# The BF Calculus and the Square Root of Negation

## TL;DR

This paper introduces the BF calculus, a four-valued logical system inspired by the concept of imaginary values, extending Laws of Form to represent complex-valued logic and operators.

## Contribution

It develops a novel four-valued algebra called BF, capable of representing imaginary values as both values and operators, and connects it to braid groups and quaternions.

## Key findings

- BF algebra is axiomatically complete and extends Laws of Form.
- BF can represent other four-valued systems like Kauffman/Varela and Belnap's algebra.
- Imaginary values are modeled using braid group representations.

## Abstract

The concept of imaginary logical values was introduced by Spencer-Brown in Laws of Form, in analogy to the square root of -1 in the complex numbers. In this paper, we develop a new approach to representing imaginary values. The resulting system, which we call BF, is a four-valued generalization of Laws of Form. Imaginary values in BF act as cyclic four-valued operators. The central characteristic of BF is its capacity to portray imaginary values as both values and as operators. We show that the BF algebra is a stronger, axiomatically complete extension to Laws of Form capable of representing other four-valued systems, including the Kauffman/Varela Waveform Algebra and Belnap's Four-Valued Bilattice. We conclude by showing a representation of imaginary values based on the Artin braid group, a representation of the braid group and a braided representation of the quaternions in this form.