The clash of universes?

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

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

Primitive concepts, terms and relations, within symbolic logic are not explained, but are simply 'taken for granted'. (167). These meanings are provided by interpretation only in the context provided. (167). The symbols for houses and related is an example as these symbolic interpretations. (167).

For clarity, philosopher, Langer writes that there is a new context assumed. (167). In this context the formal context has elements which are certain classes. (168).

Let us note that Langer adds another symbol: cls, which is the usual symbol for class.

From page 168:

K= int (interpreted) as class of houses
B = int (interpreted) as class of brick houses
W = int (interpreted) as class of white houses
-B = int (interpreted) as class of  not-brick houses
-W = int (interpreted) as class of not-white houses
B x W =int (interpreted) as class of white brick houses
---
0 = int (interpreted) as class of no houses
I = int (interpreted) as class of all houses

On page 170, Langer states that a very important point is that there is a difference between:

K = The universe of discourse (Is this context established by Langer)

&

I = The universe class

Langer warns against identifying the universe of discourse with the greatest class that is within it. (170). Langer explains that the error of equating the universe of discourse with the universe of class, was made by John Venn is his Symbolic Logic of 1881. Langer instead reasons that I does not equate with K, but rather I is an element within K. (170).

My equations

˜ (I ⊨ K)

The universe class does not entail the universe of discourse.

˜ (I ⊃ K)

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.