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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?

Jordan
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    if the domain only consist of people (two of them) Px i always true , so you can remove them, but with Pj's it is better readable. (in principle the domain cointains everything) Good luck – Willemien Apr 27 '14 at 19:32

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