An empty class

VanDusen Botanical Garden 2017

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

An empty class

The continuation of 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

Previously

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. (170).
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Generalized System of Classes

This review has progressed where we are now at the point in the textbook where philosopher, Langer explains that we have passed from a system of individuals and predicates, such as a class of white houses (wt) and a class of brick houses (bk). (171).

This leads to a system of certain classes < = Is included as in houses = white houses and brick houses. (171). Etcetera, including red houses (rd), green houses (gn), wood houses (wd).

This means that in any universe whose elements are classes there is one class having the logical properties of 'the class of no houses'. (172). This is also known as an empty class, and this class is included in every class of the universe. (172).

Langer explains that in each universe there is one 'greatest class' which is analogous to 'the class of all houses'. (172-173). This includes every class is the universe. (173). Langer means in this context, the universe of discourse for symbolic logic.

Therefore, for any class, there is at least one class 0 included.
Therefore, for any class, there is at least one class 1 included.

(∃0) (a) : 0 < a
There exists at least one class 0 that for any class a, 0 is included in a. (173).

(∃1) (a) : 0 < a
There exists at least one class 1 that for any class a, 1 is included in a. (173).

0 represents there is a class of no houses in this universe of discourse.
1 represents there is a class of houses in this universe of discourse.

This specific system. (173).

For any Universe of discourse, such as K (houses) whose elements are classes contains a 0 and a 1.(173). There are houses and non-houses. There are Christians and non-Christians, there are Canadians and non-Canadians, etcetera.

(∃!) (cr)  : 0 < cr

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

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

(∃!) (cr)  : 1< cr

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

There is a class of Christians, in this universe of discourse.