Each needs the other to complete the universe

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

The Langer philosophy text review, continues. Some key symbols from the textbook:
≡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

Fundamentum Divisionis

Whenever there is a class formed within any universe of discourse, then every individual in that universe must either belong to the class, or not belong to it. (142). A class of two-storeyed houses has every house in it that is two-storeyed, or the house is not in that class. (142).

Further from the last review of the Langer text, every class creates a dichotomy, also known as a division in two for every class. (143).

(x ε N)
Variable is N.
Variable = N.

(x ε -N)
Variable is not N.
Variable does not equal = N.

These two classes have no members in common. (143). But, this universe of discourse is divided between them and are known as complementary classes. (143). Each needs the other to complete the universe. (143).

(x ε N) ˜ ⊃ (x ε -N)
Variable equals N is not the same as variable equals not N. (143).

N = Cat

(x ε N)

x equals cat

(x ε -N)

x does not equal cat

˜ means x is unfeline. (144).

This text is from different eras (1953) (1967) but this quote is interesting in today's era as well and demonstrates how the use of the English language evolves.

'Male and female are equally "positive" notions, but in a universe of bi-sexual organisms they are complements.' (144-145). If females are represented by B, then males is -B and vice-versa. (145). In an equation the second class is the negated (negative) class. (145).