Multiple Form Logic
™
by George A. Stathis
- an enhanced generalisation of
George Spencer Brown's
“Laws of Form”
simplifying and elucidating Formal Logic
-together with theorem proving software in Visual Prolog™ and LPA Win-Prolog™. This version
includes the enhanced bit-crunching algorithm “Iphigenia” (with unification encoding) , an unusual
job-seeking ad, and a “Primordial Proof” of the “Regularity Axiom” in axiomatic Set Theory.
Note (2018): This document’s version was 1.44a in 2003, revived in July 2018, 15 years later!
Although Netfirms (the internet-host of this work) no longer exists and the Multiple Form Logic site
is down (http://multiforms.netfirms.com), it was saved in the Internet Archive, more-or-less intact:
•
https://web.archive.org/web/20110714165030/http://multiforms.netfirms.com/
Now, if you have seen the site, please bear in mind that this document preceded the site, where
additional files and improvements were added later from time to time (till about 2007) so this
document does not include those changes. However, newer versions of this document (after July
2018) will contain these (and other) enhancements...
To be honest, despite its many advantages (and many truth-values)... Multiple Form Logic never
really went beyond zero-order- (propositional) Logic! Years of effort to derive a Predicate Calculus
from it, produced no result. I found this disheartening, and also had ther things to do, so in recent
years I abandoned this work. However, my interest in it was revived after discovering that (in 2016)
someone else (Mr. George Burnett-Stuart) did a remarkable derivation of Predicate Calculus
from G. Spencer Brown’s “Laws of Form”. Now, since Multiple Form Logic is an extension and
a generalization of Spencer-Brown’s Logic, it follows that a Predicate Calculus can also be derived
within MF Logic, in a manner probably similar to George Burnett-Stuart’s, so I’m working on it...
-George Stathis, Athens July 2018
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Multiple Form Logic™
by George A. Stathis © 2003-2018
An enhanced generalisation of George Spencer Brown's
"Laws of Form"
simplifying and elucidating Propositional Logic
Together with theorem proving software in Visual Prolog™ and LPA Win-Prolog™
Web site: http://multiforms.netfirms.com
CONTENTS
(1) "Laws of Form" and the unknown history of Multiple Form Logic
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(2) The Three Fundamental Axioms of Multiple Form Logic
(3) The only Logic Operators we really need, are "OR" and "XOR"!
(4) Multiple Forms change the Philosophical Semantics of Logic Implication
(5) A Prolog Theorem Prover, simplifying Logic by using Multiple Forms™
(6) Download and run the Prolog Theorem-Proving Program
(7) The (Logic) Truth is Out There. Right in front of your eyes! ;-)
(8) What about Predicate Calculus?
(9) So, where does the alleged "efficiency" of Multiple Forms come from?
(10) Theorem Proofs in Multiple Form Logic
(11) Algorithm “Iphigenia” for Bit-Crunching Expert System Derivations
(12) Some Philosophical Aspects of Set Theory and Multiple Form Logic
(13) The Axiom of Regularity and some hints of Multiple Forms replacing Sets
(14) More Theorem Proofs (about William Bricken’s system, etc)
(15) Extending Multiple Form Logic to Other Systems of Boundary Logic
(16) Some philosophical Aspects of Multiplicity in Multiple Form Logic
(17) Revealing the secret story and the agony of Multiple Form Logic
(18) Extending and improving the Bit-Crunching Algorithm “Iphigenia”
(19) Elementary Variables and Unifications in Multiple Form Logic
APPENDIX A: Self-Biographic Humour (but all hyperlinks are serious)
APPENDIX B: "Laws of Form" (Oil Painting on Canvas)
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(1) "Laws of Form" and the unknown history of Multiple Form Logic™
George Spencer-Brown's "Laws of Form" is a revolutionary book about Logic, which influenced many
researchers and artists in the world, for about three decades. First published in 1969, "Laws of Form"
expounded a new philosophical approach to the theory of Logic, deeply challenging for the foundations of
modern Formal Logic. There are many sites in the Net about "Laws of Form", despite the fact that George
Spencer Brown does not have a site (or much sympathy for what he thinks as unnecessary publicity about
him). Many excellent studies, essays and formal extensions of Brown’s Logic have been published ever since,
in print as well as on line, by Richard Shoup, Dave Keenan, Tom McFarlane, Eddie Oshins, William Bricken, Lou
Kauffmann, Natalia Petrova, Jeff James, Francisco Varela, and others.
The "Laws of Multiple Form" (or Multiple Form Logic™ or "Calculus of Multiple Distinctions") is a Logic
Calculus that resembles Bricken's modifications of "Laws of Form", but it is much more generalised. I created
it many years ago (1982/83). To the best of my knowledge, having consulted Internet search engines about this
issue repeatedly, nobody else re-created a formal logic system identical to Multiple Form Logic™.
I studied and contemplated "Laws of Form" for many years, partly because I found it fascinating, and partly
because of finding certain technical aspects of it curiously annoying. In 1983, I came up with a new Logic
Calculus, an extension and a generalisation of George Spencer Brown's. I called this new calculus "The Laws
of Multiple Form", wrote (in the summer of 1984) an "introductory essay” about it, and sent it to the University
of Manchester (U.M.I.S.T), and other universities. As a result, I was offered a place for Postgraduate studies in
Computer Science and Logic, without having a first degree. This was an exceptional offer, made under
exceptional circumstances by an exceptionally kind person: Professor Cliff Jones, the head of Manchester
University's Computer Science Department, who persuaded the university's Admissions Committee that this
research was worth a Postgraduate Diploma, even without the prerequisite of a first degree (B.Sc.).
Note (2018): Unfortunately, for personal reasons (surviving a severe beating-up by fascists) I was unable to travel to the
UK at the time (early 1985), so this chance of an M.Sc. was lost, since in the following academic year Professor Jones
was unable to repeat his offer. However, our correspondence has been saved and may be publicized in the near future.
(2) The Three Fundamental Axioms of Multiple Form Logic™
George Spencer Brown based “Laws of Form” on two axioms of a “Primary Arithmetic”:
I1
or
I2
or
From these two arithmetic axioms, he derived two “algebraic initials” (by substitution of values):
J1
J2
3
William Bricken used three axioms in his system. His first two axioms are said to “parallel the laws of
arithmetic” and his third axiom represents the "Law of the Excluded Middle". These axioms were called
"Dominion", "Involution" and "Pervasion":
1) Dominion
A ( ) = ()
2) Involution
((A))=A
3) Pervasion
A ( A, B ) = A ( B )
Multiple Form Logic also has three axioms, which are similar, but much more generalised. Here are the Three
Axioms of Multiple Form Logic, together with their abbreviated (one word) names:
1) Oneness
1,X=1
2) Reflection
A#X#X=A
3) Perception
A , X # ( A , B) = A , X # B
The important difference is this: The variable "X", in these axioms, is not a constant operator or parenthesis (as
in Bricken's axioms), or any other "syntactic sugar glue symbol", but a Form, i.e. a "citizen of equal status" to
all other variables in these expressions, where each variable can also be another Form, i.e. another entire
expression. In another section (“More Theorems of Multiple Form Logic”) there is a formal proof that William
Bricken’s system is in fact a restricted version or a subset of Multiple Form Logic. The formal proof is followed by
graphic representations (elucidating what is going on even for people with no training in Formal Logic). Here are
the Three Axioms of Multiple Form Logic in more detail, with their full names, and some further (somewhat
"metaphysical") explanations:
The Three Fundamental Axioms of Multiple Form Logic™:
AXIOM 1: Oneness (All is One):
1,A=1
Union of Anything with "ALL" = "ALL" (where "ALL" is "One").
If "1" stands for "everything that can (ever) be distinguished", then any other thing (different than
"1") is by definition already included inside "1".
Here is a "proto-proof" of this Axiom, in a three-step "Primordial Contemplation":
1) Suppose something is outside "Everything".
2) Then, it does not exist, since "Everything" already contains it.
3) Therefore, it was not outside Everything in the first place. -QED
Primordial Verification: If “everything” does not contain something, then how can it be
“everything” (in the first place)? Primordial Corollary: There is no Existence outside "All
Existence", or: There is nothing, which is not contained in "Everything".
in Boolean Algebra: 1 OR A = 1
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NOTE: (for Sages, Seers and Mystics, mostly ;-) ): Axiom 1 arises spontaneously, by itself, out of the Unfathomable Void, by "Primordial Reasoning of the
deepest depth", at a level of Mind where imagination and reality are still One. At this level, all Gods and all Religions are still possible, since (at this stage,
which is poetic, luminously contemplative and totally spiritual), no "cancellation of the objects of perception" has taken place yet (according to Axiom 3)
and (well)... even if such objectification did take place, it is still totally reversible (by Axiom 3), so that the Primordial Purity of this Mind-State can still be
restored, in its entirety.
AXIOM 2: (self-) Reflection (is void)
A#X#X=A
To distinguish (the very same fact) that we are distinguishing, is the same as not to distinguish (it).
I.e. a finger pointing to itself, does not point to anything. Hence (by "Primordial Reasoning” of the
Deepest Depth):
-To distinguish (the fact) that we distinguish "A", is the same as "A" itself.
In Boolean Algebra: A xor X xor X = A
AXIOM 3: Perception is (reversible) internalisation
A , X # ( A, B)= A , X # B
In a (self-) Boundary of Perception X, any-thing A that exists outside the boundary, can also be
brought inside the boundary. Conversely, any-thing A that exists inside a boundary of perception X
can also be cancelled out iff (if and only if) it (or a “copy of itself”) also exists outside the boundary
X.
I.e.: Any-thing we see outside ourselves, we may assume inside ourselves. Any-thing we assume
inside ourselves, we need not assume (as something "imaginary") if we can also see it (as a "fact")
outside. In short: What is real we can imagine, but don’t need to imagine what is real.
In Boolean Algebra: A or ( X xor (A or B) ) = A or (X xor B)
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To pursue further the “metaphysical” or "psychological" essence of Axiom 3, it is note-worthy that variable "B"
(the blue human, in the above figure) represents an "Inner Reality" which does not exist in the "Outer World"
(represented by "A”, or the picture of the church).
Hence (or otherwise) the "strictly esoteric reality B" cannot be "cancelled out", inside the (self-) boundary
represented by "X" (it is esoteric, iff it doesn't exist outside ourselves). Anything that "exists outside us" is
unreliable, since (at any moment) it might be cancelled out (one way, or another). The Truth(tm) is Inner, and
there exists no "Inner Truth" in the Outer World. However, there is "Inner" Truth inside Other Minds (as well)!
Now, to elucidate these Three Axioms a bit further, and see that they are non-trivial generalisations of Bricken's
System (which had not yet been invented -by the way- when Multiple Form Logic™ was first created back in
1983/1984, in the typewritten version given to Professor Jones) please bear in mind:
1. In Multiple Form Logic™, all Forms are "relative" except logical "One", which is the
"Universal Form". This unique Universal Form "1" is defined as the Union of all
Forms in the Universe (which is "The All"). So, Axiom 1 of Multiple Form Logic™
becomes a naturally recursive representation of the (self-evident, for many people)
Truth: The union of any-thing with “the All” is (still) the All, and All is One.
(At this, point, if you're finding all this a bit too heavy, here is a relevant joke to cheer you up: What
did the hungry Buddhist say to the Hindu hot dog vendor? "Make me one with everything"!).
2. In Multiple Form Logic™ there exist only two fundamental operators or relationships
between "forms": "or" and "xor". They are almost identical in meaning to the (wellknown) Boolean operators "OR" and "XOR"; "almost" identical but not "completely
identical", because Multiple Forms are not necessarily Zero or One: By nature they
are multiple and multi-valued. Furthermore, countless “forms” can co-exist
peacefully side-by-side, in a relation we can treat formally as "logical OR". Only
when such forms are the same, do they reduce to only one. However, such (OR-)
cancellation (X or X = X) is not an axiom, but a consequence (theorem T2) of the
"Law of Perception" (Axiom 3).
The meaning of “XOR” is changed: it is now a "cancellation effect" of identical
distinctions, expressing an intuition that states "to distinguish the fact that we are
distinguishing is the same as no distinction". However, whenever different
distinctions distinguish each other, they do NOT cancel out; they can co-exist
peacefully instead. However, within any structure of Forms or Distinctions
distinguishing each other (XOR-wise), every pair of identical distinctions cancels
out. (This is an intuitive explanation of Axiom 2, above).
3. The operation "XOR", within any expression, is valid "by default", i.e. "XY" means "X
xor Y", "ABC" means "A xor B xor C", etc. This is the notation used for many years
(submitted to the university of Manchester). However, while developing Theorem
Proving software for this calculus, I decided to use the symbol '#' for XOR, hoping
for an improvement in readability (for machines, as well as humans). Furthermore,
the "OR"-operation is denoted by a comma between (Multiple) Forms. E.g. the
expression "X,Y" expresses the (Boolean) "X or Y"; "A,B,C" expresses the (Boolean)
"A or B or C"; "X Y (A,B,C)" means "X xor Y xor (A or B or C)", and so on.
4. Multiple Form Logic™ does not con-fuse the presence of parentheses as “glue
symbols”, within expressions, with the existence of Forms or Distinctions. I.e.
parentheses are mere representational tools, without "inherent essence”. Some
people discussing Brown’s work occasionally used parentheses to represent
Distinctions, so –unfortunately- many Brownians -ever since- inherited a strange
confusion about the meaning of parentheses, in the last three decades.
However, "metaphysical contemplation" is beyond the scope of practical work, or the
crux of this matter, which is: faster and more efficient Logic derivations!
In my 1983 essay about this calculus, I had included a proof that these three axioms suffice to deduce all the
Axioms and Theorems of propositional calculus. However, it is worthwhile adding an acknowledgment that (from
the point of view of rigour) there are a couple more Rules required, in order to do formal derivations in the
Multiple Form Logic™ system: Commutativity and Associativity. However (in the eighties) I had intrigued Dr.
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Tassos Patronis (a friend who was also my informal tutor in Formal Logic) by producing a proof that these
"laws" should NOT be taken as "axioms", but should instead be deduced as consequences, which "must be
inherently valid in any Space where Forms reside" by the following (intuitive?) "Primordial Reasoning":
PRIMORDIAL THEOREM 1: Commutativity holds, in (a space containing)
"Multiple Distinctions".
=
Proof:
1) Suppose that Commutativity does not hold in a space where forms or
distinctions reside. Then there must be a way to distinguish one "direction"
from another (in this Space). (“No Commutativity” means, that there must be a
distinction, between e.g. left and right, inside the piece of paper or space, or
whatever, where distinctions reside).
2) Now, since we have not assumed the existence of any other forms or
distinctions in this space, except the ones already distinguished, then there can
be no distinction between "left" and "right", or between ways of writing and
representing (existing) distinctions, in this space.
3) Hence: There is no distinction between left and right, i.e.
Commutativity holds! (Q.E.D) ;)
As an exercise, you can now construct using the same "Primordial reasoning", a "proof" of Primordial Theorem
2, wich states that:
(a,b),c = a,(b,c) -s regards the syntactic sugar of parentheses in a Space with Distinctions. (and so on…)
IMPORTANT NOTE (about this theorem and a Quantum Logic Principle, by Dr. Eddie Oshins):
At first, I thought this “primordial theorem 1” to be a kind of private mathematical joke making my friends
laugh. However, as years passed I realised that it has a value which is perhaps… grossly underestimated ;) in
our society. ;) My growing suspicion that this theorem is not a joke, but a devastatingly serious statement
about Reality™, arose very recently, while browsing Dr. Oshins’s Quantum Psychology site: He proposed a
new Law for “Quantum Logic”, which might be somehow related to “Primordial Theorem 1”. Oshins says in
“About Models & Muddles Pt I”:
The fundamental principle (Hilgard 1989; Jauch 1968, p. 106; Oshins):
“If one can not (operationally) distinguish / discriminate between two unit predicates A and B, there will
always exist a third possible contrary (unit) predicate C such that (A or B) = (B or C) = (C or A), i.e. they
are equivalent perspectives –there is no operational way to distinguish / discriminate between A, B, & C. I
was thus led to reinterpret the “liar’s paradox” as “this statement is true or false” does not imply that
“This statement is true” nor that “This statement is false”…
Well, unless I got the meaning of the “fundamental principle” very wrong, a corollary to it is this: “If one
can (operationally) distinguish / discriminate between two unit predicates A and B, then there is no
such thing as a “third possible predicate C”, such that (A or B) = (B or C) = (C or A)”… I.e. if we
can distinguish between two distinctions, then we cannot assume that there is a “third distinction”,
distinguishing the ways in which we are distinguishing A and B, i.e. a distinction defining a
“direction” in the way we are distinguishing.
Thus, the assumption of “Primordial Theorem 1” is correct (if we believe this reasoning to be valid), so
that Commutativity holds in a space where distinguishable distinctions reside.
7
(3) The only Logic Operators we really need, are "OR" and "XOR"!
What does a Boolean Algebra which uses only operators “OR” and “XOR”, look like? Well, it is the simplest
possible Multiple Form Logic. George Spencer Brown's system in "Laws of Form" then becomes a special instance
of Multiple Form Logic™, restricted by the fact that his Forms are not multiple; also restricted because Brown's
"Distinction" is interpreted as "Not” rather than "Xor". However, the meaning of the "XOR" operator is
"metaphysically" closer to Brown's fundamental "Act of Drawing a Distinction", than "NOT". George Spencer
Brown proposed in Laws of Form that "distinction is perfect continence" on a philosophical basis. Well, the
Boolean operation "XOR" is perfectly continent, as an operator. E.g. it allows one entity to exist, if and only if a 2nd
identical entity does not exist, and it allows an entity to vanish, only and only if a 2nd identical entity does not
vanish. Spencer Brown’s axioms of the “Primary Arithmetic” do not correspond to “Not” and “Or” (as he
suggested) but to “Xor” and “or”, a simple fact which passed unnoticed for over three decades, by most people who
have been extending George Spencer Brown:
George Spencer Brown:
Boolean Algebra:
Multiple Form Logic™:
1 OR 1 = 1
1,1=1
1 XOR 1 = 0
1#1=
(void)
If we also look at the relevant Truth Tables, “XOR” appears to be the simplest possible interpretation in Logic, of
Brown's "Distinction", which is an "irreducible particle" or a "Quark of the Mind". However, in Multiple Form
Logic, two Forms can be "XOR-ed" together without necessarily being mutually exclusive. The mutual exclusivity
of XOR exists only for the special cases of Logic 0 and 1: “Nothingness” and the "All".
I.e. Multiple Forms are relative and co-existent, except when there is "absence of any form" (which is Void, or
logic Zero) or when we deal with "all the forms of the Universe" combined into "The All", which is Logic "1". In
all other cases, or intermediate levels of Being, XOR is an operator expressing precisely, no less and no more than
this: "Acts of Distinction" or “Forms”, which are perfectly continent, and become textures of the Mind's
Boundaries. Unlike Brown's, such new “pluralistic” forms or boundaries are by nature "multiple" or "coloured",
and can be combined into structures of awesome complexity, if we wish. (E.g. in sophisticated computer
hardware, which are augmented images of the Mind’s topology, at a collective or universal level).
NOTE: According to some authors, "Distinction" appears to be a “building block” for
the physical world, as well: E.g. there is a treatise I found (by Ben Groetzel), which
constructs a "Clifford Algebra" by modifying Brown's axioms, applying this Algebra in
the derivation of some fundamental equations of Quantum-Mechanics. The Ultimate Goal
or "Holy Grail" of the Brownian Path to Logic Enlightenment is a "Union between Outer
and Inner World", which seems to be a Spiritual Experience that has inspired mystics,
monks, shamans and alchemists, throughout the ages.
However, only in special instances where these Boundary Structures "collapse" into the void (0) or into
the All (1), do classic Logic Proofs have "meaning" (whatever this means to you, of course) ;)
Nevertheless, my contention is that Multiple Forms have meaning in a more general sense, since their
three axioms are more consistent with the real structure of our Experience, than Boolean Algebra,
Propositional Calculus, and other such systems. This is a personalised and idiosyncratic "philosophical
Holy Grail", that has led me to the creation of this calculus, but it is by no means necessary for the
practical aspects of Multiple Form Logic™ as a computational methodology for faster and more efficient
Logic derivations.
8
(4) Multiple Forms change the Philosophical Semantics of Logic Implication
Implication can no longer be regarded as a “Causal Relation", of "something causing something”. Logicians
know this to be a fallacy or misinterpretation, in traditional Formal Logic, since long ago. The problem is that it is
hard to explain why this is a fallacy. Traditional Logic had no way of solving this comprehension problem, since
there was no interpretation of Logic Implication that could be consistent with Human Experience. (The closest
analogy was with Sets and Set Theory, where Set-inclusion was a model of Logic Implication). Now, there is a
better paradigm: The new meaning of "Logical Implication" (A -> B) is a distinction between an Inside (A) and an
Outside (B).
The implication “A -> B” means simply that inside a certain boundary of perception there is an A, and outside it
there is a B: To "imply" in logic, is in reality to perceive. So the "Law of perception" (Axiom 3) expresses natural
perception of “causes and effects”, placing assumptions "inside" and (perceived) consequences "outside"
(ourselves). “Set inclusion” is related to this process: Our minds have a natural tendency to treat the Outer
World as a subset of the Inner World, something which is expressed in a most extreme form by certain Buddhist
doctrines: “Samsara”, or the world of illusion (which is “reality”) is said to be “just a dream”, consisting of the
mind’s own projections. This is perhaps an extreme view by today’s standards, where few of us have the…
luxury ;) of doubting external realities. However, the psychological principles of our minds are still the same: We
tend to perceive “as if” the world is a subset of our minds, and “as if” the insides of our minds (our assumptions)
are a superset of the objects we see. Thus, our own assumptions seem to “imply” perceived reality. This is an
illusion, of course, but understandable illusion, given the “Law of Perception” (Axiom 3).
Another understandable illusion is the age-old tendency of the Human Mind to create or search for external
“totems”, symbols of the Inner World which have a reality outside ourselves, because we seek to externalise the
Inner, cancelling it out (discharging it) when we finally achieve the impossible: To find it outside ourselves.
However, this ancient illusion, a driving force for the Human Race’s spirituality, can cease to be mere illusion
and become a conscious process: In practical meditations, we focus on external objects, “bringing them inside
ourselves”. This is -effectively- a conscious and positive use of Axiom 3, just like… computer programming can
be a “very intense form of concentration”, which certain contemporary Indian Gurus have described as
“stronger than other meditations”. Most good programmers know this well, and this is why we’re… good! ;)
The "Law of Perception" replaces Causality, as well as traditional Logic Implication. There is no longer a need to
treat any implications as if they are axioms. Traditional Propositional Logic is superfluous, in this sense: It is like
an old bag full of unnecessary "syntactic sugar", hopefully to add "taste" for the benefit of students who find
Logic... tasteless ;). In reality, however, this unnecessary “syntactic sugar" can be harmful, poisoning the
efficiency and the clarity of both mental and computational efforts in Logic Proofs.
In fact, a "chain of implications" such as (A -> B & B -> C) -> (A -> C), can now be proved directly, by invoking
the Three Axioms repeatedly, without any need to involve implication -as such- inside the proof process itself.
(You can use the program "mflogic.exe" to prove such propositions from a library, or enter your own
propositions, and see automatic proofs). The human mind has a naturally erroneous tendency to "chase its own
tail", following big threads or chains of implications, causing unnecessary psychological stress. However, if...
Harry Potter acquires a capacity to reason in Multiple Form Logic™, he doesn't need to waste energy chasing
around chains of implications or causes and effects; All he needs to do is contemplate calmly and precisely "what is
inside and what is outside" (the boundaries of his own mind). Then, quite naturally, what is "inside" collapses
(whenever it is also seen "outside", by axiom 3); what co-exists with Everything (or "the One") collapses into the
"One", by axiom 1; and what was previously kept apart, ceases to be kept apart, when we realise our own
realisations about it, i.e. distinguish our own acts of distinction (by axiom 2). Thus the ultimate guru of such a
Radical Logic wastes no energy to do implications, being perpetually "self-liberating": Arriving at conclusions by
cancelling out unnecessary distinctions, rather than by increasing their (already prolific) number (hoping that
from such symbolic garbage, "the truth will rise, in the end").
OK, having said quite enough "philosophically" for the moment, let us proceed (as promised) to some meaty
practical consequences of this Logic System, in automatic theorem derivations by computer:
9
(5) A Prolog Theorem Prover simplifying Logic, by using Multiple Forms
The strategy of this Prolog program ("mflogic.exe", which you can download) is essentially
the same as the strategy of a (human) theorem-prover, who knows the Axioms of Multiple
Forms: As much as possible, all logic formulae are progressively reduced, by cancelling out
"Outer Parts" if these are also found in "Inner Parts" of expressions (using Axiom 3). They
are also reduced by “the All” (=One) "absorbing anything" that “exists outside itself” (using
Axom 1), or (finally) reduced by pairs of identical forms “collapsing” when they apply to
each other, “distinguishing each other” (XOR-wise, by Axiom 2).
Now, please bear in mind, that the philosophical "mumbo-jumbo" used here, does not refer
to something "vague", but to something quite formal and precise. All you need, to
understand this edifice, is to run the Prolog program accompanying this text.
I cannot guarantee that the program always works, but it has worked well till now. It includes some
extensive automatic comments, sprinkled over the derivation steps, so it can become an educational
tool for learning Multiple Form Logic™. I wrote it recently, coming back to Multiple Form Logic™ after
a long period of absence from this field, and it is still Version 2. Future versions planned may include
graphic representations of Multiple Form simplifications, which are quite spectacular, even on paper,
like watching an avalanche of "bubbles” breaking and re-organising themselves.
In the current version of the program, there are options for proving traditional Propositional Logic
formulae, by translating them into Multiple Form Logic™, and then using repeatedly the Three
Fundamental Axioms of Multiple Form Logic as re-write rules, until the resulting expression is
irreducible. (You can either pick a formula from a library, or write your own). Then, the result is
converted back into Propositional Calculus. In some cases, the conversion of the result is trivial,
since "1" or "0" are acceptable values within both calculi.
In other cases, conversions are not trivial, but crucial: -They demonstrate that Multiple Form Logic™
does better than just "prove theorems" to be "true" or "false"; It actually optimises logic
expressions, regardless of whether or not they are reducible to true or false.
In traditional Propositional Calculus, logic proofs either lead to a “true" result, or lead to a
"false" result. This is wrong, but the reasons why this is so are not entirely rejectable. It is
the philosophical interpretations that need renovation, not the formal validity. For instance,
the falseness of certain propositions is seen as evident due to the non-identity of truth
tables (between the left-hand-side and the right-hand-side) rather than as an inherent
essence of the formulae themselves; and once we abandon this old criterion for falseness,
such Forms can be seen as "relatively true”, instead of "False").
I.e. in Multiple Form Logic™, the proofs do not reject valid expressions as "false" just
because they do not reduce to "One"; such expressions are treated as "relatively true", and
they are simplifications of the original formulae (which produced them).
Thus, our "Multiple Forms Simplification Engine" offers a lot more than the (lengthier and
tedious) traditional methods of proof: - It optimises or minimises logical expressions. If they
can be minimised to “One”, they are "true"; If they reduce to “Zero”, then they are "false" or
"void" (which is rare). If they neither reduce to “1” nor to “0”, then they remain perfectly
valid logic expressions. These are equivalent in every way to the original (non-optimised)
expressions, but - very often - have fewer terms.
10
(6) How to download and run the Prolog Theorem-Proving Program
The (main) Prolog program accompanying this text was developed in LPA Win-Prolog (version 4.1), a
compiler kindly donated to me by Mr. Brian Steel of "Logic Programming Associates Ltd", after a
short period I worked for LPA, back in the spring of 2001. It has user-friendly menus and it runs in
any (32-bit) version of Microsoft Windows™. Another version of the program is currently under
preparation, written in Visual Prolog™ 5.1, a compiler kindly donated to me by Mr. Leo Jensen,
director of “Prolog Development Centre” in Denmark, after I published (on line) some Assembly
Language source code for PDC, back in the mid-nineties. (The Visual Prolog version is likely to be
more spectacular, with graphic representations and tree-views of Multiple Form Logic expressions,
but it will take time before it will be ready and you can download it from this site). The current
program is compressed and divided into three "rar" archives (because some of the hosts of these
sites don't allow downloading any files larger than 256Kb at a time). First, you should download the
three archives, from one of two mirror-sites:
Site 1 (USA): http://multiforms.netfirms.com
mflogic.part1.rar ( 186 Kb )
mflogic.part2.rar ( 186 Kb )
mflogic.part3.rar (<130 Kb )
Site 2 (UK): http://mforms.port5.com
mflogic.part1.rar ( 186 Kb )
mflogic.part2.rar ( 186 Kb )
mflogic.part3.rar (<130 Kb )
Then you have to use "winrar". (If you don't have this compression utility, download it here). Use
"winrar" to uncompress these files by double-clicking on the first file (mflogic.part1.rar). Delete the
rar-archives from your disk, and you are now ready to run "mflogic.exe". You can also place a
shortcut to "mflogic.exe" on your desktop.
Please keep this text intact when giving the program to your friends, respecting my copyrights. You
may use or give away the program, for better understanding of Multiple Form Logic™ and
Propositional Calculus, but you may not use Multiple Form Logic™ in your own software for
commercial purposes, without permission. (The algorithms used in this software are now in the
process of becoming patented).
Please notify me if you discover logic formulae known to be true, but which my program fails to
prove. A couple of such cases caused great improvements in the program's theorem-proving
strategy. The Three Axioms of Multiple Form Logic™ have been proved to be "formally sufficient", to
derive ALL the existing axioms and theorems of traditional Propositional Logic. There is truly
nothing that Multiple Form Logic™ cannot prove, iff it is also provable in the Propositional Calculus.
Nevertheless, software is… software: It contains the possibility of bugs or other drawbacks, as well
as the potential for improvements and optimisations. I hope to sustain continual upgrades of this
program, if you give me some feedback. (All criticisms welcome).
Have Fun in Multiple Forms!
George A. Stathis (Greece, 23rd of September 2003).
E-mail: omadeon@gmail.com
However, once you’ve satisfied your curiosity, read on! (This article is not finished, yet! J )
(7) The (Logic) Truth is out there, Mauldner: in front of your eyes! ;-)
Remember “Carnaugh maps”, and other methods of “Boolean minimisation” for designing more
efficient Digital Circuitry? Well, “Multiple Form Logic™” appears to get similar results with much
less effort. How much? Well, this episode (of X-files?) needs further investigation. However, by
running the program, it is easy to see at once, some visible benefits. (It is hard to deny you’re facing
an elephant, when you see him!)
11
Some of my most hilarious memories from “Computer Science Education” are those rainy (Essex,
UK) days when we were taught Logic and I asked for our lecturers' permission to demonstrate quite
shorter proofs of the same formulae. I would transliterate them first into Multiple Form Logic™,
simplify them in a couple of lines, and then translate them back into Propositional Logic. (Some
lecturers loved it; others hated it).
It has now become possible for you to do the same thing automatically, by using this program, which
is public domain, intended for general distribution through the Internet. Use it freely, to impress your
lecturers, if you are a student learning Logic. Use it to change opinions, about good old
Propositional Calculus: - It is an old seedy house full of junk, in need of renovation!
(8) What about Predicate Calculus?
It still remains to be seen, if the methods of “Multiple Form Logic™” reduce the computational effort
required for Predicate Calculus, equally well.
In my old hand-written notes (recently re-discovered inside some old dossiers) it appears that about
fifteen years ago I had managed to model Predicate Calculus in Multiple Form Logic™, by making the
assumption that Predication is a specialised form of Implication or Generalisation, i.e. P(x) was
interpreted as “x -> P” (together with certain other formal assumptions). I am in the process of
checking through the validity of these old notes, now, and will keep readers informed, after this
material is checked thoroughly. Back in 1983/84, I embarked on a self-educational journey about the
Foundations of Logic and Predicate Calculus. Later on, my attention shifted to automatic theorem
proving, while Axiomatic Logic slowly became a secondary issue. I realised that if we bring
Predicates and Existential Quantifiers inside Multiple Form Logic™, to create practical theorem
proving software, such extensions can cause some difficulties: It was like bringing “real” Existential
or Universal Quantifiers, as built-in predicates, inside a PROLOG compiler. (Most Prolog compilers
are not made this way, for reasons of efficiency). Meanwhile, I could not find a set of Predicate
Calculus Axioms which could be consistent with “everything else” (in the Universe, which is... 42, ;-)
etc.) So in 1992 I stopped this (profit-less) quest and started doing (to earn a living) something else:
dictionary software development. At that time, I believed we could apply Multiple Forms to boost up
the speed of Logic Inferences in PROLOG compilers. I still believe this, even though an e-friend
(Dave Keenan) has spent years trying to apply "Boundary Logic" to "self-designing logic circuits",
with no apparent success in the end, despite overwhelming evidence to the contrary. I am not sure
what will happen when we "model" things like Prolog Unifications in Multiple Form Logic™, but have
a strong feeling that only good things can happen. After all, a full-blown Predicate Logic in light
Multiple-Form clothing, was my favourite recursive dream, unfulfilled due to other duties for a long
time! It's time to make the dream come true.
<soapbox: ON>
Meanwhile, my old notes (literally thousands) remain buried in dusty old dossiers.
Unlike dolphins, we (humans) misuse Reason and Logic, to destroy each other’s minds
or (e.g.) to mystify further our (already mystified) Alien Babies, ex-Atlanteans etc., ;-)
who become the grown-up Students of Earthling-Science. However, Logic I.M.H.O. is
not just about "true" and "false". It’s about the Mind's ability for remarkable feats, as
constellations of Multiple Forms… “...drawn in a world where initially boundaries can
be drawn in any way we please, and where the universe - at this phase - seems like
sifting sand beneath our feet" (quoting George Spencer-Brown, from "Laws of Form").
<soapbox: OFF>
12
(9) So, where does the "efficiency" of Multiple Forms come from?
Traditional work in Boolean Algebra, is mostly based on AND, OR, and NOT. These operators appear in all
Boolean Logic textbooks, and we consider them necessary, indispensable for any serious work in computers.
Actually, these operators are (to say the least) far too many! We already know that we can reduce the number
of Boolean Logic operators to just two; For instance: OR and NOT. George Spencer Brown did precisely this:
His entire “Calculus of Distinctions” was founded on re-writing Propositional Calculus, by using only the two
specific operators OR and NOT. Of course, George Spencer Brown also “uplifted” the NOT-operator to a
“higher metaphysical level”, as the Act of Forming a Distinction™. As a matter of fact, George Spencer Brown
achieved a substantial economy in the computational efforts for proving logic formulae, solely by virtue of the
fact that he used operators OR and NOT (instead of AND, OR and NOT), but he never clarified this, nor felt
content with his other bright ideas. Instead, he criticised traditional Logic, in ways quite often justified, causing
controversy and conflict for over three decades.
As a result, some people have adored Brown’s work, and others have deeply despised him. Certain Logicians
(like Professor Turner, in Essex University) tried sincerely to find “something useful” from their point of view,
by digging into George Spencer Brown’s book, and have failed. Others conceded that his philosophy in “Laws
of Form” seems both fascinating and valid, but (at the same time) “his results in Logic don’t seem to be all that
impressive, after all”. As a rule, those researchers who felt fascinated by Brown’s ideas, his philosophical and
metaphysical viewpoint, tend to worship his work in Logic as well, often inventing all kinds of “extended
versions” for Brown’s Calculus. (So did I!)
After all, Spencer Brown was the first Logician who introduced the concept of a “distinction” in the first place,
as a Spiritual Act, which is natural for the Mind; much deeper and much more fundamental than Truth, Falsity
and Logic. This, in itself, was the work of genius! However, if we are to cultivate a balanced view of George
Spencer Brown (after all these years), we should firstly recognise that he revolutionised Formal Logic
philosophically, much more than practically. For instance, he never published his alleged proof of the Fourcolour Map Theorem. My guess is that he never finished it! It is quite certain that he believed, quite sincerely,
his own formal methods to be strong enough, so strong that they were “destined” to bring about very soon,
one way or another, such a proof. However, in every personal attempt he made to do the damned thing (the
hard work of the proof itself) it seemed seductively “close”, as well as perpetually elusive. (This is -of coursepure speculation, on my part). However, I think I know this state of mind rather well, since I also used my
calculus of Multiple Form Logic in another over-ambitious expedition, trying to prove that one can solve the
“Satisfiability Problem” of Boolean Algebra, with a method which has a complexity “better than NP-complete”.
(There is a constellation of “NP-complete problems”, all of them inter-related, where if anyone succeeds to
simplify one of them, one simplifies all of them. I.e. it has been proved, that if one "NP-complete problem" is
proved “better-than-NP”, then all of them have been proved “better-than-NP”). I remember myself, as a student,
missing… university parties, drinking plain cups of tea and coffee instead, in (Colchester, UK) coffee bars, with
pencil and paper at hand, sometimes together with other enthusiastic... Logic-crazy friends: Trying to derive
and deliver this precious result, which – alas - never arrived! I still do believe, even after so many years, that it
is possible to derive it. However, I always abstained from making hasty claims of having “derived it already”,
just because (on some crazy or rainy UK day) the result seemed “only very few proof-steps away”. ;-) (Aha! it
rhymes, as well! hmm…)
In contrast, Multiple Form Logic™ keeps no secret of the fact that it is based on yet another attempt to re-write
Boolean Algebra, using only two operators. This time: OR and XOR. This time, however, the results are
superior. Not so much because of the "Philosophy of Multiple Forms" (expressed in the “Three Fundamental
Axioms”) as much as because - quite frankly - anyone experimenting with a Boolean Algebra based on OR and
XOR (rather than other operators) is sooner-or-later bound to discover shorter and better ways for the derivation
of most Propositional Logic formulae. To see why this is so, let us compare the practical (spatial) “complexity”
required to represent a “XOR-product of logic variables”, using a Boolean or Brownian approach (without XOR
operators), with the complexity of representing exactly the same expressions in Multiple Form Logic™. The task
of representing XOR-term expressions without using the XOR-operator at all, has an increasingly high
complexity (not just in time, but also in space, e.g. on paper). In contrast, if we have already included the XORoperator in our system, and if we have access to valid derivational tools as well (such as Axiom 3) to work with it
(not just... look at it) then we can cut down most of the “spatial complexity”, to start with. We can then avoid a
great deal of computational complexity as well, if we can work with “XOR” without tediously translating it to
other operators. For example, here is the expression “A XOR B XOR C” written “graphically” using the “Laws
of Form” notation of "circles as distinctions" (in reality, NOT-operators):
13
Now look at the same expression (“A XOR B XOR C”), written in a “graphical way” akin to George Spencer
Brown’s, but where Distinctions are coloured, to indicate their newly enhanced status as “Multiple Forms”:
I.e., if we increase the number of variables, in such a XOR-term, we are bound to hit against a wall of
complexity, which is inhibitive, no matter what “intelligent” software we use in logic derivations. In
contrast, the additional amount of spatial complexity required for the same expression using
“Multiple Forms”, is very reasonable: Linear, compared to the number of variables. In a dusty old
manuscript, I had recorded the calculated complexities of representing XOR-expressions in terms of
other operators required (if we don't use XOR). The results are as follows (I did not verify them since
the mid-eighties, nor remember the formula which was used to calculate these results, so please check
them out):
XOR-ed variables:
Terms without XOR:
5
46
6
94
7
190
8
382
9
786
10
1574
11
3150
12
6302
13
12606
14
23214
15
46430
16
92862
17
185726
18
271454
19
542910
20
1085822
>=21
?? (unmanageable)
This table shows clearly that each time a new variable is added, the number of terms needed to represent the
expression (without using XOR) doubles (approximately). So far, we assumed that we could get results within
the Propositional Calculus, without any regard for the need of representing and managing "XOR" operators.
The evidence that this is not so, is beginning to show. What is not yet clarified, is the degree to which we've
managed to mystify ourselves completely into not seeing certain obvious things, to such an extent that some
of our assumed methods and their "degrees of complexity" for solving certain important problems of Computer
Science, are (very possibly) w r o n g!
14
10) Proofs of some Multiple Form Logic™ Theorems
Hyper-linked Index of Proofs:
1. Theorem T1) A , B # A = A, B
2. Theorem T2) A , A = A
3. Theorem T3) (in the original paper of 1984…)
4. Theorem T4) George Spencer Brown’s algebraic axiom J1 proved as a theorem
5. Theorem T5) George Spencer Brown’s algebraic axiom J2 proved as a theorem
6. Theorem T5.1) George Spencer Brown’s algebraic axiom J2 can be generalised
7. Theorem T6) Multiple Form Logic™ is equivalent to a Boolean Algebra
8. Theorem T7) Logic inferences are transitive
9. Theorem T8) (the Boolean Multiple Form Logic equivalent forms of “AND”)…
10. Theorem T9) (A # 1,B) = (A , B)#B#1, in Propositional Logic: A -> B
11. Theorem T10) Huntington's Axiom is a theorem in Multiple Form Logic
12. Theorem T11) A “generalised Huntington Formula” in Multiple Form Logic
(dozens more to come, in the near future…)
NOTE: Propositional Logic statements are proved by translation into Multiple Form Logic and retranslation into PC, given the following table:
Propositional / Boolean Logic
Not (A)
A or B
A and B
A=B
AaB
A xor B
Theorem T1:
Multiple Form Logic™
A#1
A,B
(A # 1, B # 1) # 1
A#B#1
A # 1, B
A#B
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:
Proof:
A,A=A
We have already proved (Theorem T1) that: A, B # A = A, B.
Replacing B by void (absence of form or distinction): A, A = A.
15
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 as1
= A, B, C, …(all possible forms). Suppose now that we examine 1 , X where X is any
(particular) form. Then, since 1 = A, B, C…,X, …, i.e. the sequence on the right-handside of this, already contains X (by definition, since “1” was constructed to contain any
form once),1, X= A,B,C…X, X,…. Now (by theorem T2, X,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): 1. Therefore: 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 discarded “as 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: Spencer Brown’s algebraic axiom J1, proved 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: (
Proof:
p # 1, p ) 1 = (void)
(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 2: void # X # X=void, applied to 1 # 1).
16
Theorem T5: Spencer Brown’s algebraic axiom J2, proved 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 # 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: 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:
17
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 # X = 1 (by Axiom 2 , which says that: A#X#X=A).
In the 2nd method we show 1)Left hand side => Right hand side, 2)Right hand side => 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)
=1 (“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 # X , 1 # X ) # X , C # 1 = ( 1 # X ) # X , C # 1 = 1 , C # 1 = 1 (“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,
18
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
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
B4. For all a, b in B, if a
b = a, then a
b’ = c
in B, and a unary
b = a.
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)
Then:
Let A , B # 1=1 = C , C # 1.
A , B = A, B # 1 # 1 = A,
B4: (Proof)
(B # 1, A) # 1 = A , 1 # 1 = A.(QED)
Let A , B = A. Then A , B # 1 = A , B , B # 1 = A # 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
19
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 , C # X , A # X , C = B # X ,X , A # X , C = B , X , A , C =
= X , A , B, C (which is the union of all the logic variables, used in the “law”).
Theorem T8: 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 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 ) # 1 = B # 1 # 1 (by axiom 2 applied to 1#1)
= B (by axiom 2, applied to 1#1).
RHS = (1 , B ) # 1 # B = 1 # 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 ) # 1 = 1 # 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).
20
Therefore: for the case A = void, LHS and RHS 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 already used Axiom
1, in the above proof, and the equation proved is still Boolean. However, we 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 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 # 1 = 1 # B # 1 = B (= LHS).
Case of A = 0: LHS = (1, B) = 1, RHS = ( B ) # B # 1 = 1 (= 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, ( A , B # ( A, B ) ) # 1 ) # 1 (by Axiom 2 applied to "1 # 1")
= (( A, B ) # 1, ( A, B # B ) # 1 ) # 1 (by Axiom 3, cancellation)
= (( A, B ) # 1, A# 1 ) # 1 (by Axiom 2, on "B # B")
21
= ( ( A, A # 1 , B ) # 1, A # 1 ) # 1 (by Axiom 3, insertion)
= ( ( A, 1 , B ) # 1, A # 1 ) # 1 (by Axiom 3, cancellation of "A", in "A # 1")
= ( 1 # 1, A # 1 ) # 1 (by Axiom 1, since "1" is in "A # 1 # C")
= A # 1 # 1 (by Axiom 2 on "1 # 1") = A (by Axiom 2 on "1 # 1") = RHS (Q.E.D.)
Theorem T11: 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" is 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 ) # 1 = A (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
RHS =A, ( B ) (Omitting "X") = LHS
11) An Outline of Bit-Crunching Theorem Prover "Iphigenia"(*)
(*) named after my mother, the late Iphigenia Stathis
(maths teacher), who died in 1998.
This is a fast bit-crunching algorithm (with inherent parallelism) for answering logic queries
in a class of Multiple Form Logic™ formulae, encoding Expert-System Rules in bit-patterns.
ALGORITHM IPHIGENIA™
Hyper-Linked Contents:
11.1)
11.2)
11.3)
11.4)
An example of the Algorithm’s operation
Rule of Vertical Cancelation (RVC)
Rule of Diagonal Transfer (RDT)
Fact-Row Extinction of type 1 (FRE-1)
22
11.5)
11.6)
11.7)
11.8)
11.9)

Fact-Row Extinction of type 2 (FRE-2)
Fact-Row Extinction of type 3 (FRE-3)
A Demo-Program you can download
Historical Note and a… Job Seeking Ad
Conclusions, Copyrights, Acknowledgments
The material that follows is of highly technical nature, but I re-wrote it recently, to make it
as simple as possible, suitable for general distribution to a variety of audiences (or…
possible employers). It concerns theorem proving for Expert Systems, but done in a
strange way that is extremely simple, elegant and parallel (in the required computations).
It is all based on the three Axioms of Multiple Form Logic™.
11.1) An example of the Algorithm’s operation:
Consider the following type of logic expression: A and B and C and ... => a or b or c ...
In "Multiple Form Logic", this translates to: (A#1, B#1, C#1, ... )#1#1, a , b , c , ...
i.e. A#1, B#1, C#1, ... a , b , c , ... (by Axiom 2, X#1#1=X)
Suppose now that our set of (logic) variables is finite (so that it can be encoded in bits). For purposes
of clear understanding, consider (for the moment) a system with eight variables:
A,B,C,D,E,F,G,H
For example, consider the following "logic system" containing these variables:
B and C => H
E and F => C or D
C => E or F
B => F
D => B
D => C
E => C or F
(where, all the statements are assumed to be “AND-ed together”)
Consider now a "logic query" (for "evaluation"):
D => H ...which turns the whole system, in Multiple Form Logic:
(B#1, C#1, H)#1,
(E#1, F#1, C, D)#1,
(C#1, E , F)#1,
(B#1, F)#1,
(D#1, B)#1,
(D#1, C)#1,
(E#1, C , F)#1,
D#1, H.
We can now easily get successive simplifications of this system (by applying the Three Axioms):
(B#1, C#1, H)#1,
(E#1, F#1, C, D)#1,
(C#1, E , F)#1,
(B#1, F)#1,
(D#1, B)#1,
(B#1, C#1)#1,
(E#1, F#1, C, D)#1,
(C#1, E , F)#1,
(B#1, F)#1,
B#1,
1,
(E#1, F#1, C, D)#1,
(C#1, E , F)#1,
(B#1, F)#1,
B#1,
1 (TRUE)
23
(D#1, C)#1,
(E#1, C , F)#1,
D#1, H
C#1,
(E#1, C , F)#1,
D#1, H
C#1,
(E#1, C , F)#1,
D#1, H
We can take advantage of bit encoding, if we represent such systems by arrays of bits. A system like
this can be represented by a "bit-array" with a left-hand side (“assumptions”) and a right-hand side
(“consequences”), where each “fact row” is a set of “AND-ed assumptions” and a set of “OR-ed
consequences” (XORed with 1):
For example, the (previous) system:
(B#1, C#1, H)#1,
(E#1, F#1, C, D)#1,
(C#1, E , F)#1,
(B#1, F)#1,
(D#1, B)#1,
(D#1, C)#1,
(E#1, C , F)#1,
D#1, H
...can be encoded as follows:
ABCDEFGH
01100000
00001100
00100000
01000000
00010000
00010000
00001000
00010000
ABCDEFGH
00000001
00110000
00001100
00000100
01000000
00100000
00100100
00000001
ß variable names
(B#1,C#1,H)#1,
(E#1,F#1,C,D)#1,
(C#1,E,F)#1,
(B#1,F)#1,
(D#1,B)#1,
(D#1,C)#1,
(E#1,C,F)#1,
(D#1,H)
ß “fact rows”
ß “query row”
11.2) Rule of Vertical Cancelation (RVC):
The bit-representations help us automate and simplify further the reductions. First of all, notice that
(because of Axiom 3) we can reset (to zero) all the bits in every column which correspond to One-bits
in the “query row”, since the query row is on the Outer Space of the fact rows (so Outer from Inner
can be cancelled –Axiom 3):
ABCDEFGH
01100000
00001100
00100000
01000000
00010000
00010000
00001000
00010000
ABCDEFGH
00000001
00110000
00001100
00000100
01000000
00100000
00100100
00000001
ß variable names
(B#1,C#1,H)#1,
(E#1,F#1,C,D)#1,
(C#1,E,F)#1,
(B#1,F)#1,
(D#1,B)#1,
(D#1,C)#1,
(E#1,C,F)#1,
(D#1,H)
ß “fact rows”
ß “query row”
24
The first stage of simplifications resets all the “fact row bits” that correspond to the fourth position
of the left-hand-side of the query and the eighth position of the right-hand-side of the query. The
resulting bit-table is:
ABCDEFGH
01100000
00001100
00100000
01000000
00000000
00000000
00001000
00010000
ABCDEFGH
00000000
00110000
00001100
00000100
01000000
00100000
00100100
00000001
(B#1,C#1)#1,
(E#1,F#1,C,D)#1,
(C#1,E,F)#1,
(B#1,F)#1,
B#1,
ß row with 1 bit
C#1,
ß row with 1 bit
(E#1,C,F)#1,
(D#1,H)
ß “query row”
11.3) Rule of Diagonal Transfer (RDT):
The next step is very interesting: Notice that there are now two rows, which contain a single bit.
Since these have become “Outer” in relation to the (“inner”) fact rows, i.e. they behave “as if” they
are already a part of the (outer) query row, they can be transferred “diagonally” to the query row at
the bottom:
ABCDEFGH
01100000
00001100
00100000
01000000
00000000
00000000
00001000
01110000
ABCDEFGH
00000000
00110000
00001100
00000100
01000000
00100000
00100100
00000001
(B#1,C#1)#1,
(E#1,F#1,C,D)#1,
(C#1,E,F)#1,
(B#1,F)#1,
B#1,
ß old row with 1 bit
C#1,
ß old row with 1 bit
(E#1,C,F)#1,
B#1,C#1,D#1,H
ß NEW “query row”
Evidently, the (strike-through-) deleted rows correspond to terms B#1, C#1, which are now "outside"
the rest of the system, i.e. they can now be safely assumed to co-exist with the query row. Therefore,
we can "transfer" these rows to the bottom “fact-row "diagonally", in a way that makes the fact-row
OR-ed with the new terms:
We can now repeat the first stage, using the Rule of Vertical Cancelation (RVC), i.e. reset all bits in
relevant columns for which the corresponding bit of the bottom line (the "query row") has recently
become "1":
ABCDEFGH
01100000
00001100
00100000
01000000
00001000
01110000
ABCDEFGH
00000000
00110000
00001100
00000100
00100100
00000001
ABCDEFGH
00000000
00001100
00000000
ABCDEFGH
00000000
1,
00110000 (E#1,F#1,C,D)#1,
00001100 (C#1,E,F)#1,
(B#1,C#1)#1,
(E#1,F#1,C,D)#1,
(C#1,E,F)#1,
(B#1,F)#1,
(E#1,C,F)#1,
D#1,B#1,C#1,H
ß “query row”
i.e.
25
00000000 00000100 (F)#1,
00001000 00100100 (E#1,C,F)#1,
01110000 00000001
D#1,B#1,C#1,H
ß “query row”
Notice now that the first line in this system has become zeroed completely. This means that the
system contains an isolated "1", or-ed with other terms.
Therefore, by axiom 1 the entire system has becomes "1" or "TRUE".
I.e. the answer to the logic query (“D => H”) is “YES”!
This is a simplification process for systems of logical formulae of the form described in the
beginning, which are Rules for a class of Expert Systems (at the moment, using only propositional
logic). Now, there are a few more things required, to build a complete Expert-System Query
Evaluation Engine, in Multiple Form Logic™:
11.4) Fact-Row Extinction of type 1 (FRE-1):
Firstly, if a bit-position in the left-hand-side (of any fact-row) is One, and the corresponding bitposition in the right-hand-side (of the same fact-row) is also One, then we can safely discard this
fact-row completely, since it is equal to (X#1,X)#1, which (by Theorem T4) is void! I.e. fact-rows like:
00010000 00010000
(where the fourth position bits are both one)
^
^
01000000 01000000
(where the second position bits are both one)
^
^
...can be discarded (erased totally, "as if they never existed")!
11.5) Fact-Row Extinction of type 2 (FRE-2):
Secondly, if a particular bit in the "right hand side of the query row" is One, and the corresponding
bit in the "left hand side of a fact row" is also One, we can again erase this fact row, "as if it never
existed":
XXXXXXXX XXXXXXXX
00010000 ........ ("fact row", with 4th bit in the left-hand-side = 1)
XXXXXXXX XXXXXXXX
=================
........ 00010000 ("query row", with 4th bit in the right-hand-side = 1)
=>
XXXXXXXX XXXXXXXX
----------------- ("fact row"- erased now!)
XXXXXXXX XXXXXXXX
=================
........ 00010000 ("query row", with 4th right-side-bit = 1)
This is because the "fact row" is: (.... X#1... ....)#1 and the "query row" is (........... ....X.....) and we can
"insert" X into ...X#1... (by Axiom 3) , to get X#X#1, which (by Axiom 2) is "1". If we also notice the
"#1" on the right of each fact-row, then we see that the entire "fact row" becomes "1#1", which (by
axiom 2, again) becomes void.
26
11.6) Fact-Row Extinction of type 3 (FRE-3):
An exactly similar process of simplification can take place in the opposite (diagonal) direction: If the
Query Row has a "1" in position N (leftwards) and there is a "Fact Row" which also has a "1" in the
same position N (but rightwards), then this "Fact Row" can be safely discarded "as if it never
existed".
This operation completes the list of very simple parallel bit operations that are required, to simplify
bit-encoded Expert System knowledge bases.
Finally, notice that everything described so far, refers to processes that can take place completely in
parallel.
I.e. we've been watching the operation of a Parallel Computation Algorithm for Logic Queries in
Expert System Knowledge-Bases, which proceeds in "bundles" of "vertical" and "diagonal"
parallel "cancellations", pruning the knowledge base, to simplify it progressively, aiming to
reduce one Fact Row to “zeroes”, which makes the value of this row equal to "1", or-ed with the
rest, giving as a final (total) result = "1", i.e. "This query is TRUE".
On the other hand, if our system does not reach this point, but appears to become stable (after a
"round of bit-operations"), we can assume that it has been "minimised with respect to the particular
query". In other words, the query was (again) answered, but answered "relatively" (to the knowledge
base) with a "relative answer".

NOTE: If you wish, you can also download a piece of software that demonstrates the
operation of this algorithm graphically, with text-mode graphics. (More details at the end
of this section)
What follows is another example, to elucidate the operation of this algorithm more clearly.
E.g. suppose we assume that
(1) "To be Cretan”, implies "to be a liar",
(2) "To be a liar”, implies "to be a sinner",
(3) "To be a sinner", implies "to go to hell"
Here we have 4 variables (Cretan, liar, sinner, going-to-hell). First, We "bit-encode" these three
expressions to:
1000 0100
0100 0010
0010 0001
OK. Now we are ready to "evaluate queries".
Suppose our first query is just one term: "to be a Cretan, implies to go to hell".
This can be bit-encoded to:
1000 0001
The system can now be progressively simplified "in parallel", to evaluate our Logical Query:
1000 0100
0100 0010
0010 0001
=========
1000 0001
27
...which becomes (by the "Rule of Vertical Cancellation"):
0000 0100 ß isolated bit, on the right
0100 0010
0010 0000
=========
1000 0001
By copying "diagonally" the isolated bit to the left-hand side of the query-line, we get ready for the
next round of “Vertical Cancellations (RVC)”:
0100 0010
0010 0000
=========
1100 0001 ß bit 2 on the left has been "copied diagonally")
which is (by application of the "Rule of Vertical Cancellation"):
0000 0010 ß isolated bit, on the right
0010 0000
=========
1100 0001
By copying "diagonally" the isolated bit to the left-hand side of the query-line, we get:
0010 0000
=========
1100 0001
0010
ß isolated bit, copied “diagonally” to the left
i.e.
0010 0000
=========
1110 0001
which is (by application of the "Rule of Vertical Cancellation (RVC)"):
0000 0000 ß empty fact-row
=========
1110 0001
Now: Since the system contains one empty row (of zero bits only), the corresponding term is "1", i.e.
"TRUE". So, the answer to our query (“Does being Cretan imply going to hell”), is “YES”. ;-)
(see
PS, at the end…)
11.7) A Demo-Program you can download:
If you have no training in Formal Logic and/or you wish to understand this algorithm more clearly,
a Prolog Program was written (in the nineties) to demonstrate such processes of simplification
graphically, albeit in text-mode (Ms DOS) graphics, with sound effects like... computer games of
the eighties. This program was kept strictly private, unpublished for more than a decade).
However (as of today) you can download it from: http://multiforms.netfirms.com/bitprover.zip.
Unzip it to a single directory and run "bitprover.exe". (examples are contained in “example.kba”).
28
There is also a Greek version http://multiforms.netfirms.com/bitproverg.zip for Greek readers.
The program will be re-written in Visual Prolog (a successor of PDC Prolog, the language in which
the nineties’ version was written), with better graphics, space for more variables, text outputs of
bit-encoded proof steps, and a “bit-operation mode” (for programmers).
I also hope to make the new program able to automatically generate Assembly Code for MMX
Pentiums, thereby becoming the world’s first “MMX Bit-encoded Expert System Assembler for
Multiple Form Logic™”.
When you run the program “bitprover.exe”, you will be asked to pick an existing Knowledge Base of
Rules. (There is one such file, but you can edit it and re-save it under different names).
The first argument in all the clauses r/3, q/3 is the internal example-number (the unique set of logic
expressions to be simplified). The clause r(n,…) is the “result clause” in the knowledge-base, or the
“query row” (if you prefer). The q(n,…) clauses are the “fact rows”. The syntax of all these clauses is
very simple: Arguments 2 and 3 are assumptions and conclusions respectively, “string lists” in PDC/
Visual Prolog, i.e. statements enclosed in inverted commas, separated by commas, in brackets [ ].
You can edit the knowledge bases to run your own example-proofs, but you should avoid the use of
very long strings in the variable names. Sometimes long variable names create bugs in the display of
the proofs, and the program crashes. If such a case occurs, edit the file and replace long strings with
shorter ones, as short as possible).
Bear in mind that the program does not display bits, but ordinary variables. However, it’s easer this
way to understand the workings of the algorithm, tracing variables and simplifications done using
the Three Axioms. Finally, don’t forget pressing “enter” (or “+” for “acceleration”) to see the show.
Another good idea is to develop Assembly Language Versions of the “Iphigenia Algorithm”
using MMX and SSE extended Assembly Language Instructions of the Pentium Processor.
(This is in fact the closest to “parallelism” we can get, in present-day personal computers).
29
11.8) A historical Note, and a… Job Seeking ad:
In the late eighties, Dr. Nikos Vainos (researcher in Optical Laser Computing) and I, both of us
students in Essex University (UK), had been contemplating to build a (hardware-based) Theorem
Prover, applying this algorithm to Quartz Crystals with directed Laser Beams of "Massive Bitencoded Parallelism", to perform derivations for Expert System Knowledge Bases.
Unfortunately, we never found the funds to build such an ambitious hardware system. So, I kept
this algorithm a secret for fifteen years(!) in the hope of finding funds for it, some time in future.
Well, after fifteen years, it appears that even if we had the funds (which we don’t) we could never
compete with the modern hardware giants in this field. So today (23rd of August 2003) I decided it
is time to release the “Iphigenia Algorithm” into the Public Domain, hoping for some kind of
support, or -at least- a... better job (for me). ;)
11.9) Conclusions, Copyright Issues and Acknowledgments
A simple but extremely efficient algorithm has been outlined, which can reduce a particular class of
logic expressions to either a "true" result, or to a "relative minimised result", Such logic expressions
(in conventional logic) are of the form "A and B and C.... => a OR b OR c....". This algorithm works by
translating such expressions to Multiple Form Logic™, and then applying the Three Axioms of
Multiple Form Logic to these expressions, while taking advantage of parallelism and additional
simplicity, achieved through a particular kind of encoding into bits (encoding Logic Variables as bit
arrays). A demo program was also provided, to make the operation of this algorithm more
understandable. This algorithm is highly suited for parallel hardware, e.g. using bit-encoded Laser
beams and crystals, storing Logic Knowledge Bases as holograms of bits.
It is extremely difficult, today, for an algorithm to be patented. (All legal help and advice on this issue,
is appreciated). This algorithm was invented by the author in the year 1986 and was kept a secret till
now. I do not preclude the possibility that someone else re-invented this algorithm, during the last
seventeen years, but extensive searches and enquiries (made both inside and outside the Internet)
showed that no such algorithm was re-invented, since 1986. To safeguard my copyrights, I have
posted this algorithm, together with the underlying theory of Multiple Form Logic, inside a stamped
addressed envelope mailed to... myself, while living in the UK, in the late eighties. I have never
opened it, so it's a “proof of copyright” (under UK law; don't really know if it also guarantees
exclusive rights of exploitation of the actual method, stripped of all secrecy, now! ;) )
I owe warm thanks to Dr. Nikos Vainos, research physicist in Lasers (now working at the National
Research Centre, Athens, Greece) for his extremely valuable contributions, as regards a possible
implementation of this algorithm in Optical Computing Hardware (even though this hardware was
never built, due to lack of funds). He showed me facts about Optical Computing, which led to
refinements of the “Iphigenia algorithm”, useful for real implementations: One such refinement is the
use of memory copies, rather than a single memory, which can be stored as bit-encoded holograms
carved into crystals under different angles; also potentially useful in future implementations of
(Prolog-) backtracking. (However, we never built the thing!)
I also thank Dr. Tasos Patronis (didactician of Maths for University students, at the University of
Patras) for teaching me all he could, in Abstract Algebra and Logic, at the time when the "Theory of
Multiple Forms" was being born, for his reference letter (of 1984), and for reading my (sometimes
unreadable!) manuscripts, with crucial suggestions for improvements during the early eighties.
Finally, I owe a lot to my... mother, the late Iphigenia Stathis (Maths Teacher) for her truly boundless
support, while being alive -as all true mothers would! (This algorithm has been named after her, for
very real reasons).
(*)B.T.W. I don't really hate Cretans. They usually... tell the truth! And I am not Cretan, either! :-)
30
12) Some Philosophical Aspects of Set Theory and Multiple Form Logic
A quotation from Jules Henri Poincaré:
"Later generations will regard Mengenlehre (set theory)
as a disease from which one has recovered".
[Whether or not he actually said this is a matter of debate amongst
historians of mathematics.]
-The Mathematical Intelligencer 13 (1991). (see source)
It is known that any formal system, equivalent to a Boolean Algebra, is also equivalent to an Algebra of
Sets. So it is quite straightforward to form a “Table of Correspondences” between Multiple Form Logic
expressions and Set-theoretic Entities:
In Set Theory:
in Multiple Form Logic:
_
A (a set’s complement)
A#1
A (a set)
A
A
B (union of A and B)
A,B
A
B (intersection of A and B)
(A#1 , B#1)#1
A
B (A includes B)
U (universal set)
(A , B)A#B by T8
A # 1, B
1 (the “One”, or “the form of all forms”).
(And so on)
Evidently Sets and Multiple Forms are isomorphic, with 1-to-1 correspondences between them
-that include some interesting analogies. E.g. Sets are analogous to Forms, “Set Inclusion” is
analogous to “Logic Implication”, the “Universal Set” is analogous to “All” or 1”, etc.
Inevitably, at some point, the following “Deep Thought Question” arose in my mind:
What is a meaningful analogy between Elements of Sets and Multiple Form Logic™?
Well, in the work I did on MF Logic back in the eighties, I had intuitively answered this
question many times in many ways, during informal discussion with Dr. Tasos Patronis. I
believed (and still believe, perhaps justifiably, or perhaps not, from a strictly formal point of
view) that there is no difference, in Multiple Form Logic, between Sets and Elements. What
seems to have happened though, is that Multiple Form Logic™ forms a “flat logic space”,
where there is no distinction between forms and collections of forms, or between sets of
forms and forms as elements of forms. Tasos thought (and still thinks) that this edifice could
be correct, probably with interesting consequences when applied to Set Theory and other
Discrete Math Theories.
Multiple Forms are exactly like Sets, except that there is no such thing as a Form which can be of an
“inferior existential status” to other Forms. I.e. there is no such thing as a member of any set which
can’t become itself (potentially) another Set (with or without members), at any moment.
“Ask the right questions, and you may get the correct answers”. The right question in this case is:
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Are there any Set-members which can’t (potentially) become Sets themselves?
My belief is that there aren’t! There is simply no distinction between a Set and a Member. In fact, the
natural tendency of the Mind to project a certain “artificial order” onto an (otherwise unfathomable)
Reality™ has created the “disease of Set Theory” by assuming that there is a “qualitative gap” that
(sometimes) divides reality into Sets and into “members of Sets”. But there is no such thing, and
once the false distinction vanishes, some paradoxes of Set Theory begin to unravel themselves too.
E.g. the paradoxes of a “set of sets which are not members of themselves” (ala Russell): If “a set is
a member of itself”, instead of being “prohibited”, it vanishes in Multiple Form Logic! Traditionally
we assumed intuitively (and falsely) that there is something like a “lower limit to Set Membership”,
like a level of “granularity” akin to atoms, or quarks, or other Physical Particles, below which there
exist “members” of sets with only “proletarian status”; not being Sets (themselves)! I could never
understand why this is so. Perhaps I continue to miss important facets of Mathematics. However,
as an A.I. programmer, I will go on striving for faster and better theorem-provers, which make no
distinction between Forms and “Elements of Forms”, except when checking that a particular form
+“does not reduce further”, i.e. it is “atomic” (or “single”). However, such “single atoms” are not
“inferior citizens”, at all. ;-) Multiple Form Logic™ is like “a family of families where every
individual is also a family”. It’s like Flatland, perhaps. Go figure! ;)
Nevertheless, I was led quite naturally to draw parallels between Sets and Multiple Forms;
between membership, the subset relation and “drawing a Distinction inside a Distinction”.
In this new light, a “set” is nothing but a Distinction: a Perceptual Boundary dividing Reality™
into what is inside the boundary and what is outside the boundary. However, precisely this
implicit “intuitive” distinction (between “sets” and “members of sets”) is what created the
mathematical edifice of Set Theory in the first place (in Cantor’s mind, etc).
Doing away with this distinction, brings about deep changes of perception about Set Theory.
13) The Axiom of Regularity, and some hints of Multiple Forms as alternatives to Sets
In the first draft of this text, I had done a serious mistake: Following older (hand-written) material about the
logic of Multiple Forms, I had mistakenly proclaimed that “there is no such thing as a Set which is not a
member of itself” My arrogance was only mitigated by the fact I acknowledged the possibility of error, by
adding the disclaimer: “…or else, I missed some important facets of mathematics…”
Well, as a matter of fact, I did ! ;-)
Recent self-educational web-browsing helped clarify such important facets. For example:
The Axiom of Regularity rules out the existence of Sets which are members of themselves.
I was surprised to (re-)discover this, recently! My school training in Mathematics had included it, at some point,
but I had forgotten it. And I had forgotten it, because it seemed irrelevant for developing real Theorem Provers!
Most Computer Science courses in the UK (during the eighties) did not include this Axiom (or Set Theory, as
such). I was taught this axiom by my mother (who was a teacher of maths) when I was adolescent, but never
became a professional mathematician. (As a “practical A.I. programmer” I can write good PROLOG code and
claim to “understand Logic well”, but only for practical goals). Discovering this fallacy (as well as the great…
arrogance) of my own mind, reminded me -once again- the need for humility in the Era of Endless Learning!
So, a Set that contains only itself as an element vanishes into the Void or into formal non-existence.
Ah well, so what?

So be it ! ( Let it vanish ! )
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
Multiple Form Logic is “Void-based” ,and the “Void” in Multiple Forms can also be interpreted
as a state of Formal Non-existence, or “Absence of Distinction”. To say that an object is “not
allowed” is the same as saying that “it vanishes”, or “collapses into” Voidness. (by Axiom 2)
If a set is effectively a distinction, what is inside it cannot be identical to itself. This realisation has a
very strong hint of similarity to Axiom 2: “A distinction distinguishing itself is no distinction”. In fact,
It turns out that the “Axiom of Regularity” in Axiomatic Set Theory (“Every non-empty set X contains
some element Y, such that X and Y are disjoint sets”) becomes unexpectedly crucial, for elucidating
deeper a newly emerging unexplored deep relationship between Multiple Forms and Sets.Suppose
that a certain set contains only one member, which is simply itself. Now, according to the “Regularity
Axiom”, such a set vanishes! When we say “vanishes”, we mean that it ceases to exist, and that
Reality™ becomes “as if” such a set never existed. Notice the subversive use of the verb “becomes”,
in the previous sentence: Both Naive and Axiomatic Set Theory rest upon (1) a hidden assumption
that “things” (or mathematical entities, such as Sets) “either exist or don’t exist”; Also (2) a hidden
assumption that “things existing, continue to exist”. (I.e. they don’t just… vanish into the Void, “as if
they never existed”). However, such a priori beliefs or assumptions are (to say the least) dubious! ;)
Consider a Set S = {S}, which contains only one element: Itself (S), as a member. We can begin to see
the “fact that it vanishes” (the Regularity Axiom of ZF Set Theory) under a totally new light, given the
following “primordial reasoning proof”, in Multiple Form Logic™:
A “PRIMORDIAL REASONING PROOF” of the “AXIOM OF REGULARITY”:
(1) To “construct a Set”, we create a distinction between what is “inside the Set” and what is
“outside the Set”. (The Set and the distinction it expresses are perceptually inseparable).
(2) Now, suppose we imagine that the Set “contains” only itself, as an “element”. Then it
expresses a distinction, which distinguishes (nothing but) itself, from… itself (as a set).
(3) Now, since there is now no distinction between itself and itself, the distinction that this set
represents, becomes non-distinction. I.e. the Set vanishes! (Since the set represents only
one distinction, now revealed to be a “non-distinction”, when we observe closer).
(4) As a result, we have to preclude the existence of such a set from the very beginning.
(5) Hence, the “Axiom of Regularity” holds, and “No Set can be a member of itself”! ( Q.E.D. )
In the previous “proof”, we can begin to see why Axiom 2 of Multiple Form Logic is strongly related (or
even identical) to the Axiom of Regularity (which holds in traditional Aiomatic Zermelo-Fraenkel Set Theory)
The consequence of this simple finding is immense, for Multiple Form Logic, as well as Set Theory:

Why bother to make the distinction between Sets and Elements, in the first place? Why don’t
we realize that we are distinguishing between distinctions, admitting to ourselves what we are
doing: - Encapsulating Reality™ by Drawing Distinctions inside it, which can contain other
distinctions. However, these multiple boundaries don’t contain any actual elements, as such!
Traditional Set Theory contains an implicit distinction between Sets and Elements. Because of this distinction,
membership and set-inclusion are quite different operations. It is impossible for a Set to be a member of itself,
whereas any Set includes itself as a Subset. So, what I should have said (in the first draft of this chapter) is that
“there is no such thing as a Set which does not include itself as a trivial Subset of itself”. I.e. individuals who
don’t like becoming… subsets of themselves” have problems about their identity, and may need some therapy
from Quantum Psychologists (like Dr. Oshins)! The rest of us can rest in peace, realising that there is no such
thing as a “set which is not a subset of itself”, since there is no distinction between “sets” and “subsets”, to
start with. It seems we have to “go to a deeper place”, to discover what is really going on. Here is one way:

Who “created” Sets in the first place (apart from Cantor, Fraenkel, and all the others)? Well, Set
Theory was created by the Mind’s ability to create or perceive abstract theories about itself, that
do not bear any true resemblance to the Mind’s True Essence! The “True Essence of Mind” is not
“Set Theory” but the ability to distinguish, creating forms. This process is Perception (Axiom 3 of
Multiple Forms ), which is the Mind’s ability to create Inner Copies of Outer Reality (within itself),
but also to forget or extinguish these copies of Outer Reality, iff it also exists Outside Ourselves.

After all this I got a strong intuition that Poincaré was right, and that Multiple Form Logic™ can
become a medicine (among others) to recover from the “disease of Set Theory”. The theoretical
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aspects of this have recently been srutinised again by Dr. Tasos Patronis -who remarked that an
alternative Set Theory already exists, probably with exciting consequences for Multiple Forms.
*14) 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.
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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 the two 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
(from older manuscript versions).
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15) Extending Multiple Form Logic™ to other “Boundary Logic” Systems
It seems to me sometimes a “trivial task”, but (in practice) ominous in volume, to start translating some other
“Boundary Logic Systems” into Multiple Form Logic™. What I mean by “translation” is (1) firstly to devise
appropriate additional definitions, with structure and properties that are derived from these other Boundary
Logics, and (2) secondly to work out derivations within Multiple Form Logic™ that take into consideration the
additional definitions, to get similar or identical results. The beauty of Multiple Form Logic™ is that it allows such
tasks to be accomplished, without discarding the validity of the three axioms, and –usually- without needing to
replace these axioms, or to use new symbols.
There is a reason for this simplicity: Multiple Forms are (by nature)… Multiple! ;)
In other words, you don’t need any extra symbols (such as “special forms of parentheses”) to denote your “special
boundaries”. You can use ordinary symbols, such as «Ψ», for any “special distinctions”. Then, after adding the
definitions of these new distinctions to Multiple Form Logic™, let it boil and simmer… E.g. Jeffrey James’s
Calculus of Number Based on Spacial Forms, a Boundary Logic with three axioms:
Invollution
Distribution
Inversion
([A]) = A=[(A)]
(A[BC])=(A[C])(A[C])
A<A>=.
If we define the distinctions Ψ, φ and Δ for “[]”, “()”, “<>” (respectively), and also drop the symbol “#” (XOR) by assuming
that it’s valid by default, then the above system becomes, in Multiple Form Logic™:
Involution
Distribution
Inversion
AΨφ=A=AφΨ
(A , (B + C)Ψ)φ = ( Α , ΒΨ )φ + (Α , CΨ )φ
A + AΔ = void
(I am working on this one). However, we can define Arithmetic in other ways, within Multiple Form Logic™. The
easiest way I found -so far- is to introduce “+”, an arithmetic operator which stands for the “sum”, or
“accumulation” of “forms in space”, and then assert an additional “Law of accumulative distributivity”:
( A + B ) # C = ((A # C) + (A # C))
What I found most challenging and fascinating, is Ben Groetzel’s use of truly different kinds of
boundaries, e.g. which either allow or disallow the process of internalisation (=to bring a form inside
the interior of another form). I think this is -just about- the only Boundary Logic which may need to
modify or discard the “Axiom of Perception”, for certain types of Distinctions. Most other Distinction
Systems I’ve seen (notably Bricken’s) can be re-written into Multiple Form Logic™ in a more-or-less
straightforward way probably without significant difficulties. Underneath all this, the philosophical
approach followed by Multiple Form Logic™ is the same as all for all the other esteemed Brownian
systems. The community of Boundary Logic researchers is bound ;-) to grow and grow, and I have
no illusion of having found the “Holy Grail” which “puts out of business every other guy’s Boundary
Logic Grail”. The basic principles of Multiple Form Logic™ are childishly simple and understandable
by kids, at primary school perhaps!
My mother (the late Iphigenia Stathis, Math teacher for High School students) and I were discussing for a
number of years, the possibility of introducing some basic ideas of Boundary Logic (and my own Multiple Form
Logic™) to experimental lessons given to school-children, before they learn other Algebraic Systems, or the
Boolean Logic of computers. Can you imagine what it would be like, if Boundary Logic was (today) a
mainstream theory for educational purposes, too, i.e. for teaching Boolean Algebra and the basics of Maths to
students? I believe that Multiple Form Logic™ is one particularly useful system for such a purpose, but it’s one
among several such systems. However, my mother died (in 1998) before we could find the time and the means
to begin such an over-ambitious project. Perhaps the most important educational project, which was
accomplished, was the one that changed our selves, our own ways of thinking and contemplating: E.g. I
stopped believing in Determinism and Mechanism. The universe was suddenly revealed a Magic Place”, where
every human is a secret “Harry Potter” at heart, and anything (well, almost anything!) is possible!
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16) Philosophical aspects of Multiplicity in Multiple Form Logic™
George Spencer Brown in "Laws of Form" had adopted a monotheistic interpretation of Forms: He identified
"God the Father" with the First Distinction itself, "God the Son" with the Outer Space which resides outside the
First Distinction, and "God the Holy Ghost" with the Inner Space inside the First Distinction. These ideas were
also expounded in "Only Two Can Play This Game" and led to Spencer Brown's strange theological doctrine
that "The Holy Spirit is Female" (identified with "Nirvana", or the Clear Light of the Void).
In contrast, when Multiple Form Logic was created, the philosophical inspiration behind it was Polytheistic,
rather than Monotheistic: created with a different metaphysical vision, it adopted a polytheistic interpretation of
Forms: Only the "Formless Void", or the Space where no distinction has yet been created, was regarded as
"unique in the Universe" and Monotheistic; just like the ancient Greek religion identified Chaos, or "the Void",
as the "Mother of all the Gods", but the Gods and Goddesses were multiple. Thus the "One God", which is the
"All", or "All the Forms in the Universe", became a "construction" generating Axiom 1 of Multiple Form Logic.
Nevertheless, axiom 1 is not essential for deriving some interesting consequences from the other two axioms.
On the contrary, I strongly suspect that if we do not take Axiom 1 for granted, we end up with a type of
"Alternative Logic" or "Quantum Logic", closer to the work of Eddie Oshins and his "Quantum Psychology". (It
is interesting to see what happens to the "generalised distributive law" in such an alternative boundary logic).
However, In private communications with another member of the "Laws of Form" forum (Mr. Philip Meguire), he
pointed out that one of George Spencer Brown's achievements was the reduction of Propositional Logic to
arithmetic axioms. I.e. the "initials" of Spencer Brown's "Primary Algebra" were not axioms "as such", but were
proved as consequences from Spencer Brown's "axioms of the primary arithmetic" (I1 and I2).
So I began to think more about this issue, and concluded that there is no reason why Multiple Form Logic
cannot be defined in a similar way (if we desire this), derived from a kind of "Primary Arithmetic". Except that
(in this case) the "new arithmetic axioms" are not going to be the same as George Spencer Brown's.
First of all, the existing axioms of the "primary arithmetic" need to be interpreted differently, within "Laws of
Form" itself: Instead of "1 or 1 = 1" and "not(1) = 0" (as Brown assumed) they must be re-interpreted as "1 OR
1 = 1" and "1 XOR 1 = 0" (as explained in the section "All we need is OR and XOR"). This is the only correct
and consistent way of interpreting the "Primary Arithmetic". Once we adopt this interpretation, the remaining
material in "Laws of Form" follows naturally as before, except that the interpretations of his logic become
absolutely consistent with Boolean Algebra. For example, Spencer Brown's "imaginary operator" is now seen
as "XOR", and it becomes consistent with conventional work in the construction of counters and pseudorandom sequence generators (etc), all of which are based in Exclusive-OR gates, for their operation.
Secondly, I sometimes suspect that there is a deeper foundation, than the Three Axioms of Multiple Form
Logic, which constitutes an "alternative primary arithmetic". I can not sustain dogmatically the romantic
polytheism of my early twenties, where Distinctions were seen as "inherently multiple". E.g. How did they
become Multiple, in the first place? There could be a kind of deeper process, something like a "primary
construction", which distinguishes between Forms at a more fundamental level than the Three Axioms: Such a
"primary construction" would then become the origin of the "distinction between distinctions", a kind of
"meta-distinction", like a "passage-way" from the "One God Universe" to the Polymorphic and Polytheistic
"Universe" of Multiple Forms. - Well, why not?
Suppose, for example, that (in a plane space of representation) we draw two distinctions, as circles within this
space (ala George Spencer Brown):
Initially, there is no distinction between these two distinctions, which means that: Unless we perform within
this space some kind of "additional distinction", these two distinctions (which we have drawn) collapse into
only one, in accordance with George Spencer Brown's Axiom 1 of the "primary arithmetic":
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Suppose now, that we require these two distinctions to "co-exist", without collapsing into only one. Then, there
must be some kind of "additional distinction" that we can make, and which distinguishes between these two
distinctions, most probably at a kind of "meta-level" to these distinctions themselves.
I.e. if they cannot be distinguished between them, they have to collapse into one. On the other hand, if they can
be distinguished between them, then there must be a "new type of distinction" at a kind of "meta-level", which
has the ability to "distinguish between distinctions". However, if our "vocabulary" (of distinctions) does not
contain any such "meta-distinction", or any other distinction than "the One" (the "Marked State" of George
Spencer Brown), it becomes by definition impossible to distinguish between (any) two distinctions, so that all
our attempts to distinguish between (any) two distinctions (such as the ones we drew) are bound to fail, from
the beginning! (These considerations should be made side by side with Primordial Theorem 1, of section 2).
Multiple Form Logic was created (partly) by realising this problem, and by attempting to solve it in a different
way than "accumulations" of Brown's "marked state": The problem was solved by assuming all distinctions to
be inherently multiple, rather than one. Thus, it becomes possible for different distinctions to co-exist naturally
at all levels. Only when they are the same, do they collapse into One (by Theorem T2), or "cancel out each
other" (by Axiom 2). Nevertheless, it is still possible to construct a special distinction ("1"), defined as "the
union of all distinctions in the universe", which generates Axiom 1 if we wish (and so on).
So, to prevent any two distinctions from collapsing into one, all we need is to assume that they are different
distinctions
However, the new question that arose naturally, from Mr. Meguire's remark about the primacy of arithmetic to
algebra, is this: Is there another way to define distinctions, e.g. using only two distinctions, one of which is the
"basic distinction", and the other one a kind of "meta-distinction" (or "distinction between distinctions"),
responsible for generating arithmetically all the different distinctions in the "universe" of Multiple Forms?
Well, other systems of Boundary Algebra indicate that there indeed exist many possible ways to define a
mechanism that generates the natural numbers, within a Logic that can be made similar to Multiple Forms. And
once we have the natural numbers, we then can associate each natural number with a unique single distinction,
so as to get different distinctions, without having to assume that they are different axiomatically. Then the
universe of Multiple Form Logic can become a construction, arising out of a very small number of other, more
fundamental distinctions, and also some modified axioms capable of accommodating a suitable construction
mechanism.
Moreover, my intuition compels me to speculate that only axiom 3 of Multiple Form Logic is fundamental in this
"universe". I.e. we can play around with the other two axioms as much as we like, generating all kinds of logic
systems with different properties, but it seems unlikely that we can play around too much with axiom 3.
An interesting example of creating a logic system which is a superset of Multiple Form Logic, and yet also
obeys the existing three axioms, is the following: -A very slight modification of Axiom 2 is to assume that
identical distinctions collapse into different kinds of Void. (Lou Kauffman has discussed extensively some
interesting consequences of assuming zero -or "voidness"- to be multiple rather than one). The natural
numbers can then be generated by assuming a succession of "multiple voids", where the first void is the
outcome of just one pair of identical distinctions cancelling out each other, the second void is the outcome of
two such pairs, and so on. The advantage of this approach is that axiom 2 of Multiple Form Logic is still valid,
as it stands, for the particular case of two distinctions collapsing into void, and it remains valid in other cases
too, if we assume that the different kinds of void do not affect logical operations within Multiple Form Logic
itself, but only affect operations in a kind of "parallel universe", co-existing peacefully with Multiple Forms. In
this universe, different voids may follow other laws, that do not affect Boolean operations, but only arithmetical
ones, as well as certain kinds of "meta-operations" that generate different distinctions. The "different
distinctions" can then be constructed within this "parallel universe", but remain invisible within Multiple Form
Logic, exactly like quarks are totally invisible within "classic" Physics, but concern Quantum Physics. I.e.
whenever we enter this "parallel universe" to examine more closely what is going on, what appears (to Multiple
Form Logic) as inherently different distinctions, is the result of certain more fundamental "hidden operations",
taking place in a parallel universe which contains mechanisms of arithmetic.
This is just one possible path of further research about Multiple Form Logic that I'm currently contemplating.
The philosophical disadvantage of this path, is of course the existence of multiple voids: It seems to rely on
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shifting the problem elsewhere, rather than solving it. Then again, perhaps the problem is partly unsolvable,
anyway: The problem of "Multiplicity versus Oneness" in metaphysics, is considered unsolvable e.g. within
Buddhism: In Buddhism, it is considered useless to speculate about the Origins of the World, while it is said
that "the World has always existed" with many beings and gods who have existed since uncountable time,
because there was no beginning of Time (or: even if there was, it is useless to think about it, since our minds
are "not capable of resolving such issues"; e.g. even if physical time can be traced back to the "Big Bang",
there is nothing to stop us from assuming a "previous phase", before the Big Bang took place, about which we
are never going to get any useful information, and so on... ) Moreover, if we assume that this universe of
multiple beings and Multiple Forms can be "explained" by another "parallel universe", where the forms are
arithmetically constructed, we may be acquiring a certain formal advantage that creates other kinds of new
problems, which we cannot imagine yet. - Anyway, if you have new ideas about all this (wild speculation) let me
know! ;)
(17) Revealing the secret story and the agony of Multiple Forms ;-)
<Soapbox, Personal Life Story: ON>
Well, when I realised all this (and more) as a "humble undergrad", more than fifteen years ago, I felt
-initially- very excited.
Soon afterwards, however, my prevailing emotion about this, at that time, was d r e a d.
- Why (you may ask)?
Because, like many others in similar rare moments, I could not stand the burden (or such mind-blowing
evidence) that... our entrusted teachers and educators at University, together with facts taught in
University textbooks were "wrong", and "I was right" (like a madman, alone in the wilderness).
I was simply too lively, too extroverted, and too sociable, to be able to put up with this deadening
isolation.
So, what happened then?
After leaving Essex University, in order to earn a living (and also protect my own sanity) I buried the
entire "Theory of Multiple Form Logic" (alive), in certain... rarely accessed drawers, back home, and
tried very hard to forget all about it.
Well, perhaps now (in 2003) the time has come to "cure the Trauma", return to the "scene of the crime"
(through Cyber-Space) and remember...
........................................................
Remember (for instance) how some open-minded Essex professors and lecturers loved this Logic
Calculus, while certain others hated it...
What actually happened to me, in Essex University during the late eighties, is probably a disgrace for
the University and for the UK Academic Community (of that era) as a whole:
- A couple of months before our Final Examinations, the "approved topic" for my "student dissertation"
(which was "Multiple Form Logic", something agreed upon from the very first year of the course) was
changed "from above", without right of appeal, to something else, which was totally different, and
which was impossible to complete -in time- before Graduation.
The "official excuse" was that "Professor Turner, the Chief Logician, was away on a journey to the
States" and that "there was nobody else suitably qualified to correct this Thesis". The real(?) reason,
however, is probably that some people in Essex did not like the whole story at all. I was just a... humble
undergraduate student, at the time. I (myself) had began to feel dread, unable to accept the situation;
unable to stand the bone-chilling fear of standing all alone against the crowd, that particular Mad Crowd
which Robert Pirsig (in "Zen and the Art of Motorcycle Maintenance") had called "The Church of
40
Reason"...
I.e., if my "Multiple Form Logic" was correct, then much of what we were being officially taught (in
Logic) had to be seen -officially- as wrong. But if that, which we were being taught was to be seen
officially as wrong, then there would no longer be an Absolute (or God-given) Academic basis, for
teaching it in the first place. (This, was the "Logic between the lines"... )
What about the Post-Graduate Course in Manchester? (I hear you asking).
A few years before these sad events happened in Essex, I had reluctantly rejected (for health reasons,
since I had been beaten up badly, by a bad coincidence, at around the same time) Professor Cliff
Jones's offer of a Post-Graduate Course in Computer Science and Logic at the University of
Manchester.
Professor Jones's exceptional offer, which had bypassed University Regulations in my favour (since I
did not have a first degree) was unfortunately NOT repeated by Manchester University on the following
academic year (1985/86), when I had finally recovered completely from the violent incident (I will write
about this incident elsewhere, in the near future), and felt ready to work and study.
NOTE: I still keep the original correspondence between me and Professor Jones intact, as evidence that
this is all true.
Well, having irrevocably lost the exceptional "once-in-a-lifetime" offer of being a Post Grad student
without a first degree, I settled for a humble undergraduate course instead, in Essex University, UK.
Unfortunately, Essex eventually became the ceremonial... Execution and Burial Ground for the Theory
of Multiple Forms (after an initial "sweet period" of about two years, during which many members of
academic staff were intrigued and impressed by it ). Since "nobody" (among Essex academic staff)
"could prove it wrong", and since "the only person who had authority to prove it correct went away", it
was thrown to the dustbin, in the very last minute.
As a result, I did not finish the (other) dissertation, which had been dictated "from above", which was
irrelevant to my interests and which I had never chosen) ...in time for the Examinations!
So, I failed! (More precisely: They "failed me").
NOTE: In the UK, at the time, there was effectively no "second chance".
<Soapbox, Personal Life Story: OFF>
( to be continued, allegro con brio, molto libero, con techno / new age etc. - ad. lib, ad. inf)
41
18) Extending and improving the bit-crunching Algorithm “Iphigenia”
Algorithm “Iphigenia” was invented in 1987 and implemented as a demo program (“bitprover”) in 1990. Soon
afterwards, it became evident that the dream of creating optical computing hardware using this algorithm (in
cooperation with Dr. Nikos Vainos) had to be postponed indefinitely due to lack of funds. So, all further work on
this algorithm was suspended, while I became involved in other types of work (article writing and software
dictionary development). However, in recent work reviving this algorithm, it became evident that the range of
logic expressions, which can be stored as “bit-patterns for logic deduction”, can be extended in a number of
ways. For example:

Expressions of the form “(A and B and C and…) => (a and b and…)”: These can be converted to
multiple (AND-ed) expressions of the form:
“( (A and B and C and…) => a) and ((A and B and C and…) => b) and… “.
In Multiple Form Logic this can be massaged by distributivity (e.g. theorem T5), as follows:
A and B and C… ) => (a and b and…)))#1 becomes:
(A#1,B#1,C#1, (a#1,b#1,..)#1)#1 which becomes:
(A#1,B#1,C#1,a)#1, (A#1,B#1,C#1,b)#1, … (of the same form as the original logic expressions allowable
in the algorithm’s bit-encoded knowledge base, i.e. “(X and Y and Y) => a or b or… “).

Expressions of the form “(A=B) and (C=D) and… ) => etc.” (with logic equalities such as X=Y) can be
converted by turning each equality X=Y into a composite expression ((X#1,Y)#1,(X,Y#1)#1)#1 and then
treating each term as a separate expression (AND-ed with the rest).
E.g. To prove an implication in Equational Boolean Logic: ((A=B) and (B=C)) => (A=C), first we translate
the premises “((A=B) and (B=C))” into:
(A#1,B)#1, (A,B#1)#1, (B#1,C)#1 , (C#1,B)#1, i.e. (in bit array format):
100
010
010
001
010
100
001
010
(A#1,B)#1
(A,B#1)#1
(B#1,C)#1
(B,C#1)#1
And then we can apply two independent logic queries: A=>C and C=>A, to evaluate “A=C”:
100
010
010
001
100
010
100
001
010
001
(A#1,B)#1 ß “vertical cancellation rule” RVC
(A,B#1)#1
(B#1,C)#1 ß “vertical cancellation rule” RVC
(B,C#1)#1
(A#1,C)
ß logic query (1): “A=>C ” ?
100
010
010
001
001
010
100
001
010
100
(A#1,B)#1
(A,B#1)#1 ß “vertical cancellation rule” RVC
(B#1,C)#1
(B,C#1)#1 ß “vertical cancellation rule” RVC
(A#1,C)
ß logic query (2): “C=>A” ?
(using two independent copies of the bit-array knowledge base). The results are easy to get:
000
010
010
001
100
010
100
000
010
001
(A#1,B)#1 ß isolated bit - row
(A,B#1)#1
(B#1,C)#1
(B,C#1)#1
(A#1,C)
ß logic query (1): “A=>C ” ?
100
010
010
000
001
010
000
001
010
100
(A#1,B)#1
(A,B#1)#1
(B#1,C)#1
(B,C#1)#1 ß isolated bit - row
(A#1,C)
ß logic query (2): “C=>A” ?
42
The “next generations” (after “transferring isolated bits diagonally” by rule “RDT”) are:
010
010
001
110
100
000
010
001
(A,B#1)#1
(B#1,C)#1 ß “zeroed row” (i.e. result = “TRUE”)
(B,C#1)#1
(A#1,C)
ß a new bit transferred diagonally
100
010
010
011
010
000
001
100
(A#1,B)#1
(A,B#1)#1 ß “zeroed row” (i.e. result = “TRUE”)
(B#1,C)#1
(A#1,C)
ß a new bit transferred diagonally
So it was straightforward to answer the query “A=C”, proving it true (given “A=B and B=C”).
The range of logic expressions which can be converted to Iphigenia-compatible bit-encoded arrays can
of course be extended further. Readers familiar with PROLOG and Horn Clause Logic may have realised
that each “bit-encoded row”, in such a logic system, resembles a Prolog Clause of the form
Goal <= A and B and C...” - except that (at this stage) such expressions do not contain variables. Now,
since (almost) any logic expression can be translated to Horn-Clauses, it can also be translated into
“bit-array patterns” for bit-crunching algorithm “Iphigenia”.
The only real problem is how to deal with variables and predicate logic expressions, rather than
propositional logic expressions. Well, here are some crucial steps, on the road to arriving there:
19) Elementary Variables and Unifications in Multiple Form Logic™
Suppose we express a set of equalities (e.g. “A=B and C=D”) in Multiple Form Logic. Such (AND-ed)
bundles of logic equalities, become Multiple Form Logic expressions, such as “(A#B, C#D) # 1”, i.e.
inside a Universal Boundary (“1”) are gathered together a number of XOR-ed terms, which can take a
number of possible values, by consecutive applications of Axiom 3 (followed by Axiom 2):
The proposition “A = B and B = C” becomes: (A#B#1#1, B#C#1#1)#1, i.e. (by Axiom 2):
(A#B, B#C)#1
And this can also be rewritten (by inserting “A#B” into “B#C“, and cancelling out “B#B”):
(A#B, A#C)#1
as well as:
(B#C, A#C)#1 (by inserting “A#C” into “A#B“, and cancelling out “A#A”):
(and so on)
In such examples it becomes evident that “equational logic implications”, such as “A=B and B=C) =>
(A=C)” are implicitly valid by default, due to the interpretation of equality “A=B” as “A xor B xor 1”, in
Multiple Form Logic. Furthermore, it becomes evident that certain types of unification are implicitly
contained within Multiple Form Logic, “as if” they are hidden (and -so far- unused) potentialities.
E.g. suppose we encode PROLOG-like Predicate-expressions, as “sets of equalities”, between a
functor’s “distinct arguments” and certain “values” (which can be known or unknown):
“F(X,Y)”
is encoded as “(F1=Y) and (F2=Y)”, i.e. (in Multiple Form Logic): “(F1#X, F2#Y)#1”.
Suppose now that ( given “F(X,Y)” ) we wish to evaluate the “query”: F(2,2).
43
Now , expressing the fact and the query together, in Multiple Form Logic, we get:
( (F1#X, F2#Y)#1, X#2#1, Y#2#1 )#1
i.e.
( F1#X, F2#Y, X#2, Y#2 )#1
Now, this expression (as before, by axiom 3 and axiom 2) is “almost by default” equivalent to
several possible (in fact, ALL possible) “unification-containing variants” such as:
(F1#2, F2#2, X#2,Y#2 )#1
ß in conventional logic: (F1=2) and (F2=2) and (X=2) and (Y=2)
(F1#2, F2#2, X#Y,Y#2 )#1
ß in conventional logic: (F1=2) and (F2=2) and (X=Y) and (Y=2)
(F1#Y, F2#Y, X#Y,Y#2 )#1
ß in conventional logic: (F1=Y) and (F2=Y) and (X=Y) and (Y=2)
(and so on...)
I.e. it looks like we’re getting all the possible unifications as a “bonus” quite effortlessly!
This process, of transliterating PROLOG-like predicates, such as “foo(Arg1,Arg2,...)” into
Multiple Form Logic “sets of equations” such as “( (foo1 # Arg1), (foo2 # Arg2), ... ) # 1” can
be extended in many ways, to arrive at the implementation of Prolog implication engines and
Prolog interpreters with an “inherent parallelism” (using algorithm “Iphigenia”).
However, this specialised application of Multiple Form Logic will not be pursued more, here,
within this introductory presentation: One reason is that it’s still not complete (at the time of
writing). Another reason is that it’s an extremely hot topic, from an industrial point of view.
Contact me privately if you are an employer (or venture capital
owner) interested in commercially viable and profitable results.
E.g. one area where these techniques can perhaps be applied successfully, is the area of
Rules associated with “Ontologies” and “Information Extraction”: Existing “Ontologies” (e.g.
about DNA structure) can be annotated with expert-system rules, expressed as bits (with
variables this time) and manipulated by Assembly language implementations of the “Iphigenia
Algorithm” to achieve simplifications and/or derive intermediate results, which simplify the
Ontology’s Knowledge base.
The inherent parallelism of such processes also suggests that we can get “Prolog inference
engines” working in parallel, or in ways that outperform existing “WAM” implementations.
What we are doing here, is an Formal Logic analogue to “RISC”, except that our “reduced
instruction set” consists of three Logic Axioms applied to Boolean operators OR and XOR.
44
APPENDIX A
Self-Biographic Humour
(but all the hyper-links are serious)
Purpose of this Appendix: Surrealist / Post-modernist Metadata Self-Profiling Art Manifesto
Earthling’s official Earth-name: George Alexander Stathis
Web: http://www.geocities.com/omadeon .
Web site motto: “Renaissance = (Art + Technology) 2”.
Curriculum Vitae: http://www.geocities.com/omadeon/CV.html .
Source-code: http://www.geocities.com/omadeon/gs_sourcecode.html .
Identity: Alien (Atlantean humanoids’ descendant) temporarily “Greek resident” ;-)
Professions: Logician, programmer, electronic dictionary maker, music composer, painter,
cartoonist, writer, worker, translator, sculptor, poet, etc.
Drawings: http://www.geocities.com/omadeon/drawings.html
Collages: http://www.geocities.com/omadeon/_collages.html
Paintings and Sculptures: http://www.geocities.com/omadeon/art.html
Writings and poetry translations: http://www.geocities.com/omadeon/gs_writings.html
Software dictionary “HyperLEX”: http://www.geocities.com/omadeon/hyperlex.html
Samples of electronic music composed (mp3):
Rebirth of the Dolphins (genre: New Age/Trance, CD: "Magic of Atlantis",
2000). Christodoulos Techno (genre: Satire / Techno, 2002).
Music Knick: “O M A D E O N”. (In Greek, "omada" means "team" – I am a team-worker).
More recent music works: CD "Children of Atlantis" (14 danceable tracks), CD "777" (9 songs).
Place in World Top 40 Music Charts (of I.U.M.A): 6th Place (no joke; this is trueueueue….)
(Summer 2000, web-page copy: http://www.geocities.com/omadeon/charts2000.html)
Process of “Life-Cycle Erasure”: Ended; rebuilding M.F. Logic memory successfully.
DNA: Greek, officially: 100% ;-) Defective DNA: Less than 0.00003% (ISO2003-compliant).
Formerly Resident in: London, Essex, Oxford (UK); Alexandria (Egypt); Switzerland, etc
Memory of previous Abductions by Aliens: None reported. ;-)
Chronological Age: 49.
Biological Age: stabilizing at 35 “bio-years”, anticipating Anti-Aging Mutation in 2007. ;-)
Anti-aging medicines: None. (a natural Vitamin user, from supplier who makes them).
(Pre-)destination: High Energy Global Creativity due to “Selfless(?) Love for the Universe"
Drugs: None. (100% a "Natural High person"; experienced Winter Swimmer, etc.)
Religion(s): Buddho-Christian, Neo-Pagan, Post-Atlantean, under Self-Debugging X-phase.
Spiritual source-code: under major self-upgrade (in personal dream-space, etc.)
Spirit’s Origins: City of Atlantis and... Constellation of Sirius, approx. 3000 B.C. ;-)
Astral Memory Data: Chose my own parents (reported from “Akashic Dreamscapes”) ;-)
Favourite Comedians: Woody Allen, Jimmy Panousis.
Disability: Can’t fly in Astral Planes, the way others do, whenever they’re acting so stupid.
......
What? - Would I hire someone with THIS personal profile? - Oh Yes, most definitely!
45
APPENDIX B
“LAWS OF FORM”
oil painting (50x70 cm.)
(inspired by George Spencer Brown’s Axioms of the “Primary Arithmetic”)

History: I created this painting during the early phases of contemplating "Laws of Form". The original
painting was sold (to an unknown client) at a one-man Exhibition in "New Thought Gallery" (Athens,
May 1980). The blue pyramid in the painting depicts George Spencer Brown’s two axioms of the
"Primary Arithmetic", in "Laws of Form", corresponding to Multiple Form Logic statements "1 # 1 =
0" and "1 , 1 = 1", or to the Boolean expressions "1 xor 1 = 0","1 or 1 = 1").

This poor quality slide (taken with an instamatic camera) is all that remains today:
LAWS OF FORM
Jules Henri Poincaré quotation: A scientist worthy of his name, above all a mathematician,
experiences in his work the same impression as an artist; his pleasure is as great and of the
same nature. (quoted in N. Rose “Mathematical Maxims and Minims”, Raleigh N C 1988).
46