The Circle Of Twelvefold
Systemization
Beginning and end are part of a single ring and no one can {comprehend its principle.
Chuang Tzu [C525:27.305]
6.1 The Systematic Twelvefold Form of Relation
Twelvefold systemization is the operation of combining four sets of
simple synthetic (threefold) relations in such a way that they form a second level
analytic relation. Any system which exemplifies this pattern (viz. "3.4 = 12"),
whether it be natural or theoretical, can be described as a perfectly balanced
logical system, inasmuch as it makes equal use of the two types of fundamental
logical operation, analysis and synthesis. Several examples of such systems,
taken from both natural and theoretical constructions, will be given in 6.4. But
first we must bring out the formal similarity between all such systems by
specifying the twelve compound forms of relation and by exploring (in 6.2-3)
the various ways in which they can be mapped onto geometrical figures.
The expressions in a twelvefold compound form of relation can be
derived quite simply by successively combining each of the three expressions in
a simple synthetic form of relation (viz. "+", "-", and "x") with each of the four
expressions in a second level analytic form of relation (viz. "++", "-+", "+-",
and "--"). The results are listed in Table 6.1 on the following page. (Note that if
the third expression in each column, the synthetic expression, is dropped, the
eight remaining expressions constitute a list of the third level analytic forms of
relation.) These forms of relation can be represented either on "static" maps,
such as rectilinear plane and solid figures, or on "dynamic" maps, such as
circles with arrows indicating the direction of
Table 6.1: The Compound Expressions in a Twelvefold
Form of Relation
the logical flow for each relation. The latter is the preferable method, since the
former tends to ignore the dynamic implications of any synthetic operation [see
4.4]. Nevertheless, each provides valuable insights of different sorts, so we
will deal with them in turn in the following two sections.
6.2 Static Maps of Twelvefold Systemization
In rectilinear terms the problem of all compound systemization on the
level of the twelvefold form of relation is expressed in the question: How does
the triangle of simple synthesis fit into the cross (or square) of second level
analysis? The most obvious answers to this question are fairly simple ones: by
either connecting the four poles of the cross with straight lines, or connecting
the vertices of a square with two diagonals, we can construct a figure which
consists of four triangles united together in the form of a square. Labelling
either the edges or the vertices of each triangle with the twelve expressions listed
in Table 6.1 yields the results shown in Figure 6.1.@ In Figure 6.1a the centre
of the cross stands for the starting-point (the "0") and the perimeter stands for
the conclusion (the "1") of all four syntheses, whereas in Figure 6.1b the centre
is the conclusion, and the starting-point is beyond the perimeter. This difference
has important implications for the dynamic flow implied by each of these
(otherwise static) diagrams. For it implies that the synthetic "direction" (i.e. the
movement from "0" to "1") of the synthesized cross is
Figure 6.1: A Second Level Analytic Map of the
Twelvefold Forms of Relation
(a) Based on the Cross(b) Based on the Square
Figure 6.2: A Third Level Analytic Map of the
Twelvefold Forms of Relation
(a) Based on the Double Cross(b) Based on the Inverted Double Cross
outward, while that of its inversion, the synthesized square, is inward.
The same form of relation can be represented rather more clearly in
terms of the map for third level analysis, the double cross. For the double cross
separates the two sides of each line (and so also the two angles at each vertex)
of the single cross with a space. This enables the same labels to be assigned, but
manages to avoid the risk of equivocation. Rather than using any of the
inversions of the double cross suggested in Figure 2.18, we can simply rotate
the figure by 45\ [cf. note 2.11], thus maintaining a more obvious parallel
between the maps in Figure 6.2 with those in Figure 6.1.@ The fact that this
same double cross can be used to represent not only twelvefold relations (3.4),
but also eightfold relations (2.4) [see Figure 2.17b] and ninefold relations (3.3)
[see Figure 4.3b] could be used to highlight various similarities between these
rather different forms of logical relation. But to do so here is unnecessary [see
4.2]. Instead we shall proceed directly to a discussion of several ways in which
the twelve compound expressions can be mapped onto solid rectilinear figures.
In all there are nine regular polyhedra, each of which can be employed
as a map of formal relations in the Geometry of Logic. Five of these are convex
and four are non-convex. The latter are too complicated to be easily labelled in
their two-dimensionalrepresentation, to which we are limited. So, although they
may be of some use in a three-dimensional context (e.g. in the classroom, or in
the privacy of one's study), the four regular non-convex polyhedra will be
ignored in this book. The relevant properties of the five regular convex
polyhedra are summarized in Table 6.2. The number of edges (N), of vertices
(N), and of faces (N) in any polyhedron are related according to "Euler's
theorem", which states that "N-N+N = 2" [B520:74; cf. C521:89]. (Note the
ubiquity of the number "12" or its submultiples, "6", "4", "3", and "2", in
Table 6.2.) The "dual" of a polyhedron is
Table 6.2: Properties of the Five Regular Convex
Polyhedra
Number of: Numerical analysis of: sides Dual
Nameedgesverticesfacesper face edges verticesfaces
Tetra- 6443Tet.2x322 hedron
Hexa- 12864Oct.2x3 2 2x3 hedron
Octa- 12683Hex.2x3 2x32 hedron
Dodeca- 3020 125Ico. 3x102x10 2x3 hedron (=3+3)(=2+2)
Icosa- 3012 203Dod.3x10 2x3 2x10 hedron
the polyhedron in which the number of vertices and faces is reversed. The
tetrahedron is its own dual, so these five solids form a "1+2+2" pattern.
Since the tetrahedron has four faces, each of which has three edges,
these edges can be used in much the same way as the cross to map out the
twelve expressions currently under consideration [see Figures 6.3a,b]. Indeed,
the suitability of the tetrahedron as a map of this compound form of relation is
confirmed by an interesting perspectival shift which occurs in the two-
dimensional representation of the figure when the angle of view (i.e. the
perspective) changes. For if we look at any one of its faces straight on, the
figure will be represented as a triangle divided into three triangles, as in the map
of the second level synthetic form of relation [see Figure 6.3c; cf. Figure 4.4].
The same result can be obtained by looking straight at one of its vertices,
centred in relation to its opposite face. But if an edge becomes the centre of our
attention, then the figure will be represented as a second level analytic cross
inscribed in a sideways square [see Figure 6.3d; cf. Figure 6.1a].@ Another
equally ambiguous way of mapping the twelve compound expressions onto the
tetrahedron would be to regard the
Figure 6.3: The Tetrahedron as a Twelvefold Map
(a) The "Solid" Perspective(c) The Synthetic Perspective
(b) The Net(d) The Analytic Perspective
vertices, each of which has a valency of three (i.e. three edges extending from
it), as the matrix for each synthetic relation. But in this case each line would be
defined differently, and would have a different logical "direction", at each end.
All these maps suffer from the same problem of equivocation, since each
edge is a side for two different triangles, or leads into two different vertices [cf.
4.2]. One solution is to chop off the four corners in such a way as to construct a
regular "truncated tetrahedron". The angles (or the sides [cf. Figures 4.7a and
b]) of the four new triangles can then be labelled without equivocation as in
Figure 6.4.@ Even in its truncated form, however, this map is less than ideal,
for it requires four hexagonal faces to be employed with no logical function
corresponding to them. Moreover, the rules for mapping the twelve expressions
onto any of the above tetrahedral figures are not sufficiently unambiguous.
The twelve edges of the cube or of the octahedron can also be used to
represent the expressions for twelvefold compound systemization. The rules for
mapping in these instances are slightly different from those considered up to this
point. For the twelve expressions in Table 6.1 must be grouped not according to
their columns (of three), but according to their rows (of four). In both cases the
four edges at the top of the figure represent the analytical relation between the
four "positive" elements, those at the bottom of the figure represent the
equivalent relation between "negative" elements, and those in the middle
correspond to the "synthetic" (x) elements [see Figure 6.5].@
The reciprocal ("dual") relationship between the cube and the octahedron
is not very obvious in the diagrams given in Figure 6.5, especially since the
former uses squares for its faces, while the latter uses triangles. (The nets of
each figure only obscure the systematic relationships between the edges.) But a
different perspective on each figure can reveal their high degree of similarity. If
the cube is
Figure 6.4: The Truncated Tetrahedron
(a) The "Solid" Perspective
(b) The Net
Figure 6.5: Synthetic Maps of the Twelvefold Form of
Relation
(a) The Cube
(c) The Octahedron
(b) The Cube's Net
(d) The Octahedron's Net
viewed in such a way that two diagonally opposite corners are aligned, then its
edges form a hexagon [see Figure 6.6a]. Likewise, if the octahedron is viewed
from a point perpendicular to any one of its faces, its edges form precisely the
same shape [see Figure 6.6b].@ The two resulting figures can be distinguished
only by looking at the shapes formed by the edges which fall within the (two-
dimensional) perimeter. These two figures [6.6a and b] are strikingly similar to
the ones used in 5.3 to represent the first two stages in the synthetic
differentiation of the hexagon into its triangular elements [see Figure 5.3]. In
fact, the only differences are that some of the solid lines are here dotted in order
to give the illusion of depth, and that the lines connecting the midpoints of
opposite sides in Figure 5.3a are omitted in Figure 6.6a. The implication of this
similarity for the Geometry of Logic is that it confirms the thoroughly
systematic nature of sixfold distinctions: from the geometrical sixfold distinction
which defines the hexagon we can construct solids whose faces have either
three (synthetic) or four (analytic) sides each. Moreover, this "3+4" pattern
inherent in the hexagon makes for an interesting reciprocal relationship with the
"3+3" pattern inherent in the heptagon [see Figure 5.6], thus linking the logical
symbolism of the numbers "6" and "7" even more closely.
Finally, the twelve forms of the dodecahedron, or the twelve vertices of
the icosahedron, could also be used as maps of the twelvefold form of relation.
The problem in both cases, however, is that the elements are related in fivefold
patterns: the twelve faces of the dodecahedron are all pentagons, and the twelve
vertices of the icosahedron all have a valency of five. The rules we have
developed for the geometrical mapping of logical forms of relation do not give
us the capacity to relate twelvefold forms in groups of five. Indeed, the
difficulty in constructing a truly consistent map is evident when the
dodecahedron is depicted in its "projected" form (as we did with the
Figure 6.6: The Dual Relationship Between Cube and
Octahedron
(a) The "Corner" Perspective
(b) The "Face" Perspective on the Cubeon the Octahedron
Figure 6.7: The Dodecahedron
(a) The Solid(b) Its Projection
(c) The Net
cube in Figure 2.20b). The best that can be done at this point is to mark out
groups of three by shading the edges of the pentagon which are not intended to
stand in a special systematic relation, and to place each pair of opposite
expressions on opposite faces [see Figure 6.7b].@
As with the former pair of duals, the reciprocity between this latter pair
could be represented by picturing the dodecahedron from a point perpendicular
to one of its twelve faces, and the icosahedron in such a way that two of its
vertices are perfectly aligned. The outcome in both cases is a ten-sided figure
with a ten-pointed star inscribed in it. (The icosahedron would also include lines
connecting all opposite points through the centre.) But there is no need to draw
attention to these diagrams here (they can be viewed in C521:87-8), since such
tenfold (5.2) distinctions are more relevant to natural than to theoretical systems,
as is exemplified by the flower shape of the dodecahedron's net [see Figure
6.7c].
These figures, like all the others used so far in this chapter, suffer to
some extent from the same drawback: they are all so complex that it is often
difficult, if not impossible, to see the direction of logical flow implicit in
twelvefold systemization because of its synthetic aspect. In order to rectify this
situation, we will turn in the next section to a rather different type of figure,
which has not yet received the attention which it deserves.
6.3 The Rectilinear Inscription of the Dynamic Circle
The Geometry of Logic would be grossly incomplete without a
thoroughgoing account of the symbolic power of the circle, for the circle has
proved to be among the most common geometrical figures in the religious and
psychological symbolism of all men and in all cultures. In most cases it seems
to be connected with a predominantly dynamic and/or synthetic content. Thus,
for example, Lao Tzu represents the unfolding of the Dao (i.e. the creation of
the "One" and its differentiation into the "ten thousand things") as turning "in a
circle". Until now we have mentioned the circle only on several occasions. In
1.2 we used the distinction between the circumference and the centre of a circle
to represent the logical distinction between "everything" and "nothing" in a
given universe [see Figure 1.1b]. (A similar distinction was noted in 5.3 in our
discussion of the "6+1" structure of the heptagon.) And in 4.4 the 3:4 ratio
between the circle and the square was mentioned as a hint concerning the
symbolic value of the circle for synthetic integration. But it is now time to focus
our attention more systematically on the perfect symmetry of the circle, and on
its power as a dynamic map of the detailed interrelationships between the
elements of a twelvefold relation.
Mapping the twelve expressions produced by compound systemization
onto a circle will at first proceed along quite different lines than those followed
when using rectilinear plane and solid figures. Rather than letting the form of
the figure determine the placement of the logical expressions we will begin by
letting the form of the expressions determine how the figure itself is to be
divided up. The perfect symmetry of the circle can be accounted for in two
opposite ways. On the one hand the circle can be regarded as a single,
continuous line enclosing a space, the simplest of all geometrical figures in two
dimensions. But if, on the other hand, the process of constructing a two-
dimensional figure is assumed to be "atomic"--i.e. to consist always in a series
of straight strokes--then the circle turns out to be "a curve of infinite
complexity", made up of an "infinite number of infinitely short strokes"
[M536:41; cf. note 4.16]. These two perspectives correspond exactly to the
distinction between "unity (1/1)" and "infinity (1/0)", which was mapped onto
the horizontal axis of the cross in Figure 3.4. The former represents synthesis
(1) viewed synthetically (1), and the latter represents synthesis (1) viewed
analytically (0). But whichever method we use to construct a circle, the perfect
symmetry of the end product makes it divisible into equal parts in an infinite
number of ways. For our purposes, we shall divide the circle into twelve equal
arcs. We have already seen in 3.3 how simple synthetic integration can be
analysed in terms of four elements connected with three relations (or steps). If
we flatten out their ordinary triangular map, we can arrange these elements in a
single linear order as follows:
G10 + - 1.
Some combination of the use of two different rules must then be adopted
in order to apply this linear pattern to each of the expressions in a second level
analytic division. First, the analytic expressions at the four points of the cross
can be viewed as either actively dispensing the synthetic third terms of each
compound expression (thus giving rise to the analytic progression "1,-,+,0"),
or as passively receiving this synthetic element (thus assuming the standard
synthetic progression, "0,+,-,1"). This perspectival difference is crucial. For in
the former case, each of the four points of the cross is regarded as the starting-
point for one of the synthetic stages, so the first two terms in each of the four
cardinal (i.e. second level analytic) expressions will be identical to the first two
terms in the following two expressions; in the latter case, by contrast, these two
terms will correspond to those in the preceeding two expressions.
Consequently, in the former case [see Figures 6.8a and d] the dominant
expression at each cardinal point will be the one containing a "0" in the third
term (e.g. "--0" in Figure 6.8a suppresses the "+-1" expression which also
belongs there), while in the latter case [see Figures 6.8b and c] the "1" is always
dominant (e.g. "--1" in Figure 6.8b suppresses the "+-0" which also belongs
there). The second rule is, quite simply, that the compound expressions can be
mapped around the circle in either a clockwise or a counterclockwise direction.
(The latter direction is
Figure 6.8: The Four Forms of the Dynamic Circle
(a) Clockwise and Analytic(b) Counter-Clockwise and Synthetic
(c) Clockwise and Synthetic (d) Counter-Clockwise and Analytic
based on an inversion in the rules for mapping and applying the former version
of the dynamic circle.) These rules give rise to the four maps of twelvefold
systemization which are presented in Figure 6.8.@ (Leaving a gap in each case
between the first and last expressions indicates that the system does have a
specific staring-point and goal. The full significance of these gaps will become
clear in 7.1; but for now it is sufficient to say that any twelvefold system
without a definite start or finish can be mapped more appropriately onto a
complete circle--i.e. one without such a gap.
The position of the expressions in Figures 6.8a and c are identical to
those in Figures 6.8b and d, respectively, with the exception that every "0"
changes to "1" and every "1" to "0". The only other difference is in the direction
of development. As a result, we could define all the alternatives using only two
maps--either maps (a) and (d), which follow the analytic method, or maps (b)
and (c), which follow the synthetic method--so long as we specified that a
change of method requires an interchange of all 1's and 0's. What this means is
that either experience (i.e. outward, clockwise movement, or evolution) or
reflection (i.e. inward movement, or involution) can be described in terms of
two complementary twelvefold systems, and that the pattern followed by each
pair is identical, except that what is real (1) in one is ideal (0) in the other.
The significance of mapping the twelve compound expressions onto the
circle is that it represents more adequately than any other geometrical figure the
unity of the diverse elements mapped onto it--and this unity is precisely what is
required by the operation of compound systemization. At first sight there also
appears to be a corresponding disadvantage, inasmuch as each element mapped
onto the circle seems to be explicitly related only to the elements immediately
preceeding and succeeding it on the circle. If this were the case, the circle would
not be an entirely adequate symbol for compound systemization, since every
element of a perfect system is related to every other element in that system. Such
thoroughgoing interrelationship is what makes a system into a system, and not
just a collection of unrelated or semi-related elements. Therefore, in order to
perfect the symbolic power of the circle we must find a way of depicting such
relationships between the elements mapped onto its perimeter.
The obvious way of showing the relations between a set of points on a
circle is to connect them with lines. The various ways in which this can be done
can be shown most simply and consistently by adopting as the standard map for
the remainder of this section the circle as mapped in Figure 6.8c (but without the
gap; hence "0" and "1" will be combined as "x"). The trouble is that if we
connect all the points all at once we end up with a figure containing sixty-six
lines, which is far too complex to serve as a graphical aid for grasping any
given type of relations within the system, though its complexity does have a
certain measure of beauty [see Figure 6.9].@ The solution is to categorize the
various types of relation which hold between the twelve elements and to
represent them separately in the form of various rectilinear figures inscribed in
the circle. This will demonstrate how the circle can be used as a general,
dynamic context in which each of the main rectilinear figures we have discussed
can be fit. In the process of examining how each of these represents a specific
type of relation within the framework of the formal system, the beauty and
suitability of the circle will become increasingly apparent.
Six different types of logical relation can hold between the elements of a
twelvefold logical system. These can be categorized quite simply in terms of the
distance between two points on the circle. Thus there are twelve relations in
which two given points are next to each other on the circle; twelve in which
there is a point in between the two given points; twelve each in which there are
two, three, or four intervening points; and six in which the two given points
stand opposite each other, with five points intervening on either side.
Connecting all the points in these various ways requires one, two, three, four,
five, or six discrete figures to be inscribed into the circle, respectively.
Figure 6.9: The Sixty-Six Relations in a Twelvefold
System
Figure 6.10:
Figure 6.11:
We shall deal with each of these types of systematic relation by grouping
them into complementary pairs. In geometrical terms a pair of complements can
be represented by the circumference-centre distinction: i.e. complementary types
of systematic relation will be represented by figures which are of relatively equal
distance from the circumference on the one hand and from the centre on the
other. The first and last pair of complementary types both contain a total of
seven discrete figures, while the middle pair contains only three; they can be
listed as follows:
1 dodecagon + 6 lines = 7
22 hexagons + 1 dodecagram = 3
3 triangles + 4 squares = 7. &1
Once again this exemplifies the systematic connection between "7" and "12"
[see 4.4,5.3]. But perhaps the most significant point of all is that the seventh
figure in the series, which stands over and against the six rectilinear
inscriptions, is the twelvefold circle itself (or alternatively, its complement, the
point at the centre).
Inscribing a regular dodecagon (twelve-sided polygon) into a circle will
highlight the "one-place" relations which hold in our twelvefold formal system
between each element and its two neighbors [see Figure 6.10].@ Now if the
twelve points of the circle are left unlabelled, and the circle is rotated randomly
so that we do not know which end is up, how much information can be gained
from one-place relations alone if the appropriate position for one of the
expressions is given? In the present case the third term of the following
expression can be determined by plugging the third term of the given expression
into the pattern for simple synthetic integration. As long as this does not entail
crossing the boundary between two sets of simple synthetic operations (i.e. as
long as the third term in the given expression is not "x") we can also assume
that the first two terms will stay the same. For instance, if we are given the
expression "+++", then we know that the next expression will be "++-", since
"-" always follows "+" in a standard synthetic progression. But if the given
expression is "++x" then all we know for sure is that the next expression's third
term will be "+". Thus only one set of three expressions can e fully defined.
Nothing about the one-place relations in a twelvefold system enables us to
determine or even describe how the analytic (first and second) terms of the
various expressions are related, because the rule relating each expression is
entirely synthetic.
The next simplest type of systematic relation, the "six-place" relation, is
the complement of the one-place relation, because it tells us information about
all three levels in a given relation. It connects opposite pairs of expressions
according to the rules of first level analytic division, represented in Figure 6.11
by each of the six lines passing through the centre.@ Given the position of one
expression in any of these six first level analytic relations, we can determine all
three terms in the expression opposite it: the first and third terms will always
remain the same, and the second term will switch to its opposite. Thus "-++"
defines for us "--+". (If we were to map the twelve expressions onto the circle
using the rules relevant to the inversion of the cross [see note 6.12], then the
first term would also change to its opposite, and only the third term would
remain the same. So in this case "-++" would define "+-+".) However, we are
unable to determine the relation between any two adjacent expressions, since
there are no rectilinear connections between the six lines in the figure for six-
place relations.
Thus neither one-place nor six-place relations on their own are sufficient
to describe the entire system, the former being entirely synthetic and the latter
entirely analytic; nevertheless, each has important functions when comparing
specific parts of some actual system [see 6.4]. Even if these complementary
types of relation are considered together, they do not enable us to define the
whole system on the basis of one given expression. Only half the system can be
defined in this way, since no principle is available to enable us to pass, for
example, from "+-x" to "--+" and so to define the other two sets of simple
synthetic relations.
The twelve "two-place" relations can be mapped onto two regular
hexagons inscribed into the circle [see Figure 6.12].@ But the rule for such
mapping does not yield nearly as much information as do the rules for one-place
and six-place relations, because two-place relations do not follow the synthetic
process as closely as one-place relations, yet no analytic substitute is made, as
in six-place relations. Within either hexagon only the third term in each
expression can be determined if the position of a single expression is given, for
in both hexagrams the third terms always follow the reverse order of the
standard synthetic pattern (i.e. they follow the analytic pattern "1,-,+,0") ; hence
their status can be regarded as "psuedo-analytic". Since the two hexagons are
always exactly one place out of step with each other, the third term in each
expression on the other hexagon could also be determined--assuming that a
knowledge of the normal pattern for simple synthesis is allowed. The patterns
for the other terms are either "+,+,-,+,+,-" or "-,-,+,-,-,+" for the first term,
and "+,+,+,-,-,-" for the second term, which means that it is never possible to
be certain whether a given "+" will be followed by another "+" or by a "-" in the
following two-place relation.
Without a doubt the least regular of the six types of systematic
twelvefold relation which can be inscribed in the circle is the complement of the
two-place relation, the five-place relation. The figure
Figure 6.12: Two-Place Relations
Categorization 16of Expressions 6 8++x-++ 8-+--+x 8+-++-- 8+-x--+ 8---
Figure 6.13: Five-Place Relations
Categorization 6of Expressions 6 <++x <+-- <+++ <-+x <--- <-++ <+-x <++-
<+-+ <--x <-+- <--+ 2
Figure 6.14: The Dodecagram and the Pentagram
(a) Analysis of the Pentagram(b) Analysis of the Five-Place 1+Dodecagram
(c) An Irregular (Six-Pointed) Pentagram
which represents five-place relations is a twelve-pointed star, or dodecagram
[see Figure 6.13].@ The rule followed by five-place relations is much like that
of its complement: given one expression we are able to determine the third term
of all twelve expressions [see note 6.17], based on the same pseudo-analytic
"1,-,+,0" pattern; but the first terms proceed in groups of three (i.e. +,+,+,-,-,-
,+,+,+,-,-,-) and the second terms proceed in a self-reflecting alternating pattern
(i.e. +,-,+,-,+,-,-,+,-,+,-,+), so neither can be determined with certainty.
Because of the similarity between this pattern and that followed by two-place
relations, no additional certainty is gained by considering the pair of
complements together.
Knowing the patterns according to which each term develops through
the series of expressions in a given type of twelvefold relation could be used to
determine the probability of a given term appearing as the first or second term of
the expression which follows it. Thus in the case of five-place relations the
probability that a "+" in the first term will be followed by another expression
with a "+" in the first term is 2/3 (67%), while the probability that a "+" in the
second term will be followed by another expression with a "-" in the second
term is 5/6 (83%). The same method of prediction can be applied to any type of
twelvefold relation which does not guarantee absolute certainty. But it is
perhaps no accident that the pair of complementary types of relation which have
to depend most on probability are the ones whose number of discrete diagrams
adds up, as we noted earlier, not to seven, but to three.
This rather mysterious five-place figure bears a striking resemblance to
the pentagram. It can be broken down into segments, any two of which could
actually be used to construct a pentagram, albeit an irregular one [compare
Figures 6.14a and b]. Indeed, the map for any of the twelve positions from
which five successive five-place relations can be traced is itself an irregular
pentagram [see Figure 6.14c].@ This is undoubtedly the closest we can get to
forcing a fivefold relation into the systematic mould of twelvefold relations. It
suggests once again the importance of approximation in natural (e.g. fivefold)
systems, especially when constructing theories of such systems.
As it turns out, rather appropriately, the type of relation which conveys
the least amount of certainty to the theoretical systemizer is symbolized by the
most intricate, indeed most beautiful, rectilinear inscription of the circle. For the
five-place dodecagram has a number of other features which help to define its
place within the Geometry of Logic. For instance the three squares (3.4 = 12),
the four small triangles (4.3 = 12), or the four larger triangles which interlace at
the centre of the figure [see Figures 6.15a-c] all point us in the direction of the
next pair of complementary types of relation, where systematic interrelationship
reaches its highest level. But the most interesting pattern hidden in the maze of
dodecagramic relations is the presence of three double crosses [see Figure
6.15d].@ Needless to say, the rules for mapping the five-place forms of
relation onto these three double crosses will be rather different in each case than
those for mapping the standard third level analytic forms of relation, a difference
which is symbolized by the gradual rotation of the double cross from its
standard upright position.
A detailed comparison between the three double crosses in Figure 6.15d
is facilitated by extracting them from their context in the circle and standing them
up beside their analytic counterpart. In Figure 6.16, each of the three synthetic
double crosses is defined by noting which two synthetic (i.e. third) terms are
used in each set of expressions. The "+-" double cross [Figure (d)] is mapped
in a way remarkably similar to its analytic prototype [Figure (a)]: the
expressions on each pair of poles are the same, though their order is reversed in
two of the four cases. Rotating Figure (c) clockwise by 45\ reveals that the
pattern followed by the first two terms in each
Figure 6.15: Components of the Five-Place Dodecagram
(a) Three Squares{(b) Four Small Triangles
(c) Four Larger Triangles(d) Three Double Crosses
Figure 6.16: Analytic and Synthetic Double Crosses
a)Third Level Analytic(b)Synthetic "-x"
(c) Synthetic "x+"(d) Synthetic "+-"
expression is the same for each of the four maps, since each follows the pattern
of first level analysis, with two occurrences of each expression. That is, the first
two terms in each of these four maps follows the pattern "++, ++, -+, -+, +-,
+-, --, --", the duplicated occurrences being distinguished by an alternating third
term (either "+" and "-" or "+" and "x" or "-" and "x"). Upon close examination
even more interesting patterns and sub-patterns emerge; but to follow them up
here would be superfluous to the main task of this section, the examination of
the types of inscription appropriate to the twelvefold systemization of the
dynamic circle.
The three-place relation is symbolized by the inscription of three squares
into the circle [see Figure 6.17].@ Each of the three squares represents a unique
second level analytic form of relation, as can be seen from an examination of the
first two terms in each expression. And the three squares stand in a simple
synthetic relation to each other, as can be seen from an examination of the third
term in each expression. This means that the entire system of twelve interrelated
expressions can be determined on the basis of a given knowledge of the position
of any single expression. These four three-place relations are of utmost
importance, for they can be used to clarify the direct parallels between the
corresponding parts of the four synthetic "stages" in a twelvefold system.
The compliment of the three-place relation is equally simple, and has
equally significant systematic implications. The map of four-place relations is
the inscription of four triangles within the circle [see Figure 6.18].@ The
presence of four triangles is a good indication that they will be related according
to the standard second level analytic form of relation, and the fact that each
triangle has three vertices is a good indication that each will exhibit a simple
synthetic relation. As usual, the validity of these geometrical hints is confirmed
when the logical structure behind them is examined. For the order of the third
Figure 6.17: Three-Place Relations
Categorization of Expressions
Figure 6.18: Four-Place Relations
Categorization of Expressions
terms in each triangle (proceeding clockwise) follows the standard form for
simple synthesis; and if all the third terms are dropped, each row in the
"categorization of expressions" is revealed to be a perfect second level analytic
division. (Moreover, if each expression containing a third term "+" is taken as
the primary expression for a triangle (as in the above categorization) then each
column could also be regarded as a second level analytic division--though an
imperfect one, since the expression containing an "x" would need to be analysed
into its "0" and "1" constituents. For example, in the first column of the
categorization, the expression "-+x" could be split into "-+0" and "-+1" to
complete the analytic form. Note that even the order of the analytic terms in each
triangle corresponds to the standard given in 2.2.) On the basis of these rules,
all the expressions in a twelvefold system can once again be defined when a
single expression is given.
Obviously, three- and four-place relations are the most powerful ones in
the operation of twelvefold systemization. (Taken together, as a pair of
complements, their description of the system is no more complete, though it
does provide for an unambiguous means of transition between squares (via
triangles) or between triangles (via squares), in case the rules of analysis alone
were assumed in the former case, or those of synthesis alone in the latter.)
Nevertheless, their completeness does not take anything away from the
importance of the other types of systematic relation, each of which serves to
define a unique set of relations within the system. Indeed, if the rules for
mapping both synthesis and analysis are taken as given in every case, as they
were in the cases of the three- and four-place relations, then these other types
would have just as much explanatory power (i.e. from one given expression, all
the others could always be determined, even without the help of any inscribed
figures!) We denied this right to these other figures for two reasons--one
geometrical and one logical. The geometrical reason is that the only figures
whose combination of shapes and number suggested both analysis and
synthesis are the three- and four-place relations. The others suggest either
synthesis, analysis, or a sort of "pseudo-" version of one of these on its own.
The logical reason for not assuming that the rules for mapping both
analysis and synthesis were known in every case is that the particular way in
which we have chosen to map out the twelve expressions onto the circle is by
no means the only legitimate way of doing so. Indeed, examples of alternative
ways have already been given in Figure 6.8. Other alternatives could have been
explored as well [see e.g. note 6.12]. But in each case the particular way in
which the twelve expressions were mapped onto the circumference of the circle
(i.e. in which analytic and synthetic operations were combined) is irrelevant to
the functions of the inscribed figures. For the rules stated in each case for the
formal relations implied in each type of systematic relation are applicable to all
methods of mapping twelvefold systematic relations onto the dynamic circle. As
a review of these rules, and of the contents of this section in general, Table 6.3
on the following page summarizes the six ways we have discussed of inscribing
the circle with rectilinear figures.
6.4 Applications of Twelvefold Systemization
In a material system (i.e. one with some real content) we are not usually
faced with the situation of knowing only one element and a minimum number of
rules. More often we find ourselves in the position of knowing many or even
most elements and of having to use the rules to help us fill in the missing
elements (a task which often involves first determining what the rules are). Thus
when choosing applications for our twelvefold formal system their is no reason
to avoid using rules which we know will hold, as we did in the previous
section. Rather, the rectilinear relations defined in 6.3 are of use primarily in
highlighting similarities and differences between particular elements of a given
system, in drawing various parallels between the way major stages develop, and
in guiding the course of our overall interpretation of such a system.
Numerous twelvefold systems could be chosen as examples of how the
compound systemization we have been discussing actually operates. The ancient
quasi-science of astrology has already been mentioned in this regard in note
6.18 [s.a. 7.3]. Other obvious examples are: the division of the day into two
twelve-hour periods of time; the division of the year into twelve months,
grouped into four seasons and composed of 364 days; the twelve points of the
compass (N, NNE, ENE, E, ESE, SSE, S, SSW, WSW, W, WNW, NNW)
grouped into the four cardinal points (N, E, S, W) and divided into 360 degrees
(cf. the geometrical division of the circle). Less obvious examples also abound,
such as the description of electricity in terms of its twelve constituent functions
or powers, grouped into the primary functions of amperage, wattage, voltage,
and resistance. Such applications of our formal system can be mapped with
perfect accuracy onto the twelvefold map of the dynamic circle; and when this is
done, the interrelationships between the parts can be made crystal clear with the
help of the six rectilinear types of inscription. Thus it will be helpful to choose
two applications and examine them both in some detail. These examples will be
taken from traditional Western logic, and from Kant's Critical philosophy.
Traditional logic puts a good deal of emphasis on the "four syllogistic
figures", each of which is expressed most concisely by a three-step argument.
The logical system of syllogistic argumentation can therefore be summarized
(using an arrow in the place of the usual "is", and arranging each expression so
that only the direction of the arrows differs) as in Table 6.4. Each "figure"
(column) is an example of simple synthetic integration; and the four figures
altogether are related according to a second level analytic division. The first
terms of each analytic expression given in the top row of Table 6.4 represent the
direction of the arrow in the first and second symbolic propositions for each
"figure" ("+" = "to the right"; "-" = "to the left").
Table 6.4: The Four Syllogistic Figures
I (-+)II (++)III (--)IV (+-)
1+P M P M P M P M
-S M S M S M S M
xS P S P S P S P
It may seem odd at first to relate the syllogism to synthetic logic, since I
have argued in 3.3 that all simple synthetic relations are based on the laws of
non-identity and contradiction, the very opposites of the laws upon which the
syllogism is generally assumed to be based. Yet a closer, or more open-minded,
look reveals that this is in fact the case. Only the enumeration of all possible
types of syllogism depends on the analytic laws of identity and non-
contradiction. The syllogism itself is synthetic. For the validity of the
conclusion depends upon establishing a perspective from which we can equate
two words or phrases which were not previously related. That the words
"Socrates" and "mortal" do not have the same meaning is obvious. Thus we
could say that in an analytic sense, "'Socrates' = 'mortal'". Yet 15/ by relating
both these terms to the concept of "man" (e.g. by pointing out that all men are
mortal, and that Socrates is a man) we are led to the point where we can affirm
the contradiction and conclude "Socrates is mortal". The argument only
succeeds if it engages with the opponent's former perspective (here assumed to
include Socrates' manhood and man's mortality) and forces it to be transformed
into a new perspective, from which the contradiction of the synthetic
proposition ("Socrates = mortal") takes on the self-evident character of an
analytic truth. So the syllogism--i.e. the systematic, rational argument--depends
upon both analytic and synthetic laws of logical relation.
The radically systematic character of the form of syllogistic figures is
best revealed when these static forms are placed side by side in a series of actual
arguments. Such a series, called a "polysyllogism", provides a perfect example
of the way simple synthetic relations are interconnected in a compound synthetic
relation: "In such a series, the syllogism whose conclusion becomes a premiss
in the next syllogism is called a prosyllogism; a syllogism one of whose
premisses is the conclusion of a preceeding syllogism is called an episyllogism"
[S515:109]. Thus if we construct a four-stage polysyllogism in which each of
the four syllogistic figures is used once (in the order I, IV, III, II), then we will
end up with a system of exactly the same form as the purely formal system
discussed in 6.3:
Figure 6.19: The Perfectly Systematic Prosyllogism
The syllogism is often played down or even ignored altogether by
logicians and logic textbooks nowadays. In light of the changes which have
occurred in logic over the past century [see Introduction, ?], this should come as
no surprise. For logicians are becoming more and more tied down to an analytic
perspective on the empirical world (in particular on language), and are less and
less concerned with the equally legitimate synthetic perspective; they often deny
the very possibility of viewing the formal structure of thought and action as
such, or at least refuse to regard this as a legitimate part of their task. The
syllogism, viewed from this point of view, is merely "one important form of
reasoning", and is most useful for its educational value [S515:102]. Yet from
the point of view of a more holistic treatment of logic, such as is adopted by the
Geometry of Logic, the syllogism must be raised to a rather higher position and
viewed as the archetype of all forms of systematic argumentation. To be sure,
not many actual arguments can claim to have attained the rigorous systematic
heights of the perfect polysyllogism; and many cogent arguments bear little or
no resemblance to the syllogism in any of its forms; but the formal structure of
this classical basis for rational arguments clearly cannot be safely ignored by
anyone interested in formal logic, even if its application to actual arguments is
not stressed.
The ideal of many--perhaps most--Western philosophers has been to
construct a philosophy which can boast of "systematic unity". Our findings so
far in Part Two equip us with the tools necessary to assess such claims.
Naturally, any such assessment could be done properly only in the context of a
full-length interpretation of a given philosophical system, which would of
course be out of place here. What is possible, however, is to give a summary of
one such interpretation; for I have made a thoroughgoing study of the systematic
coherence of the philosophy of Immanuel Kant, who was certainly no exception
to the above-mentioned tendency among philosophers. I shall therefore devote
the remainder of this section to a brief account of the extent to which this great
thinker was able to construct a material system on the basis of the pattern
provided by formal logic, as described above.
Kant's System of Critical philosophy is composed of three Critiques,
each of which is based on a line of argument which corresponds with surprising
accuracy to the archetypal form of twelvefold systemization, as exemplified by
the syllogism. In particular the theory of knowledge developed in the first of his
three systematic treatises, the Critique of Pure Reason, purports to be highly
organized and intentionally structured on the basis of the "architectonic" form of
reason itself [see P1:Ch.6]--a claim at which most interpreters have scoffed
without even giving serious thought to its possible implications. However, if
the interpreter gives Kant the benefit of the doubt and interprets his philosophy
according to the systematic and other presuppositions which he himself held,
then many of the apparent ambiguities and contradictions in his writings can be
resolved, and his three Critical systems revealed to be profoundly consistent
with the compound form of relation we have been discussing. What follows is a
brief summary of his "theoretical system", or theory of knowledge, which I
have developed in full in P1:Ch.7.
The starting-point of Kant's system is the presupposition of a radically
unknowable foundation upon which the material of all knowledge is based. This
foundation, called the "thing in itself", fulfills the logical function of the
synthetic "0" at the beginning of all compound systemization. The ultimate goal
of the system (the synthetic "1") is "knowledge", or what Kant calls the
"complete determination" of an object by certain "forms" imposed upon it by the
subject (i.e. by the knower). Kant puts forward a twelve-step argument which
leads from the unknowable thing in itself to the fully known object. As we
would expect from such a systematic thinker, he groups these twelve steps into
four stages: sensibility, conceptual understanding, judgment, and rational
inference. These stages are related to each other according to a perfect second
level analytic division, so they can be mapped onto the cross (using roman
numerals to designate the logical order of their counterclockwise development),
as shown in Figure 6.20. Hence the three steps which compose each stage are
related to those of the other stages according to the three-place relation depicted
by squares in Figure 6.17.
Figure 6.20: The Four Stages in Kant's System of
Knowledge
rational inference (++) #
IIV # sensibility judgment (--)(-+) IIIII # conceptual understanding (+-)
To follow the development of Kant's argument through each of these
twelve steps would be to burden ourselves with unnecessary detail. All the
relevant details are, in any case, given in P1:Ch.7. Instead it will suffice to
summarize the system by plotting its twelve "elements" onto our twelvefold
dynamic circle [see Figure 6.21; cf. circle (b) in Figure 6.8].@ The twelve
numbers labelling each curved arrow in Figure 6.21 represent the operations
which transform one "element" (i.e. one step in the determination of the object)
into the next. We could just as well have used these operations as examples of
the systematic unity of Kant's system, but they would have been difficult if not
impossible to describe without reference to the twelve elements. Specifying only
the elements makes for a slightly less complicated discussion.
At this point a reminder of how our formal notation is actually used to
define each expression in the map given in Figure 6.21 will be a helpful
preparation for examining how this map exemplifies the various types of
twelvefold relation discussed in 6.3. The first term of each expression is a "+"
for all expressions in the second and fourth stages, and a "-" for all expressions
in the first and third stages. This represents Kant's (first level analytic)
distinction between functions in which the object takes the active role, while the
subject is primarily passive or "receptive" (-), and those in which the subject
takes the active role (+), while the object is primarily passive. This fundamental
subject-object duality is then split into the four primary "faculties" of the
knowing subject, as given in Figure 6.20. The distinction between "+" and "-"
on this second level is between the intuitive (-) versus the discursive (+)
function of each faculty. Thus sensibility (--) deals with the intuition (-) of the
object by a passive subject (-); conceptual understanding (+-) deals with the
subject's active (+) intuition of its own categorial forms (-); judgment (-+) deals
with the subject's passive recognition (-) of the discursive reality (+) of the
object; and rational inference (++) deals with the subject's active (+) treatment
of the judged object in discursive theorizing (+) about its implications. Each of
these four stages is defined by a simple synthetic relation, as represented by the
third term in each expression. Combining these three levels of logical relations
into a single system results in the twelve steps to knowledge listed in Figure
6.21.
The true logical test of the accuracy of any attempt to interpret a system
by mapping it onto the circle of twelvefold systemization is to examine whether
or not the interrelationships between the elements as mapped agrees with the six
forms of rectilinear inscription discussed in 6.3. Thus the test for all one-place
relations in Kant's system would be to determine whether or not Kant actually
intended each element used in the interpretation to have the specific synthetic
character which has been assigned to it. In the first stage, for example, the
question is: Does Kant intend to depict the transcendental object (--+) as the
undifferentiated material (+) upon which the forms of space and time are placed
(operation 2) to produce an appearance (---), i.e. an object which is thereby
available for sensation (--x)? (I have argued in P1:Ch.7 that this and the other
synthetic relations given in Figure 6.21 are indeed precisely what Kant was
arguing.) The complementary six-place relations yield a more obvious result.
For the pairs of elements which stand opposite each other on Figure 6.21 are in
each
Figure 6.21: The Twelve Steps in Kant's System of
Knowledge
case suitable candidates for such polar opposition. The relevant (second level
analytic) opposition is between the intuitive versus the discursive nature of the
two elements in question. Thus the discursive complement of the intuitive
transcendental object is the schematized object; the appearance when projected
into a discursive context is a phenomenon; a sensation is for intuition precisely
what a judgment is for discursive reasoning; etc. In each case the parallels
implied by six-place relations are indeed illuminating, as befits a well-ordered
system.
Two- and seven-place relations are not so important until the material
elements in a theory have already been mapped onto the formal structure.
Nevertheless, one could ask whether the third terms in two successive relations
of these types are in fact related according to a reversed synthetic pattern. Thus
we could ask whether sensation (--x) is somehow further along the path of
simple synthetic integration than the transcendental object (--+), the self-
conscious perception (+--), the unconditioned object (+++), or the phenomenon
(-+-). Clearly it is, since a sensation is, for Kant, the culmination of a whole
process of transcendental and empirical functions, whereas all these other
elements in the constitution of knowledge are in no sense "resting-points" in the
dynamic development of his theory, but rather all function as transition steps to
the full determination of a given faculty.
Examining the three- and four-place relations also yields happy results.
The three-place relations are best grasped by viewing them in terms of their
synthetic element. Thus the four three-place relations are based on the elements
of sensation (--x), concept (+-x), judgment (-+x), and knowledge (++1). Each
four-place relation is valid if the two other elements in the "triangular" relation
are in some sense positive and negative elements whose synthesis yields one of
the above four. This only holds true, however, when both the "+" and the "-"
elements come before the "x" element, since the system is not perfectly circular
(i.e. it has definite starting and ending points). In the two cases where this
holds--knowledge as the synthesis of a conscious perception (+-+) with a
phenomenon (-+-) and judgment as the synthesis of the transcendental object (--
+) and the self-conscious perception (+--)--the results are satisfactory (i.e. the
relations described make sense, though they do not tell the whole story). By
contrast, the similarity of function between the three sets of four-place relations
is quite remarkable, as is best evinced by arranging them in columns according
to their synthetic value:
Table 6.5: Four-Place Relations in Kant's System of
Knowledge
Third term (synthetic) value Stage + -x one transcendental
objectappearancesensation | two conscious perceptionself-conscious perception
concept | threeschematized object phenomenon judgment | four unconditioned
object ideaknowledge
If Kant's agreement with these correlations is established after a
thorough examination of his texts (as I have done in P1:Ch.7), then surely there
is no longer any justification for regarding his epistemology as lacking in
systematic unity. On the contrary, those who ignore the grounding of Kant's
philosophy in the foundations provided by the formal structure of compound
systemization are more than likely to misinterpret the bulk of what he says, both
in his particular arguments and in his general conclusions.
The detailed examples of twelvefold systemization which we have
considered in this section demonstrate the applicability of the principles we have
developed to the systematic description of actual (not just formal) relations.
Rather than citing more examples, we must now extend our discussion of
systemization from the rather narrow realm of the logical order evident in
specific, self-sufficient systems, to the order implicit when such systems are
themselves related with other such systems in higher orders of Systematic
relation.
NOTES TO CHAPTER SIX
1. Since the vertical is to the horizontal as "+" is to "-" [cf. 2.1], the third term
of every expression mapped onto the vertical axis is "+", and those mapped
onto the horizontal axis is "-". Also recall from 2.2 that the first two terms of
each expression are derived, respectively, from the second term of the
preceeding, and the second term of the following, analytic expressions,
proceeding clockwise.
2. Note that the analytic expressions at the four corners of the square can be
derived from the middle terms of the two synthetic expressions at the same
corner, proceeding clockwise. Note also the symmetry between Figures 6.1a
and b: the first two terms of each synthetic expression in each of the triangles in
Figure 6.1a is the same as the analytic expression at the corresponding corner in
Figure 6.1b; and the first two terms of each synthetic expression in each of the
triangles in Figure 6.1b is the same as the analytic expression at the
corresponding pole in Figure 6.1a. This suggests that the operation of inversion
converts analysis to the position of synthesis, and vice versa, in a given set of
geometrical figures.
3. The early Greeks were well acquainted with the properties of the five regular
convex polyhedra, but it never occurred to them that regular polyhedra could be
non-convex as well [B520:78-9].
4. The centre of the net is the base of the solid in Figure 6.3a; each triangle is
then folded upwards so that the three extreme vertices of the net meet together as
the apex of the tetrahedron (the origin ("0") of the synthetic relation).
5. This polyhedron is the first (and simplest) of the thirteen "Archimedian"
polyhedra [C521:101]. A similar figure, called the "half-cube", was used in
Figure 4.7; the only differences are that in the truncated tetrahedron the
triangular angles are all 60\ (rather than 90\, 45\ and 45\) and that four (rather
than three) triangular segments are cut out.
6. In this net the second hexagon from the left represents the base of the
truncated tetrahedron and the hexagon at the far right represents its front face.
The other elements are folded upwards accordingly.
7. The central square of this net (i.e. second from the top, second from the left)
represents the base of the cube. The other squares are folded upwards, so that
the lowest square in the net represents the top of the cube.
8. Joining the lower edge of the left-most triangle to the left edge of the lower-
most triangle forms the four triangles at the left of the net into the bottom have
of the octahedron. The same procedure holds for the four triangles at the right
which form the top half.
9. Jung's work on the "mandala" (Sanskrit for "circle") as an archetype
common to the art and religions of East and West and to the dreams of modern
man is of particular significance in this regard [see e.g. J473: ].
10. L532:25; cf. W533:21. Again at L532:40 he says: "Return is the movement
of the Dao." Wilhelm adds that "'Return' signifies a cyclical movement"
[W533:131]. Thus for the Daoist, the deepest secret of the universe "is
represented by a simple circle" [116].
11. A fully explicit map would include two expressions at each of the four
cardinal points. For example, in Figure 6.8a these four pairs would be: "--
1/++0", "++1/-+0", "-+1/+-0", and "+-1/--0".
12. Four other maps could be defined by adding the further option of using the
order of expressions appropriate to the inversion of the cross (i.e. the square),
and submitting them to the above rules. This possibility should be kept in mind
by anyone who wishes to choose an appropriate map for a given material
system, but there is no need to specify the results here, since they are derived
quite simply by replacing the "++, -+, +-, --" order of the cardinal expressions
with the "++, +-, --, -+" order.
13. This change in direction has an interesting parallel in nature. A star-gazer in
the northern hemisphere sees the night sky as if it were turning counter-
clockwise (in relation to the north pole), while his colleague in the southern
hemisphere seen a clockwise motion (in relation to the south pole).
14. Of course, these relations are no different from the simple synthetic relations
which are defined by the circle itself. But specifying the dodecagon serves to
highlight just these twelve relations, whereas the circle is intended to symbolize
all possible interrelations between its elements, as explicated in Figure 6.9.
15. This assumes that the rules for simple synthetic integration are known--a
valid assumption since one-place relations are defined purely in terms of these
rules. Moreover, if the rules were not known, no information about the other
points could be gained by assigning an expression to one of them.
16. The direction of synthetic flow is not reversed in two-place relations.
Rather, the direction is the same; but skipping every other point as one proceeds
around the circle clockwise yields the same result as does a simple reversal of
direction. This can be demonstrated by setting out these two progressions side
by side: Clockwise, skipping: + - x + - x + - x + - x etc.
Counterclockwise, successive: x - + x - + x - + x - +etc.
17. Both the pentagram and the five-place dodecagram can be drawn without
lifting one's pen from the paper. Of the six inscribed figures we are
considering, the only other one which shares this quality is the dodecagon used
to represent one-place relations [see Figure 6.10].
18. By "all methods" I mean all methods which continue to give priority to the
circumference, rather than to the area of the dynamic circle. Hence other
variations could include changes in the direction of logical development, or in
the position chosen as the starting-point, or in the order in which the synthetic
expressions are given relative to the direction assumed. If, on the other hand,
changes were introduced which subordinated the logical development around
the circumference of the circle to that of the figures inscribed within its area,
then the rules would at least have to be interpreted differently. For instance, if
we mapped the twelve expressions onto the four triangles in such a way that
each triangle represented the simple synthesis of three expressions whose first
and second terms all agreed (e.g. "+++", "++-", and ++x", all on a single
triangle), then the logic which guides the mapping of expressions around the
perimeter of the circle would be less obvious, if not totally obscure; for in such
a case the triangles would constitute the essential system, and the circle would
merely be one of the variations for stating the relations between its elements.
This is, in fact, how the twelve signs of the zodiac are sometimes
regarded (i.e. in such a way that the triangular, four-place relations are of
primary, or "cardinal", importance). In such a case the twelve forms of relation
would be mapped onto the circle as follows:
In any case, the zodiac is clearly constructed on entirely logical
principles. (Even its historical development reflects the logical development
which begins with unity, progresses through analysis to duality, and ends in the
threefold unity of synthetic integration. For instance, the "male" and "female"
constellations of Scorpio and Virgo were discovered long before the mediating
constellation of Libra [P538:24].) Hence the misuse of astrology in the modern
Western form of horoscopes, supposedly useful in guiding the day-to-day
decisions of all men, represents a fundamental perspectival error: the logical is
applied to the empirical without any recognition of their formal difference. In its
true form, astrology has as much to do with the geometrical stars which can be
inscribed in a circle as it does with the astronomical stars inscribed in the
heavens; but as a means of setting forth general platitudes which are supposed
to assist Everyman in his starry-eyed quest for wealth, happiness and success,
its use is dubious to say the least.
19. Another related fact is that the division of each pentagonal face of a
dodecahedron into its thirty constituent triangular elements [see 5.3] yields a
total of 360 triangles on the surface of this regular solid. Hence each triangle in
the differentiated dodecahedron correlates to one degree of a circle.
20.Cf. S515:84. The same pattern holds for a number of related distinctions as
well. The four possible modes of compound syllogism, traditionally referred to
as "modus ponendo ponens" (++), "modus tollendo tollens" (--), "modus
tollendo ponens" (-+), and "modus ponendo tollens" (+-) [see S515:104-5],
also constitute a second level analytic relation. Moreover, even the invalid
modes of inference fit this same pattern [106]: [+][-] [x] {1 1. Implicative: If p,
then q; but q; .. p.[++]
2. Implicative: If p, then q; but -p; .. -q. [--]
3. Alternative: Either p or q; but q; .. -p. [+-]
4. Disjunctive: Not both p and q; but -q; .. p.[-+]
(The brackets over each column indicate the synthetic steps in each invalid
inference; those to the right of each inference indicate its analytic relation to the
others, based on the values assigned in the second premise and the conclusion.)
The form of all possible dilemmas is yet another example of the same twelvefold
pattern [see 107-8].
21. The "transcendental logic" which guides Kant's philosophical System as a
whole differs from ordinary (+-) formal logic in that it is not analytic, but
synthetic; so it is best correlated with the "-+" pole of second level analytic
division. We must therefore map it onto the counterclockwise (reflective) and
involuting (synthetic) circle given in Figure 6.8b. Moreover, as we shall see in
7.1 [see esp. Figure 7.2] this circle is itself correlated with the "-+" pole of
second level analytic division.