- This new web-page will contain formal proofs of theorems in Multiple Form Logic, proposed by
other people, or dictated
by necessity.
Current list of theorem proofs:
1) Theorem
ART-1 (about the "#" operator)
2)
An algebraic proof of "Theorem 9"
(more to
be included)
LINKS
- (new blog-article) The
“Extended XOR Operator” as a Consistent interpretation of George
Spencer-Brown’s "distinction"
1) Theorem
ART-1 (about the "#" operator)
Recently, Mr. Art Collings expressed some
interesting doubts and objections about certain aspects of Multiple
Form Logic, particularly with regard to the "#" operator (interpreted
as logical "XOR"). E.g. he remarked:
"In
LoF, the typical way to express XOR is via the form ( ( A B
) ( ( A
) ( B ) ) ) [ which basically translates as 'A or B but not A
and B'.
It is easy to verify this expression corresponds to XOR]. In your
notation, the corresponding expression is ( ( A, B)
#1 , ( A#1 ,
B#1)#1 ) #1 . This being the case, we can easily prove
arithmetically
that the following theorem is true:
A#
B = ( ( A, B) #1 , ( A#1 , B#1)#1 ) #1
The
question then becomes whether you can demonstrate this as a consequence
in your algebra. GSB proves that LoF is complete by proving that any
theorem that is true in the arithmetic can be demonstrated as a
consequence in the Algebra. In order for your algebra to be complete in
the same sense as LoF, it must be the case that expression such as this
can be demonstrated as a consequence from your three axioms: (1. A,1 =
1 ; 2. A#A = 0; 3. A, (A, B)#X = A, B#X.
[ i.e., 'demonstrated',
as distinct from 'proved'! ]
ANSWER: Evidently, the
'instinctive guess that such a demonstration is not possible from the
axiom set" (etc.) is wrong.
Here is a complete
algebraic proof:
