13) More Theorems of
Multiple Form Logic^{™}
Theorem T12:
William Bricken's Calculus is a special
instance of Multiple
Form Logic.
Proof:
Observe that Bricken's
logic implicitly contains two ways of combining forms: (1) sidebyside and (2) one inside the
other.
Multiple Form Logic makes these two ways of
combining forms explicit:
(1)
"side by side" is a <,> operator,
and
(2)
"one inside another" is a <#> operator.
Furthermore, Multiple Form Logic has a special form, called
"1" (or "the All"), which has the property 1,X=1 (axiom 1).
To model William
Bricken's axioms in Multiple Form Logic, it is
sufficient to substitute "1" for every instance of "()", in
such a way that:
(1) the
"sidebyside"relation is modelled by the operator ","
and
(2) the "oneinsideanother"relation is modelled by the operator "#"
Given these definitions, the proof becomes almost
trivial:
1) Dominion: A , 1 = 1.
Proof: Axiom 1 of Multiple Form
Logic.
2) Involution: A # 1 # 1 = A.
Proof: Axiom 2 of Multiple Form
Logic: A # X # X = A,
and the special instance X
= 1.
3) Pervasion: A , 1 # ( A, B ) = A , 1 # B.
Proof: Axiom 3 of Multiple Form
Logic: A , X # (A, B) = A, X # B
and
the special instance X = 1.
Thus, William
Bricken's system of axioms
expresses a particular instance, or a subset of Multiple Form Logic, where
some of the variables (in the Multiple Form Axioms) have been replaced by the special
distinction "1".
To see the situation more clearly with graphics, here is the
difference between these two logic systems:
William Bricken's system: 

Multiple Form Logic: 

Dominion: 


Oneness: 

Involution: 

Reflection: 


Pervasion: 


Perception: 

In the above figures, it becomes visually evident that
Bricken's system is a restricted version
of Multiple
Form Logic: In Bricken's system, all the boundaries
are equal to a particular
constant (the red circle, in the diagram). In contrast, Multiple Form Logic has variables everywhere (except
in Axiom 1, where it uses the constant "1", redcoloured to show the correspondence with Bricken's system). However,
instead of the constant (red) distinctions, variable "X" (in Multiple From Logic) is a variable which can be an entire expression, just like any other variable, i.e. it constitutes (in relation to Bricken's system) a generalisation.
NOTE:
In Multiple Form Logic, distinctions are multiple, so a particular
boundary can take any colour or logic value. (It might even be an entire expression ).
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":
Theorem T13:
All the axioms of Propositional
Calculus are theorems in Multiple
Form Logic^{™}.
Proof: to be filled in.
(UNDER
CONSTRUCTION)
Hyperlinked References (for this section):
Next Section: Extending
Multiple Form Logic to other Boundary Logic Systems