Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors
Boundary algebra [BA] is a simpler notation for SpencerBrown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] is built up from the atoms, ‘()’ and the blank page, by enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction, and concatenation. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are “()()=()” (A1), and “(()) [=⊥] may be written or erased at will” (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),⊥} called its simplification. If (a)b [dually (a(b))] ⇔ a=b, then ⊥=() [()=⊥] follows trivially, so that B is a poset. (a) has two intended interpretations: (a) ⇔ a' (Boolean algebra 2), and (a) ⇔ ~a (sentential logic). BA is a selfdual notation for 2: () ⇔ 1 [0] so that B is the carrier for 2, and ab ⇔ a ⋃ b [a ⋂ b]. The BA basis abc=bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies the usual normal forms and Quine’s (1982) truth value analysis. () ⇔ true [false] yields boundary logic.
Subjects
G. Spencer BrownCollections
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