At Most Two Distinct Members of A The quantified predicate logic statement that describes at most two distinct members of A, where A, is some arbitrary set is:
$\forall$xyz( (Px $\land$ Py $\land$ Pz) $\Rightarrow$ (x=y $\lor$ x=z $\lor$ y=z) )
I can parse this quantified statement into three cases:


*

*There are No Members of A

*There is 1 Member of A

*There are 2 Members of A
In case 2, x must equal y, then in turn for any z candidate, z is either equal to x, or to y, but x=y.
(So all three "or" statements are true.)
In case 3, x does not equal y, therefore for any z, z will either be equal to x, or z will be equal to y. (The first "or" fails, but the the other two are true.)
In case 1 I don't see how the quantified predicate statement guarantees that no membership can exist, i.e. there are non such members of A. If we don't choose any x, y, z then there are no such x,y, or z in A and the quantified statement "short circuits"? How do I interpret this or reason about it?

In the case 1 (no elements) the antecedent $(Px ∧ Py ∧ Pz)$ is  is false and the implication is true.
