From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188
Diamond Bracket Forms
and How to Count to Two
by Nathaniel Hellerstein
This paper extends G. Spencer Brown’s “Laws of Form” notation to four-valued “diamond” logic. Diamond logic is a wave-form logic that I adapted from Louis Kauffman’s work; it contains the paradox values “true but false” and “false but true”. These resolve the liar paradox and other paradoxes of self-reference.
This paper follows the practice of denoting Brown’s crossing symbol by parentheses; in this case brackets: [x]. I denote the marked form  as 1, and the unmarked “doublecross” form [] as 0; to these, add two new forms; 6 and 9. These forms follow laws similar to Brown’s form laws. These laws are complete; but proof of this is left as an exercise for the student. This paper gives a full proof that diamond logic supports self-reference. Any system of statements referring to each other in diamond logic has a lattice of solutions.
This paper ends by analyzing “modulators” – that is, circuits that count to two. Brown gave out one at the end of “Laws of Form”; Louis Kauffman gave another in his paper “Knot Automata”. Here I apply “diffraction” – an operation unique to diamond logic – and reveal both to be circular “rotor” circuits.