$(\forall x)[Mx \to (\forall y)(My \to Kxy)]$, can $x\;\&\; y$ be the same thing?

Question

$(\forall x)~[Mx \to (\forall y)~(My \to Kxy)]$ 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!

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.)