Discovering boundary algebra: A simple notation for Boolean algebra and the truth functors
Philip G. Meguire · International Journal of General Systems · 2003
Boundary algebra is a new and simple notation for 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 boundary algebra [BA], a simpler notation for Spencer-Brown’s (1969) primary algebra [pa]. The BA axioms are “()()=()”, and “(()) [=⊥] may be written or erased at will.” Repeated application of these axioms to a PA formula yields a member of B={(),⊥}, its simplification. If (a)b [dually (a(b))] ⇔ a≤b, then ⊥≤() [()≤⊥] follows trivially, so that B is a poset. BA is a self-dual notation for the Boolean algebra 2: (a) ⇔ a′, () ⇔ 1 [0] so that B is the carrier for 2, and ab ⇔ a∪b [a∩b]. The basis abc=bca (Dilworth 1938), a(ab) = a(b) (Bricken 2002), and a(a)=() 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.