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) side-by-side 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 "side-by-side"-relation is modelled by the operator ","

and

(2) the "
one-inside-another"-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particular instance, or a subset of Multiple Form Logic, where some of the variables (in the Multiple Form Axioms) have been replaced by the special distinction "1".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", red-coloured 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)