10) Proofs of some Multiple Form Logic Theorems

 

Hyper-linked Index of Proofs:

·         Theorem T1)   A , B # A = A, B

·         Theorem T2)   A , A = A

·         Theorem T3)   (in the original paper of 1984…)

·         Theorem T4)   George Spencer Brown’s algebraic axiom J1 proved as a theorem

·         Theorem T5)   George Spencer Brown’s algebraic axiom J2 proved as a theorem

·         Theorem T5.1)  George Spencer Brown’s algebraic axiom J2 can be generalised

·         Theorem T6)   Multiple Form Logic™ is equivalent to a Boolean Algebra

·         Theorem T7)   Logic inferences are transitive

·         Theorem T8)   (the Boolean Multiple Form Logic equivalent forms of “AND”)…

·         Theorem T9)   (A # 1,B) = (A , B)#B#1, in Propositional Logic: A -> B

·         Theorem T10)   Huntington's Axiom is a theorem in Multiple Form Logic

·         Theorem T11)   A “generalised Huntington Formula” in Multiple Form Logic

·         The Three Axioms of Multiple Form Logic (for quick reference)

·         More Theorems (in another web page)

·         A bit-crunching theorem prover for Expert Systems (in another web page)

 

(dozens more to come, in the near future…)

 

NOTE:  Propositional Logic statements are proved by translation into Multiple Form Logic and re-translation into PCgiven the following table:
 

Propositional / Boolean Logic

Multiple Form Logic

Not (A)          

A # 1

A or B

A , B

A and B

(A # 1, B # 1) # 1

A = B

A # B # 1

A a B

A # 1, B

A xor B 

A # B 

 

Theorem T1:                        A , B # A = A, B

Proof:            Axiom 3 states that A, B # (A, C) = A, B # C.

Now, we can replace C by void (absence of form or distinction).


Therefore A, B # A = A, B.

Theorem T2:                        A , A = A

Proof:            We have already proved (Theorem T1) that: A, B # A = A, B.


Replacing B by void (absence of form or distinction):
A, A = A.

Theorem T3: (in the original paper of 1984):               

1 , X = X  (iff “1” is defined as “the Union of All Possible Forms”).

IMPORTANT NOTE 1 (and a “proof”):

This “ theorem” is in reality Axiom 1, in this presentation of Multiple Forms. However, in the older paper of 1984 (handed over to professor Cliff Jones of Manchester University) this formula was considered a “theorem”. Here is a “proof” of why this had happened:

Constant “1” was defined “constructively” as the “Union of All Possible Forms”, and was introduced much later, while the only assumed axioms were (today’s) Axiom 2 and Axiom 3. Then, the following “derivation” was made: -Suppose “1” is defined as1A, B, C, …(all possible forms). Suppose now that we examine 1 , X where X is any (particular) form. Then, since 1A, B, C…,X, …, i.e. the sequence on the right-hand-side of this, already contains X (by definition, since “1” was constructed to contain any form once),1, XA,B,C…X, X,…. Now (by theorem T2X,X=X) we can cancel out one of the two X’s in the right-hand side of *this, to get: A,B,…,X, …, which is (again): 1Therefore: 1, X=1 -“proved” now, as a “theorem”! ;)

 

NOTE 2:         From an A.I. programmer’s point of view, it is not so important if we see this as a theorem (T3), or an axiom (axiom 1) if our aim is to prove theorems automatically. However, philosophically speaking, I prefer the older (1984) formulation, since it assumes less and creates more. Furthermore, recently (after writing the first part of this presentation) I browsed Eddie Oshins’s site about Quantum Psychology, and speculate whether axiom 1 should (again) be discardedas an axiom” and kept only as a theorem (T3). In this case, perhaps we can use the other two axioms, creating a “Quantum Logic” of some kind, compatible with Quantum Psychology, e.g. perhaps through some additional axioms or constructions. (I am curious for Eddie Oshins’s comments about all this (in the LoF forum), and will shortly provide a hyperlink to such comments (here), if he does bother to comment).

Theorem T4: George Spencer Brown’s algebraic axiom J1proved as a theorem:

George Spencer Brown’s First “Algebraic Initial” (J1) of the “Primary Algebra” in “Laws of Form” is:

This corresponds precisely to the Multiple Form Logic theorem: ( p # 1, p ) 1 = (void)

Proof:            (p # 1, p) # 1(1, p ) # 1(by Theorem T1, applied to p#1,p)

1 # 1(by Axiom 1, or –if you prefer- Theorem T3)

void  (by Axiom 2void # X # X=void, applied to 1 # 1).

Theorem T5: George Spencer Brown’s algebraic axiom J2proved as a theorem:

George Spencer Brown’s Second “Algebraic Initial” (J2) of the “Primary Algebra” in “Laws of Form” is:

 

 

This corresponds precisely to the Multiple Form Logic theorem:

( ( p , r ) # 1 , ( q , r ) # 1 ) # 1 = ( p # 1, q # 1 ) # 1, r

Proof:  LHS ( (p , r) # 1, (q , r) # 1) # 1 , 1 # 1 (adding “1 # 1”, which is void, by Axiom 2)

( (p , r) # 1, (q , r) # 1) # 1 , ( 1 , 1 ) # 1 (changing “1” to “1 , 1”, by Axiom 2)

                        =( (p , r) # 1, (q , r) # 1) # 1 , ( r # 1 , 1) # 1 (changing “1” to “r # 1, 1”, by Axiom 3)

                        =( (p , r) # 1, (q , r) # 1) # 1 , ( r # 1 ,( (p , r) # 1, (q , r) # 1) # 1 # 1) # 1 (by Axiom 3)

                        =( (p , r) # 1, (q , r) # 1) # 1 , ( r # 1 ,( (p , r) # 1, (q , r) # 1) ) # 1 (by Axiom 2)

                        =( (p , r) # 1, (q , r) # 1) # 1 , ( r # 1,( (p , r # r # 1 ) # 1, ( q , r # r # 1) ) # 1 (by Axiom 3)

                        =( (p , r) # 1, (q , r) # 1) # 1 , ( r # 1,( (p , 1 ) # 1, ( q , 1) ) # 1 (by Axiom 2)

                        =( (p , r) # 1, (q , r) # 1) # 1 , ( r # 1,( # 1,# 1 ) ) # 1 (by Axiom 1)

( (p , r) # 1, (q , r) # 1) # 1 , ( r # 1 ) # 1 (by Axiom 2)

( (p , r) # 1, (q , r) # 1) # 1 ,r  = RHS (Q.E.D.)


We now demonstrate a 
generalised version of this “distributive law”:

Theorem T5.1: George Spencer Brown’s algebraic axiom J2 can be generalised:

If we replace “1” with “X” (an arbitrary form), we get a Generalised Distributive Law in Multiple Form Logic™:
                         

(((A , C) # X), ((B , C) # X)) # X = (A # X , B # X) # X , C

Now, is this formula valid in Multiple Form Logic? Well, yes, it appears to be valid iff we also assume Axiom 1. We shall prove it in two different ways. In the first method, we shall use the Boolean Substitution Rule:  “If a logic equation reduces to identical left-hand and right-hand sides, when substituting (1) a variable by “1” and (2) the same variable by “0”, then the equation is valid”. (A proof of this rule can be found in most Boolean Algebra textbooks). If we use the three axioms, the Multiple Form Logic system is “equivalent to a Boolean Algebra”, as proved in theorem T6. So, this rule is also valid in (the Boolean form of) Multiple Form Logic™. So it can be used in derivations, too:

            Proof(1): Examine the two complementary cases of C = 1, and C = [no form] (or “void”).

Case 0: If C=0 (or void), (A # X, B # X) # X=((A # X), (B # X)) # X (true).

Case 1: If C=1, (A # X, B # X) # X, 1=(((A, 1) # X), ((B, 1) # X)) # X

Now the left-hand side of this formula needs Axiom 1, to be reduced to “1”, i.e.
Z , 1 = 1. In this case, iff we assume Axiom 1, the 
right-hand side of the expression is:

(((A, 1) # X), ((B, 1) # X)) # X = ((1 # X), (1 # X)) # X (by Axiom 1, for A,1 and B,1) 

(1 # X)) # X (by Theorem T2, applied to (1#X),(1#X)(1#X))

1 # X # X1 (by Axiom 2 , which says that: A#X#X=A).

In the 2nd method we show 1)Left hand side a Right hand side, 2)Right hand side a Left hand side:

Proof (2a):  LHS => RHS, using the transliteration “a => b” to “a # 1 , b” (again, assuming Axiom 1):

(( ( ( A , C) # X) , ( ( B , C ) # X ) # X ) # 1 , ( A # X , B # X ) # X , C 

=(( ( A # X ) , ( B # X ) # X ) # 1 , ( A # X , B # X ) # X , C (by Axiom 3)

= (“True”)  (by Axiom 3, applied to “( A # X , B # X ) # X ”).

Proof (2b):  RHS => LHS, using the transliteration “a <= b” to “a , b # 1”:

( ( A , C) # X ) , ( ( B , C ) # X ) # X , ( ( A # X , B # X ) # X ,C ) # 1 

( ( A , C) # X) , ( ( B , C ) # X ) # X , ( ( ( A , C )# X , ( B , C ) # X ) # X ,C ) # 1 

( ( A , C) # X ) , ( ( B , C ) # X ) # X , C # 1 (by Axiom 3 for “( ( A , C )# X , ( B , C ) # X ) # X “)

( ( A , C, C # 1) # X ) , ( ( B , C , C # 1 ) # X ) # X , C # 1 (by Axiom 3, inserting “ C # 1”)

( ( A , 1) # X ) , ( ( B , 1 ) # X ) # X , C # 1  (by Axiom 3 and Axiom 1, for “C , C # 1”)

1 # X1 # X ) # X , C # 1 ( 1 # X ) # X , C # 1 1 , C # 11 (“True”)   (Q.E.D.)

 

On reflection, after seeing this, I speculate that perhaps the existence of “1” forces the distributive law to become valid, i.e. If we assume there is such a thing as “the All” (or God, or Allah, whatever you like to call “it”) then we enter the World of Classical Logic, where the distributive law holds. Whereas, if we avoid “the All”, ignoring it, or pretending that it does not exist, etc., then we get a kind of Quantum Logic, where the distributive law does not hold, and where childhood and… schizophrenia, Art (etc.), all become possible! Now, I do not wish to take advantage of Eddie Oshins’s precious time, who replies to so many students’ queries, but I feel strongly that it’s worth his time to deal with this. If this kind of speculation is correct, then we can combine Classical Logic and Quantum Logic in one system. If not, I still have work to do, as a programmer! ;)

Theorem T6: Multiple Form Logic is equivalent to a Boolean Algebra

NOTE:           This is provable only if we assume all three axioms to be true, having defined “1” as equal to “all Forms in the Universe”. If -on the other hand- if we do not assume axiom 1, then it remains to be seen what actually happens; In the latter case, the resulting system is not equivalent to a Boolean Algebra, but perhaps it is equivalent to a “Quantum Logic” of some kind (Dr. Oshins?)

Proof:          In a university textbook (“Set theory and Boolean Algebra”, pp. 254-258) it is stated that a definition for a Boolean Algebra is the following:

 

For an unspecified set B of at least two elements, a binary operation  in B, and a unary operation ’ in B, the following axioms define a Boolean Algebra:

B1.  is a commutative operation.


B2. 
is an associative operation.


B3.
For all a, b in B, if a  b’ = c  c’, for some c in B, then a  b = a.


B4. For all a, b in B, if a  b = a, then a  b’ = c  c’, for all c in B.

If we choose the operation <> to be <,> (“OR”), and the operation <> to be the result of a “XOR” with the Universal Distinction “1” (X X # 1), then:

B1 and B2: True “by definition” (see the “Primordial Theorem 1”).

B3: (Proof)   Let A , B # 1=1C , C # 1.

Then: A , BA, B # 1 # 1A,      (B # 1, A) # 1A , 1 # 1A.(QED)

B4: (Proof)   Let A , BA. Then A , B # 1A , B , B # 1A # 1 = 1.(QED)

The proof is complete.

Theorem T7: (propositional calculus) 

Logic inferences are transitive, i.e. if A => B and B => C, then A => C

Proof: (translating PC into MF( (A # 1 , B) # 1 , (B # 1 , C) # 1) # 1 ) # 1) , A # 1 , C =

( (A # 1 , B) # 1 , (B # 1 , C) # 1, A # 1, C (by axiom 2, applied to 1#1)

                        B # 1((B # 1) , C) # 1, A # 1, C (by axiom 3, applying A#1 to A#1,B)

                        (B# 1, C # 1, A # 1 , C (by axiom 3, applying B#1 to B#1,C)

                        (B# 1,1, A # 1 , C (by axiom 3, applying C to C#1)

                        1 (by axiom 1, applied to “… , 1 , …”)

 

                        I.e. it reduces to “True”, or “All possible forms”.

NOTE(1):           If you download and run the Prolog theorem-prover “mflogic.exe”, you will see similar proofs. This theorem is among many examples stored in a knowledge base. All the examples are shown in a menu, from which you can pick one, to see its proof. However, the proofs are not stored; Only theorems are stored. (I.e. what you see each time is a “fresh proof”). You can also type your own theorems (in another program option) to see other proofs -or bugs. ;)

NOTE(2):           If you look closely into the above proof, and into the one automatically generated by the program (mflogic.exe), you may discover some differences in proof steps followed. In fact, the program gives a slightly lengthier proof, since the present algorithm used searches “blindly” to apply the three axioms wherever possible, without using much intelligence. (Still, it does find the solution, even “blindly”).

NOTE(3):           I have sought a generalised version of this “transitive implication law” which holds in Multiple Form Logic™ and probably in some “Quantum Logic” as well. The generalised form of this, has an arbitrary form “X” instead of “1”, and becomes (by successive applications of Axiom 3):

 

A # X , B) # X , (B # X , C) # X , A # X , C = B # X , (B # X , C) # X , A # X , C

B # X , # X , A # X , C B # X ,X , A # X , CB , X , A , C =

X , A , B, C (which is the union of all the logic variables, used in the “law”).

Theorem T8: (the Boolean Multiple Form Logic™equivalent forms of “AND”):

( A # 1, B # 1 ) # 1 = ( A , B ) # A # B= (in Boolean Algebra:A&B )

Proof:            In this proof, we shall use the following Boolean Substitution Rule:

If a logic equation reduces to identical left-hand and right-hand sides for each case of substituting (1) a particular variable by “1”, and (2) the same variable by “0”(void), then the equation is valid. (A proof of this can be found in most Boolean Algebra textbooks). If we use the three axioms, then the Multiple Form Logic system is equivalent to a Boolean Algebra, as shown by theorem T6. So this rule is also valid in (a Boolean) Multiple Form Logic, and can be used in derivations:

Case (1): Let A = 1. 

LHS = (1 # 1, B # 1 ) # 1B # 1 # 1 (by axiom 2 applied to 1#1)


B (by axiom 2, applied to 1#1).

 

RHS = (1 , B ) # 1 # B1 # 1 # B (by axiom 1 applied to 1,B)

 

B (by axiom 2 applied to 1#1

Therefore: for the case A = 1, both LHS and RHS are the same.

 

Case (2): Let A = void.

LHS = ( 1 ,B # 1 ) # 11 # 1 (by axiom 1, applied to 1 , B#1)

void (by axiom 2, applied to 1#1).

RHS = ( B ) # B(void) (by axiom 2, applied to B#B)

void (by axiom 2, applied to 1#1).

Therefore: for the case A = void, LHS and RHS are again the same, so the proof is complete.

 

The importance of this theorem, philosophically as well as derivation-wise, is that we can express logical conjunctions (AND) without using “1” at all. Of course we have already used Axiom 1, in the above proof, and the equation proved is still Boolean. However, we slowly begin to realize that Multiple Form Logic is a superior system to Boolean Algebra, for more than one reason; and one (small) reason is this.

 

 

Theorem T9: -An interesting simple theorem about Logic Implication & Equality:

( A # 1, B) = ( A , B ) # B # 1 ( in Propositional Logic: A => B(A or B) = B )

Proof:            The Boolean “Substitution Rule” can be used, together with the Axioms:

Case of A = 1: LHS = (1 # 1, B)B, RHS = ( 1 , B ) # B # 11 # B # 1B (= LHS).

Case of A = 0: LHS = (1, B)1, RHS = ( B ) # B # 11 (= LHS). (Q.E.D.)

This theorem expresses an interesting idea of Philosophical importance: Logic Implication is an equation between (1) the Union of Assumptions and Consequences, and (2) the Consequences.  (Noting that A=B is A # B # 1 ).

 

Theorem T10: Huntington's Axiom is a theorem in Multiple Form Logic™

Huntington's Axiom, in George Spencer Brown's notation, is: ( ( A B ) ( A ( B ) ) ) = A

This axiom is important: E.g. Lou Kauffman uses it to prove all the other axioms of Propositional Calculus.

It translates to the Multiple Form Logic formula:

( (A , B) # 1, (A , B # 1) # 1 ) # 1 = A

Proof:  LHS    = ( ( A, B ) # 1, ( A , B # 1 ) # 1 ) # 1

( A, B ) # 1, ( A , B # ( A , B ) # 1 # 1 ) # 1 ) # 1 (by Axiom 3, insertion)

( ( A, B ) # 1, ( A , B # ( A , B ) # 1 # 1 ) # 1 ) # 1 (ready for Axiom 2...)

(( A, B ) # 1, ( , B # ( A, B ) ) # 1 ) # 1  (by Axiom 2 applied to "1 # 1")

(( A, B ) # 1, ( AB # B ) # 1 ) # 1 (by Axiom 3cancellation)

(( A, B ) # 1A# 1 ) # 1 (by Axiom 2, on "B # B")

( A, A # 1 , B ) # 1 A # 1 ) # 1 (by Axiom 3, insertion)

A, 1 , B ) # 1 A # 1 ) # 1 (by Axiom 3cancellation of "A", in "A # 1")

1 # 1 A # 1 ) # 1 (by Axiom 1, since "1" is in "A # 1 # C")

# 1 # 1 (by Axiom 2 on "1 # 1")

A  (by Axiom 2 on "1 # 1")

= RHS (Q.E.D.)

Theorem T11There is a “generalised Huntington Formula” in Multiple Form Logic

 

Like the "Generalised Distributive Law" (Theorem T5.1), there is a "generalised version” of Huntington's formula, where every occurrence of "1" has been changed to an arbitrary Form X. However, the Right hand side of this formula is slightly different now; It is not "A", but "A or (B or X) xor X".

As far as I know, this theorem is a kind of small novelty in Logic (like Theorem T5.1). Here it is:

( (A , B) # X, (A , B # X) # X ) # X = A, (B, X) # X

Proof (by the use of the "Boolean Substitution Rule"):

1) Case of X = 1:

LHS     = ( ( ( A, B ) # 1, ( A , B # 1 ) # 1 ) # 1 ) # 1A  (by Theorem T10, Huntington)

RHS    A, ( B , 1 ) # 1  = A , 1 #  1 (by Axiom 1 applied to "B,1" = "1")

A = LHS (by Axiom 2 applied to "1 # 1")

2) Case of X = void: LHS =( ( ( A, B ), ( A , B ) ) )A , B  (Omitting "X", and using Theorem T2, or Axiom 3)


RHS =A, ( B ) (Omitting "X") = LHS

 

The Three Axioms (for quick hyperlink reference):

 

(1)   Oneness 1 , X = 1  (All is One, and Union of Anything with “the All” is still “the All”).

 

(2) Reflection A # X # X = A (to distinguish the very fact of distinguishing, is no distinction)

 

(3) Perception A , X # ( A , B)=A , X # B (what is real we can imagine, but don’t need to imagine the real)

 

 

Next Section: A bit-crunching Expert System Deduction Algorithm

 

   Another relevant Section: More Theorems of Multiple Form Logic