Discovering boundary algebra: A simple notation for Boolean algebra and the truth functors

Boundary algebra ( BA ) consists of Spencer Brown's [ Laws of Form , 1969 (Allen & Unwin, London)] primary arithmetic ( PA ) and primary algebra ( pa ), restated here in a more convenient notation. The PA consists of an ordered set B ={( ), “ =(( ))} of primitive values cum atomic formulae, formulae generated recursively by enclosure between '(' and ')' and juxtaposition, logical equivalence denoted by '=', and the axioms ( )( )=( )= “ ( )=( “ )=( ) “ , (( ))= “ = “ “ . Inserting variables with domain B into a PA formula yields a pa formula. The pa is a d B ,(·)·, B ¢ algebra of type d 2,0,0 ¢ , whose [ pa , calculus of truth values (CTV), dual CTV, Boolean] intended interpretations are: [( ), T , F ,1], [ “ , F , T ,0], [ ab , a b , a # b , a ? b ], [ ( a ),¬ a , ¬ a , a ' ], and [( a ) b , a M b ,¬( b M a ), a h b ]. The pa is thus a bracket-free notation for the Boolean algebra 2 and CTV, embodying the expressive adequacy of { / # ,¬} and { M ,¬/ F } . ( a ) a =( v ) a =( v ) and abc = bca form a basis. The pa simplifies duality, normal forms, natural deduction, Quine's truth value analysis and the decision procedure for monadic predicates. Syntactically, BA resembles Peirce's [Peirce, C.S. (1993) Collected Papers of Charles Sanders Peirce (Harvard University Press, Cambridge, MA), §§4.372-384] and Byrne's ["Two brief formulations of Boolean algebra", Bulletin of the American Mathematical Society , 52 , 269-272] Boolean notation; semantically, BA resembles Peirce's entitative graphs (Shin [ The Iconic Logic of Peirce's Graphs , 2002 (MIT Press, Cambridge, MA)]) and the system of Anderson and Belnap ["A simple treatment of truth functions", Journal of Symbolic Logic , 24 (1959), 301-302].

Keywords:

• Boundary Algebra
• Boundary Logic
• Primary Algebra
• Primary Arithmetic
• G. Spencer Brown
• Truth Functors