Wikipedia’s new references to Multiple Form Logic, etc.
“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 http://multiforms.netfirms.com, which is still valid, although there is now a preferred mirror-site, hosted in my personal domain: (http://omadeon.com/logic).
These Wikipedia references were apparently written fairly recently, 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)
Furthermore, leaving aside my Theorem Proofs in Multiple Form Logic and More Theorem Proofs (about William Bricken’s system, etc)…
- Mr. Ralf Barkow (a Computer Scientist) showed some interesting possibilites for unifying Multiple Form Logic with the Pile System (invented by Erez Elul).
- 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 beginning – i.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 the “Laws of Form Forum“archives).
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 theologize: –Is 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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