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**F**.__E__.__D__.
__Vignette____ ____#13__ --

*William of Ockham*’*s *__Dialectic__* **of** **Science** *--

*A
*__Dialectical__ Episode in Early Medieval Philosophy

*by*
*Aoristos Dyosphainthos*

__Author____’__**s Preface**. The purpose of __F__.__E__.__D__.* *Vignette **#13**** **is to present
an __E__.__D__. *‘*__Dialectical__ **Model** **Equation**’
for a *systematic *__dialectical__ method of presentation of the *Early Medieval Philosophy of Science*
of William of Ockham.

This *‘*__Dialectical__ **Model** **Equation**’
also serves as a “worked”, “cook-book” example, and sample, of the application
of the _{N}__Q__ __dialectical__ algebra
to help organize, and to compactly encode, *systematic* expositions for *‘‘‘***synchronic** **snapshots**’’’ of the structure/process
of the living self-sustenance of systems -- ‘*physio-*systems’, *‘ideo-*systems’,
and *hybrids*
of the two, such as the system of *Science* -- that are *presently in existence*; expositions that
explicate and assess the *present*
content of such systems, without explicitly expositing their *‘“***diachronic chronicles**”’
-- neither explicitly *reconstructing *the *prior*, *diachronic history* that constituted that
content, nor explicitly *‘pre-constructing’ *any *presently**-*predicted
*future*, *successor
system* of such. [__Note__: the ordering __Mythopoeia__, __Religion__, __Philosophy__, __Science__, __Psychohistory__ is, per
the __E__.__D__. __first__
*Psychohistorical**-*__Dialectical__ ‘**Meta**-**Equation**’,
that of *Human Ideology**/***Knowledge **‘**Meta**-**Evolution**’. See http://www.dialectics.org/dialectics/Aoristoss_Blog/Entries/2012/5/19_The_F.E.D._Psychohistorical_Equations.html ].

__A Note about the On____-____Line
Availability of Definitions of ____F____.____E____.____D____.____ Key Technical Terms__. Definitions
of __Encyclopedia Dialectica__ technical terms, including of __E__.__D__. ‘neologia’, are available on-line via the following URLs --

-- by clicking on the links associated with each
such term, listed, in alphabetic order, on the web-pages linked-to above.

Definitions of the __Encyclopedia Dialectica__ special terms
most fundamental to this vignette are linked-to below --

«*archÃ©*»

«*arithmos*»
and «*arithmoi*»

«*aufheben*»

__Dia__*chronic *vs.* *__Syn__*chronic*

_{N}__Q__ __dialectical__ *arithmetic**/**algebra*

*Systematic *or* Synchronic** *__Dialectics__

-- and we plan to expand these definitions resources
as the __Encyclopedia Dialectica__* * *Dictionary* *Project* unfolds.

[__Note__:
__‘‘‘Arithmetical Quantifiers’’’
vs. ‘Arithmetical Qualifiers’__. In the phrase “**3** apples”, we
term “**3**” the “arithmetical [“pure”-]__quant__ifier”, and “apples”
the *‘‘‘**ontological**’’’* -- or __kind__ of thing -- ‘‘‘__qual__ifier’’’. In the phrase “**3** pounds of apples”, we term “pounds” the *‘**metrical*[-unit] __qual__ifier’
-- or ‘‘‘unit of measure __qual__ifier’’’
-- *quant*ified by
the **3**, which, together, *‘*__quant__*o**-*__qual__*ify**’* the *‘**ontological*** **__qual__ifier’, “apples”.
A key use-value of the __dialectical__* arithmetics* is to provide
algorithmic, ideographical-symbolic systems for the various kinds of
‘arithmetical __qual__ifiers’,
both with and without the co-presence of ‘‘‘arithmetical __quant__ifiers’’’.].

__I____. __**Introduction**__ to the Subject __*Systematic Dialectic* within *Medieval Occidental Philosophy*. The
present __F__.__E__.__D__.* *vignette sets forth a __dialectical__*-***ideographical
model** of William of Ockham’s *circa* **1323** C.E. *philosophical theory* / *‘***ideo**-**taxonomy**’ of *the sciences*.
Ockham lived *circa* **1290** to **1349** C.E. [__C__ommon __E__ra].

This vignette contains this
example of __dialectical__*-**mathematical models* in __gene__ral, formulated using a __speci__fic __dialectical__*-**mathematical language* -- that of the “purely-__qual__itative”, “purely connotative” mathematics of
the *‘**algorithmic**-**heuristic algebra**’* of the _{N}__Q__ *‘**arithmetic of *__dialectics__*’*, which is an *axiomatized algebraic *__dialectical__* logic*.

For further background on
the algorithmics and the axiomatics of its *ideographical*, or *“**symbolic**” *__dialectical__* logic* -- or *‘**algebraic
logic**
**of *__dialectics__*’’* -- see the www.dialectics.org website’s Briefs Page as well
as section **II**. of this vignette.

The core of Ockham’s *philosophical theory **of** **the sciences* is as follows.

Ockham gave to terms of
language that refer to ‘non-term’ *“**physical** **facts**”*, the descriptor [the term] *“**categorematic**”*.

Thus, the “term”, or word,
“rock” refers to that “external-to-mind __r__eality”, and is therefore a *“**categorematic**”* term.

Ockham gave to __t__erms of language that refer
to, e.g., *“**categorematic**”* terms, i.e., to other terms
of language itself, the descriptor [the term] *“*__syn__*categorematic**”*.

Thus, the “terms”, or words,
“all”, “some”, “none”, and “not” are all examples of such “word-referring
words”, or “term-referring-__t__erms”.

Ockham also labeled terms
that refer to things belonging to the __r__eality outside of [the reality of] language, by the
phrase *“terms of the* __f__*irst* *intention**”*.

He labeled __t__erms that refer to *terms
of the *__f__*irst intension* by the phrase “__t__erms *of the *__s__*econd intention**”*.

In the most direct sense,
then, *“*__syn__*categorematic**”* __t__erms are “__t__erms *of the *__s__*econd
intention**”*.

However, Ockham
distinguishes *mere* “__t__erms” of language, in the sense
of *individual words*, from those higher-level /- higher ‘qualo-fractal
scale’ language constructions [constructions that we of __F__.__E__.__D__. call *‘**meta**-**terms**’*, or *‘**meta**-**words**’*, *each one made up out of
a *[usually]* heterogeneous
multiplicity of* *mere* “terms”, of *mere* individual words] which are named *“**complete *__sentences__*”*, or *“**well**-**formed *__propositions__*”*.

Ockham does not *“*__reduce__*”* the latter *‘**ideo**-**ontology**’* to the former.

Thus, for Ockham, *categorematic* __t__erms, in ‘sentence-ial’ *combination* with __syn__*categorematic* __t__erms, in the form of well-formed *sentences* or *propositions*, and, thus, *functioning together*, refer to, and describe, *more adequately*, typically, the things belonging to the __r__eality outside of human
language, than do single, isolated, *“mere”* __t__erms, or single, isolated words, such as those __t__erms/words which are named
“names”, or “nouns”.

The *sentence*/*proposition* -- “All natural objects are transitory.”
-- is thus __f__*irst intentional*. Note: this
sentence “contains” [*combines*] both *categorematic* __t__erms, like “natural objects”, and also __syn__*categorematic* __t__erms, like “All”.

The *sentence*/*proposition* -- “All second-order-formal-logic axiomatizations of “**N**atural
Numbers” arithmetic are
either formally inconsistent or formally incomplete.” -- is thus __s__*econd intentional* [at least!]. Note that this sentence “contains” no __t__erms which refer to __non__*-*linguistic, *“**physical**” **objects*, but only __t__erms which refer to other __t__erms, or to the names of *‘‘‘**mental** **objects**’’’*/*‘‘‘**idea**-**objects**’’’*, to ‘meme-ic’ or ‘memetic’,
human-Phenomic, *‘**psychohistorical** **materialities**’*.

On the basis of these
distinctions, Ockham divides the «__Genos__» of the *“*__Sciences__*”* into two «*species*», or *‘**sub*^{1}*-*«__Gene__» --

__«____species____» ____1__: __«____Scientia ____rea____l____is____»__, encompassing the *sciences* of “__r__ea__l__ things” [i.e., of __ex__ternal-to-mind, physical
things].

For example, the science of *biology* would form a __sub__^{1}*-*«*species*» of this «*species*».

This «*species*» is __f__*irst intentional*.

__«____species____» ____2__: __«____Scientia ____ratio____n____alis____»__, encompassing the *sciences* of ‘__n__oetic*-*things’ [i.e., of __in__ternal-to-mind, ‘Phenomic’ things].

For example, *mathematics* would form a __sub__^{1}*-*«*species*» of this «*species*».

This «*species*» is __s__*econd intentional*.

For further background on
Ockham’s theory, see W. L. Reese, __Dictionary of Philosophy and Religion__, *Eastern
and Western Thought*, Humanities Press [Atlantic Heights, NJ:
**1980**], pp. **627** ff.

The two passages from the
source cited above that are most relevant to this vignette are extracted below:

**“**...Ockham makes an important
distinction between *categorematic* and *syncategorematic* terms [A.D.: a distinction in logic which harks back at
least to the Stoics; cf. also the work of Ockham’s predecessor, William of
Sherwood, *circa* **1210** to **1270** C.E.].”

“Terms which refer to
reality are *categorematic*, while terms which refer to categorematic
terms are *syncategorematic*.”

“Most common nouns are *categorematic*,
while words such as “not,” “all,”, and “some” are *syncategorematic*.”

“...Terms that refer to [A.D.: external-to-language/-mind] things are called terms of
*first intention*.”

“Terms referring to terms of
first intention are called terms of *second intention*.”

“In one sense the
syncategorematic terms mentioned above are second intentional; but, as Ockham
wishes to use the distinction, categorematic terms and syncategorematic terms,
functioning together, refer to [A.D.: external-to-language/-mind] things. “All men are mortal” is thus first
intentional.”

“But when we use terms such
as “genus,” “species,” and “difference” we are using terms of second
intention.”

“Propositions utilizing such
terms refer not to the world [A.D.: i.e., to the physical world,
external to the human-language/-mind “world”] but to terms of first intention.”

“...On the basis of the
distinction just mentioned, Ockham divides the sciences into two types [A.D.: into two «*species*»].

*“Scientia rationalis*, or rational science, is
second intentional. Logic is a science
of this type.”

*“Scientia realis* is first intentional.
Physics is an example of a science of real [A.D.:
external-to-language/-mind] things.**”**

The same source gives,
elsewhere [p. **563**], a useful excursion into
the ‘philosophical etymology’ of the term “syncategorematic”:

**“**From the Greek *syn*
(“together”) and *categorema* (“predicate”).”

“The derivation may refer to
those terms which go together with the predicates or [to] those terms which
hold the predicates together; and thus, [to] those terms which cannot be
[intelligibly] used by themselves but only in conjunction with other
terms: *e.g.*, “all,” taken alone,
has no definite reference to anything, but “all men” has.

“ “All” is syncategorematic
and “men” is categorematic.”

“In addition to the
quantifiers “all,” “some,” and “none,” included among syncategorematic terms
are “not,” “if...then,” “either...or,” and “both...and.**”**

__II____. ____E____.____D____.____ ____Standard Interpretations for the __*Initial *__Gene____ric Ordinal ____Qual____ifiers____ of the__** **_{N}__Q__** **__Arithmetic__. Perhaps a bit surprisingly, upon first apprehension,
the first four, first-order-logic, Dedekind-Peano Postulates for the “**N**atural” Numbers focus on their *ordinality*, __not__ on their *cardinality*, viz. --

**1**. **1** is a “Natural Number”.

**2**. The *successor* of any “Natural Number” is also a
“Natural Number”.

**3**. No two, distinct “Natural Numbers” have the
same *successor*.

**4**. **1** is __not__ the *successor* of any
“Natural Number”, i.e., **1** has __no__ *ancestor* within
the “Natural Numbers”.

These postulates thus
identify the essence of the “**N**atural Numbers”, explicitly,
in terms of [apparently purely-] __quant__*itative** ordinality*.

In keeping with this focus on *the ordinal*, Seldon defines the system
of the _{N}__Q__ __dialectical__* arithmetic* -- the first *‘*__anti__*thesis**-**system**’*, or *‘*__contra__*-**system**’*, to the “**N**atural Numbers” as «*archÃ©*»*-***system** -- in terms of __qual__*itative***
ordinality**.

The_{ N}__Q__, which he also calls the *‘**meta**-***N***atural* *meta**-**Numbers**’*, are,
in their simplest, least-interpreted essence, a consecutive sequence of
‘meta-numeral’ ideograms representing the successive __qual__*ities*, __not__ the __quant__*ities*, of *ordinality* -- the __qual__*ity* of *‘**first**-*ness’, followed by the __qual__*ity* of *‘**second**-*ness’, followed by the __qual__*ity* of *‘**third**-*ness’, and so on... --
satisfying the four first-order-logic *‘*__contra__*-**Peanic**’*, *‘*__Qual__*o**-Peanic**’* axioms:

**1****q**. The *ordinal *__qual__*ifier* for the __qual__*ity *of *‘**first**-*ness’ is an element of the *‘**consecuum**’* of generic *ordinal *__qual__*ifiers*.

**2****q**. The *successor* of any element of the *‘**consecuum**’* of generic *ordinal *__qual__*ifiers* is also an element of same.

**3****q**. Any two, distinct *ordinal
*__qual__*ifiers* have __qual__*itatively* __un__equal *successors*.

**4****q**. The *ordinal *__qual__*ifier* for the __qual__*ity *of *‘**first**-*ness’ is «*archÃ©*»: __not__ the *successor* of any element of its *‘**consecuum**’*.

The symbols, or *‘**meta**-**numerals**’*, that stand for the *‘**meta**-**numbers**’ *of the _{N}__Q__ *‘*__arche__*onic** **consecuum**’* are derived, *syntactically*, in a way which represents
the *semantic* *‘**self**-**subsumption**’*, *‘**self**-**subordination**’*, or *‘**self**-**demotion**’* [__dialectical__, *self**-*«*aufheben*» *self**-***negation**] of the “**N**atural Numbers”.

That derivation is part of the positive
fruition of the __dialectical__, *immanent* __self__*-**critique *of the “**N**atural Numbers”, which
divulges the _{N}__Q__ as the implicit, most extreme known *opposite*, *“*__Non__-Standard
Model” of the “Standard”, Peano “**N**atural Numbers”.

It involves the turning of the *‘**gene**ric** **ordinal** *__quant__*ifiers**’* of the “**N**atural Numbers” into the *‘**gene**ric** **ordinal *__qual__*ifiers**’* of the _{N}__Q__ *‘**meta**-***N***atural* *meta**-**Numbers**’*.

The conceptual *‘**self**-**subsumption**’ *of the __quant__*itative
ordinality* intended by the Dedekind-Peano Postulates surfaces the _{N}__Q__ as their hitherto hidden,
implicit *‘**intra**-**dual**’*, based upon the __gene____ric__ __qual__*ity
of** **ordinality*, a *‘‘‘**generic**ity’’’* which we represent by the *‘**meta**-**numeral**ic**’* ideogram ‘__q__’.

That *‘**meta**-**numeral** component**’* represents *‘*__q__*ual**itative** ordinality**’*, or *‘**ordinal *__q__*ual**ity**’*, __in__* *__general__: just ‘__q__’, or, more fully expressed,
just __q___{N}.

To fully express, *‘**meta**-**numeral**-**y**’*, or ideographically, the *‘**consecuum**’* of __speci____fic__ *ordinal** *__qual__ities, namely --

_{N}__Q__** ****=****
****{***‘*__first__*-*ness’; *‘*__second__*-*ness’; *‘*__third__*-*ness’, etc.**}**

-- we must add a second *‘**meta**-**numeral** component**’*, via *‘‘‘*__sub__*ordinating**’’’* __speci____fic__ “**N**atural
Numbers”, as __speci____fic__ *‘**ordinal** *__quant__*ifiers**’*, to the __gene____ric__ *ordinal
*__qual__*ifier
*symbol ‘__q__’,
by *‘*__sub__*scripting**’* those __speci____fic__ “**N**atural Numbers” to a ‘__script__-level’
‘__q__’, ‘‘‘above’’’ them, yielding --

_{N}__Q__** ****=****
{ **__q___{1},** **__q___{2},** **__q___{3},** ... }**

[in which each *‘**meta**-**number**’* is a minimal, «*genos*»**/**«*species*» «*arithmos eidetikos*»
in its own right],

vs.

**N ****=**** ****{ 1**,** 2**,** 3**,** ... }**.

Note that this *opposition* of an arithmetical system of *purely**-*__quant__*itative** ordinality*, based upon the **N**, __versus__ an arithmetical system of *purely**-*__qual__*itative** ordinality*, based on the _{N}__Q__, is not a *radical dualism*, imagined as an *absolute*, *irreconcilable diremption* between an *absolute* __quant__*itative* and an *absolute* __qual__*itative*.

This *opposition* is, on the contrary, a __dialectical__* **anti**thesis**-**sum*.

The **N** __quant__*ifiers* are still there, as *‘**speci**fiers**’* -- still present -- in, or ‘‘‘under’’’,
the generic __q__s of the _{N}__Q__ __qual__*ifiers*, though __sub__sumed, __sub__ordinated, __de__moted -- demoted to being their mere __sub__scripts or *denominators*: The **N** __quant__*ifiers* are still *“**contained**”* in[side] [‘‘‘beneath’’’]
the _{N}__Q__ __qual__*ifiers*.

That is, each _{N}__Q__ __qual__*ifier* is an «*aufheben*» *determinate* *negation* / *conservation* / *elevation**-into-one-step-higher-***gene**rality of an individual **N** __quant__*ifier*.

The joint «*aufheben*» *elevation* of the __quant__*itative***
ordinal**__s__ «*species*» yields the «*genos*» ‘__q__’, of the __qual__*itative** ordinal*__s__.

For this first layer of
interpretation of these *“purely-*__qual__*itative**”* _{N}__Q__ *‘**meta**-**numbers**’* -- which does not yet make explicit their universal
interpretability for the modeling of __dialectical__* progressions* -- this is all that they
represent: *abstract
temporality*; [*abstract chronological*]
*order*; *gene**ric* *‘**order**ed**-ness**’*; the consecutive succession
of *‘*__qual__*itative** ordinality**’*; the *‘**consecuum**’* of *order *__qual__*ity* or of *order** *__qual__*ities*.

But even here, at this
minimally-interpreted stage of the construction of the _{N}__Q__ __dialectical__* arithmetic*, there is already a kind of
*gene**ric* *‘**connotative entailment**’* at work.

True, it is but a shadow, and but a
‘pre-vestigial’ harbinger, of the richness of the kind of particularity of *categorial followership* that drives forward,
intuitively, the __dialectical__, *purely**-**qual**itative** **logic* of the more concrete, more *speci*fic __dialectical__*-**algebraic* interpretations thereof.

A case in point is exemplified in the very _{N}__Q__ *algebraic
model* Ockham’s
* *__dialectic__* **of** **Science* constructed herein.

This *gene**ric* *‘**connotative entailment**’ *can be formulated as
follows: *‘**second**-*ness’ follows -- and even
follows __from__ --*‘**first**-*ness’; *‘**third**-*ness’ follows [__from__] *‘**second**-*ness’, and so on.

In the next section, the
construction, by iterated interpretations layering, of the Seldonian *first** *__dialectical__* **arithmetic* will advance from this
harbinger of *‘**connotative entailment**’* to the following, still __gene____ric__, but at last also explicitly
__dialectical__, form of *‘**connotative entailment**’* *ordinality*: *‘* *first* *full* *anti**thesis* follows from the __self__*-**interaction* of [«*archÃ©*»*-*]*thesis*; *first* *full synthesis* follows from the *mutual interaction* of *first full thesis* and *first full antithesis*’, and
so on.

__III____. ____Dyad____ic Seldon Function____ ____Interpretation of the __*Initial *__Generic__** **_{N}__Q__** **__Ontological Category ____Qual____ifiers__.

The *gene*ric *‘*__dialectical__* model meta**-**equation**’* form for the functions-family
of the Seldon Functions is that of a *gene*ric __cumulum__ symbol [‘__|-|-|__’] on the LHS [__L__eft-__H__and
__S__ide] of the *‘*__dialectical__* **meta**-**equation**’*, equated to an RHS expression
representing *‘**self**-**reflexive operation**’* of an [«*archÃ©*», *‘‘‘**seed**’’’*, *‘‘‘**cell**-***form**’’’, or *‘**ultimate ancestor**’* *ontological category* symbol [represented, *gene*rically, by [‘__q__**1**’] -- indicating its
recurring *‘**self**-**reflexion**’ *via a *‘**meta**-*exponentiated’, *monotonically
increasing* whole-number-valued ‘‘‘independent variable’’’ [‘**h**’ ] -- on the RHS of the *gene*ric *Seldon Function
equation*, viz. [with
*‘**gene*ric-ness’ connoted by the
“rectangular” *motif* of the
symbols-set]:

__|-|-|___{h} **=** **|****[** __q__**1** **]****|v**^{h}, for **h** in **{** **0**,** ****1**,** ****2**,** ****3**, **.** **.** **.** **}**, **v** in **{****2**,** ****3****}**.

If **v = ****2**, the *Gene*ric *Seldon Function*
above is said to belong to the __Dyad__ic Seldon Function sub-family.

If **v
= ****3**, the *Gene*ric *Seldon Function*
above is said to belong to the __Triad__ic Seldon Function sub-family.

Our remarks herein are concentrated on the __Dyad__ic Seldon Functions,
because the main *‘*__dialectical__-**mathematical meta**-**equations**’, modeling the* *__dialectical__*-***systematic*** **presentation* of *Ockham**’***s ****Theory of Science**, exposited herein, are
of the **v = ****2 **variety.

With
**v = ****2**, and selecting that special generic
__Dyad__*ic **Seldon Function*
__form__ that we reserve for a __systematic__ __dialectic__ -- for
the __dialectical__*-***systematic method of presentation**,
the __form__ of the *‘**model meta**-**equation**’* to be
constructed herein becomes, more specifically --

__)-|-(__**s****** **=** **(**) __q__**a** (**)****2**^{s}^{ }****, for **s** in **{** **0**,** ****1**,** ****2**,** ****3**, **.** **.** **.** **}**.

-- wherein the symbol **s**, replacing the more *gene*ric symbol **h**, takes on ‘‘‘__s__ystematic’’’, ‘‘‘taxonomic’’’ connotations, representing
the well-ordered, __s__uccessive __s__teps of a __s__ystematic exposition, and wherein, in general, the “curvilinear”
*motif* of the entire symbols-set
used is to connote the __systematic__ __dialectic__ domain of *‘*__dialectical__ **modeling**’.

The *Seldon
Functions* bring with them a further, *second* layer of
interpretation of the _{N}__Q__ __q__*ual**ifiers*, ** **

**{ **__q___{1},** **__q___{2},** **__q___{3},** ... }**,

by which they are interpreted as __q__*ual**ifiers* that stand for __gene__*ric *__dialectical__* **ontological
categories*, e.g., for *‘‘‘*__thesis__*’’’* *categories*, or for full or partial *‘*__contra__*-**thesis**’* *categories*, or for full or partial *‘*__uni__*-**thesis**’* *categories*.

If we *assign*
[ ‘**[----)**’ ] the «__a__*rchÃ©*»*-***thesis** category, __q__**a**, to the *generic* _{N}__Q__ __q__*ual**ifier* *‘***meta**-**number**’, __q___{1}, as signed by ‘__q__**a**_{ }**[----)
**__q___{1}’, and if we can discern that __q__**a**, and all of its successor-*categories*, and their *cumula*, as generated by
its successive, *cumulative*,
‘Seldon-functional self-operations’, connote «*aufheben*» *operators*, that is, __dialectical__ __negation__ operators,
then the __Dyad__ic Seldon
Function is seen to
signify, under the axioms of the system of arithmetic of the _{N}__Q__** ***‘***meta**-**numbers**’
[
http://www.dialectics.org/dialectics/Correspondence_files/Letter17-06JUN2009.pdf
], a *‘*__self__-**iterated**’, *cumulative* recurrence of __dialectical__* “***negation**__s__ of [*the*]*
negation*__s__*”*.

With every [unit] increase in **s**, the __Dyad__ic Seldon
Function ‘formulaic recipe’ calls for the *dyadic* *self**-***operation**
of the result of the previous *dyadic* *self**-***operation**,
i.e., for a *negation of the
negation* of the result of the previous *negation of the negation*.

Only for **s ****= ****0** -- only for the case in which __no__ *self**-***operation** occurs -- is the
“result” a singleton [ideo-] *ontological
category* *symbol*,
the *symbol* for
the «__a__*rchÃ©*»
[ideo-]*ontological category*
alone, instead of that “result” taking the form of a *cumulum* of [powers-of-]*two* [or more] such
symbols, i.e., a “non-amalgamative sum” [cf. MusÃ¨s], or an «__a__*-*__sum__bletoi»
sum [cf. Plato], of ‘[ideo-]*ontological
category**’* *symbols*,

since **2**^{0} **=** **1**,

and since __q__**a**^{1} **=** __q__**a** --

__)-|-(__**0** **=** **(**) __q__**a** (**)****2**^{0} **=** **(**) __q__**a** (**)****1** **=**

**(**) __q__**a** (**)** ** =** __q__**a**.

For example, if we take epoch **s ****= ****1**, and
denote the «__a__*rchÃ©*» *ontological
category* simply by __a__,
for syntactical convenience, then the __Dyad__ic Seldon Function calls for the following, as
per the _{N}__Q__
axioms, since **2**^{1} **=** **2** --

__)-|-(__**1** **=** **(**) __a__ (**)****2**^{1} **=** **(**) __a__ (**)****2** **=** **(**) __a__
(**)** **x** **(**) __a__
(**)** **=**

__a__** ****+**** **__b__.

-- wherein ** **

__a__** ****[----)**_{ }_{q1 }connotes the «__a__*rchÃ©*» *category*
or *‘‘‘***thesis**’’’,

and where

__b__** ****[----)**_{ }__q__**2 **connotes the *first* *‘***contra**-**category**’ or *‘‘‘***antithesis**’’’,

with ‘**+**’ standing for a *gene*ralized *addition* operation, that encompasses
the *addition*
of __q__*ual**itatively* distinct terms, and with ‘**x**’ standing for a *gene*ralized *multiplication* operation,
that encompasses the *multiplication*
operation *speci*fic
to the _{N}__Q__
__q__*ual**ifiers*.

If we take ‘**(**) __a__
(**)** **x** **(**) __a__
(**)**’ with **(**) __a__
(**)** connoting the *category* *to be* __dialectically__ negated, and with ‘**(**) __a__
(**)**’ connoting the *category* *doing that* __dialectically__ [__self__*-*]*negating*, with ‘**(**) __a__
(**)**’ as the __dialectical__, *determinate* *‘***negation**-sign’ *speci*fic to **(**) __a__
(**)** as the object of the so-indicated
__dialectical__,
*determinate negation
operation*, then we have ‘**(**) __a__
(**)** **x** **(**) __a__
(**)**’, as a whole, as connoting
the *first *__dialectical__ negation,* by the negation**-***operation ****(**) __a__
(**)** *of the self**-***same negation
operation**,* ***(**) __a__
(**)**,* that is also the *__operand__
of the negation*-***operator ****(**) __a__
(**)**, forming what Seldon calls an
ideographical

‘subject [**(**) __a__
(**)**]-verb[**(**) __a__
(**)**]-object[**(**) __a__
(**)**] identical’,

and yielding the
*first dyad*:

‘‘‘__thesis__** ****+**** **__antithesis__’’’, or

‘__«__**archÃ©**» **category** **+** __first____ contra-category__’, viz. --

**(**) __a__
(**)** **x** **(**) __a__
(**)** **=** __q__**a** **+** __q__**aa**** **** ****=**

__q__**a** **+** __q__**b**** ** **=** __a__ **+** __b__** **** **

-- which, in terms of the *gene*ric, minimally-interpreted _{N}__Q__ *arithmetic*, is a __dialectical__* interpretation* of the *gene*ric --

__|-|-|__**1** **=** **|****[** __q__**1** **]****|****2**^{1} **=**
**|****[** __q__**1** **]****|****2**** ****=**

** **

**|****[** __q__**1** **]****|** **|**__x__**|** **|****[** __q__**1** **]****|**
**=**

** **

**|****[** __q__**1 ****|**__+__**|**** ** __q__**1****+****1** **]****|**** ****= **

__q__**1** ** ****|**__+__**|** __q__**2**.

The *second*
iteration of this __dialectical__ negation of the negation,
corresponding to the consecutively *next* value of **s** namely, **s ****= ****2**, for
the _{N}__Q__ *arithmetic* interpreted for [*psycho*]*historical* __dialectics__, yields the following,
ontologically-expanded *cumulum*
of *ontological categories*
-- a *‘***dyad of dyads**’,
consisting of **4** consecutive *ontological categories*:

__)-|-(__**2** **=** **(**) __a__ (**)****2**^{2} **=** **(**) __a__ (**)****4** **=** **(**) **(**) __a__ (**)****2** (**)****2** **=** ** **

**(**) __a__ **+** __b__
(**)****2** **=** ** **

**(**) __a__ **+** __b__
(**)** **x ** **(**)** **__a__ **+** __b__ (**)**** =**

__a__** ****+**** **__b__** ****+**** **__c__** ****+**** **__d__.

The additional **2** ontological category-symbols above are __dialectically__ interpreted, per the __E__.__D__. standard, as follows:

__c__** ** **=** *third** ontological
category*,
*first* __full__ *uni**-**category*;

** **

__d__** ** **=** *four**th ontological
category*,
*second* *contra**-**category*.**
**

We will not here pursue this __E__.__D__. standard __dialectical__ interpretation of the *ontological categories* generated by the
generic __Dyad__ic Seldon
Function beyond **t ****= ****2**, because the ‘‘‘solution’’’
-- or *‘***semantification**’
-- of the category-terms generated by the *‘*__dialectical__-**mathematical meta**-**model meta**-**equation**’ constructed in this vignette
[whose terms are generated, initially, as __algebraic__ __un__knowns, terms of __un__known meaning], as presented herein, does not
extend beyond that *second*
*‘***self**-**iteration**’
for that *‘***meta**-**model**’.

The *‘***purely**-**qual**itative **calculations**’ illustrated above
describe our expectations for this *‘***meta**-**model**’ in terms of __gene__*ric* characterizations of the successive, consecutive __dialectical__ categories.

The next section addresses the heart of this *‘***meta**-**model**’ -- the __speci__*fic* meanings of the *gene**ric*** **__dialectical__ categories as applied to
the __speci__*al* *case*
of *the systematic *__dialectic__ of the Early*-***Medieval**** Philosophy of Science **of
William of Ockham.

To begin our model of Ockham’s __dialectical__ taxonomy of this universe-of-discourse of
human ideas, we use the __dialectical__ logic axiomatized in an earlier letter
available via the www.dialectics.org Correspondence
Page --

-- and we enact the ‘‘‘mental action’’’ / ‘‘‘mental
operation’’’ signified by the ideogram ‘**(----]**’. That is, we “interpret” or “assign” __q___{1}, the *first* of the *gene*ric ‘meta-**N**atural meta-Numbers’ of our _{N}__Q__,*‘*__unquant__ifiable __Q____ual__ifier’ *‘*__dialectical__ ideography’, or *‘*__dialectical__ symbolic logic’ [*‘*__dialectical__-**ideographic logic**’], as
follows --

__q___{1 }**(----] **__q___{r}, or __q___{1 }**(----] **__r__

-- such that __r___{ }**(----) **“__r__eality”, the universe of “__r__eal things”, outside of human language, outside of the human mind.

Our *‘*__dialectical__-**ideographic model**’ then
captures a sequential, systematically-ordered, ‘‘‘ordinal-ized’’’ *presentation* of Ockham’s *theory of the sciences*, using the
connotations of the series of ideographical “shorthand” symbols that it *gene*rates algorithmically.

It does so via __r__ as the __recurrently__ ‘self-operating
operator’, ‘self-operation’, ‘self-function’, ‘self-refl__u__xive function’,
‘‘‘self-refl__e__xive
function’’’, or __recurrently__ self-applied, connotationally
specific, determinate __dialectical__,
«*aufheben*»*-***negation
operation** at the heart of the *‘*__self__-**iteration**’ formula --

__)-|-(__**s****** ** =** **(**) __r__ (**)****2**^{s}****

-- wherein **s**, as the higher exponent of the exponent **2**, counts the “__s__teps” or “__s__tages” of that
*systematic presentation*
of the *‘idea-*ontology’, or *‘ideo-*ontology’-- of the *‘ideo-taxonomy’*
-- of the philosophical categories, and wherein __)-|-(__**s** denotes the *‘cumul*um’
-- meaning the connotative and non-amalgamative, *‘‘‘*__qual__itative sum’’’,
or ‘‘‘superposition’’’ -- of the *categories* presented / ‘ac*Cumul*ated’ as of __s__tep **s** of this “shorthand” ‘‘‘__systematic__*-*__dialectical__’’’ *‘***present**-**ation**’.

Let us, then, unfold this model __s__tep-by-__s__tep, starting
with __s__tep **s**** ****=**** ****0**, then moving on to __s__tep **s**** ****=**** ****1**, and thence onward to __s__tep **s ****=**** ****2**, and, finally, to __s__tep **s ****=**** ****3**, a __s__tep which *immanently extends*
Ockham’s theory to a *category*
of *The Sciences*
that Ockham, as far as we know, did not envision.

__Step ____0__. The **0**th __s__tep of this *‘***presentation**-**model**’, i.e., for **s**** ****=**** ****0**,

since ** **

**2**^{0}** ****=**** **

**2**^{+}^{1}**/2**^{+}^{1}** ****=**** **

**2**^{+}^{1}** x ****2**^{-}^{1}** ****=**** **

**2**^{+}^{1}^{-}^{1}** ****=**** 1**,

yields
--

__)-|-(__**0** **=** **(**) __r__ (**)****2**^{0}^{ } **=**

__r__^{1} **=** __r__

-- which simply reiterates the starting point -- the «*archÃ©*» *‘***ideo**-**taxonomic** **category**’ -- of this *‘***present**-**ation**’.

*¿*Does this categorial-singleton of __s__tep **0**** **grow into a ‘categorial-*cumulum**’* in __s__tep **1***?*

__Step ____1__. For __s__tep **s**** ****=**** ****1** of this *‘***presentation**-**model**’, since **2**^{1}** ****=**** 2**^{+}^{1}** ****=**** 2**, we
have --

__)-|-(__**1** **=** **(**) __r__ (**)****2**^{1}^{ } **= ** __r__^{2}^{ } **=**

__r__**(**) __r__ (**)**^{ } **= ** __r__ *“***of**”
__r__^{ } **=**

__r__ *“***squared**”

-- which connotes the *‘*__self__-**refl**__e__xive
function’, or *‘*__self__-**operation**’, of the __dialectical__ [«*aufheben*»] *operation*
denoted by __r__.

Syntactically, per the axioms of the _{N}__Q__ ‘meta-numbers’, we have the following values as the
content of __s__tep **1**:

__)-|-(__**1** **=** **(**) __r__ (**)****2**^{1}^{ }**= ** __r__^{2}^{ } **=**

__q__**r**^{2}^{ } **= **__q__**r** **+**** **__q__**r****r**

**[----)**

__q__**1** **|**__+__**|**** **__q__**1+1** **= **__q__**1** **|**__+__**|**** **__q__**2**.

Our challenge, for this __s__tep, is to
“solve for” the new category, __q___{rr}, that is, to determine the meaning, the connotation,
of that new ‘categorogram’, given the already known/assigned meaning, or connotation,
of the ‘categorogram’ __r__** ****=****
**__q___{r}.

So, let us apply the __gene__ral ‘‘‘canon of interpretation’’’ of
such __dialectical__ categories to the __speci__fic case of this
example.

Per that ‘‘‘canon’’’, the «*archÃ©*» ‘categorogram’, __r__, is to be
interpreted, as stipulated, as connoting an «*arithmos*» of «*monads*» -- an *assemblage* of *units* -- each one of which is a *“***fact**”, a *physical*, empirical “__r__eality”,
“outside of” language and of the human mind.

Also per that ‘‘‘canon’’’, a symbol like __q___{rr} connotes the new *category*/«*arithmos*» that results
from the __self__*-*«*aufheben*» of the __q___{r} *category*/«*arithmos*», and, quite often,
that results from the __self__*-*«*aufheben*» *‘*__meta__-**monad**ization’ of some of the «*monads*» of the __q___{r} *category*/«*arithmos*».

That is, each «*monad*» or *unit* of the __q___{rr} «*arithmos*» / *category */ *assemblage* of *units* should be a *‘*__meta__-«*monad*»’, made up out of a
multiplicity -- made up out of a ‘sub-«*arithmos*»’ -- of the «*monads*» of the __q___{r} «*arithmos*» / *category*.

So, the *units* of the __q___{rr} *assemblage* should be, in
this __speci__fic
case, ‘meta-__r__ealities’, ‘__r__ealities *of second degree**’*,
each one made up out of a heterogeneous multiplicity of ‘__r__ealities
*of first
degree**’*.

Also, the *category* __q___{rr} should be *an*__other__ *category* -- __other__ than and to the
*physical*/empirical
“__r__ealities” *category*; a *‘*__meta__-**category**’ to, and a *‘*__contra__-**category**’
of, the __r__ category.

Moreover, the __q___{rr} symbol should point to a *category* that is already
known to us, that is part of our *synchronic*, *present* conceptualized experience of the totality of *The Sciences*, and to a
category that is ontologically different than, and that is richer in
determinations than -- that is “more determinate” than -- that is more “complex”
and more ‘thought-concrete’ than, the __r__ category.

On the bases set forth above, we propose that our
solution -- our ‘semantification’ -- of the syntactic term __q___{rr} should be __t__, connoting the «*arithmos*»/*category* of __t__erms
of human language:

__q___{rr}** ****=**** **__q___{t} **=**** **__t__ **[----)
**__q___{2}.

We propose this solution, in part, because the __t__erms
of human language, as *units*,
are __meta__*-***units**
of the *units*
of empirical/*physical*
“__r__eality”.

Each __t__erm
of human language refers to, and represents, a heterogeneous multiplicity of
similar *physical*
*“***facts**”.

The __t__erm “**apple**” maps to a large,
heterogeneous multiplicity of similar, physical
**apples**.

The __t__erm “**orange**” maps to a large,
heterogeneous multiplicity of similar, physical
**oranges**.

That is, “__t__erms”
are *gene*ral relative
to *physical* *“***facts**.

One “__t__erm” typically
stands for a large, although still finite, set of similar *physical* situations.

Each “__t__erm” is an «*aufheben*» *‘*__meta__-«*monad*»*’* of a large number of similar “*fact**”*-«*monads*».

Our solution for __q___{rr} is that __q___{rr} connotes the «*arithmos*»/*category* which has __t__erms
of human language as its «*monads*».

In an *historical*, __dia__chronic
sense, __r__ *“***of**” __r__ also connotes
the natural-historical result of the ‘self-reflexion’, of the
‘bending-back-upon-self’, of “__r__eality”; of the turning
back, upon itself, of physical “__r__eality”, to form *physical* objects -- new
parts of that *physical*
“__r__eality” -- that are capable of conscious reflection
upon *physical* “__r__ealities”,
including upon *themselves*.

In that *historical*, __dia__chronic
context, this would mean the immanent emergence, within pre-human[oid] *Nature*, from out of
pre-human[oid] *Nature*
-- as part of the __dialectic__,
or of *the self**-***development**,
of *Nature*
itself -- of “observing eyes”, and of “observing sensoria” in *gene*ral; of observing and
thinking minds, as also “things” / objects / bodies within *Nature* -- i.e., the
emergence of natural ‘[ev]entities’ which produce languages, hence “__t__erms”,
to name / communicate their observations / experiences among themselves, as a
crucial part of their “struggle for existence” -- of their struggle for
successful expanded human[oid]-social reproduction.

But this consideration does not belong to *synchronic*, *systematic *__dialectics__ in
itself.

It only represents the *diachronic* implications -- the pointers to __past__ *history*, and to __future__ *history* -- that are __im__plicit in *synchronic* __dialectics__, but which
do __not__, as such -- as
*diachronic* --
form an __ex__plicit
component of the explanatory apparatus of *synchronic*, *systematic *__dialectics__.

Marx’s treatises on the __dialectical__, *immanent critique* of the
capitalist, ideology-ridden *science*
of “political economy”, are works of *synchronic*, *systematic *__dialectics__.

But, in Marx’s view, *systematic
*__dialectics__
is __not__ absolutely
dirempt from *diachronic*,
*historical *__dialectics__.

Marx discussed this *interconnexion* between *systematic *__dialectics__ and *historical *__dialectics__, as it arises in his extension of previous
concepts of __dialectics__
-- an *interconnexion*
which is, e.g., inaccessible to the __eternally__ *systematic*, *Parmenidean*
__dialectics__
of Plato’s __immutable__
«*arithmos eidetikos*» --
in a crucial methodological passage from the __Grundrisse__:

**“**...our [A.D.: *systematic**-*__dialectical__] method indicates the points where *historical* investigation
must enter in, or where bourgeois economy as a merely *historical* form of the
production process *points
beyond itself to *__earlier__ *historical* modes of production.”

“In order to develop the laws of bourgeois economy,
therefore, it is __not__
necessary to write the *real ***history**
of the relations of production.”

“But the correct observation and deduction of these
laws, *as having themselves *__become__
in history, always leads to primary equations -- like the
empirical numbers e.g. in natural science -- which *point towards a *__past__
lying *behind*
this system.”

“These *indications* [*Andeutung*], together with a correct
grasp of the __present__,
then also offer the key to the understanding of the __past__ -- a work in its
own right which, it is hoped, we shall be able to undertake as well.”

“This correct view likewise leads at the same time to
points at which the *suspension*
[at which the «*aufheben*» -- A.D.] of the *present* form of production relations gives signs of its __becoming__ -- *foreshadowings of the *__future__.”

“Just as, on one side the pre-bourgeois phases appear
as *merely ***historical**,
i.e. *suspended*
[i.e., ‘«*aufheben*»-ed’ -- A.D.] pre-suppositions, so do the *contemporary* conditions of
production likewise appear as engaged in *suspending** themselves* [i.e., in ‘«*aufheben*»-ating themselves’ -- A.D.] and hence in positing the *historic**
presuppositions* for a new state of society.**”**

[Karl Marx, __Grundrisse__: *Foundations of the Critique of
Political Economy* (*Rough Draft*), M. Nicolaus, transl.,
Penguin Books [Middlesex: **1973**], pp. **460**-**461**, *italics*
only emphasis by Marx; __combinations__
of *italic*, __underlined__, **bold**, and color emphasis have been added by A.D.].

Accepting this solution, our __s__tep **s**** ****=**** ****1 **__dialectical__*-**model equation* thus becomes [with ‘__D__’ for the *systematic**-*__dialectic__ context’s
‘curvilinear, ontological __delta__
operator’, signifying the addition of an increment of new ‘ideo-ontology’] --

__)-|-(__**1** **=** **(**) __r__ (**)****2**^{1}^{ }**= ** __r__^{2}^{ } **= **__q___{r}^{2}^{ } **= **

__r__ **+**** **__D____r__^{ } **= **__r__ **+ **** **__t__

**[----)**

__q___{1} **|**__+__**|**** **__q___{2}

-- and we have the meaning that, as of __s__tep **1**, **s**** ****=**** ****1**, our model,

__)-|-(__**s** **=** **(**) __r__ (**)****2**^{s},

posits the *possible*
existence of *two*
philosophical categories -- **(1)** the category of physical “__r__ealities”,
‘‘‘plus’’’ [signed by ‘**+**’, for connotatively “interpreted”, or “assigned”, *‘***ontological **__qual__ifiers’, in
the ‘curvilinear symbols *motif’* context
of *systematic *__dialectics__], **(2)** its
‘first contra-category’, of ‘‘‘__t__erms of language’’’, *both* together comprising **(1)** the «*physis*»*-*«*monads*», or *‘***fact**-**units**’, of the __physi__cal __r__ealities’
«*arithmos aisthetos*», or sensorially-perceived
*“***facts**”-**assemblage**, superposed
with, and *‘‘‘*__named__’’’
-- in a *many**-*to*-***one** relationship -- __by__ **(2)** the ‘trans-Platonian’
«*arithmos eidetikos*»
of __t__erms-*units*,
or of __t__erms-«*monads*».

*¿*How does this *two**-*dimensional
‘possibility-space’, or categorial ‘ideo-ontology cumulum’, expand in __s__tep **2***?*

__Step ____2__. For __s__tep **s**** ****=**** ****2**** **of this
presentation-model, since **2**^{2}** ****=**** 2 ****x**** ****2**** ****=**** 4**, we have
--

__)-|-(__**2** **=** **(**) __r__ (**)****2**^{2}^{ } **= ** __r__^{4}^{ } **= ** __r__^{2}^{ }^{x}^{ }^{2}^{ } **= ** ** **

**(**) __r__^{2} (**)**^{2}^{ } **=** **(**) __r__ **+ **** **__t__
(**)**^{2}^{ }

-- which connotes the *‘*__self__-**refl**__e__xive
function’, or *‘*__self__-**operation**’, of the *collective* __dialectical__ *operation* -- of the *two**-*category, *two**-*dimensional
*categories**-*__cumulum__ operation -- denoted by

**(**) __r__ **+**** **__t__
(**)**.

Syntactically, per the axioms of the _{N}__Q__ ‘meta-numbers’, we have the following values as the
content of __s__tep **2**:

__)-|-(__**1** **=** **(**) __r__ (**)****2**^{2}^{ }**= ** **(**) __r__ **+**** **__t__
(**)**^{2}^{ } **= **** **

**(**) __r__ **+**** **__t__
(**)** **x**
**(**) __r__ **+ **** **__t__ (**)**^{ } **=**

** **

**(**) __r__ **+ **** **__t__
(**)** **+**** ** __D__**(**) __r__ **+ **** **__t__ (**)**^{ } **=**^{ } ** **

**(**) __t__ **x** **(**) __r__ **+**** **__t__
(**)** (**)** ^{ }[using
Miguel’s shortcut] **=**

** **

__q___{r} **+**** **__q___{rr} **+**** **__q___{tr} **+**** **__q___{tt} ^{ } **=**^{ }

__r__ **+**** **__t__
**+**** **__q___{tr} **+**** **__q___{tt} ** **

**[----)**

__q___{1} **|**__+__**|** __q___{2} **|**__+__**|** __q___{2}_{+}_{1} **|**__+__**|** __q___{2}_{+}_{2}^{ } **= **

__q___{1} **|**__+__**|** __q___{2} **|**__+__**|**
__q___{3} **|**__+__**|** __q___{4}.

Our challenge, for this __s__tep, is to
“solve for” the *two*
new categories, __q___{t}_{r} and __q___{tt}, that appear for the first time in this __s__tep, that is,
to determine the meanings, the connotations, the definitions, of each of these
new ‘categorograms’, as ‘connotative entailments’ of the already solved / known
/ assigned meanings / connotations /definitions of the ‘categorograms’ __r__ **=** __q___{r} and __t__ **=** __q___{rr}.

So, let us apply the __gene__ral ‘‘‘canon of interpretation’’’ of
such __dialectical__ categories to the __speci__fic case of this
example.

Per that ‘‘‘canon’’’, the ‘categorogram’ __q___{t}_{r} connotes a new *‘*__uni__-**category**’/*hybrid* «*arithmos*» -- the *synthesis* category --
that results from the «*aufheben*»
__of__ the __q___{r} *category*/«*arithmos*», __by__
the __q___{t} *category*/«*arithmos*».

This often means a *category*/«*arithmos*» each of whose «*monads*» is a *hybrid* of the «*monads*» __of__ the __q___{r} *category*/«*arithmos*» __with__
the «*monads*»
of the __q___{t} *category*/«*arithmos*».

Also per that ‘‘‘canon’’’, a ‘categorogram’
like __q___{t}_{r} might also
connote *‘‘‘*__conversion__’’’,
__of__ «*monads*» of the __q___{r} *category*/«*arithmos*», __into__ «*monads*» of the __q___{t} *category*/«*arithmos*».

Also, per that ‘‘‘canon’’’, the __q___{t}_{r} symbol should
point to a *category*
that is already known to us, that is part of our *synchronic*, *present* conceptualized experience of the
totality of *The Sciences*,
and to a category that is ontologically different than, and that is richer in
determinations than -- that is “more determinate” than -- that is more “complex”
and more ‘thought-concrete’ than, all of the categories evoked in the preceding
__s__teps.

And, per that ‘‘‘canon’’’, a symbol like __q___{tt} connotes the
new *category*/«*arithmos*» that results
from the __self__*-*«*aufheben*» of the __q___{t} *category*/«*arithmos*», and, quite
often, that results from the __self__*-*«*aufheben*» *‘*__meta__-**monad**ization’ of some of the «*monads*» of the __q___{t} *category*/«*arithmos*».

That is, each «*monad*», or *unit*, of the __q___{tt} «*arithmos*»/*category* should be a *‘*__meta__-«*monad*»*’*, or *‘*__meta__-**unit**’, of «*monads*» or *units* of the __q___{t} «*arithmos*» / *category*.

So, the *units* of the __q___{tt} *assemblage* should be, in
this __speci__fic
case, ‘meta-__t__erms’, ‘__t__erms *of second degree**’*,
each one made up out of a heterogeneous multiplicity of ‘__t__erms
*of first
degree**’*.

Also, the *category* __q___{tt} should be *an*__other__ *category* -- __other__ than __all__ of the categories previously
evoked, in the preceding __s__teps; a *‘*__contra__-**category**’
of each of those categories, and a *‘*__meta__-**category**’ **/ **‘__meta__-«*arithmos*»’ to the __q___{t} *category * / «*arithmos*».

Moreover, the __q___{tt} symbol should
point to a *category*
that is already known to us, that is part of our *synchronic*, *present* conceptualized experience of the
totality of *The Sciences*,
and to a category that is ontologically different than, and that is richer in
determinations than -- that is “more determinate” than -- that is more “complex”
and more ‘thought-concrete’ than, all of the categories evoked in the preceding
__s__teps.

On the bases set forth above, we propose that our
solution -- our ‘semantification’ -- of the syntactic term __q___{t}_{r} should be __f__, connoting the «*arithmos*»/*category* of Ockham’s __propositions__ of the __f__*irst** intention*:

__q___{t}_{r}** ****=**** **__q___{f}** ****= **** **__f__ **[----) **__q___{3}.

We propose further that our solution/‘semantification’
of the syntactic term __q___{tt} should be __s__, connoting the «*arithmos*» /* category* of Ockham’s __propositions__ of the __s__*econd** **intention*:

__q___{tt}** ****=**** **__q___{s}** ****= **** **__s__ **[----)
**__q___{4}.

In proposing these two ‘semantifications’, we are in
resonance with Ockham in invoking here, __not__
just a subdivision of the universe of __t__erms of language, i.e., of
*single* __words__ of language,
into two sub-universes, of *“categorematic”* words vs. of *“syncategorematic”* words, and
with *“***categorematic**”
*words*
corresponding to __q___{t}_{r}, and with *“***syncategorematic**” *words* corresponding to __q___{tt}, so that __q___{t}_{r} and __q___{tt} would fail to
differ, in terms of the __ontology__
of their «*monads*»,
from __q___{t}.

We hold that the *‘‘‘*__conversion__’’’, i.e., the *apt* *description*, of
the «*arithmos*»/*assemblage* of *physical* *“***facts**”, by *“***terms**”, __q___{t}_{r}, as of the «*arithmos*»/*assemblage* of *“***terms**”, by *“***terms**”, __q___{tt}, requires, in each case, something more than mere *single* __words__.

Human language is far more than the mere utterance of *single* __word__-sounds. Human language is, for starters, an «*arithmos*»/*assemblage* of __sentences__ -- of __propositions__, that is,
of well-formed *‘***meta**-**words**’,
each one made up out of a heterogeneous multiplicity of *words*.

We thus hold that __q___{t}_{r} must connote a
multiplicity of “__t__erms” -- both “categorematic” and “syncategorematic”
alike -- combined, into *well**-***formed sentences**,
i.e., into __propositions__,
that describe our knowledge of the *physical world*, __q___{r}.

So:

__q___{t}_{r}** ****=**** **__q___{f}** ****=**** **__f__ **[----) **__q___{3}.

[cf. Boole’s
category of “Primary Propositions”, __Laws of Thought__,
Chapter IV, Proposition I.].

We thus hold also that __q___{tt} must connote a
multiplicity of “__t__erms” -- both “categorematic” and “syncategorematic”
alike -- combined, into *well**-***formed **__sentences__,
i.e., into __propositions__,
that describe our knowledge of the *secondary world* of our own creation, of that core
component of ‘The Human Phenome’ which is *the world of human language itself*, __q___{t}.

So:

__q___{tt}** ****=**** **__q___{s}** ****= **** **__s__ **[----)
**__q___{4}.

[cf. Boole’s “Secondary
Propositions”, __Laws of Thought__, Chapter IV, Proposition
I.].

Thus, __f__
connotes an «*arithmos*»/*assemblage* of “true” __proposition__*-***units**,
or of __proposition__*-*«*monads*», that are *about*
-- of *empirically-apt* __propositions__ which *describe* -- *the physical world*.

And, __s__
connotes an «*arithmos*»/*assemblage* of “true” __proposition__*-***units**,
or of __proposition__*-*«*monads*», that are *about*
-- of *cognitively-apt* __propositions__ which *describe* -- *the world of human language*, as an expression of *the world of human thought*.

But the *assemblage* of “true” __propositions__ about the *“***real**” world, external to
human thought, __q___{t}_{r}, is what *‘‘‘***the **__sciences__ of the rea__l__*’’’*
“contain”; is the ‘‘‘content’’’ of those sciences; is what Ockham means by *‘‘‘***the **__sciences__ of the rea__l__*’’’*, by the
category he names «*scientia
rea*__l__*is*», which we will connote by __l__: __f__** ****=**** **__l__.

And the assemblage of “true” __propositions__ about the *“***terms**” world, the world
human language/thought, __q___{tt}, is what *‘‘‘***the sciences of the ratio**__n__*al**’’’*
“contain” for Ockham; is the ‘‘‘content’’’ of those sciences; is what Ockham
means by *‘‘‘***the sciences of
the ratio**__n__*al**’’’*, by the
category he names «*scientia
ratio*__n__*alis*», which we will connote by __n__: __s__** ****=**** **__n__.

Accepting these solutions, our __s__tep **s**** ****=**** ****2**** **__dialectical__*-**model equation* thus becomes --

__)-|-(__**1** **=** **(**) __r__ (**)****2**^{2}^{ }**= ** __r__^{4}^{ } **= ****(**) __r__ **+ **** **__t__
(**)**^{2}^{ } **= **** **

**(**) __r__ **+ **** **__t__
(**)** **x**
**(**) __r__ **+ **** **__t__ (**)**^{ }**= **

**(**) __r__ **+**** **__t__
(**)** **+**** ** __D__**(**) __r__ **+**** **__t__
(**)**^{ } **=**^{ }

**(**) __t__ **x** **(**) __r__ **+**** **__t__
(**)** (**)** ^{ }[using
Miguel’s shortcut] **=**

** **

__q___{r} **+**** **__q___{rr} **+**** **__q___{tr} **+**** **__q___{tt} ^{ } **= **^{ }

__r__ **+**** **__t__
**+**** **__l__ **+**** **__n__ ** **

**[----) **

__q___{1} **|**__+__**|** __q___{2} **|**__+__**|**
__q___{3} **|**__+__**|** __q___{4}.

-- and we have the meaning that, as of __s__tep **2**, **s**** ****=**** ****2**, our model,

__)-|-(__**s** **=** **(**) __r__ (**)****2**^{s},

posits the *possible*
existence of *four*
philosophical categories -- **(1)** the «*archÃ©*» category of *physical* “__r__ealities”,
‘‘‘plus’’’; **(2)** its *‘***first** __contra__-**category**’,
of ‘‘‘__t__erms of language’’’, ‘‘‘plus’’’; **(3)** the *‘***first** __uni__-**category**’
of the preceding two categories, whose *units* or «*monads*» are, as __propositions__, *hybrids*, *combinations*, or __syntheses__ of
“categorematic” and “syncategorematic” __t__erms, and, in
a sense, as *“*__synthetic__”
[cf. Kant] __propositions__,
*hybrids* of
the *units* of *physical* “__r__eality’
with the *units*
of __t__erms, thus constituting the category of __propositions__ of the __f__*irst** intention*, constituting Ockham’s «*scientia rea*__l__*is*», ‘‘‘plus’’’; **(4)** the *‘***second** __contra__-**category**’,
the category of __propositions__ of the __s__*econd** **intention*, constituting Ockham’s «*scientia ratio*__n__*alis*», *‘*__meta__-**monad**ic’ with respect to
the «*monads*»
of the *‘***first** __contra__-**category**’,
of ‘‘‘__t__erms of language’’’, given that these __propositions__, which are
the *units* or «*monads*» of the «*scientia ratio*__n__*alis*», constitute, each one, a *‘*__meta__-**term**’,
or *‘***term of the second degree**’, each one made up out of a
heterogeneous multiplicity of *‘*__t__*erms** of the first degree**’*, which are
the *units* or «*monads*» of the *category*/«*arithmos*»/*assemblage* of *units* herein connoted by __t__.

All *four*
categories together, as an *‘***ideo**-**cumulum**’,
comprise --

**(1)** the «__physi__s»*-*«*monads*», or *‘***fact**-**units**’, of the __physi__cal __r__ealities’
«*arithmos aisthetos*», or
sensorially-perceived *“***facts**”-**assemblage**, superposed
with, and __named__ -- in a *many**-*to*-***one**
relationship -- __by__;

**(2)** the
‘trans-Platonian’ «*arithmos eidetikos*»
of __t__erms-*units*,
or of __t__erms-«*monads*», *both* together superposed with;

**(3)** the
‘trans-Platonian’ «*arithmos eidetikos*»
of the __propositions__*-***units**,
or of __propositions__*-*«*monads*», of the Ockhamian «*scientia rea*__l__*is*», all *three* together superposed with;

**(4)** the
‘trans-Platonian’ «*arithmos eidetikos*»
of the __propositions__*-***units**,
or of __propositions__*-*«*monads*», of the Ockhamian «*scientia ratio*__n__*alis*».

*¿*But is there a *next* step, a __s__tep **3**, for this *classificatory*, *taxonomic*, *systematic**-*__dialectic__
*presentation* of
*the Sciences? *

Our __s__tep **2** models where this
__dialectic__
completed for Ockham, and for his times; for what was *synchronic*/extant in his
experience, and in the experience of his *contemporaries*.

Ockham saw, as far as we know, no other sub-categories
of *Science*
besides «*scientia rea*__l__*is*» **&** «*scientia
ratio*__n__*alis*».

*¿*But is there another sub-category of *Science* that was __not__ *synchronic*/extant for Ockham, but that __is__ *synchronic* for us*?*

*¿*How could this now *four**-*dimensional ‘possibility-space’ of
categorial ‘ideo-ontology’ expand in a possible __s__tep **3***?*

Let’s see what arises if we iterate this model one
more __s__tep, i.e., for **s**** ****=**** ****3**.

__Step ____3__. For __s__tep **s**** ****=**** ****3**** **of this *presentation**-***model**, since **2**^{3}** ****=**** 2 ****x**** ****2** x** ****2**** ****= **** 8**, we have --

__)-|-(__**3** **=** **(**) __r__ (**)****2**^{3}^{ } **= **__r__^{8}^{ }** = **__r__^{4 }^{x}^{ }^{2}^{ } **= ****(**) __r__^{4} (**)**^{2}^{ } **=** ** **

##
**(**) __r__ **+ **** **__t__
**+ **** **__l__ **+**** **__n__ (**)**^{2}^{ }

-- which connotes the *‘*__self__-**refl**__e__xive
function’, or *‘*__self__-**operation**’, of the *collective* __dialectical__ *operation* -- of the *four**-*category, *four**-*dimensional
*categories**-*__cumulum__ operation -- denoted by

**(**) __r__ **+ **** **__t__
**+**** **__l__ **+**** **__n__ (**)**.

Syntactically, per the axioms of the _{N}__Q__ ‘meta-numbers’, we have the following values as the
content of __s__tep **3**:

__)-|-(__**3** **=** **(**) __r__ (**)****2**^{3}^{ }**= ** **(**) __r__ **+**** **__t__
**+ **** **__l__ **+**** **__n__ (**)**^{2}^{ } **=**

** **

**(**) __r__ **+ **** **__t__
**+ **** **__l__ **+**** **__n__ (**)** **x**
**(**) __r__ **+ **__t__ **+**** **__l__ **+**** **__n__ (**)**^{ } **=**

** **

**(**) __r__ **+ **** **__t__
**+**** **__l__ **+**** **__n__ (**)** **+**** ** __D__**(**) __r__ **+**** **__t__
**+ **** **__l__ **+**** **__n__ (**)**^{ } **=**

** **^{ }

**(**) __n__ **x** **(**) __r__ **+ **** **__t__
**+ **** **__l__ **+**** **__n__ (**)** (**)**^{ }[using Miguel’s shortcut] **= **

__q___{r} **+**** **__q___{rr} **+**** **__q___{tr} **+**** **__q___{tt} **+**** **__q___{nr} **+**** **__q___{nt} **+**** **__q___{n}_{l} **+**** **__q___{nn} ^{ } **=**^{ }

__r__ **+**** **__t__
**+**** **__l__ **+**** **__n__ **+**** **__q___{nr} **+**** **__q___{nt} **+**** **__q___{n}_{l} **+**** **__q___{nn}

**[----) **

__q___{1} **|**__+__**|** __q___{2} **|**__+__**|**
__q___{3} **|**__+__**|** __q___{4} **|**__+__**|** ^{ }__q___{4}_{+}_{1} **|**__+__**|** __q___{4}_{+}_{2} **|**__+__**|**
__q___{4}_{+}_{3} **|**__+__**|** __q___{4}_{+}_{4} **=**

__q___{1} **|**__+__**|** __q___{2} **|**__+__**|** __q___{3} **|**__+__**|** __q___{4} **|**__+__**|** ^{ }__q___{5} **|**__+__**|** __q___{6} **|**__+__**|**
__q___{7} **|**__+__**|** __q___{8}.

Our challenge, for this __s__tep, is to
“solve for” the *four*
new categories, __q___{n}_{r}, __q___{n}_{t}, __q___{n}_{l}, and __q___{nn},
that is, to determine the meanings, the connotations, the definitions, of each
of these new ‘categorograms’, as ‘connotative entailments’ of the already solved
/ known / assigned meanings or connotations of the ‘categorograms’ __r__** ****=**** **__q___{r}, __t__** ****=**** **__q___{rr}, __l__ **=** __q___{t}_{r}, and __n__** ****=**** **__q___{tt}.

Here, at this juncture, we wish to venture a
conjecture as to the ‘connotatively-entailed’ meaning-solutions, or
definition-solutions, of these *four*
new categories, __q___{n}_{r}, __q___{n}_{t}, __q___{n}_{l}, and __q___{nn}.

In general, we see each of them as connoting
the fruition of the __application__
of the «*scientia ratio*__n__*alis*» -- i.e., of advanced *formal*,* mathematical logic */ __dialectic__ -- to each
of the already-defined domains, resulting in their *‘‘‘***subsumption**’’’ by the «*scientia ratio*__n__*alis*».

We conjecture, more specifically, as follows --

__q___{n}_{r} **=**
the «*scientia ratio*__n__*alis*» *‘‘‘***conversion**’’’
of the «*arithmos aisthetos*» of *physical facts*; the *rational**-***scientific**
immanent __critique__, __explanation__ [including *theory of* [past] *perceptual* *error*], and __reconstruction__ of human perception
of the «__r__*ealis*» realm, extending to the formation of a *universal*, *systematic**-*__dialectical__
*taxonomy* of
this realm, related to the project of the «*Natur*» division of Hegel’s __Encyclopedia
of the Philosophical Sciences__;

__q___{n}_{t} **=**
the «*scientia ratio*__n__*alis*» *‘‘‘***conversion**’’’
of the «*arithmos eidetikos*»
of __t__erms of human language; the *rational**-***scientific**
immanent __critique__, __explanation__ [including *theory of* [past] *terminological* *error*], and __reconstruction__ of human language,
including the formation of a *rational*
*universal* *language*, comprising a* *__dialectical__ *phonetic language*, or *‘*__dialectical__ phono__gram__y’,
as well as of a *‘*__dialectical__ picto__gram__y’,
and of a *‘*__dialectical__ ideo__gram__y’,
related to Leibniz’s vision of a «*characteristica universalis*»; also related to the project
of the «*Logik*»
division of Hegel’s __Encyclopedia of the Philosophical Sciences__;

__q___{n}_{l} **=**
the «*scientia ratio*__n__*alis*» *‘‘‘***conversion**’’’
of the «*arithmos*»
of the __proposition__s of the «*scientia rea*__l__*is*»; the *rational**-***scientific** immanent __critique__, __explanation__ [including *theory of* [past] *scientific theories* *error / ideology*], and __reconstruction__ of the «*scientia rea*__l__*is*», via the formation of *‘***theories**’ accounting for the *differences* in the history of *different* *theories/ideologies*,
or *systems of *__propositions__,
regarding *the same* *“***real**” *facts */ *physical phenomena*, e.g.,
of *different* *theories/ideologies* of the same *facts* arising from different historical
epochs/human-social formations, related to the project of the «*Natur*» division of
Hegel’s __Encyclopedia of the Philosophical Sciences__, and to
that of Marx’s __Grundrisse__, __A Contribution to the Critique of Political
Economy__, and __Capital__: *A Critique of Political Economy*.

__q___{nn} **=**
the «*scientia ratio*__n__*alis*» *‘‘‘***conversion**’’’
of the «*arithmos*»
of the __propositions__
of the «*scientia ratio*__n__*alis*» themselves; the *rational**-***scientific** *immanent *__self__*-*__critique__,
__self__*-*__explanation__
[including *theory of* [past]
*scientific theories*
*error / ideology*], and __self__*-*__reconstruction__
*of* the «*scientia ratio*__n__*alis*» *by*
the «*scientia ratio*__n__*alis*» themselves, including the formation of *‘*__meta__-**theories**’
accounting for the *differences*
in the history of *different*
*theories/ideologies*, or *systems of *__propositions__, regarding *the same* *mathematical *and other *“***noetic**” *facts*/*phenomena*, e.g., of *different* *theories/ideologies* of “the same” *‘‘‘***noeta**’’’ , or objects of thought, as they arise
in different historical epochs/human-social formations, extending to
development of a comprehensive *mathematics
of *__dialectics__
itself; related to the project of Hegel’s __Science
of Logic__, and especially to its second volume.

Note that Marx’s __A Contribution to the
Critique of Political Economy__, and __Capital__: *A Critique of Political Economy*,
provide a *speci*fic
example, and, mostly implicitly, a *gene*ric paradigm, for a core component of __q___{n}_{l} as
conjecturally defined above.

Nothing
like Marx’s works was extant in Ockham’s time, but those works __are__ *‘***synchronic**’ for us, today.

In those works, Marx addresses “...the
evolution of the economic formation of society ... as a process of natural history”
[Marx, Preface to the First German Edition
of __Capital__], and,
thus, addresses the human species and its ‘Phenome’ as a *natural object*, as a “__r__*eal*” *fact*/*object*, a part of the
purview of «*scientia *__r__*ealis*».

The
subsidiary objects which Marx addresses in those two treatises -- values, *commodities*, *monies*, *capitals*, and their “law
of [surplus-]value” -- are tied to *objects* of *physical* “__r__*eality*” -- *packages of tea*, *bank notes*, *steam**-***powered looms**,
etc. -- but they are __not__
simply *physical objects*.

They are also *noetic objects*, ‘meme[t]ic objects’, ‘‘‘mental
objects’’’, ‘‘‘ideal objects’’’, ‘cultural objects’; *‘***psyche**-ic objects’, parts of the human-social *“***intersubjectivity**”; parts
of ‘The Human Phenome’ -- human, “social relations of production” [Marx].

Such ‘category-[idea[l]-]objects’, such ideas, such
memes, residing within ‘The Human Phenome’, the very «*arithmos *[*eidetikos*]» of meme «*monads*», are *“***material**”
not only in the sense that they are *germane*,
that they are *salient*, for *Science*, but also in that,
being *“***inter**__subjective__”
-- held mentally, and behaviorally, *in common* among human actors -- they are also __objective__.

Such is the paradigm that we call *‘*__psycho__historical materialism’, pioneered by Marx.

Note also that this conjectured solution models the
research programme actually pursued by our co-founder, Karl Seldon.

The universe of discourse of our model of the
Ockhamian __dialectic__ of Science, as updated, for our time, above --

__)-|-(__**3** **=** **(**) __r__ (**)****2**^{3}^{ }**= **

__r__ **+ **** **__t__
**+ **** **__l__ **+**** **__n__ **+ **** **__q___{nr} **+**** **__q___{nt} **+ **** **__q___{n}_{l} **+ **** **__q___{nn}

-- is that of ‘*the
*__elements__ [ *f*__acts__ and *words *], *and the sub**-***divisions**,* of knowledge* [*science*]’.

If we abbreviate this universe of discourse,
restricting it to just *“***The
Sciences**”, leaving its __elements__ [ *f*__acts__ and *words *] __im__plicit therein, instead of making
them __ex__plicit, as
before, then perhaps we can encompass the most essential content of this *systematic *__dialectic__, or *‘***synchronic** **ideo**-**taxonomy**’, by means of a *Triad*ic Seldon Function
of only three terms --

__)-|-(__**1** **=** **(**) __L__ (**)****3**^{1}^{ }**= **

__L__ **+**** **__N__
**+ **__Y__

-- instead of the earlier eight algebraic terms, but one
whose third and final algebraic term, __Y__, we must now define.

For the terms of this new model, we make the following
assignments to, or interpretations of, the terms of the old --

**1.** __L__** ****(----) **** **__l__;

**2.** __N__** ****(----) **** **__n__;

**3.** __Y__** ****(----) **** ****(**) __q___{n}_{r} **+**** **__q___{nt} **+**** **__q___{n}_{l} **+**** **__q___{nn} (**)**.

The *‘***cumulum**’
-- the superposition -- of the conjectured definitions of the ideographical terms
__q___{n}_{r}, __q___{n}_{t}, __q___{n}_{l}, and __q___{nn},
taken together, amount to a definition of the *knowledge**-***field** which we name ‘[*socio**-***politico**-**economico**-**]***ps*__Y__*chohistory*,
or *‘***ps**__Y__**chohistorical** __dialectics__’, for
short, and ‘__Y__’ is our standard symbol for that *knowledge**-***field**.

It is conjectured here as a *‘‘‘***complex unity**’’’, or __dialectical__ synthesis, of __N__** ****&**** **__L__, of *logico**-***mathematical**-cognitive-__ps____Y____che__ological
**science**
**& ***physical **science*, including *human social science* as part of *the science of natural history* in
Marx’s sense:

__Y__ **=** __q___{N}_{L}.

The fuller development of __Y__ is the intended
target and focus of the work of **Foundation **__Encyclopedia__** **__Dialectica__.

We can illustrate this proposed abbreviation, *‘*__dialectical__-**pictogramically**’,
as --

The connotations of this __Y__ encompass the
connotative content of the earlier model’s algebraic terms __q___{n}_{r}, and __q___{n}_{l}, and also __q___{n}_{t}, since __L__
, itself ‘proxying’ __l__
-- and as the operand of __N__,
‘proxying’ __n__, in the formation of __Y__ -- implicitly
connotes / "contains" the elements __r__** **and __t__.
However, it can be well-argued that the *‘self-refl*__e__xive moment’ of the earlier
model’s **s**** ****=**** ****3** __s__tep -- namely, its final algebraic term, __q___{nn} -- is missing from, this “abbreviated” model,
which thus has a gaping ‘homeomorphic defect’ as a model of that earlier model.
To get back this crucial component of the
Seldonian concept of *psychohistory*, while still retaining
some of the second model’s succinctness, we can convert the __Triad__ic Seldon Function model back into a __Dyad__ic Seldon Function model, still using the same
«*archÃ©*».
Then the new __dyad__ic model’s **s**** ****=**** ****1**** **__s__tep gives us Ockham’s __dialectic__, and its **s**** ****=**** ****2** __s__tep gives the Seldonian ‘extention’, with
__Y__ **= **__Y___{1 }**+**** **__Y___{2},
viz. --

__V____. ____Solution____-____Implicit Issue__:** ***¿*__Did Ockham __*Intentionally* Construct a *DIALECTIC of Science**?*

Ockham was certainly no
stranger to concepts of *“**the *__dialectic__*”*.

In the first place, medieval scholarship was permeated
by notions of, and discourses upon, __dialectic__, to an
extent which has yet to be fully explored and appreciated by modern scholarship.

In the introduction to her book __Dialectic and
Its Place in the Development of Medieval Logic__, Eleonore Stump
wrote:

“Since 1975 my
work in medieval logic has concentrated on dialectic.”

“I have tried to
trace scholastic treatments of dialectic to discussions of it in the work of
Aristotle, the Greek commentators on Aristotle, and the Latin rhetorical
tradition. But I have been especially
interested Boethius [**F**.__E__.__D__.: **480**-**525** C.E.], whose discussions of dialectic were among the most
important influences on scholastic treatments of the subject.”

“Accounts of
dialectic based ultimately on Boethius’s views continued to play a fundamental
role in philosophy through the fourteenth century.”

“The earliest
scholastic logician whose work we know, Garlandus Compotista [**F**.__E__.__D__.: __Dialectica__, *circa*
**1054** C.E.], devoted a
great deal of attention to Boethian dialectic, and I have tried to follow the
development of scholastic dialectic from Garlandus through various
twelfth-century logicians (including Abelard) and thirteenth century terminists
into the fourteenth century... .”

“The scholastic
literature on dialectic is a vast treasure for scholarship, which even now is
largely unmined ... .”

[Eleonore Stump, __Dialectic
and Its Place in the Development of Medieval Logic__, Cornell University Press [Ithaca: **1989**], p. **1**].

In the second place, Ockham himself was a major
fourteenth century *innovative* contributor to medieval scholastic __dialectic__, principally via his *circa*
**1323** treatise __Summa
logicae__, or *“*__Summa__[*ry*]__
of All Logic__*”*, including to the logic of “obligations”*****.
Some of his innovative contributions were reviewed by Eleonore Stump in
the following terms --

“... there are
also contexts (notably that of dialectical disputation, discussed in
obligations treatises) in which types of inferences or rules for inferences
that would otherwise be perfectly acceptable produce paradoxical results.”

“...for instance, Ockham discussed cases in which the
otherwise acceptable rule ‘From the impossible anything follows’ does not hold.”

“In the
burgeoning study of consequences and the growing interest in obligations the
rather narrow formalization of logic brought
about by the terminists was stretched and
broadened. Ockham included a discussion
of obligations within his treatment of inferences, and he also considered a
broad variety of dialectical inferences.”

“In fact, he
blurred the boundary between dialectic and demonstration, thereby elevating
dialectic above the second-class status accorded it by the terminists.”

-- And --

“...contrary to the views of some contemporary
scholars, there is nothing in Ockham’s criteria for valid inferences which
corresponds to the notion of material implication in contemporary analytic
philosophy.”

“... syllogisms,
Topics, obligations, and insolubles are woven together in Ockham’s work into
one general account of inferences which is at once richer and broader than that
of the thirteenth-century terminists.”

“The terminists
tend to force all nonsyllogistic arguments, including dialectical arguments,
into syllogistic form. By contrast,
Ockham’s account of inferences recognizes a variety of nonsyllogistic
inferences, including dialectical and obligational inferences, without trying
to mold them into syllogistic form; and it includes syllogisms as one species
of consequences among others. The
result, which clearly represents an advance, is a development toward a logic
which is both formal and able to account for many sorts of inferences.” [Eleonore Stump, *ibid*., pp.
**3**; **8**-**9**].

*[We have an
‘‘‘obligation’’’ here to define “obligation” as a term of logic. A respondent in
a formal disputation is said to hold an __obligation__ to avoid
being forced into formally contradictory [*“*__im__possible”] assertions *due to respondent
logical error*, given that the original assertion upheld by the respondent
is a “possible” one.].

However, Ockham’s work on __dialectics__
was informed primarily by the __dialectics__ of
Aristotle, specifically that of Aristotle’s __Topics__ and __Rhetoric__,
and thus only indirectly, not directly, by the __dialectics__
of Plato.

We therefore cannot infer, with any certainty, that
Ockham consciously intended his ‘ideo-taxonomy’ of *the Sciences* to be anything
like a Platonian «*arithmos eidetikos*», especially
given Ockham’s “nominalist” views, rejecting Platonian and other hypostatizations
or reifications of *“***universals**”.

More likely, Ockham’s ‘ideo-«*genos*»’ of *the Sciences*, and its two
‘ideo-«*species*»’
of «*scientia realis*» and «* scientia
rationalis* », were conceived within the prevailing *Aristotelian* «*genus*»/«*species*»
‘ideo-taxonomical’ tradition, which, though distantly related to Plato’s «*arithmoi eidetikoi*» __dialectic__,
represents a very attenuated form thereof, in the aftermath of Aristotle’s
critique and critical rejection of Plato’s «*arithmoi eidetikoi*».

Links to definitions of additional __Encyclopedia Dialectica__* *special
terms deployed in the discourse above --

«*arithmos aisthetos*»

«*arithmos eidetikos*»

*categorial*

*category*

*‘***cumulum**’

__dialectical__ *categorial
progression presentations*

*homeomorphic defect*

[*The*]* Human Phenome*

*immanent*

*immanent critique*

«*monad*»

*ontological category*

*ontology*

*psychohistory*

*psycho**historical** *__dialectics__

__qual__*o**-**fractal*

__qual__*o**-**Peanic*

*Seldon
Functions*

*‘***self**-**meta**-**monad**-**ization**’ or *‘***self**-**meta**-**individual**-**ization**’
or *‘***self**-**meta**-**holon**-**ization**’