Sunday, June 24, 2018

Part 12: Seldon Presents Series -- ‘ ‘Oppositional Products’ and ‘‘‘Products of Oppositions’’’, Boolean vs. ‘Contra-Boolean’. ’










Part 12:  Seldon Presents Series --
  
Oppositional Products and ‘‘‘Products of Oppositions’’’, Boolean vs. Contra-Boolean. ’







Dear Readers,



It is my pleasure, and my honor, as an officer of the Foundation Encyclopedia Dialectica [F.E.D.] Office of Public Liaison, to share with you, from time to time, as they are approved for public release by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.

The twelfth such release in this new series is entered below [Some E.D. standard edits have been applied, in the version presented below, to the direct transcript of our co-founder’s discourse].


For more information regarding, and for [further] instantiations of, these Seldonian insights, please see --




ENJOY!




Regards,


Miguel Detonacciones,

Member, Foundation Encyclopedia Dialectica [F.E.D.],
Participant, F.E.D. Special Council for Public Liaison,
Officer, F.E.D. Office of Public Liaison.







...Perhaps the starkest way to portray the difference -- even the opposite-ness -- of the original Boolean algebra in relation to our contra-Boolean’ algebra, is this:

(1)  In Boolean logic, the generic, algebraic representation for physical opposites [primary propositions] or for propositional opposites [secondary propositions] is the following:   

(x)  vs(1 x).”


“In Boolean logic, there is no reconciliation of specific opposites, or of opposition-in-general. “

“The product of two opposites produces precisely NOTHING, 0 -- a ‘‘‘lose-lose situation’’’ in which both sides are lost --

(x)  x  (1 x)  =  0,

which is also the Boolean expression of Aristotle’s ‘‘‘Law of NON-Contradiction’’’;

This expression also algebraically implies that --

x - x2  =  0,

which further algebraically implies --

x  =  x2

-- which is algebraically equivalent to --

x2  =  x

-- which is Boole’s Fundamental Law of Thought, or Law of Duality.  It
implies that logical NON-linearity reduces immediately to absolute linearity.
There is no expanded reproduction of ideas/memes modeled here.
For Boolean logic, there is only the ‘‘‘simple reproduction’’’ of ideas/categories        /classes
[Boolean ‘‘‘self-(s)election’’’ is gainless].”


(2)  In the Foundations contra-Boolean, dialectical Logic, on the contrary,
  
x2  =  x + Delta x,  

modeling a qualitatively expanding reproduction of [ideo-]ontological categories, via the second term on this equations Right-Hand Side, Delta x, as well as conserving the Boolean moment via the recurrence of the x variable as the first term, also on the Right-Hand Side of this equation.  The product of categorial opposites always possibly yields, again, yes, the ‘‘‘simple reproduction’’’ of those opposite categories as free-standing, unreconciled opposites, but also, plus [+], a possible reconciliation, a ‘‘‘synthesis’’’, or ‘‘‘complex unity’’’, of the two thus become one, a <<tertium quid>> category-unit named --
q       Delta x; x.   

Using the meta-genealogical evolute product rule axiom, we have --
x x Delta x   =   x + Delta x + q            Delta x; x .























Tuesday, June 19, 2018

Introducing ‘The Hegelian Triadic-Tetradic Product Rule’ for Seldonian ‘Dialectical Algebras’.







Introducing The Hegelian Triadic-Tetradic Product Rulefor SeldonianDialectical Algebras.







Dear Readers,


Hegel wrote, in Part I [The Science of «Logik»] of his Encyclopaedia of the Philosophical Sciences, as follows, regarding the ‘‘‘numerology’’’ of the rational [«vernunft»] division of contents, or of ‘‘‘subject-matters’’’, in the “philosophical sciences” -- Any division is to be considered genuine when it is determined by the Concept.  So genuine division is, first of all, tripartite; and then, because particularity presents itself as doubled, the division moves on to fourfoldness as well.[G.W.F. Hegel, The Encyclopedia Logic, translated by T.F. Geraets, W.A. Suchting, and H. S. Harris, Indianapolis:  Hackett Publishing Company, Inc., 1991, p. 298, §230, Addition [«Zusätze»]].

While we of Foundation Encyclopedia Dialectica [F.E.D.] in no way wish to claim any Hegelian orthodoxy for any of our work, let alone to import any of Hegel’s ‘‘‘massive’’’ mystifications into that work, we call attention, here, to the passage above, because of certain resonances between Hegel’s description, in that passage, of his version of dialectical «diairesis» [ontological-categorial division, per Platon], and some recent results of our ongoing research into possible alternative axioms, for Seldonian dialectical algebras, that go beyond our classical ‘‘‘Seldon Functions’’’, dyadic and triadic alike, in their efficacy for dialectical meta-modeling.

Those recent results center upon what we are provisionally calling The Hegelian Triadic-Tetradic Product Rule, which is showing signs of more apt dialectical meta-modelingwith regard to a certain super-Domain of dialectical instances, both at the ontological-categorial level, e.g., at the «species-arithmos» level, and also, with unprecedented efficacy, in our work, at the «monadic» level, that of ‘‘‘individual historical objects’’’.

I have included, here, below, a pictogramical/ideogramical/phonogramical illustration, in the form of a new kind of E.D. dialectogram, applied to a particularly relevant case in point:  that of the currently consensus-hypothesized cosmo-historical genesis of the Earth-Moon system in which we have our existence at this very moment.

As an axiom, that can be substituted for axiom §9 of the core axioms of the NQ axioms-system for the Seldonian First Arithmetic for Dialectic, the formal essence of this Hegelian Triadic-Tetradic Product Ruleis the following --

A.  If the current ontology-state of the dialectic is expressed via a single dialectical meta-number value, call it

qn,

then the next (E[_])ontology-state of the dialectic is expressed via the non-amalgamative sum of the next two, consecutive, qualitatively distinct dialectical meta-number values,

E[qn]  =  sqn + ssqn  = qsn + qssn  =  qn+1 + qn+2,

wherein s denotes the successor function for these dialectical meta-numbers’, and wherein s denotes the Peano successor function for the Peano Natural Numbers;

B.  If the current ontology-state of the dialectic is expressed via a non-amalgamative sum of two, consecutive, qualitatively distinct dialectical meta-number values,

qn+1 + qn+2,

then the next (E[_])ontology-state of the dialectic is expressed via a single dialectical meta-number value, namely, that value which is the next consecutive value after the value of that one value of that summed pair of values which bears the larger Peano Natural Number subscript: 

E[qn+1 + qn+2]   =  s(max-subscript(qn+1, qn+2))  =  qn+3.’


An illustration of the generic dialectical progression that arises per this candidate axiom is also pictured below.

The candidate axiom above represents what we call the convolute product version of the Triadic-Tetradic Product Rule for dialectical algebras.  Per it, all previous ontology uniformly vanishes when each next, incremental new ontology makes its irruption.

We are also investigating evolute product versions of this candidate axiom, wherein previous ontology is conserved, at least possibilistically, external to and coeval with eachnewly-irruptant increment of ontology.

So far in our research, the convolute product version, given above, appears to work better for the dialectical meta-modelingof individual «monads», units, or ‘‘‘holons’’’.  The evolute product versions appear to work better for certain «arithmos»-level, categorial dialectics.



FYI:  Much of the work of Karl Seldon, and of his collaborators, including work by “yours truly”, is available for free-of-charge download via --



Regards,

Miguel Detonacciones,
Member, Foundation Encyclopedia Dialectica [F.E.D.],
Officer, F.E.D. Office of Public Liaison






















 

Monday, June 18, 2018

‘‘‘Dialectical Taxonomy’’’ -- E.D. Definition.







‘‘‘Dialectical Taxonomy’’’ -- E.D. Definition.







Dear Readers,


This blog-entry is for the purpose of sharing, with you, the definition of the term ‘‘‘Dialectical Taxonomy’’’ from our forthcoming Encyclopedia Dialectica, volume 0, entitled Encyclopedic Dictionary for a Unified Theory of Universal Dialectics --


A ‘‘‘dialectical taxonomy’’’ is, first of all, a human-cognitive divisiondiairesis»/sorting -- of the «monads» or units/individuals ‘‘‘populating’’’ a given Domain of ontolog(y)(ies), or universe of discourse -- their sorting into . . . «Genos» ontological categories, and into their «species» ontological categories. . ., thus including the partial «synagôgê»/collection of each group of «species» category-units into their «Genos» category-unit, i.e., as an instantiation of that which we term vertical «aufheben» dialectic [the aspect of F.E.D. dialectics with the clearest roots in our demystification of Plato’s dialectic] -- that is  . . . the meta-unit-izationof «species» categories-as-units into their «Genos» category-as-unit . . . .”

A ‘‘‘dialectical taxonomy’’’ is, further, an ‘«aufheben» taxonomy in the sense of that which we have termed horizontal «aufheben» dialectic, in our unified theory of universal dialectics.  This means a taxonomy -- a system of ‘‘‘class-ification’’’ of individualsmonads» together with their similants -- structured in accord with the «aufheben» meta-«monad»-ization relationships among its self-hybrid ontological categories, and their self-hybrid «monads» -- that is, among their self-hybrid’ “taxa, i.e., among their ‘‘‘dialectical synthesis’’’ oruni-«physis»’,taxa.      


FYI:  Much of the work of Karl Seldon, and of his collaborators, including work by “yours truly”, is available for free-of-charge download via --



Regards,

Miguel Detonacciones,
Member, Foundation Encyclopedia Dialectica [F.E.D.],
Officer, F.E.D. Office of Public Liaison