Absolute particularity:

beyond the problem of universals

The essentially new form of thinking now actually occurring for the first time is precisely and demonstrably beyond the threefold of the "problem of universals” that belongs to thinking essentially in the past. The concept of absolute particularity1 is for the first time beyond the universal-singular divide that hitherto sustained both realism and nominalism and the merely relative particularity that was conceptualism.

1)
(= Ordinary Logical Order2
[OLO] 1) Let the first solution to the problem be the realist relative
singularity S_{r}* _{
}=* U|S, where S = UP, and thus S

2)
(= OLO 2) The nominalist *alternative* to such a realist singularity is a
relative universality U* _{r}* = U|U, where U = SP, and thus U

3)
(= OLO 4) *Tertium quid absolutum*: Absolute particularity (beyond the
realist-nominalist divide) P* _{a}* = U|P, where P = US, and thus P

4)
(= OLO 3) Where the fundamental form of all logical statements4
is pq + p̅q + pq̅ + p̅q̅ (= OLO 1 + 2 + 4 + 3), the complex element missing from
the sequence, U|UP (= S* _{r }*= OLO 1)
+ U|SP (= U

It is clear from the foregoing that absolute particularity P* _{a}*
(OLO 4) is not only beyond the realist-nominalist S

Beyond understanding the universal/the concept as an abstraction (beyond the
universal-singular divide mediated by the particular) it is understood for the
first time that there is no universal (concept) that is not the singular
absolutely identified with the particular (the percept),^{7}
not the singular identified with the singular identified with the particular:
that the particular is beyond the universal-singular divide mediated by the
particular: the particular beyond beyond *x*: the real whole (the reality)
absolutely named: absolute particularity for the first time.

**Notes**

^{
1}
Cf. D.G. Leahy, *Beyond Sovereignty: A New Global Ethics and
Morality* (Aurora, 2010).

2
Cf. D.G. Leahy, *Foundation: Matter the Body Itself* (Albany,
1996), Section III.1.

3 Cf., above, nn. 1 and 2; also, D.G. Leahy, Novitas Mundi: Perception of the History of Being (Albany, 1994), "Prolegomena" and "Appendices α, β, and γ".

4 Leahy, Foundation, Section III.1.

5
The reduction of the sequence pq + p̅q + pq̅ + p̅q̅ to the
mathematical framework of the Cartesian coordinate system, + +, − +,
+ −, − −, makes it possible to elucidate the deep structure of the
conceptualist equation S_{r}^{2} = SS = (UP)^{2}.
In the sequence UP + SP + US + SS, when + and – are
predicated of the respective elements qua unity, the last term SS is
equivalently
−1^{2}.
This S, the singular as such, is then, qua
−1, the square of the imaginary number *i*
= √−1:
*i*^{2} = (√−1)^{2
}=
−1 (cf. the equivalents of the real trinary logic elements (1,
0,
, 1) in Leahy, *Beyond Sovereignty*, Backnote 4, Table
4, et passim, and, on this web,
Real Trinary Logic, the Imaginary Number *i*, & the Cyclic
Series of Complex Numbers.
The singular as such −1, here, corresponds to absolute
placedness* i*^{2}, on the scale employed
loc. cit.,
the square of absolute singularity).
Thus the coherence of the conceptualist notion of the
exhaustion of the generic *in mente* as the nameable whole
radically imaginary is rendered patent: SS =
−1^{2} = (√−1)^{4}
= *i*^{4}
= (UP)^{2 }= 1^{2} (cf.,
ibid.,
where *i*^{4 }corresponds to Omnipotence).

6
Transposing the OLO 1, 2, 4, 3 to the minimum order (MO) 4, 1, 2, 3
(cf., above, n. 2), the proportion U|US (Absolute Particularity) :
U|UP (Realism) :: U|SP (Nominalism) : U|SS (Conceptualism)
simplified by reduction to either S^{−1} : P^{−1} ::
(SS)^{ −1} : (SP)^{−1} or −1 : 1 :: 1 : −1 proves in
both forms to Unity (= *i*^{4}, as above, n. 5).
Note that the penultimate form of the proportion confirms
that Nominalism (the Real Unnameable) and Conceptualism (the
Nameable Imaginary) are each the other’s reciprocal, respectively,
(SS)^{−1} and (SP)^{−1}.
The ratio (SS)^{−1} : (SP)^{−1 }further
simplified produces the ratio S^{−1} : P^{−1}—transforming
the reciprocal relation in the ultimate form of the proportion above
(−1 : 1 :: 1 : −1) to identity (−1 : 1 :: −1 : 1) at once the form
of the original proportion unreduced—precisely
the form of the ratio of Absolute Particularity (the Real Absolutely
Named) to Realism (the Real Nameable), the elements of which ratio
are related as befits the ‘absolute originality’ of the MO
‘fourth/fourthness’ (cf. Leahy, *Foundation*, Section IV.1) not
as essential reciprocals the one of the other but indeed as
contradictories. See,
also, on this web,
Real Trinary Logic, the Imaginary Number *i*, & the Cyclic
Series of Complex Numbers where Omnipotence, on the scale there
employed, corresponds to absolute singularity* *identified with
absolute particularity as *i*^{4} = *i*^{1}* ∙* *i*^{3}.
The conceptual correspondence here (OLO 4): U|US: the universal
restricted to the *singular* identified with the universal,
i.e., identified with *the singular identified with the particular*.*i*^{4}/*i*^{2},
=
Absolute Particularity
(U|US)*, i*^{2}.
This expanded version of the table(s)
loc. cit.
summarizes the complex (!) complex/noncomplex relationship between
the two aforementioned scales of analysis

*i*^{0}
= *i*^{4}) to neither of whose differentiated factors (Absolute Singularity [*i*^{1}]
and Absolute Particularity [*i*^{3}]) does it or any
other element in the (self-consciousness) Universals Problem Analogue
column correspond as such (cf., above, passim). Thus is highlighted the radical removal
foundational to self-consciousness at once from Immediacy (*i*^{1
}= *i*) and Beginning (*i*^{3 }= *-i*).
Cf. G.W.F. Hegel's *Gesamtwerk*; also,
D.G. Leahy, *Faith and Philosophy: The Historical Impact* (Aldershot
and Burlington, 2003), "Appendix: Thinking in the Third Millennium:
Looking Without the Looking Glass."

Vis-à-vis the ordinary logic of mathematical proportion note the
following departure therefrom shared by the logic of complex
numbers and the real trinary logic of the thinking now occurring:

1) It is ordinarily true for all proportions composed of four discrete
elements that the product of the lesser elements divided by the
greatest element = the square of the least element.

2) Where these discrete elements are related to one another by
discrete powers of the antecedent in the ratio of the second to the
first the underlying form of this constant is that of all logical
statements pq, p̅q, pq̅, p̅q̅, where p = the first element not
multiplied by *x*, and q = the first element not multiplied by
*y*. Thus, the
aforementioned constant mathematical relation is logically
equivalent to the relation (pq × p̅q × pq̅)/p̅q̅ = (pq)^{2}.

See, for example, on this web,
Epitome or Food for Thought, the sequence, where *x* =
82944 and *i*(*x)* = the integral product of *x *
(note: here *i* = integral product is not to be confused with
*i* = imaginary number),
[*i*(*x*)/9]/*i*[*i*(*x*)/9],
*i*(*x*)/9, *i*(*x*), (*i*[*i*(*x*)/9][*i*(*x*)])
= 113.7̅7̅7, 9216, 82944, 6718464 = pq, p̅q, pq̅, p̅q̅, where p =
113.7̅7̅7 not multiplied by 9^{2} and q = 113.7̅7̅7 not
multiplied by 9^{3}. In
accord with the stated rule, (pq × p̅q × pq̅)/p̅q̅ = (pq)^{2},
(113.7̅7̅7 × 9216 × 82944)/6718464 = 113.7̅7̅7^{2}.

Likewise,
ibid.,
the sequence M (mass), M/T (mass flow rate), M/T^{2}
(surface tension), M/T^{3} (Poynting vector) = pq, p̅q, pq̅,
p̅q̅, where p = M not multiplied by T^{−1}, and q = M not
multiplied by T^{-2}.
In accord with the stated rule, (pq × p̅q × pq̅)/p̅q̅ = (pq)^{2},
(M × M/T × M/T^{2})/(M/T^{3}) = M^{2}.^{2}, cf. gravitational coupling constant
αG =
Me^{2}/MP^{2},
as well as the imaginary mass of the Higgs field −M^{2} and the
real mass of the Higgs
boson +M^{2}.

3) But
where *i* is the imaginary number, the sequence
i,
i^{2},
i^{3},
i^{4} may be logically
equivalent to pq, p̅q, pq̅, p̅q̅ (as above), but it may as well be
equivalently pq, p̅q, p̅q̅, pq̅, where p =* i* not multiplied
by *i*, since not previously so, or, since previously so
multiplied, by -*i*, and q = *i *not multiplied by *–i*,
and the relation between the last term and the product of its
predecessors, productive of the square of the first term, (i
×
i^{2
}× i^{3})/*i*^{4
}= i^{2},
will have the form (pq × p̅q × p̅q̅)/pq̅ = (pq)^{2}.

a)
On the analogy of the division in extreme and mean ratio whereby
uniquely the whole line is divided but once to provide the elements
of a geometrical proportion, so where p̅
and q̅
each ordinarily provide one of two multiples of *i* necessary
to produce *i*^{4}
the latter in the case of the cycle of imaginary numbers is
nevertheless uniquely also able to be produced by multiplying but
once by q̅.

b)
The relation of the real trinary logic elements 0,
**1,** whose shared absolute value = 1 (cf. Leahy, *
Foundation* III.1), may be compared to the complex numbers
i,
–1, –*i*,
and 1, whose shared
absolute value |z| = 1 in the unit circle of the complex plane,
where, with respect to the complex numbers, p = relative value not
negative, and q = relative value y > x, as here illustrated:

c)
The departure of the cycle of complex numbers from the ordinary
logico-mathematical order of the form of proportions (above, para.
3), while real, is not unlimited.
While
|z| = 1 is
the shared absolute value of the complex numbers
i,
–1, –*i*,
and 1^{ }in
the unit circle of the complex plane, the latter itself is not
beyond the + and – framework of the x/y plane.
Where the complex numbers are grounded so, real trinary
logic, on the other hand, is by analogy grounded in the always
positive x > y pq̅
that is 1 = *i*^{4}
and is, so grounded, consistently beyond the + and – framework of
the x/y plane.

d)
That the complex plane falls short of the absolute indifference of
the elements of real trinary logic to the + and – framework is
demonstrable when real trinary logic
*i* (complex notation 0,–1) is positive (complex notation
1,0) and consistently so as 1
**1** (complex notation 1,0) when placed analogously alongside
the y > x negative form –*i*
×
–1
×
i
= –1 (complex notation –1,0).

e)
In the case of the complex numbers where
–*i*
×
–1
×
i
=
–1 the end product is
the included middle
–1,
i.e., the end product is not the excluded middle
–i
×
i
= 1. Were the middle
here the excluded middle then the end product would be
–i
×
1
×
i
=
1 = *i*^{4}
=
pq̅.
But,
as it is, where the end product is the included middle
–1, the end product is the return of the cycle itself upon itself in
the form of the included middle, prematurely folding back upon
itself perpetually foreclosing arriving at positive foundational
unity 1.

In contrast, in the case of real trinary logic, where
1
**1**,
the middle actually
is the excluded middle, i.e., 1
0 =
**1 **= *i*^{4}
=
pq̅.

f)
Where **1** = *i*^{4}
=
pq̅, the real trinary logic end product immediately preceding, p = 0
not multiplied by 1, since not previously so, or, since previously
so multiplied, by 0, and q = 0* *not multiplied by
*i*, viz., *i *and
–*i*,
the latter appearing twice, function in place of three multiples,
since, as demonstrated above, there is no excluded middle in the
threefold sequence of complex numbers.
But in the real trinary logic analogue to the aforementioned
construction such a repetition of a second multiple in lieu of such
a nonexistent excluded middle is absolutely obviated: the excluded
middle

g)
Summarily: (–*i*
×
–1
×
i
= –1) : (1**
**0

For mediation and omnipotence, and relative vs. absolute
discontinuity of the continuum, cf. Leahy, *Beyond Sovereignty*,
passim, for the former and the latter, and Backnote 4, Tables 1-2,
for the former. For the excluded vs.
included middle, cf. Leahy, *Faith and Philosophy*, Chapter 7.
For dyadic ternary vs. real trinary logic, cf. Leahy,
*Foundation* III.1, *Afterword* 2, and “Appendix: The *
De Trinitate* of Augustine and the Logic.”

4)
Where the cyclic relation of the powers of *i* is understood as
this logico-mathematical departure from the ordinary form a
discretionary contextual relation surfaces that makes a real
logico-mathematical
difference as p̅, whose configuration here is fully consonant with
the real trinary logic fundamental operator
‘since’
(cf. Leahy, *Foundation*, Section III.1, and, above, passim), at once with the first
appearance of p̅ in the "Index of the Ethics of Simplicity" indeed in
association with Discretion p̅q (see, on this web,
Beyond the Superego: The
Pneumasomatic Human Person).

5) Cf. also the table above, where, on the analogy to *i* in the
cycle of complex numbers, the logical equivalent of Omnipotence is
pq̅ =
i^{4},
related as such to Immediacy pq = *i*, Mediation p̅q = *i*^{2},
and Beginning p̅q̅ =* i*^{3},
and compare this with, on this web,
ibid., "The Index of the Ethics
of Simplicity," where, likewise, the logical equivalent of
Omnipotence is pq̅, related as such to Readiness pq, Discretion p̅q,
and Beneficence p̅q̅, where p = not discriminating, and q = not
doing.

^{
7}
For the identity of concept and percept, cf. Leahy, *Foundation*, passim.