Appendix 1

From the notion that since a cut denies its content—i.e., what lies within the cut is severed from the rest of the sheet—Peirce derived his five “Alpha Rules of Transformation” (see Roberts 1973:40-45). The first rule is the Rule of Erasure (R1), “Any evenly enclosed graph may be erased,” written thus:

which is a counterpart to Spencer-Brown’s (1979:5) “form of cancellation” by way of his Axiom 2, “The value of a crossing made again is not the value of a crossing,” written:

In other words, in the Peirce equation, Q lies within two cuts, and if a cut denies its content, a double cut is the same as if there had been no cut.

Spencer-Brown’s equation read right to left yields what he calls “compensation,” which is comparable to Peirce’s Rule of the Double Cut (R5), “The double cut may be inserted around or removed (where it occurs) from any graph in any area,” such as:

Peirce’s Rule of Iteration (R3), “If a graph P occurs on the sheet of assertion or in a nest of cuts, it may be scribed on any area not part of P, which is contained by the set of all Ps,” can be written:

Since a cut denies its content, the Rule of Iteration is closely related to the Rule of Insertion (R2), “Any graph may be inscribed on any oddly enclosed area,” written:

That is, it makes no difference what graph is inscribed in the area in question, for that very graph is denied by the cut anyway. If read from right to left, the rules of iteration and insertion are comparable to Spencer-Brown’s (1979:5) “form of condensation” by application of his Axiom 1, “The value of a call made again is the value of a call,” which can be written:

In other words, P can be inscribed again or X can be inscribed alongside P in any area where P is already inscribed in alternately nested cuts, and the graph will remain the same as it was. Reading Spencer-Brown’s equation from right to left is “confirmation,” which is a counterpart to Peirce’s Rule of Deiteration (R4), “Any graph whose occurrence could be the result of iteration may be erased,” which is equivalent to reading the above graph representing the Rule of Iteration from right to left.