(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 qualification of a first degree (B.Sc.).

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:

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 informative 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^™:

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) Universal 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 (well-known) 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).

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).

4.      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. 

5.      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.

 -

 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 early eighties) I had intrigued Dr. Tassos Patronis (a friend, who was also my informal tutor in Formal Logic) by producing a strange 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":

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)    - as 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”:

 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.

(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:

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".

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.

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:

(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 1. 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 very 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 logical "One"; such expressions are treated as "relatively true", and they are simplifications of the original formulae (which produced them). Thus, our "Multiple Form Logic 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.

(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, the 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 some time before it will be ready and you can download it from this site).

Main download location:   http://www.omadeon.com/logic/mflogic.zip ( 540 Kb )

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).

However, once you’ve satisfied your (logic theorem-proving) curiosity, read on:

next section> The (Logic) Truth is Out There, Maulder: Right in front of your eyes!

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