(a response to
William
Bricken’s paper on “the advantages of Boundary Logic...”)
23^{rd}
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):
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)
Reference: http://multiforms.netfirms.com/more_theorems.html