sense? Rather than telling students 'how it is' (boring!), have them share in the struggle to make sense of things, what works, what doesn't, what works better, etc. It's exactly these kinds of reflections that can be more engaging and thus pedagogical. – Bram28 Jan 22 '25 at 16:03
  • @ryang Yeah, #3 is weird ... not recommended :) – Bram28 Jan 23 '25 at 21:52

1 Answers1

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  1. Suppose that for all $n {\in} \mathbb{N}$, $P(n)$ is true.
    Then $Q(n)$ is true.
  2. Suppose that $P(n)$ is true for all $n {\in} \mathbb{N}.$
    Then $Q(n)$ is true.
  3. For all $n {\in} \mathbb{N},$ suppose that $P(n)$ is true.
    Then $Q(n)$ is true.

To me, for 1 and 2, the scope of $n$ is closed at the end of the first sentence, so it doesn't make sense to refer to $Q(n)$ afterwards.

All the above constructions translate to the open formula $$\big(\forall \color\red m{\in}\mathbb N\,P(\color\red m)\big)\implies Q(n),$$ whose bound occurrence of $n$ (here renamed to $\color\red m)$ is essentially a different variable from the consequent's copy of $n.$ This isn't your intended meaning.

  1. Suppose that for some arbitrary $n {\in} \mathbb{N}$, $P(n)$ is true.
    Then $Q(n)$ is true.
  2. Suppose that $P(n)$ is true for some arbitrary $n {\in} \mathbb{N}.$
    Then $Q(n)$ is true.
  3. For some arbitrary $n {\in} \mathbb{N},$ suppose that $P(n)$ is true.
    Then $Q(n)$ is true.

In these constructions, the wording "for some arbitrary" may be misunderstood by some readers as existential quantification.

  1. Let $n \in \mathbb{N}.$
    Suppose $P(n)$ is true. Then $Q(n)$ is true.

This construction most cleanly communicates your intended meaning that $$\forall n{\in}\mathbb N\,\big(P(n)\implies Q(n)\big).$$ Equivalently:

  • Let $n$ be an arbitrary natural number. Suppose that $P(n)$ is true. Then $Q(n)$ is true.
  • Take an arbitrary natural number $n.$ Suppose that $P(n)$ is true. Then $Q(n)$ is true.
  • Consider an arbitrary natural number $n$ such that $P(n)$ is true. Then $Q(n)$ is true.
ryang
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