Understanding what equates to zero

Vancouver

The book review continues

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

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 last review back on August 18th presented:

Here addition and multiplication are being expressed through commutation, as equal, therefore,

(a x b) + (a x c) is equated as (a x (b + c).

As confusing as this work from Langer can appear for me without prayerful consideration and serious concentration, I actually can make sense of this as I am learning. Using (a x b) + (a x c) and then (a x (b + c) Consider (3 x 3) + (3 x 3) equals 18 (3 x (3 + 3) equals 18

In earlier reviews, a, b, c were all equated.
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Today further, I document Langer for this November 20 entry...

She writes

Assume that 02 has the properties of 0. (213).

Langer then equates 02 with 01 (213), therefore by implication both equating to 0.

(a ). a + 02 = a (213).

a in brackets is therefore equal to a + 02 equals a.

In other words a+ 0 = a

It is according to philosopher Langer a simplified (from the much more complex she states) equation that presents 0 as unique. (214). She notes this as Theorem Ia. (213).

There is some clarification to why this is even valuable within philosophy and symbolic logic where she writes that...

'Most of our proofs, however, involve the elements 1 and 0, and require that there should be just one element such as 1 and just one such as 0. (212).

Therefore there is at least one class 1 and at least one class 0.

My examples from a previous entry

I mentioned in a previous entry in December 2017:

(∃!) (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.

There exists a (cr) Christian class, therefore there is a no Christian class.

Logically implied here is that Christians are a type of human being and not every human being.

(∃!) (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.

Again, at least Langer demonstrates that logic and non-contradiction is essential in philosophy.