A Breakthrough in the Algebraic Representation of Dialectical Negation

01 May 2012 --

Dear Reader,

The F.E.D. generic model of dialectical progressions in general, holds as follows --

Suppose that we have a "sum" of mutually qualitatively different qualitative determinations, collectively denoted by Hh, characterizing a given system, as of step h in the dialectical presentation / systematic dialectic of that theoretical system, or as of epoch h in the self-presentation / historical dialectic of that natural system, where h is a discrete variable ranging over the "Whole" Numbers, { 0, 1, 2, 3, . . . }.

If that "sum" of characteristics is a qualitative, heterogeneous [or inhomogeneous], irreducible, and non-amalgamating, or <<asumbletoi>> [Plato:  unaddible] sum, then stage h+1 will exhibit the following, expanded non-amalgamative sum, with additional possible determinations/characteristics describable as follows --

Hh   --->  Hh+1    =    Hh "of" Hh    =    Hh(Hh)    =    ~(Hh)    =     Hh x Hh    =    HhHh    =  

 Hh "squared"   =   Hh^2    =    Hh + Delta(Hh)

-- such that the term Delta(Hh) "contains" all of the revolutionary, new "ontology" -- all of the newest possible determinations / characteristics -- all computed from the old determinations / characteristics of stage h.

For example, if h = 1, and if the possible features of the maximal natural system known as the cosmos, as of that epoch, epoch 1, are --  

( n  +  s )   =   ( sub-nuclears  +  sub-atomics )   =   ( bosons & non-composite fermions  +  hadrons ) 

-- such that the underscores indicate the "contra-Boolean" computational logic exhibited by these terms as [dialectical] logic elements, or as [dialectical] logic symbols [i.e., x^2   ~=   x], then, as of the next discretely representable epoch, epoch h + 1  =  1 + 1  = 2, the possible features of the cosmos are "predicted" by the F.E.D. "Dialectical "Theory of Everything" Equation" to be --

 H1  =  ( n  +  s )   --->  H2  =  ( qn  +  qs  +  qsn  +  qss )   =   H1 "of" H1   =    

( n  +  s ) "of" ( n  +  s )  =  Hh(Hh)    =    ( n  +  s )(( n  +  s ))  =   

~(H1)  =  ~( n  +  s )    =     H1 x H1    =     ( n  +  s ) x ( n  +  s )   =    H1H1    =  

 ( n  +  s )( n  +  s )   =    H1 "squared"   =    ( n  +  s ) "squared"   =   

H1^2    =    ( n  +  s )^2    =    H1 + Delta(H1)    =    ( n  +  s ) + Delta( n  +  s ).

=  ( n  +  s)  +  (qsn  +  a )  =  ( n  +  s  +  qsn  +  a )  =  H2   

=  ( sub-nuclears  +  sub-atomics  +  real subsumption of sub-nuclears by sub-atomics  +  atomics  )   

=  ( sub-nuclears  +  sub-atomics  +  sub-atomic / sub-nuclear hybrids  +  atomics  ). 

Note especially the "sub-equation" extracted from the original serial equation above -- 

Hh(Hh)  =  ~(Hh).

Its Left-Hand-Side [LHS] expression, read off as "Hh of Hh", denotes the "self-function[ing]" of Hh, that is, the "self-movement" of Hh -- the "autokinesis" [Plato] of Hh. 

Its Right-Hand-Side [RHS] expression, read off as "[dialectical] negation of Hh", or as "[determinate] not Hh", denotes the dialectical, determinate negation of Hh, that is, the <<aufheben>> negation of Hh.

So, the sub-equation --

Hh(Hh)  =  ~(Hh)

-- asserts that the dialectical, determinate, <<aufheben>> negation operation symbol, for a given system-characterization symbol, generically denoted by Hh, itself formulated as a function / operation / action / activity, is, precisely, itself --

~   =     Hh [for Hh].  What more determinate, specific, negator could there be, for Hh, than Hh itself?

That is, the effect of the inherent, immanent, "intra-duality", or "self-duality", of such a system, as of stage h in its [self-]development, denoted Hh -- the "internal dialectical contradiction" or "dialectical self-contradiction" of / immanent to that system -- can be expressed as a dialectical, determinate, <<aufheben>> self-negation of that system-stage --

Hh  ---> ~(Hh)  =  Hh(Hh)  =  Hh x Hh  =  HhHh  =  Hh^2  =  Hh + Delta(Hh) =  Hh+1.

[Note that  ~(Hh)  =  Hh(Hh), so ~(~(Hh))  =  ~(Hh+1)  =  Hh+1(Hh+1)  = 

Hh^2(Hh^2)  =  Hh^4, not  ~(~(Hh))  =  Hh(Hh(Hh))  =  Hh(Hh^2)  =  Hh^3.  

That is, the "tilde" symbol, '~', acts like a pair of "horizontalized" ditto marks]. 



The series of equations --

 Hh  ---> ~(Hh)  =  Hh(Hh)  =  Hh x Hh  =  HhHh  =  Hh^2  =  Hh + Delta(Hh) =  Hh+1    

 --  expresses the core breakthrough discovery of F.E.D. in achieving the first algebraic representation of dialectical processes in general, the first algebraic dialectical logic -- 

http://www.dialectics.info/dialectics/Dialectic_Ideography_files/5_Dialectics-Part1b-WhyDI_OCR.pdf

-- [F.E.D., Dialectical Ideography, Prolegomena, Part I.b., pages I-73 through I-75].

F.E.D. thus achieved, in 1999 [or even as early as 1996, judging from the day-counter in their document version-numbers], for algebraic dialectical logic, the counterpart and <<surpassement>> of what Leibniz, from 1666 [though he did not publish his algebra of logic], and Boole, from 1847 [who published, in that year, his booklet entitled The Mathematical Analysis of Logic:  Being an Essay towards a Calculus of Deductive Reasoning], achieved for algebraic formal logic --


Leibniz / Boole:  x^2  =  x, i.e., 0^2  =  0, and 1^2  =  1.

F.E.D.:   x  --->  ~x   =  xx   =   x^2   =   x + delta_x, i.e., for every n in N  =  { 1, 2, 3, ... }, 

qn  --->  ~qn   =   qn x qn   =   qn^2    =    qn + delta_qn    =    qn + qn+n    =    qn + q2n.  

The iteration of such ideographically expressed [algebraically expressed] dialectical negation --

 Hh   --->  ~(Hh)  --->  ~(~(Hh))  --->  ~(~(~(Hh)))  ---> . . .

--  which is equivalent to the following self-iteration of dialectical self-negation --

  
Hh^1   --->  Hh^1(Hh^1)  =  Hh^2  --->  Hh^2(Hh^2)  =  Hh^4  --->  Hh^4(Hh^4)  =  Hh^8    ---> . . . 

--  is the process that F.E.D. names "The Dyadic Seldon Function" , which can be succinctly summarized as --

Hh   =   H0^(2^h).  



Regards,

Miguel