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?

P is a property that defines A.
For Case 1:
Objects x,y, and z exist in the Universe but since (Px∧Py∧Pz) was false, no candidate was held true for the property P that defines A thus their existence has no identity in A because they are not members of A; therefore, A is empty if the antecedent is false always -the consequent has no effect on defining membership in A. 
The consequent only ensures that IF there is a member in A for no matter how many candidates greater than or equal to one either they either all the same object or they are two distinct objects.
For Case 2:
(Px∧Py∧Pz) is true implying that it must be the case where at least one element is in A. The consequent checks to see what object in A it is- Since this is case 1 x=y and y=z and by transitivity of equivalence x=z. Therefore there is only one distinct member in A for case 1.
For Case 3:
(Px∧Py∧Pz) is true implying that it must be the case that two distinct elements are in A. The consequent checks to see what object(s) in A it is but in this case x $\neq$ y, so x=y check is irrelevant in the disjunctive statement so x and y are distinct and for any other candidate z checked it must be an alias for x or y.  
