Shape

193 Shapes in Algebras and Algebras in Rows
are combined, the results of these operations compare in the following way
(In the few places I use symbols to denote shapes in part I, they’re the letters S, M, and
P in accordance with William James’s description of the syllogism. But henceforth I
revert to my usual practice of taking letters from the beginning of the alphabet—in
particular, A, B, and C. It’s a habit that’s hard to break. Rules and expressions don’t
look right otherwise.)
Keeping to symbols—I’ll use A a B when the shape A is part of the shape B, and
I’ll use 0 to denote the empty shape—it’s easy to state the standard Boolean relationships.
A few of these are in table 6. Evidently, facts 8 and 9 show that product and symmetric
difference include both sum and difference. So starting with sum and difference
is arbitrary, at least for the purpose of defining Boolean operations. My second reason
for starting with sum and difference, however, isn’t so easy to dismiss. It has to do with
rules and how they work.
Algebras of shapes are perfect examples of Barzun’s preposterism—most formal
devices are. When I started playing around with shapes, I wanted to use rules to generate
them. I tried this in many ways drawing with pencil and paper, until I discovered
Table 6
A Few Boolean Facts
(1) A a B if and only if A þ B ¼ B Aa B if and only if A B ¼ A
(2) A a B if and only if A B ¼ 0
(3) A ¼ B if and only if A l B ¼ 0
(4) A þðB þ CÞ ¼ðA þ BÞþC AðB CÞ ¼ðA BÞC
(5) A þ B ¼ B þ A A B ¼ B A
(6) A þðB CÞ ¼ðA þ BÞðA þ CÞ A ðB þ CÞ ¼ðA BÞþðA CÞ
(7) C ðA þ BÞ ¼ðC AÞðC BÞ C ðA BÞ ¼ðC AÞþðC BÞ
(8) A þ B ¼ðA l BÞ l ðA BÞ
(9) A B ¼ A l ðA BÞ