The BF Calculus and the Square Root of Negation

Louis H. Kauffman; Arthur M. Collings

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arXiv · 1905.12891

The BF Calculus and the Square Root of Negation

Louis H. Kauffman·Arthur M. Collings

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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.

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Louis H. Kauffman, Arthur M. Collings. 2019-05-30. The BF Calculus and the Square Root of Negation. https://arxiv.org/abs/1905.12891

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