Laws of Form

***Laws of Form*** is a book by George Spencer-Brown published in 1969.  It is about logic, mathematics, and philosophy.  The mathematical systems that Spencer-Brown presented in the book are known by the names "calculus of indications", "distinction calculus", and often just "LOF".

*Laws of Form* grew out of the author's work in electronic engineering.  The book has been published in several editions and translations and has never gone out of print. A short book, its mathematical part is only 55 pages long.

Spencer-Brown's philosophy was influenced by Ludwig Wittgenstein, R.D. Laing, Charles Sanders Peirce, Bertrand Russell, and Alfred North Whitehead.

## Reception
*Laws of Form* was listed in the *Whole Earth Catalog* in 1969 and quickly became a cult classic.  The calculus of indications and the primary algebra may be regarded as a way to think about a fundamental activity of the mind, namely, the ability to *distinguish* or to *draw distinctions*.  The book argues that this ability is the foundation of human cognition and consciousness.  According to Spencer-Brown, the primary arithmetic and primary algebra reveal new connections among logic, mathematics, the philosophy of language, and the philosophy of mind.

## Mathematical ideas
Let 0 and 1 be the two basic primitive values of Boolean algebra. Let *AB* denote a binary operation of Boolean algebra. Let (*X*) stand for the Boolean complement of *X*. Then the calculus of indications is simply Boolean arithmetic reduced to the two equations 11=1 and (1)=0. These are the only "axioms" in *LoF*.

The primary algebra is mainly a simpler notation for Boolean algebra, except for one thing. In Boolean algebra, () is not defined. () is "empty" complementation (the complementation of "nothing"). On the other hand, in the primary algebra () is defined, and stands for one of 0 or 1. (()) stands for the other primitive value, and is the same thing as the blank page.

Let *A* and *B* be any two expressions of the primary algebra. The primary algebra is made up of equations of the form *A*=*B*, and these equations are treated in the same way as the equations of the number algebra taught in all schools. Standard methods of logic rarely use equations. *LoF* argues that doing elementary logic with the primary algebra is easier. In particular, if *A* is a tautology in logic, then one of *A*=() or *A*=(()) holds in the primary algebra.

*Laws of Form* proves the following fact about the primary algebra:

* Cannot prove both *A*=*B* and *A*/=*B*. Hence the primary algebra is free of contradiction (is *consistent*);

* Can always prove whichever of  *A*=*B* and *A*/=*B* happens to be true. (The primary algebra is *complete*.)
Hence the primary algebra is a useful piece of mathematics. It can be useful even if the philosophy and cognitive science of *LoF* are wrong or uninteresting.

## Reference
* Spencer-Brown, George, 1997 (1969). *Laws of Form*. E. P. Dutton.

## Other websites
*Laws of Form Web Site
*Spencer-Brown Form : MathWorld
*Logical Graph  : PlanetMath
*Logical Graph : Commens Dictionary of Peirce's Terms
*Louis H. Kauffman, *Box Algebra, Boundary Mathematics, Logic, and Laws of Form*