$(\forall x)~[Mx \to (\forall y)~(My \to Kxy)]$ where M = "is a man" and K = ".. killed .."

Could x and y be the same man or since I used two different variables they have to be different?

I'm studying logic, and my book says that you have to symbolize statements of the form: "There are at least two students" Like this:

$(\exists x)~(Sx \land \exists y~(Sy\land x\ne y))$

So in this case it specifies that x and y are not the same. In the former case instead, since it's not specified, i was wondering if x and y could be the same thing.

Thank you!

  • 2
    $\begingroup$ Yes; "for all" means for all. $\endgroup$ Jan 21, 2020 at 14:28

1 Answer 1


Yes, distinct variables can be substituted with the same name in first order logic. So that sentence would translate as 'every man killed every man', where that entails that every man also killed themselves. (What an unpleasant example.)

  • $\begingroup$ Thanks for you answer! I understand now. One more question to check if i got it, (∀x)(Mx->(∃y)(Mx & Kxy)) in this case also they can be the same right? $\endgroup$
    – Abcd
    Jan 21, 2020 at 14:47
  • $\begingroup$ @Abcd The second Mx was probably intended to be My, but in any case yes, x and y can have the same value. $\endgroup$ Jan 21, 2020 at 14:50
  • $\begingroup$ Yeah I meant "y" my bad. One more doubt though (in sorry) even if both were existential quantifiers they could be the same yeah? $\endgroup$
    – Abcd
    Jan 21, 2020 at 15:55

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