# Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors

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## Abstract

Boundary algebra [BA] is a simpler notation for Spencer-Brown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms ‘()’ and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. 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. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. 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 normal forms and Quine’s (1982) truth value analysis. () ? true [false] yields boundary logic.

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File URL: http://www.econ.canterbury.ac.nz/RePEc/cbt/econwp/0702.pdf

## Bibliographic Info

Paper provided by University of Canterbury, Department of Economics and Finance in its series Working Papers in Economics with number 07/02.

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 Length: 97 pages Date of creation: 01 May 2007 Handle: RePEc:cbt:econwp:07/02 Contact details of provider: Postal: Private Bag 4800, Christchurch, New ZealandPhone: 64 3 369 3123 (Administrator)Fax: 64 3 364 2635Web page: http://www.econ.canterbury.ac.nzMore information through EDIRC

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