**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

-- since --

-- which is the Boolean formulation of the traditional

This

". . .Second, for the "additive decomposition" of the two "factors" of

**(****x****)**x**(****1****B****-****x****)**--**(****x****)****+****(****1****B****-****x****)****=****1****B**-- since --

**(x)****+****(****1****B****-****x)****=****x****+****1****B****-****x****=****x****-****x****+****1****B****=****0****B****+****1****B****=****1****B**-- which is the Boolean formulation of the traditional

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

*"*-- "multiplicative", and "additive" -- and**dual**"*radically**"*, "mutually-**dual**istic"**", or***external*

__exo__*-*, thus justifies, I think, the revised name "

**duality**"

*law of**for the Boolean equation*

**-**__exo__**duality**"**xx = x**, as expressive of the

*"*of

**Fundamental Law**"**-**

__sub__

__dialectical__**.**

*thought*On the contrary, the

**F**.

**.**

__E__**.**

__D__*"*

**first**

__dialectical__

*system of arithmetic**"*, as constructed from an <<

**>> of the**

*aufheben***W**-based "Standard" arithmetic, rather than from an <<

**>> of the**

*aufheben***N**-based "Standard" arithmetic -- that is, the axioms-system of arithmetic whose

*"*, or

**meta**-**number space**"*"*, is --

**meta**-**number set**"**W**,

__Q____=__{ q/0**,**

__q__/1**,**

__q__/2**,**

__q__/3**,**

__q__/4**. . . }**

-- is a

*"*, with a correspondingly

**-**__contra__**Boolean****arithmetic**"*"*

**contra**-**Boolean**"**, and the generic such**

*algebra**"*value

**meta**-**number**"**-- the algebraic "variable" or "unknown" -- denotes an unknown value of/belonging to that**

__x__*"*of that

**meta**-**number space**"*"*, as signified by its underscore [

**-**__contra__**Boolean****arithmetic**"**denotes, or "ranges over", every value in that space but**

__x__**q/0**].

The generic "multiplicative" operation that

**, as an**

__x__

*ontological category**-*, stands for -- as an "<<

__unit__**>> operator", or as**

*aufheben**"*[

**immanent**

__self__*-*]

**operator" -- of the**

*critique***W**, is

__Q__**the "**

__not__**lection" operation of Boole. The**

__E__**are**

__x__**"**

__not__**lectors", but**

__E__*"*of [former, mere]

**", or "**__act__ualizers__kinetic__izers**potentials**"

*; "*,

**-**__de__**"**__im__plicitizors*"*,

**-**__de__**occultors**"*"*, or

**-**__de__**"**__sup__pressors*"*.

**-**__de__**"**__re__pressorsThat is, the generic "mental operation" modeled by, or simulated symbolically by, the

__x__*"*operators, is the human "mental operation" named "<<

**dialector**"**>>**

*aufheben**-negation*", the

*"*of the [formerly]

**plicitization"**__ex__**plicit; the**

__im__*"*of the [formerly]

**ering"**__out__**hidden**, or "

**occult**"; the

*"*of the [formerly] "

**ternalization"**__ex__**internal**"; the

*"*pression" of the [formerly]

__ex__*"*

**pressed", or**

__sup__*"*

**pressed", and/or the**

__re__*"*ization", or

__act__ual*"*ization", of the [formerly merely] "

__kinetic__**potential**".

The generic ideogramic name for that which is thereby

*"*plicitized",

__ex__*"*ered",

__out__*"*ternalized",

__ex__*"*pressed", and/or

__ex__*"*ized", is

__act__ual**-**

__delta__**.**

__x__Thus, the fundamental rule/description of

__trans__*-*formal-logical human thought; of

__meta__*-*formal-logical human thought; of

__meta__*-*<<

**>>, or of <<**

*verstand***>>,**

*vernunft*

__dialectical__*-logical*"[human] mental operations", is --

__x__*"*

**of**"**-**

__x__=__x__(__x__) =__x__+__delta__**.**

__x__The

__self__*-*operation expression

**, or**

__x__(__x__)

__x__*"*

**of**"**, thus, per our generic**

__x__*"*/

**intuitive**

__dialectical__

*interpretation**"*, expresses the "outering" of

**-**

__delta__

__x__**[**

*from***]**

*out of***-- from "inside"**

__x__**--**

__x__

*by***itself, thus**

__x__**, [ideo-]**

__qual__itatively

*ontologically*

__expanded__ly*-*

**reproducing****[along with the <<**

__x__**>>-**

*aufheben***/**

*conservation**"*

**simple**

*reproduction**"*

*of***,**

__x__

*by***]; the**

__x__**of the formerly invisible/unnoticed, of the formerly only**

__ex__ternalization*"*,

**"**__in__ternal dual*"*, or

**"**__in__ner dual*"*[

**-**__in__tra**dual**"*"*]

**-**__self__**dual**"

*of***itself,**

__x__

*as***-**

__delta__**, also**

__x__*by*

**itself, so that, indeed,**

__x__**-**

__x__(__x__) =__x__+__delta__**.**

__x__The [

*self-*]critique [

*by*

**]**

__x__*of*

**posits, in addition to first re-positing**

__x__**itself -- i.e., in addition to re-affirming**

__x__**as a possibly still**

__x__*partially*apt,

*partially*adequate,

**, or**

*ontological category*

*system**-*-- also posits, explicitly and externally, the possibility of that of and about itself which

**category****[previously] avoided / diminished / de-emphasized / minimized / neglected / omitted / refused to recognize / denied / suppressed / silenced / repressed: a [formerly] missing / absent categorial**

__x__**[y] that, once made present/extant, is denoted generically by**

__unit__**-**

__delta__**.**

__x__For instances --

__N___(

__N___**)**........

**=**........

**.....**

__N___**+**.....

__NQ___-- wherein

**is**

__NQ___

*the***,**

*denial of the denial***plicitly asserted by**

__im__**, of the possibility of**

__N___**[**

*arithmetical***]**

*ordinal***, of a[n even**

__qual__ifiers*"*]

**purely**"*"*[

__qual__itative*ly*]

**ordinal**

*arithmetic**"*, or of an

*"*[

**arithmetic**of**]**

*ordinal*

__qual__ities*"*,

and --

__WE___(

__WE___**)**........

**=**........

**.....**

__WE___**+**.....

__WQ___-- wherein

**is**

__WQ___

*the***,**

*denial of the denial***plicitly asserted by**

__im__**, of the possibility of an**

__WE___**[or even of a**

*expanding*

__self__*-*] logical

**expanding****Universe**[-of-discourse].

Regards,

Miguel

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