There’s a Principle of Bivalence that applies to anything relative to any set. It goes:

For anything μ, and for any set *S*, μ is either a member of *S* or else μ is not a member of *S*, and μ is not both a member and not a member of *S*.

In set notation, it appears as follows: that for anything μ, and for any set *S*:

((μ ∈ *S*) v (μ ∉ *S*)) & ~((μ ∈ *S*) & (μ ∉ *S*)).

This is what I call * The Principle of Bivalence of Sets*. Notice how this formulation and all its implications are of the same form as those of The Principle of Bivalence. For instance, if it’s not the case that μ is a member of

*S*, then μ is not a member of

*S*(and vice versa), and if it’s not the case that μ is not a member of

*S*, then μ is a member of

*S*(and vice versa). That is, in set notation:

~(μ ∈ *S*) <–> (μ ∉ *S*) and ~(μ ∉ *S*) <–> (μ ∈ *S*).

Recall from my previous post on How Do Sets Have Members? that things are members of sets by either being named by the set or else by the set’s providing a property that the thing has.

Such ways presume that the set will either name or provide a property. If the thing doesn’t have the name, it won’t be a member of the set; and if the thing doesn’t have the property, it won’t be a member of a set.

If this is right, what are we to say about things in relation to things that aren’t sets? For example, is a cup a member of a chair (cup ∈ *chair*)? No. But is it therefore not a member of a set, in the same sense as above, of following the Principle of Bivalence of Sets? That is, is it because a cup has some other name or property than *the one that is provided by a chair*? Obviously not, since a chair gives no name or property. Thus, just because a cup is not a member of a cup, it will not follow that it is not a member in the sense that it has a different name or property than the one a chair gives. The lesson is that being not a member in the sense of failing to be a thing named or failing to have a property specified is a positive attribute, one that follows from the negation of membership in relation to a set, since sets either name or give a property, but one that cannot follow from the negation of membership of something in relation to a non-set, since by definition no name or property is given. Let’s call this positive property of not being a member for failing to have the name or property already provided, the property of non-membership. This is what was denoted by ∉ all along. The fact that a chair specifies no name or property at all can now be accounted for in the following way:

For anything μ, and for anything that is not a set μ_{n}, it is not the case that μ is a member of μ_{n} and it is not the case that μ is a non-member of μ_{n}. That is,

~(μ ∈ *S*) & ~(μ ∉ *S*).

With things in relation to non-sets, the following implications hold:

~((μ ∈ *S*) v (μ ∉ *S*))

*This clearly shows that what we are negating is precisely the Principle of Bivalence of sets.*

~(~(μ ∈ *S*) -> (μ ∉ *S*))

and

~(~(μ ∉ *S*) -> (μ ∈ *S*)).

Let’s call this principle * the Principle Non-Sets*. Notice how all the above follows beautifully the Principle of Bivalence and then the Principle of Nonsense, both of which you can refer to in The Principle of Bivalence.