In line segment notation, visual proofs in calculus of expressions
that SpencerBrown calls the primary arithmetics look like:
By denoting Øp by , and pÙq by pq, working in line segment notation, with squarefree polynomials, and interpreting unmarked state as 0, various tautologies are easy to prove. For example,
The proposed method is very powerful even in direct proofs of more complex tautologies, e.g., a crosstransposition The left side gives:
The right side results in:
so this proves the tautology.
Duality (De Morgan laws) holds for Ù and Ú, after
replacing 0 by 1 and vice versa, and each variable p by
its inverse . For example, a dual tautology for the crosstransposition
can be proved in the same way, working with =1p and pÚq=p+qpq. All proofs and calculations hold in the extended continuous set [0,1], i.e., in the polyvalent Boolean logic. In order to simplify, we can calculate squarefree polynomials corresponding to other logic operations and work with them in the same way as before. For example, pÞ q = 1  p + pq, pÛ q=1  p  q + 2pq, pq = p+q2pq, etc. The most interesting for a future use in logical gates can be the Sheffer stroke NAND, pq=[`pq], that can be algebraically expressed as pq=1pq. Operations (Ø, Ù), or (Ø, Ú) are a base of Boolean polyvalent logic: it is possible to express all other logical operations by them. Sheffer operation NAND ( or in his original notation) and its dual, Lukasiewicz's operation NOR (¯ ), are singleoperation logical bases.
After reducing a complicated logical expression and obtaining
corresponding squarefree polynomial, we can be interested to find
from it a simplified logical expression. For this, it is
sufficient to take all possible values for its variables that
belong to the discrete set {0,1} and write the corresponding
conjuctive or disjunctive normal form (CNF or DNF).
For example, for the reduced squarefree polynomial 1p+pq, by
taking for (p,q) values from the set
{(0,0),(0,1),(1,0),(1,1)} we obtain DNF ØpÚq.
