Friday, September 23, 2011

Algebras of Logic, Boolean versus "Contra-Boolean"


CONTINUED AND CONCLUDED FROM IMMEDIATELY-PREVIOUS POST, # 49



Algebras of Logic, Boolean versus "Contra-Boolean"



Dear Readers,

. . .Second, for the "additive decomposition" of the two "factors" of (x) x (1B - x) --

(x) + (1B - x)  =  1B

-- since --

(x) + (1B - x)   =   x + 1B - x   =   x - x + 1B   =   0B + 1B   =   1B

-- which is the Boolean formulation of the traditional "law of the excluded middle", i.e., the proposition that any "class",
x, "plus" or "together with", its "exo-dual" opposite/complement "class", (1B - x), [re-]constitutes the entire Universe[-of-discourse], 1B; the proposition implying that there is never any "third"[or "fourth", or "fifth", . . .], or "mediating", "class(es)", i.e., that <<tertium non datur>>; that the "Universe", 1B, is divisible into a strict dichotomy or "exo-duality" of classes for/by any class that it "contains".

This "dual" -- "multiplicative", and "additive" -- and radically "dualistic", "mutually-external", or
"exo-duality", thus justifies, I think, the revised name "law of exo-duality" for the Boolean equation xx  =  x, as expressive of the "Fundamental Law" of sub-dialectical thought.


On the contrary, the F.E.D. "first dialectical system of arithmetic", as constructed from an <<aufheben>> of the W-based "Standard" arithmetic, rather than from an <<aufheben>> of the N-based "Standard" arithmetic -- that is, the axioms-system of arithmetic whose "meta-number space", or "meta-number set", is --

WQ  =  { q/0, q/1, q/2, q/3, q/4, . . . }

-- is a "contra-Boolean arithmetic", with a correspondingly "contra-Boolean" algebra, and the generic such "meta-number" value
x -- the algebraic "variable" or "unknown" -- denotes an unknown value of/belonging to that "meta-number space" of that "contra-Boolean arithmetic", as signified by its underscore [x denotes, or "ranges over",  every value in that space but q/0].


The generic "multiplicative" operation that
x, as an ontological category-unit, stands for -- as an "<<aufheben>> operator", or as "immanent [self-]critique operator" -- of the WQ, is not the "Election" operation of Boole.  The x are not "Electors", but "actualizers", or "kineticizers of [former, mere] potentials"; "de-implicitizors", "de-occultors", "de-suppressors", or "de-repressors".


That is, the generic "mental operation" modeled by, or simulated symbolically by, the
x "dialector" operators, is the human "mental operation" named "<<aufheben>>-negation", the "explicitization" of the [formerly] implicit; the "outering" of the [formerly] hidden, or "occult"; the "externalization" of the [formerly] "internal"; the "expression" of the [formerly] "suppressed", or "repressed", and/or the "actualization", or "kineticization", of the [formerly merely] "potential".

The generic ideogramic name for that which is thereby "explicitized", "outered", "externalized", "expressed", and/or "actualized", is
delta-x.

Thus, the fundamental rule/description of trans-formal-logical human thought; of meta-formal-logical human thought; of meta-<<verstand>>, or of <<vernunft>>, dialectical-logical "[human] mental operations", is --

x "of" x   =   x(x)   =   x + delta-x.



The self-operation expression
x(x), or x "of" x, thus, per our generic "intuitive / dialectical interpretation", expresses the "outering" of delta-x from [out of] x -- from "inside" x -- by x itself, thus qualitatively, [ideo-]ontologically expandedly-reproducing x [along with the <<aufheben>>-conservation / "simple reproduction" of x, by x]; the externalization of the formerly invisible/unnoticed, of the formerly only "internal dual", "inner dual", or "intra-dual" ["self-dual"] of x itself, as delta-x, also by x itself, so that, indeed,

x(x)   =   x + delta-x.

The [self-]critique [by
x] of x posits, in addition to first re-positing x itself -- i.e., in addition to re-affirming x as a possibly still partially apt, partially adequate, ontological category, or system-category -- also posits, explicitly and externally, the possibility of that of and about itself which x [previously] avoided / diminished / de-emphasized / minimized / neglected / omitted / refused to recognize / denied / suppressed / silenced / repressed:  a [formerly] missing / absent categorial unit[y] that, once made present/extant, is denoted generically by delta-x.



For instances --

N_( N_ )........=........N_.....+.....NQ_

-- wherein
NQ_ is the denial of the denial, implicitly asserted by N_, of the possibility of arithmetical [ordinal] qualifiers, of a[n even "purely"] "qualitative[ly ordinal] arithmetic", or of an "arithmetic of [ordinal] qualities",

and --

WE_( WE_ )........=........WE_.....+.....WQ_

-- wherein
WQ_ is the denial of the denial, implicitly asserted by WE_, of the possibility of an expanding [or even of a self-expanding] logical Universe[-of-discourse].



Regards,

Miguel

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