How do I express this sentence in predicate logic, when I can pick the domain of discourse myself?
Friends of Michelle's friends are her friends.
I was thinking of picking the domain of discourse Michelle's friends:
So that I would get $\forall x (Fx \rightarrow x) $
$F =$ friends of
Is this correct?
How would I express this?
Thanks in advance, Rope.
After the back and forth with GitGud in the comments I got the following answer.
You're allowed to put forth multiple for-all variables (which I didn't consider before).
The domain of discourse has to be all People or all humans.
It follows that for the Friends of relation we can put $Fxy$ as x is a Friend of y. Hence we get:
$$\forall x \forall y((Fxm \wedge Fyx)\rightarrow Fym)$$
$Fxm$ All humans that are not y are friends of michelle.
$Fyx$ all humans that are not x are friends of friends of michelle(stated in the previous predicate)
$\rightarrow$ thus it is the case
$Fym$ that all friends of friends of michelle are friends of michelle