Interpretations of Laws of Form

The laws of calling and crossing reduce broad and deep structures in a manner consistent with logical arithmetic. Different laws produce other mathematical systems. Eliminating calling changes the system from logic to number, while eliminating crossing instead changes it to sets, as illustrated in the table below:

Among these four, the logical system has been most thoroughly studied and is described in some detail below. The number system requires additional distribution law but still runs into trouble; ways of dealing with it are also described below. Sets and multi-sets have not been fleshed out and are not covered here.

The logical arithmetic has a corresponding algebra. In Laws of Form, Spencer-Brown gives one axiomatization of this algebra but more interesting versions have been devised by Bricken and Kauffman.

Each of these systems support the following mapping from standard notation to boundary notation:

The boundary form of logic, called boundary logic, has a one to many mapping to standard notation. This collapse of many forms into a single one eliminates many of the transformations necessary in the traditional notation. For example, "a IFF b" is equivalent to "IF a OR b THEN a AND b".

Spencer-Brown's axiomatization has historical relevance due to its inclusion with the arithmetic in Laws of Form. He uses two axioms, position and transposition, as follows:

Bricken's version uses three axioms which are computationally convenient and conceptually clear. The first two axioms parallel the laws of the arithmetic, while the third represents the law of the excluded middle. Bricken gives different names to the axiom and to each direction of application of the axiom but here we will call them dominion, involution, and pervasion.

Unlike the logic, the numerical applications of Laws of Form use different structures as well as different axioms. The first approach holds true to the boundary structures of the logic while the other deviates from the simplicity of the form by introducing boundary types (i.e. distinctions among distinctions).

Spencer-Brown's approach was to restrict the law of calling and to introduce distribution as an axiom (in the logic, distribution is a theorem).

Kauffman and Engstrom have each tried to solve problems with Spencer-Brown's numbers. Kauffman (1996) modifies Spencer-Brown's version of the natural numbers by restricting null power to odd spaces and transfer to even spaces, thus eliminating the need for the ill-defined colon. Engstrom (1996) takes a similar tactic, restricting the application of transfer to prevent invalid distributions.

James's solution (1993), in contrast, is to introduce new boundary types. A subset of these types map onto the even-odd distinction offered by Kauffman, above, but extend the numerical representation well beyond the natural numbers. This algebra uses three types of boundaries: two that are each other's inverse and a third that acts as the spatial inverse. These three forms are sufficient to build forms that map onto real, complex, and transcendental numbers.

James's first four axioms mirror define logic when the boundary types are made identical. The inversion axiom carries the system beyond natural numbers but in doing so it introduces problems associated with infinity, division by zero, and phase redundancy.

Such an element leads to a contradiction with the real axiom set. We can relax Bricken's axiom set by replacing pervasion, which enforces the law of the excluded middle, with a distribution axiom or an occultation rule as in the following set:

Kauffman has shown a mapping between distinction laws and knot theory. In knot theory, string crossings draw a distinction:

Knot theory classifies equivalent knots with unravelling rules based on where these crossings occur. These rules are variations on calling, crossing, and distribution of the other forms.

Idempotency
Crossing
Distribution