A well-established system of logic is essential

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

In Chapter IX 9: The Algebra of Logic

Philosopher Langer, explains that within symbolic logic an assumed  Universe of Discourse exists whose elements were certain with specific classes of A, B, C etcetera; this includes class I and a null class 0. (206).

For each (positive) class of A, B, C, etcetera, there is a negative class. (206). For every two classes (as described by Langer) there is a sum and a product. (206). Therefore what is a produced is what Langer names a calculus. (2006).

Facts about specific elements are expressed. (206). If these facts are not expressed then a correct calculus cannot be reached within a Universe of Discourse in symbolic logic. For example, if the sum of A and (plus) B were not equal to B (In other words, if A and B were not the same), then the product of A and -B could not be 0. (206). She reasons that we might instead calculate the sum of A x -B and B as the element of A + B. (206). Every structure is composed of elements. (49).

Notice, she is stating might...

This is rules of formal reasoning as opposed to necessarily truth. But, the truth there is requires logic and reason for correct presentation.

For arguments sake, reviewing Langer, if from her example the sum of A and  B did not produce the product of B, then perhaps the product in a Universe of Discourse would instead be C,or Z, etcetera, produced by multiplication. Instead of a general theory of classes, rather the properties of each class may be learned. (206). For me, Langer's example here lacks clarity.

What exactly does she mean by A x -B and B?

This textbook requires more footnotes with explanation.

A key to understanding any calculus, any calculations would be to have a well-established system within the Universe of Discourse. A well-established system of logic is essential in symbolic and also with the presentation of premise (s) and conclusions within academic disciplines.

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