A collection of essential equations from LoF.

A pdf version is available.

$\require{enclose}$ $ \newcommand{\CR}[1]{\enclose{actuarial}{#1}} \newcommand{\st}[1]{\small{\text{#1}}} \newcommand{\v}[1]{\vphantom{#1}} \newcommand{\b}[1]{\vphantom{#1}\hphantom{#1}} $

This is an equation reference for other Laws of Form posts. For axioms and consequences, we use Spencer-Brown’s original labels. For most other equations, we use variations on the following transformations (where we simplify every $\CR{\CR{x}}$ to $x$):

  1. Let the complement of a form $p$ be the crossed form $\CR{p}$. Then the complement of an equation $E$, denoted by $En$, equates the complements of the LHS and RHS of $E$. For example, if $E$ is $p=q$, then $En$ is $\CR{p}=\CR{q}$. Since $Enn=E$, $E \leftrightarrow En$.
  2. The contradual of a form results from crossing every variable. For example, the contradual of $\CR{p\,\CR{q}}$ is $\CR{\CR{p}\,q}$. The contradual of an equation $E$, denoted by $Ec$, equates the contraduals of the LHS and RHS of the equation. Since we are effectively substituting the same value for each instance of a given variable, $E \leftrightarrow Ec$.
  3. The dual of a form results from crossing every variable and the entire form (i.e., a combination of contradual and complement). The dual of an equation $E$ is denoted by $Ed$. As above, $E \leftrightarrow Ed$.

Remark. All three transformations are involutions: whether $\theta=n,c,$ or $d$, $E\theta\theta=E$. Together with the identity transformation, these form a Klein 4-group. Among the transformations, duals are particularly significant — interpreted logically, $Ed$ means the same as $E$ with the truth values reversed. Occasionaly, the same equation can be the result of more than one transformation. In such cases, we make an arbitrary choice for the label.

We may also resort to selective duals and contraduals, where not every variable is crossed. An equation will typically have multiple selective [contra]duals.

\[\begin{array}{l l} \textbf{Axioms} & \\[1ex] \CR{\CR{a}\,a} = & \st{(Position J1)} \\ \CR{\CR{ac\v{b}}\;\CR{bc}} = \CR{\CR{a\v{b}}\;\CR{b}}\,c & \st{(Transposition J2)} \\ & \\ \textbf{Consequences} & \\[1ex] \CR{\CR{a}} = a & \st{(Reflexion C1)} \\ \CR{a b} \,b = \CR{a\v{b}}\, b & \st{(Generation C2)}\\ \CR{\b{a}}\, a = \CR{\b{a}} & \st{(Integration C3)}\\ \CR{\CR{a}\,b}\, a = a & \st{(Occultation C4)}\\ aa = a & \st{(Iteration C5)}\\ \CR{\CR{a\v{b}}\;\CR{b}}\;\CR{\CR{a\v{b}}\,b} = a & \st{(Extension C6)}\\ \CR{\CR{\CR{a}\,b}\,c} = \CR{ac\v{b}}\; \CR{\CR{b}\,c} & \st{(Echelon C7)}\\ \CR{\CR{a\v{b}}\; \CR{b r}\; \CR{c r\v{b}}} = \CR{\CR{a\v{b}} \;\CR{b}\; \CR{c\v{b}}}\; \CR{\CR{a\v{b}}\; \CR{r\v{b}}} & \st{(Modified transposition C8)}\\ \CR{\CR{\CR{a\v{b}}\; \CR{r\v{b}}} \;\CR{\CR{b}\; \CR{r\v{b}}} \;\CR{\CR{x\v{b}}\, r} \;\CR{\CR{y\v{b}}\, r}} = \CR{\CR{r}\, a b}\; \CR{r x y} & \st{(Crosstransposition C9)}\\ & \\ \textbf{Corollaries} & \\[1ex] \CR{a}\, a = \CR{\b{a}} & \st{(J1d)}\\ \CR{\CR{a}\, a b} = & \st{(J1.1 generalization of J1)}\\ \CR{a c\v{b}}\; \CR{b c} = \CR{\CR{\CR{a\v{b}}\; \CR{b}}\, c} & \st{(J2n)}\\ \CR{\CR{a\v{b}} \,c\v{b}}\; \CR{\CR{b}\, c} = \CR{\CR{a b}\, c} & \st{(Combination K5)}\\ \CR{\CR{a\v{b}}\,\CR{b}}\;\CR{\CR{c\v{b}}\,\CR{d}} = \CR{\CR{a c \v{b}} \,\CR{a d}\, \CR{b c}\, \CR{b d}} & \st{(Distribution K9)}\\ \CR{\CR{\b{a}}}\, a = a & \st{(Meguire B2)}\\ \CR{\CR{\b{a}}\, a} = & \st{(C3n)}\\ \CR{a b} \;\CR{a\v{b}} = \CR{a\v{b}} & \st{(C4c)}\\ \CR{\CR{a}\, b} \;\CR{a b} = \CR{b} & \st{(C6c)}\\ \CR{\CR{\CR{a}\, b} \;\CR{a b}} = b & \st{(Robbins C6d)}\\ \CR{\CR{a\v{b}}\; \CR{b r}} = \CR{\CR{a\v{b}}\; \CR{b}}\; \CR{\CR{a\v{b}}\; \CR{r\v{b}}} & \st{(C8.1 special case of C8)}\\ \CR{\CR{\CR{a}\; \CR{r}} \;\CR{\CR{x} \,r}} = \CR{\CR{r}\, a}\; \CR{r x} & \st{(C9.1 special case of C9)}\\ & \\ \textbf{General theorems} & \\[1ex] \CR{\CR{a_1r}\; \CR{a_2r} \dots \CR{a_n r}} = \CR{\CR{a_1} \;\CR{a_2} \dots \CR{a_n}} \,r & \st{(J2*)}\\ \CR{a_1 r}\; \CR{a_2 r} \dots \CR{a_n r} = \CR{\CR{\CR{a_1}\; \CR{a_2}\dots \CR{a_n}}\, r} & \st{(J2n*)}\\ \CR{\CR{\CR{\CR{a_n b}\dots }\, a_2}\, a_1}\, b = \CR{\CR{\CR{\CR{a_n}\dots }\, a_2}\, a_1}\, b & \st{(C2*)}\\ \CR{\CR{a\v{b_1}}\;\CR{b_1r}\;\CR{b_2r}\dots \CR{b_nr}} = \CR{\CR{a\v{b_1}}\;\CR{b_1}\;\CR{b_2}\dots \CR{b_n}}\;\CR{\CR{a\v{b_1}}\;\CR{r\v{b_1}}} & \st{(C8*)}\\ \CR{\CR{\CR{a_1}\; \CR{r}}\; \CR{\CR{a_2}\; \CR{r}}\dots \CR{\CR{a_n} \;\CR{r}} \; \CR{\CR{x_1}\, r}\; \CR{\CR{x_2}\, r}\dots \CR{\CR{x_m}\, r}} & \\ \quad = \CR{\CR{r}\, a_1 a_2\dots a_n}\; \CR{r x_1 x_2 \dots x_m} & \st{(C9*)}\\[1.5ex] \small{\text{For all even $n ≥ 2$,}} & \\ \CR{\CR{\CR{\CR{a_n}\dots }\, a_2}\, a_1} = \CR{\CR{a_n}\, a_{n-1}\dots a_3 a_1}\dots \CR{\CR{a_4}\, a_3 a_1}\;\CR{\CR{a_2}\, a_1} & \st{(C7.1*)}\\[-1ex] \small{\text{and,}} & \\ \CR{\CR{\CR{\CR{\CR{a_{n+1}}\, a_n}\dots }\, a_2}\, a_1} & \\ \quad = \CR{a_{n+1} a_{n-1}\dots a_3 \,a_1}\; \CR{\CR{a_n}\, a_{n-1}\dots a_3 a_1}\dots \CR{\CR{a_4}\, a_3 a_1}\;\CR{\CR{a_2}\, a_1} & \st{(C7.2*)}\\ \end{array}\]


Notes:

  1. Proofs of the general theorems are given in this post.
  2. Corollary B2 is from Meguire’s Boundary Algebra: A Simpler Approach to Boolean Algebra and Sentential Logic (2020) preprint.
  3. Corollaries K5 and K9 are from Kauffman and Varela’s Form Dynamics (1980) J. Social Bio. Struct.
  4. This post was updated in September 2021.