(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 so than practically. For instance, he never published his alleged proof of the Four-colour 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 course- pure 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 XOR-operator 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):

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):

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!

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