Wikipedia’s new references to Multiple Form Logic, etc.

Inclusion-exclusion illustrated for three sets
Image via Wikipedia

Multiple Form Logic is my own extension of George Spencer-Brown‘s Laws of Form. Wikipedia’s recent references to it have been good news. They include two links to my older site, which is still valid, although there is now a preferred mirror-site, hosted  in my personal domain:  (

These Wikipedia references were apparently written by some (unknown) people (probably researchers from the official Laws of Form Forum). Here they are:


The Multiple Form Logic, by G.A. Stathis, “generalises [the primary algebra] into Multiple Truth Values” so as to be “more consistent with Experience.” Multiple Form Logic, which is not a boundary formalism, employs two primitive binary operations: concatenation, read as Boolean OR, and infix “#”, read as XOR. The primitive values are 0 and 1, and the corresponding arithmetic is 11=1 and 1#1=0. The axioms are 1A=1, A#X#X = A, and A(X#(AB)) = A(X#B).


Τhe Multiple Form Logic, by G.A. Stathis, owes much to the primary algebra.

BTW, in case you got the… wrong idea(!) I did not write these references! Oh no! 🙂 Besides, although I  feel greatful towards these… nice people, I do not agree with… everything they wrote! 🙂 E.g. as regards their reference to Multiple Form Logic, which is not a boundary formalism”… I believe that -on the contrary Multiple Form Logic IS a “boundary formalism”; a very fundamental and radical one, in fact! My view is also that Multiple Form Logic changes  the way we think of boundaries, as such; enhancing the ontological nature or (if you prefer) the existential fabric of  (our Reality consisting of -) boundaries, in (at least) two ways:

  • 1) To start with, boundaries are multiple (rather than the Unique, one-and-only Form in “Laws of Form”). Other multi-valued extensions to “Laws of Form” have been proposed by others, e.g. Ben Goertzel’s “Ons Algebra”.
  • 2) Secondly, to our amazement -perhaps- boundaries are also (or can also be) mathematical “first class citizens”; i.e. boundaries are entire expressions (inside their own  system) themselves! (This entails  a conceptual difficulty of a need for a paradigm shift, as well as a practical difficulty of  visual representation; a challenge for Visualisation software; e.g. “DreamProver”  -my own visual theorem prover, totally un-funded, delayed, long overdue; more about it… soon! – hehe)

dreamprover430x.gifDream-Prover Snap-shots (click to see a short article about it)

Furthermore, leaving aside my Theorem Proofs in Multiple Form Logic and More Theorem Proofs (about William Bricken’s system, etc)

  • Some cool alternative proofs of Multiple Form Logic theorems have been produced by Mr. Art Collings, a professional mathematician.

Now, although Mr. Art Collings is  an uncompromising critic of my work (!) I think that his criticisms have been of immense benefit! His professional mathematical expertise has helped me clarify some very important issues, while his objections to some of my (occasionally… wild) philosophical claims (hehe) have been extremely valuable and thought-provoking, despite the fact we (usually) don’t agree. E.g. Some time ago he had expressed strong doubts (HERE) about the “completeness of Multiple Form Logic, saying  that it would be (probably) impossible to prove. However, soon afterwards, I produced a verifiable formal proof  of  the contrary, which he then verified (which… was nice of him). He  subsequently produced his  own (different) proofs, after doing  some research. Interestingly, these communications took place (more-or-less) in public: -In the (semi-)public “Laws of Form Forum“, a Yahoo Group where William Bricken also participates. Some interesting logical problems have been solved (with mathematically sound answers) in those public communications. However, there is still, a… tiny open problem:

  • My proof of Theorem T12 (here), stating that William Bricken’s “Boundary Algebra” is “a special instance of Multiple Form Logic” is still not accepted by William Bricken himself, on (more-or-less) philosophical grounds!Nevertheless, this proof has been accepted by other researchers, e.g. Ralf Barkow. Also, Mr. Tasos Patronis (Ph.D),  a Greek mathematician who -I must admit- is also a good friend of mine (OK, so maybe he is a bit biased -hehe)

  • The real problem however, in this case, is not the proof itself (which is undoubtedly consistent) but whether or not certain philosophical and mathematical (meta-)criteria are also satisfied, validating my… wild claim that Multiple Form Logic is more generalised and more fundamental (as a “theory of boundaries”) than Bricken’s  “Boundary Logic”.

William Bricken’s main objection is that Multiple Form Logic is a “higher abstraction” than his “boundary logic”. So, philosophically (he argues) it would be wrong to regard it as “more fundamental” than his system. However, my contentions are:

  • 1) that this is not a drawback, but a formal advantage (since Bricken’s logic follows, as a “special instance” -provably-only one type of Form in precisely by assuming M.F. Logic) and

    2) that all Forms are multiple – from the very beginningi.e. that Multiplicity is fundamental, in this Universe!…( Go(d) figure… 🙂 )

Ah well – all this is a loooong  story, beginning here:

(and the rest is in theLaws of Form Forumarchives).

P.S. Art Collings (much to the delight of William Bricken) made use of the (well-known) fact that the “XOR” relation can be re-expressed as a composite expression containing only ORs and NOTs, to prove that one can -indeed- construct (without adding any new axioms or unproved assumptions) a Multiple Form Logic system by using the XOR relation as a new abstraction,  inside Bricken‘s system. However, it appears that Bricken’s system itself follows (provably) as a special instance of Multiple Form Logic, if (and only if) all different forms are fused into one .

-So… which is the “chicken” and which is the “egg”?

  • Well, devotees of the “Simplest Egg” in the Universe (that can probably make the smallest… omelet) say “boundary logic” is “more fundamental”.
  • Proponents of Simplicity… NOT necessarily being associated with “the One” but probably (a) being of a strange new Fundamental Quality (that can be) called “A Priori Multiplicity”… can say that (omelets being in need of many eggs, anyway) Multiple Form Logic is “more fundamental”.

Still others, may blatantly theologizeIs there one God, or many Gods?

  • Hm… does it… matter, how many Gods you imagine? 🙂

– Ah Well, …It may turn out, actually… . . .  . that it doesn’t matter!!! 🙂

  • What does appear (to matter) however, is not “the number of Gods” but the number of… sacrifices(!):
  • Worshipping the Logic of Only One God (=Truth Value) …it turns out that you need an exponentially larger amount of Deductions (Proof steps, i.e. logic computation) than for the Logic of Multiple Gods ….er… Forms. More about this astonishing fact will be explained further, during the course of events to come. (Appetizers are here).

I rest my case…  🙂

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~ by Omadeon on February 14, 2009.

7 Responses to “Wikipedia’s new references to Multiple Form Logic, etc.”

  1. what could be done diffrently in multiple form logic.
    if it’s just another way to interept boolean logic. what the use of it?

    in order to stand on it’s own as a field of resarch in mathematics.
    it should have consequences outside it’s own borders. so as to be considerd as a “method” a “new way” or somtething of that sort.
    it at least have to be sepearte, in the sense. that a full mapping of the field
    to another well understood field could not happen.

    but lof could be mapped to boolean logic
    and so do multiple form logic.
    so as a field of mathematic they are useless.

    still it has some justification as a philoshpy
    or am i wrong?

  2. Well, ever since 1969 when G. Spencer-Brown first published “Laws of Form” these objections (or very similar ones) have been raised, again and again. There is SOME element of truth in objections of this type, which can be a big or a small element, depending on one’s viewpoint, and -especially- on WHAT is considered to be of higher priority.

    It is e.g. shortness / conciseness of proofs (lesser proof-steps)?
    Is it Number of redundant axioms?
    It is exponential growth in certain expressions depending on their number-of-terms?
    It is a philosophical issue?

  3. I will come back on this in much more detail, as it’s perhaps the most important reason that Laws of Form was not adopted by others as a mainstream way of doing Logic (even though Bertrant Russell – among other great people) had expressed his wholehearted approval of it.

  4. actul math isn’t done by the axiom-inference schemea but in a rather free style. it is said that the axiom-inference scheme capture the nature of this sort of natural activity which is in itself a statement that could not have exact meaning before it is translated into some other formalism.

    a proof is only formally a list of statment which are axioms theorems or statement arrived at by one elementary step on any former of the last in which the last statement is the conclusion.

    something like this (take or leave a word)

    it is actully something very diffrent.

    what do you mean by “higher priority”?
    offcourse if lof is to be considerd as a theory at all.
    it shouldn’t happen that a reduction to another theory is possibole.

    this is so obvious principle that no one bother to mention it.
    but offcoure. a new theory should be new.
    if you want to have a new way to represent and manipulate in the usual manner matrixes,number,shapes,functions, or anything else.
    that’s ok. but its not a new theory. its just another interpetation
    of an exsiting theory.

    you want to say that this interpetation is in some way better.
    but it is actully irrelevant as long as the “space of meaning” is the same as another that of another system (boolean algebra)

    in the terms of G. Spencer-Brown what you need here is a distinction between lof and b.a which couldn’t be made

  5. Again, your objections are (in principle) correct; many other people have expressed similar objections, in the last 30 years or so…

    What I tried to do, partly, is to attempt a GENERALISATION of LoF which hopefully answers such objections.

    However (above all), I look at these issues very practically, from the point of view of more elegant, computationally less “expensive” theorem – proving.
    If LoF (or my own generalisation of it) can be regarded as yet another interpretation of Boolean Algebra, this is (to a certain extent) irrelevant, if THIS interpretation (including the specific computational _methods_ enabled by it) leads to gains in elegance or lower complexity of proofs.

    It MAY be possible, that Boolean Algebra itself is supplemented with such methods, without resorting to alternative theories or alternative interpretations (such as LoF or Multi-form Logic). However, it may also turn out that Multiple Form Logic is more general, a system where Boolean Algebra is only a special case. By relaxing the system of axioms this appears possible, though not yet certain, by any means.
    E.g. if you consider that logic “1” is only a _construction_, NOT a “given”, and if you change the 3 axioms to only 2 (or even 1, the only “necessary one” which is axiom 3) then such issues may be clarified…

    However, “shorter and more elegant proofs” are not the _only_ benefit of THIS “interpretation” of Boolean Algebra; there is also elegance of principles, axiomatic conciseness (reminding us of Occam’s razor, applied to Boolean Algebra).
    – WHAT is the most minimal set of axioms for Boolean Algebra, which ALSO ensures the most elegant, minimal computational effort in the consequences / theorem proofs? -It may turn out that THIS is LoF (or, indeed, M.F.Logic).

    My intent, in coming weeks, is to finish (at last) the remaining work forDream-Prover“, a theorem-proving engine (software) that uses Multiple Form Logic and it also automatically displays (in visible, graphic form) the proofs generated by it. Now, _if_ this work can be seen as (yet another) Boolean Algebra application, e.g. akin to SAT-provers (which can prove almost anything, as you know) then… so be it. I am NOT going to get…defensive, about this! 🙂

    We shall find out, EXACTLY what is going on, during the next few months, hopefully.

  6. suprisingly, i’m satisfied with your answer.
    but if your work is to be concerned with problem of idealizying method of proofs you should make it more apparent as part of the theory

    i have recently scribbled here
    about 3 state boundary arithmatic.

  7. @οευ
    Well, I am very pleased to have answered your valid objections with honesty and clarity!
    Also, very interested in your “scribbles”. It’s nice that you are familiar with the “lawsofform” Yahoo Group, evidently also a member, since you posted something there.

    So, I would suggest that you raise similar questions inside this Yahoo group, itself. It’s full of very talented and qualified people (a lot of them much more so than myself) and their answers are likely to be even more satisfying! 🙂

    (I haven’t checked out the group myself, in recent weeks; will do so in a minute…)

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