Suppose I asked: "Does every X have property Y?"

Then I want to rephrase the question for the reader, in order to make use of some special terminology...

In other words, "Can an X exist without property Y?"

But, of course, even though I am essentially asking the same thing, these two questions are not the same and have different answers (yes for the first question and no for the second)

How can I introduce the second version? What is the relationship called?

I think you have answered your own question.

The best course of action is to just say "In other words ...". It implies that it's the same question and the answers to both questions actually mean one thing and it's easy to use.

Those two questions you asked might have different answers but only in wording because both answers mean the same thing. Or you could say Alternatively before the second question.

  • I like your "alternatively, ..." suggestion – Forever Mozart Nov 20 '17 at 18:41

I would call the second question the negation of the first. In symbols, the first question asks whether ∀X Y(X), and the second asks whether ∃X ¬Y(X). These statements are the logical negations of each other.

I'm ignoring your use of "can", which suggests the possibility–necessity distinction of modal logic, because I presume you don't mean to invoke that.

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