Redefining Multiple Form Logic for «PILE» (English text with Greek abstract)

Summary in Greek: Το Μάϊο του 2006 ο Ralf Barkow έγραψε άρθρα στο blog του για μία πιθανή σχέση μεταξύ της «Λογικής Πολλαπλών Μορφών» και του Σχεσιακού Συστήματος Οργάνωσης Πληροφορίας «PILE». Ακολουθεί ένας επανορισμός της «Λογικής Πολλαπλών Μορφών« έτσι ώστε να διευκολυνθεί μία πιθανή ενοποίησή της με το σύστημα «PILE».
Summary in English:
In May 2006, Ralf Barkow wrote articles in his blog
about a possible relationship between «Multiple Form Logic» and a «Relationist information system» called «PILE». In this article, a redefinition of Multiple Form Logic will be attempted, to facilitate the possibility of unification between this Logic and the PILE system.

Multiple Form Logic is an enhanced generαlisation of George Spencer Brown’s «Calculus of Distinctions» (expοunded in his book «Laws of Form»). English readers who are haven’t heard about «Laws of Form» may browse the Laws of Form site (maintained by Dick Shoup). I also wrote a short Greek introduction about Laws of Form and Multiple Form Logic, here. However, the new presentation of Multiple Form Logic in this article, will be attempted differently, based on simplified and radically different principles, which are -perhaps- more appropriate for unification with the Pile system.

Τhe philosophical foundation of Multiple Forms is a simple property of every sentient being’s Inner World, traditionally often identified as the realm of «the imagination». In this Space, which is traditionally regarded as being «inside ourselves», the Outer World is introjected by perception. This fundamental activity of introjection or internalisation, which is also the essence of perception, can be illustrated graphically through the following diagram, which depicts Inner Space «before» and «after» a particular internalisation of the Outer through a «Boundary of Perception» B (distinguishing between Inner reality A and Outer object X):

axiom3.jpg

 «What is real we may imagine«

Οn the left-hand-side of the above diagram, the red circle marked «B» represents a boundary of perception, the blue human labelled «A» represents an arbitrary «Inner Reality» (which does not exist «outside ourselves») and the object labelled X is an arbitrary External Object of the «Outer World, which exists «outside the boundary of Perception».

On the right-hand-side of the diagram, we see the same situation, after Outer Object X is «internalised» or «introjected» by the Inner World (or by the realm of «imagination»): An exact replica of the external Object X now exists also inside the «Inner World».

Τhe Fundamental Principle of Multiple Form Logic is:

  • Τhe two sides of the previous diagram (or «equation») are equivalent, in the sense that the two states are interchangeable. I.e. if we read the diagram in the reverse direction, it is always possible to «cancel out» (from the Inner World) any «internalised object X», iff (if and only if) it also exists on the Outer World. I.e.

axiom3b.jpg

«Whatever is real we may imagine,

but need not imagine what is real«

Τhe «next step» in this reformulation of Multiple Form Logic, a formally complete and consistent basis for Propositional Logic as well, in fact, which -however- does not require any more connections to «Laws of Form» (but may facilitate unification with «PILE») is:

  • The «perceptual boundary» B is a (potentially unlimited) construction of further «acts of perception»

  • In other words, philosophically speaking, our «perception» is precisely (nothing but) such «acts of perception», constructed and accumulated (by «the Mind») without limit:

  • Our «perception» is the «boundary of perception», constructed by an unlimited number of (such acts of) perception; no less and no more than all this.

The consequences of this «next step» are far-reaching: First of all, we can new proceed to express a formal expression of this «step», combining it with the Fundamental Principle:

  1. Let the symbol «B / X» denote an «act of perception», where B stands for a set of «acts of perception» forming a «boundary of perception» and X denotes the «content» or «Inner Part» of B, where (X) is -of course- further acts of perception

  2. To signify multiple «acts of perception» that co-exist inside a given boundary of perception (side by side) we may use brackets or parenteses, as required.
  3. Then the Fundamental Principle of Perception, together with the «next step» (expressed earlier) can be re-written more formally as follows:

X (B/A) = X (B/(A X))

(where «B» is any expression of the form «α/β»).

Consider now a particular boundary of perception P containing other acts of perception X, such that P is equal to itself (as the entire expression):

(P/X) = P

Then, this «P» can be re-written as:

((P/X)/X) = P
(((P/X)/X)/X) = P
[ . . . ]

This «P» is an «infinite regress» where all the (above) expressions are equivalent, since there is no limit in (more and more complicated) re-writing of the same expression (P) by replacing each P with each entire expression, as itself. For the moment, however, let’s not delve any deeper into this phenomenon, except to mention -briefly- that it may be formally associated with a «self-referent» or «imaginary» state of perception.

Some notes now, before we go on:

  • The term «imaginary» has been used by some authors for similar self-referent states of «infinite regress», but it will be avoided in Multiple Form Logic, since the realm of imagination and the contents of perception are regarded as philosophical «cousins«, and since imagination is not necessarily self-referent; so the «self-referent state» is a more appropriate name that can be used for the state of «infinite regress», later on…
  • It is important to remember that a «boundary of perception» can be (effectively) a composite «conglomerate» of multiple acts of perception,nature of all such boundaries is the same as the nature of their contents, without any distinctions of «type». I.e. without limit, and that
  • The conceptual space of Multiple Forms is inherently «flat», just like «Pile objects»: All Multiple Form Logic entities or boundaries are «equal citizens».

Summary of material that follows later on, in this blog:

  1. The Construction of Logic truth-value «1», through «pseudo-axiom 1», as a Universal Distinction, which is the Totality of all Possible Distinctions in all Possible Universes
  2. The Assumption of Cancellation («pseudo-axiom 2»), as the Voidness of a Distinction distinguishing itself
  3. The derivation of all Propositional Logic (of order Zero) as a result of these assumptions
  4. A demonstration that George Spencer-Brown’s calculus, as well as William Bricken’s Algebra, are special instances of this (more general) system of Logic (upgrading previous material here).
  5. A discussion of alternative assumptions (instead of «pseudo-axioms» 1 and 2) leading to entirely new («relativistic») Alternative Logic Consequences.
  6. I have translated «Laws of Form» in Greek many years ago, but could not persuade any Greek publisher to publish it (at the time). George Spencer-Brown’s (scanned) type-written authorisation to publish my Greek translation of his book, stating his required copyright royalties may soon be scanned and placed on-line, as a historically relevant document (for George Spencer Brown’s biographers)

(to be continued)

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