The Principle of Bivalence of Sets

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 μn, it is not the case that μ is a member of μn and it is not the case that μ is a non-member of μn. 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.

 

 

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