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Here's an example of a general problem I have sometimes.

As an example, lets take the theory of prosets (aka 'pre-ordered' sets). We begin with a transitive and reflexive relation $\leq$, and we extend our theory with new definitions which we can then prove propositions about. Begin by defining a new relation $\sim$ such that $x \sim y$ iff $x \leq y$ and $y \leq x$. We can then go ahead and prove:

Proposition 1. In the theory of prosets, TFAE.

  1. $x \leq y,$ and $x \not\sim y$
  2. $x \leq y,$ and $y \not\leq x$

Note that, what we're really trying to convey here is: for all $x$ and all $y$, statements 1 and 2 are equivalent. So more explicitly, we might have written the following.

Proposition 1'. In the theory of prosets, we have that for all $x$ and all $y$ TFAE.

  1. $x \leq y,$ and $x \not\sim y$
  2. $x \leq y,$ and $y \not\leq x$

Okay, now lets define that $x < y$ means that either 1 holds, or 2 holds (in which case both hold). With this further definition, we can prove:

Proposition 2. In the theory of prosets, TFAE.

  1. if $x \sim y$, then $x = y.$
  2. if $x \leq y,$ then either $x = y$ or $x < y$.

What we're trying to convey here is that these two ways of stating the antisymmetry principle are equivalent. So more explicitly, we might have written the following.

Proposition 2'. In the theory of prosets, TFAE.

  1. For all $x$ and all $y$, if $x \sim y$, then $x = y.$
  2. For all $x$ and all $y$, if $x \leq y$ and $x \neq y,$ then $x < y$.

Now here's the point. Notice that Propositions 1 and 2 are in the same format. However, if we compare Propositions 1' and 2', well the universal quantifiers are in different positions. So, this is ambiguous.

This general issue comes up fairly often in my writing. What's a good, systematic way of resolving the issue?

goblin GONE
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    I don't see this as ambiguous, only delicate. I think the problem is that I would not interpret Proposition 2 and Proposition 2' the same; I would read Prop. 2 as saying that "for all x and y, TFAE", versus the true statement, which you correctly stated in Proposition 2'. – Eric Tressler Jul 21 '13 at 16:17
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    So to summarize my thoughts, I would take your omission in Propositions 1 and 2 to mean what you were explicit about in Proposition 1', and never to mean what you intended in Proposition 2'. – Eric Tressler Jul 21 '13 at 16:18
  • I would instead say that Proposition 2 needs to change. Your lack of quantifiers implies to me that $x$ and $y$ are simply arbitrary elements. – Eric Tressler Jul 21 '13 at 16:20
  • @EricTressler, so in your opinion, it is the phrasing of Proposition 2 that needs to be changed? – goblin GONE Jul 21 '13 at 16:20
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    Yes; and it never hurts to be explicit, if you're concerned. – Eric Tressler Jul 21 '13 at 16:32

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