Laws of Form

First published in 1969 and currently out of print, The Laws of Form is a book by the British mathematician G. Spencer-Brown. The book, a formal math text, is a challenge for the non-mathematician to understand. This is unfortunate because the mathematics are not all that difficult. In addition, applications of the mathematics and the philosophical implications of the LAWS do not become apparent until the end of the book. In the second appendix the author shows how the LAWS OF FORM can be applied to problems of logic.

The author also made claims in the book that his system could be extended to solve many other problems such as the four color theorem. These claims were never fully realized and Spencer-Brown’s reputation, as well as the book’s went on the decline. However, the book does present, as Bertrand Russell said “a new calculus, of great power and simplicity. While ostensibly a book of formal mathematics, it became somewhat of a cult classic, mentioned in places like the WHOLE EARTH CATALOG. To some it represents “a mathematics of consciousness”.

Critics of the LAWS OF FORM often dismiss it because (as asserted in a paper by Banaschewski) the system can simply be reduced to a conventional logical system. Proponents counter that the beauty of the LAWS is that the conventional logical system can be reduced to the LAWS OF FORM -- a system with has only one symbol. In addition that singular symbol is seen as representing the foundation of consciousness and language.

The Form

The form is the essence of the LAWS OF FORM. It is represented by the symbol:

which is also called a CROSS. The form represents drawing a distinction. It can be thought of 3 ways:

The act of drawing a boundary around something and separating it from everything else
That which has been defined by drawing the boundary
Crossing from one side of the boundary to the other

All three of these implies the person who is making the distinction. As Spencer-Brown writes “the first distinction, the mark and the observer are not only interchangeable, but, in the form, identical”. Draw a distinction can “well be expressed in such ways as

Let there be a distinction
Find a distinction,
See a distinction,
Describe a distinction,
Define a distinction,

Or

Let a distinction be drawn.”

To use a biblical analogy, “Let there be light” is the same as

 “and there was light” –- the light itself
 “morning”
 “and evening” -- the boundaries of the light

Or

 “one day” -- the manifestation of the light.

The form is the observer, and the observed . It is the creative act of making an observation.

Along with the CROSS is the unmarked state. This is simply nothing. No distinction has been made and nothing has been crossed. The unmarked state is represented by a blank space. It is simply the absence of a cross.

The Two Axioms

1) The law of calling. If you make a distiction twice it is the same as doing it once. For instance, if you say “Let there be light” and then you say “Let there be light” again, it is the same as saying it once. This is represented using symbols as the equation:

		File:Image:Laws of Form - cross.gif File:Image:Laws of Form - cross.gif = File:Image:Laws of Form - cross.gif

2) The law of crossing. If you cross the boundaries of a distinction twice, you are back where you started and no distinction has been made. For instance if you cross from night to day, that is morning and there is light. If you cross back into night, there is no light. Using symbols this is represented as:

The Form as it Applies to Logic

A cross means that a distinction has been made and the absence of a cross means that no distinction has been made. In the biblical example there is light, or there is the void – the absence of light. This binary system also represent the Boolean values of 1 and 0, or TRUE and FALSE. Since the form also has the meaning of crossing the boundary of a distinction, it also means NOT. This may seem odd on the surface, however TRUE is equivalent to NOT FALSE and both TRUE and NOT FALSE are represented the same way -- with a cross.

			= TRUE  (or not False)

			= FALSE

 			=  NOT TRUE  =  FALSE

The Primary Algebra

Variables can be used to represent unknowns. It can have one of two states, It can represent a cross or it can represent the absence of a cross. Building on this the LAWS OF FORM derives simple rules for simplifying complex statements.

Since these two states represent TRUE and FALSE, Variables can be used to represent any statement who’s truth is unknown.

Thus:			represents NOT A.

	a b		represents A OR B .

	    b	represents   NOT A OR B which is the same as IF A THEN B.

	   	represents  NOT ( NOT A or NOT B ) 
			which is the same as  NOT ( IF A THEN NOT B) 
			which is the same as  A AND B.

The external links below has more detail about how the LAWS OF FORM relate to logic.

External Links

lawsofform.org

memory forms -- an overview of the LAWS OF FORM as it relates to Logic

pancakemath.org -- an adaptation of the LAWS OF FORM suitable for teaching to children