I want to symbolize with the predicate calculus the sentence "There are exactly two people"
Let, P = x is a person.
Then the sentence would go:
$(\exists x)(\exists y)(((x \neq y) \land Px \land Py) \land (z)(Pz \rightarrow (x = z) \lor (y = z)))$
Would this change if I specified my universe of discourse as all people? Would I need the predicate letter P then?