(a response to William Bricken’s paper on “the advantages of Boundary Logic...”)



George A. Stathis

23rd of July 2007


Multiple Form Logic is a generalized Boundary logic where the boundaries or containers are themselves Logic expressions. Thus the “nested containers” of Boundary Logic are themselves entire expressions (in Multiple Form Logic), expressing logic and computation in a new way.


Think of any physical container such as a glass, a box, or a room.  A container has an inside and an outside, but the container itself is a configuration of “insides” and “outsides” (and other containers, ad.inf).  In text, any delimiter is a container.  Eg:


Outside (  inside  )


        delimiter = ANOTHER expression (of insides and outsides)


William Bricken’s Boundary logic assumed that all Boolean logic is “simply configurations of nested containers”. The relationship of containment in Boundary Logic was said to “expresses everything about logical forms”:


(in Boundary Logic):

( ( ) )  two containers, one inside the other
( ) ( )     two containers in the same space

“Common sense example:  How many ways can a box and a glass be arranged?  They can be outside each other, or one can be inside the other.  The two ways they are arranged can be used as two different logic values”. (William Bricken’s description)


If we assume that there is only one truth value, “the Form” (as George Spencer Brown calls it), then the two arithmetic axioms of George Spencer Brown are (de-mystified as being) precisely identical to the Boolean relationships “OR” and “XOR”:


George Spencer-Brown:






Boundary Logic:


()()   =  ()



(( ))  =

Boolean Logic:

1 OR 1 = 1



1 XOR 1 = 0

Multiple Form Logic:


1 , 1 = 1



1 # 1 =


In Multiple Form Logic, as in George Spencer Brown’s and William Bricken’s logic, there is no need to use a special symbol, “0”, for the absence of form. An empty space is used instead (i.e. nothing on paper).



·        If we imagine George Spencer Brown’s system to be like a “conglomerate of boxes”, where each box can contain other boxes, then the corresponding visualization paradigm for Multiple Form Logic is more like a multi-dimensional system of “hyper-boxes”, where each “hyper-box” can contain other hyper-boxes, but at the same time it can represent another universe or constellation of hyper-boxes, containing other boxes, etc.


E.g. the Multiple Form Logic expression "A #(X#1,Y),B"  can be depicted as follows, with a (green) boundary around A, which is (in reality) an entire expression, "X#1,Y":

At this stage we will discuss the particular instance of only one Form, equivalent to Boolean Logic 1. In this case, Boundary logic was (correctly) said to be “much simpler than Boolean logic” since “one container is the same as two Boolean values. Boolean logic uses two different tokens or values, while boundary logic uses one container and two different spaces” (as William Bricken wrote). Well, the same argument is (more-or-less) valid in Multiple Form Logic as well, since -again- no special symbol is used for Logic Zero, which is written as an absence of form. However, there is also an important additional sense, in which Multiple Form Logic is even simpler than Boundary Logic: In the minimally simple depiction of Exclusive Or expressions, which in Boolean (as well as Boundary-) logic require a steeply increasing number of “indirect representations”, in terms of “and” and “not” (in Boolean Algebra) or in terms of composite expressions in Boundary Logic (also rising steeply in size); whereas in Multiple Form Logic, exclusive-OR expressions are of the lowest possible complexity, since XOR is an inherently embedded operator in this system. (In fact, XOR is also present as a fundamental element or building block in “Laws of Form”, but only implicitly, without… anyone realising this fact for an astonishing number of years, ever since “Laws of Form” was written - in 1969).


(to be continued)