The universe is exhausted between them

LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York. (Philosophy)

The continuation of text review: I am half way through this text review.

Key symbols

≡df = Equivalence by definition
 : = Equal (s)
 ε = Epsilon and means is
⊃ = Is the same as
 ⊨ is Entails
˜ = Not
∃ = There exists
∃! = There exists
∴ = Therefore
· = Therefore
< = Is included
v = a logical inclusive disjunction (disjunction is the relationship between two distinct alternatives).
x = variable
. = Conjunction meaning And
0 = Null class
cls = Class
 int = Interpretation
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The author summarizes on page 193. In symbolic logic, every class includes itself; every class includes the sub-classes of the sub-classes; every class is included in a universe class and includes an empty class. (193). The universe class is not the same as the universe of discourse. In other words the universe of discourse contains the universe class. The universe class does not contain the universe of discourse. Every class has a complement and the class and its complement are mutually exclusive. The universe is exhausted between them. (193). In other words, every class has a non-class.

Langer explains that by definition, mutual inclusion is called equality the sum of two elements a and b, which is written a + b and the product is shown as a x b. (193). Together they are the whole system of classes. (193).

Any two classes have a sum and product, if they are mutually exclusive (In symbolic logic) the product is 0. (193).

I reason based on Langer in regard to every class:

0 + 0 = 0
0 x 0 = 0

Based on my interpretation of Langer, if we created a universe of discourse  with a universe class of those within the Kingdom of God, the non-class would be those not within the Kingdom of God. This would equate to all of humanity. Or in other words, 0 persons would not be within the universe class.

There exists at least one class 0 that for any class cr (Christians) 1, 0 is included in a. (a + b).

There is a class of no Christians, in this universe of discourse.

In other philosophical words:

In these classes, Langer never assumes a universe class where there is not a non-version within.