ANALYTIC FORMS OF RELATION
AND THEIR PERSPECTIVAL
TRANSFORMATIONS
The ability to think depends upon the power of seeing connexions. Reflective
thinking consists in pondering upon a given set of facts so as to elicit their
connexions. --Stebbing [S515:4]
Transformation is the distinguishing marks of one thing changing to the
distinguishing marks of another. --Anonymous [C529:A45; a.q.i. G522:295]
2.1 The Levels and the Rules for Mapping
In Chapter One I introduced the two fundamental laws of analytic
logic, the law of identity (unity) and the law of non-contradiction
(everything and nothing). Mapping these laws onto several geometrical
figures provided an intuitive grasp of the way they operate. I then related
these laws to the analytic-synthetic distinction as it applies to abstract
logical "forms of relation", and concluded by suggesting a notation using
"+" and "-" to represent the formal relations of positivity and negativity and
their application to appropriate sets of concrete elements. The task which
will occupy us for the remainder of Part One is to develop this notation into
a calculus of the forms of relation by investigating how positive and
negative logical relations operate in more complex types of analytic division
[Chapter Two] and in synthetic integration [Chapters Three and Four].
As we saw in 1.3, analytic division at its simplest is the
transformation of unity into duality. It is the splitting of an undifferentiated
whole into its positive and negative elements. What was only briefly hinted
at in Chapter One is that analytic division does not operate solely at this
simple level. On the contrary, the same process can be repeated over and
over indefinitely, by continually dividing the contents of a given unity (or
universe class) into more and more reciprocal pairs. The further one
proceeds in analytic division, the more complex is the form of relation
which results, and the more difficult it becomes to find applications for it in
the empirical world.
I shall limit the scope of this chapter to a discussion of the first four
"levels" of analytic division. The constituents of these levels can be
represented in a number of ways, each of which is helpful for different
reasons. The first way demonstrates why "level" is such an appropriate
word to use in this context. By replacing the notation appropriate to the I
Ching [see Figure 7.7a] with that suggested in 1.4 we can adapt the set of
tables given in S493:10 in such a way as to depict the progressive
complexity of analytic division when the forms of relation for the first four
levels are "stacked" successively on an original unity:
Table 2.1: Levels 0 - 4 of Analytic Division
Sherrill and Chu refer to the law which determines the course of this
process as "the law of evolution" [10]. This is an accurate (though possibly
misleading) description, so long as it is kept in mind that such evolution
operates only on conceptual objects (such as thoughts), and not on the
external objects we experience in the natural world--a caveat which Sherrill
and Chu, following the typical approach to the I Ching [see 7.2], tend to
ignore.
Several points must be clarified concerning the use of "+" and "-" in
constructing the expressions listed in Table 2.1 before we can use them
consistently to label logical maps. On its own each occurrence of one of
these signs will be called a "term". But when it is used, either on its own or
in combination with other terms, to define an entire constituent of an
analytic form of relation, the combined set of terms will be called an
"expression". Some logicians are satisfied to use the word "constituent" in
this context [see e.g. B506:5.11]; but the mathematical meaning of
"expression", as "A collection of symbols together expressing an algebraic
quantity" [O543:5.447], is more suggestive of this logical function. In
successive levels of analytic division, more and more terms combine
together to define more and more expressions, which in turn define the
particular form of relation under consideration.
The rules guiding the construction of the expressions in Table 2.1
are as follows. First, the number of terms in an expression stands in a
direct, one-to-one proportion to the level of division in which it is a part.
Thus, Level 0 has expressions with no terms (in fact, it has no
expressions), the first level has one-term expressions, the second, two-
term expressions, etc. Secondly, the various expressions given at each
level represent all the possible combinations of terms for that level of
analytic division. This can be proved simply by noting that the progression
of analytic forms of relation follows the same pattern as does the
mathematical equation of "2 = x", where "n" corresponds both to the level
of analytic division and to the number of terms used in each expression,
and "x" corresponds to the total number of possible expressions in each
analytic form of relation. The arithmetical description of the levels of
division given in Table 2.1 can therefore be stated as:
Level 0: 20 = 1 First Level: 21 = 2 Second Level: 22 = 4
Third Level: 23 = 8 Fourth Level: 24 = 16 etc.
Unfortunately, many logicians who employ analytic forms of relation never
state the simple rule which determines their pattern. For instance, Boole
uses the first three levels of analytic division when he assigns a value of
either "1" or "0" to each variable in the functions "f(x)", "f(x,y)" and
"f(x,y,z)" [B506:5.10-1]. These functions play an important role in his
Algebra of Logic [see e.g. 5.12,6.2,10.5], yet he never so much as hints
that the pattern which determines the number of alternatives in each case is
common to innumerable similar relations, inasmuch as it is based on a
simple algebraic formula. Perhaps such logicians assume that this most
basic logical fact is too obvious to be stated explicitly.
A point which is mentioned even less frequently is that the levels of
analytic division also correspond directly to the operation of raising the sum
of two variables to a power. In this analogy the power corresponds to the
level and the variables "A" and "B" correspond to "+" and "-".
Accordingly, first-level division corresponds to the equation "(A + B) = A
+ B", because the algebraic operation yields a single occurrence of the two
variables, just as "+" and "-" each occur only once in the first-level form of
relation. Likewise, second-level division corresponds to the equation "(A +
B) = A + 2AB + B", where the power signs in the second half of the
equation correspond to the number of times a term is repeated, and the
numerical prefix corresponds to the number of expressions in which a
given combination of terms occurs. Thus "A" represents "++", "B"
represents "--", and "2AB" represents "+-" and "-+". All subsequent levels
follow the same pattern:the third-level expressions can be categorized in
terms of the equation "(A + B)3 = A3 + 3A2B + 3AB2 + B3" as:
+++ (A3)
++- +-+ -++ (3A2B)
+-- -+- --+ (3AB2)
--- (B3);
and the fourth-level expressions can be categorized in terms of the equation
"(A + B)4 = A4 + 4A3B + 6A2B2 + 4AB3 + B4" as:
++++ (A4)
+++- ++-+ +-++ -+++ (4A3B)
++-- +-+- +--+ -++- -+-+ --++ (6A2B2)
---+ --+- -+-- +--- (4AB3)
---- (B4)
For the logician to ignore the algebraic basis of the logical forms of relation
is to run the risk of failing to categorize the different types of expression in
the proper ways. But paying attention to these and other patterns will help
us to determine appropriate maps for each form of relation.
The third and fourth rules guiding the derivation and use of the
expressions listed in Table 2.1 have to do with the order of the terms in
each expression. The third rule is that, when mapping a set of expressions
onto a geometrical figure, the left is to the right as "-" is to "+", and
likewise, the bottom is to the top as "-" is to "+" [cf. Table 2.1]. These
correlations can be regarded as arbitrary, though they do have a basis in the
conceptual schemes of most cultures. (Two good examples are the common
tendency to regard the left hand as weak (-) in comparison to the strength
(+) of the right hand, and the modern discovery that the left side of the
brain governs a person's conscious and analytic functions (-), while the
right side governs the unconscious and synthetic functions (+) which
subtly control the former in many ways. Indeed, Purce goes so far as to
say: "In all traditions, and now in science, the left side of the body is
passive" [P538:44].) Each successive level is established by dividing each
expression of the previous level into its positive and negative poles. On the
new level a further "+" or "-" is therefore appended to the positive end (i.e.
either the right or the top) of each expression in the former level. Arranging
the third- and fourth-level expressions vertically (to be read from the
bottom up) in Table 2.1 not only enables tables (d) and (e) to fit onto the
page more neatly, but also makes it easier to visualize the alternating
patterns which are created by following this rule.
The fourth and final rule is that the relations expressed by each
successive level of analytic division are cumulative. In other words every
form of relation contains within it the relations defined by all of the
previous levels, as well as those newly defined by the level in question.
Thus, for example, fourth-level division produces a form of relation
containing sixteen four-term expressions. But within each of these
expressions, only the fourth term is uniquely defined at this level:the first
three terms are each taken directly from the first three levels of analytic
division. We can generalize this rule by stating that the "nth" term in any
expression represents the unique relation contributed by the "nth" level of
analytic division. Taken together, the "nth" terms of all the expressions
which define a given form of relation represent what will be called the "nth-
level relation". So the "fourth-level relation" is symbolized by the fourth
terms in all sixteen of the fourth-level expressions. But if these fourth terms
are dropped, we are left with two identical sets of third-level expressions;
likewise if all the third terms are dropped as well, we are left with four sets
of second-level expressions; and if only the first term in each expression is
considered, we are left with eight sets of first-level expressions. This
participation of the lower-level relations in the higher levels can be
demonstrated in simple arithmetical terms:
24 = 2(23) = 4(22) = 8(21) = 16.
The logical distinction between the levels of analytic division correspond
directly to the geometrical distinction between dimensions. As we saw in
1.1-2, the geometrical representation of Level 0 is a simple,
undifferentiated point, the only zero-dimensional figure in geometry [see
notes 1.3 and 1.5], and the standard map for the first level of analytic
division is a line. Applying this analogy to the second, third and fourth
levels of analytic division can be done most straightforwardly by mapping
them onto two-, three-, and four-dimensional geometrical figures,
respectively. In order for the correspondence to be exact, we must in each
case choose a figure which has the same form as the level in question. That
is, the number of points, vertices, lines, angles, and/or faces must exactly
match the number of expressions defined at the given level. As well as
depicting the dimensions as changing with the levels, it will be helpful to
map the expressions at every level onto the endpoints or vertices of
increasingly complex two-dimensional figures. This, as we shall see, will
facilitate a more consistent comparison between the levels than is possible if
the dimension is allowed to increase at each higher level. In the three
remaining sections of this chapter I shall deal in turn with the second, third
and fourth levels of analytic division, and with the diagrams which can
serve as their geometrical maps. In the Appendix I will show how the
standard maps used in this chapter fit into the overall pattern of alternatives
which exists when none of the rules in this section are taken into
consideration. Before embarking on these tasks, however, it will be helpful
to reiterate the general function of these maps and to cite an example of how
the analytic form of relation functions in a particular logical application.
Like a geographical map, or indeed a geometrical one, a logical map
"will never be a complete picture..., but it can be perfectly true on the scale
intended. [Such maps] are useful precisely because they do not copy the
whole, but only the significant relations" [C492:101]. Whereas
geographical maps represent the concrete form of relations such as
distance, logical maps represent the abstract, logical form of such real
relations [see 1.3]. Logical maps are therefore set apart from their empirical
counterparts because they represent not just a selection of important
relations, but "the assertion of exhaustive possibilities" [C492:54]; hence
they can be called "tautologies" in both their logical and geometrical forms.
"It is because...tautologies exhaust the field of possibility that they serve as
the necessary though not sufficient condition for materially significant
inquiries in the field of nature" [55]. Because the logical forms of relation
are void of all content, they are always tautological; and for this very reason
they are useless apart from their application to some concrete content.
Examples of how the various levels of analytic division serve as the
basis for less abstract logical operations abound in virtually every textbook
on beginning logic. One of the most familiar examples is the "truth table".
A truth table is an exhaustive list of the possible combinations of truth or
falsity of a set of variables, which can be used to determine the overall truth
value of any proposition which performs an operation on those variables. It
"presents the essential scope of an operation, and may be constructed to
show the scope of an entire propositional system, wherefore it is
sometimes called the 'matrix' of such a system" [L483:354-5]. As such--
i.e. apart from any given operation--a truth table expresses precisely the
same relations as the corresponding analytic form of relation. The only
difference is that the application of the former is limited to the task of
determining the truth value of propositions, while that of the latter is more
generally applicable to logical relations of all sorts. This identity of form
can be made explicit simply by listing the three simplest truth tables [see
Table 2.2]. Just as in Table 2.1b-d, a truth table with one variable (p) gives
rise to two alternatives, one with two variables (p and q) yields four
alternatives, and one with three variables (p, q and r) has eight possible
arrangements of truth value. Obviously, the truth values represented in
Table 2.2 by "T" and "F" correspond exactly to the relational values for "+"
and "-" given in Figure 2.1.
(a) With One (b) With Two (c) With Three Variables
and
Variable Variables a
Propositional Operation
Table 2.2: Three Simple Truth Tables
In themselves truth tables can be called "tautologies" (listings of all logical
possibilities), for they "simply describe the performance of the system,
they do not imply any particular principle of operation" [A489:1.8]. But
when the variable(s) in a truth table are related by means of a specific
operation, so that the truth value of the overall proposition is included (as
exemplified in the fourth row of Table 2.2c), the table defines a tautology
only when all possible combinations of variables yield a value of "true" for
that proposition. Analytic division abstracts from all questions of truth or
falsity, but follows essentially the same rules: it is always tautological in its
purest form, apart from its application to some empirical content; but when
it is applied, a form of relation is tautological only when it relates all the
members of a set according to the logical pattern defined by the form of
relation. (The latter is called a "perfect" application [see 2.2].) Thus, the
analytic forms of relation determine the pattern for the "input" of a truth
table, but the "output" can be determined only when a concrete operation of
some sort (usually in the form of a proposition) is imposed upon the formal
structure.
2.2 Polar and Contradictory Opposition
in Second-Level Analytic Division
Our discussion of the splitting of unity into duality in 1.2 was
actually a
discussion of the operation which was defined in 2.1 as "first-level analytic
division". The straightforward application of the law of non-contradiction
which characterizes this operation is the fundamental basis of all coherent
thinking. It is for this reason that such a multitude of applications of the
first-level form of relation are evident in our conceptual description of
empirical reality:thinking about the world naturally gives rise to such
twofold divisions. By the same token, when the philosopher or logician
takes a step back from empirical thinking and begins thinking about
thinking, his tendency is to make fourfold distinctions. The logical
explanation for this phenomenon is that such second order mental activity is
based on an application of the law of non-contradiction to itself. Like all
generalizations, this explanation is only partially valid, for it refers more to
a tendency than to a hard and fast rule. But the extent of its significance can
perhaps be seen more readily if we turn again to our geometrical maps.
The form of relation produced by first-level analytic division was
symbolized in Figure 1.2 by mapping its two one-term expressions onto a
line segment. Altering the notation of that diagram to correspond with the
new conventions outlined in 1.4 gives us the following map:
Figure 2.1:First-Level Analytic Division
Another new feature added to this map is the arrowhead at the left, which
indicates the active character of the "+" pole in relation to the passivity of
the "-" pole. (As we shall see shortly, this same relation could also be
depicted by putting the arrowhead at the opposite end and reinterpreting its
function.) This is not intended to imply that the "+" expression is logically
or chronologically prior to the "-" expression, but only that they are
dependent on each other in a certain fixed way. Arrows will be used
throughout this chapter to highlight various sorts of logical relations
between the parts of geometrical maps.
In order to construct a corresponding diagram upon which the four
expressions produced by second-level analytic division can be mapped, it
will be necessary to expand this one-dimensional figure on a two-
dimensional surface. This can be done in several ways, the clearest of
which is by analyzing, or "expanding" each endpoint of the first-level
(horizontal) line into a new twofold division, represented by its own
(vertical) line [see Figure 2.2]. One of two rules must be followed in
constructing this expansion of the first-level form of relation to its second-
level counterpart. The first rule would be to regard the two dualities arising
out of the first-level expressions as distinctions between polar opposites. A
"polar opposition" is a distinction between two expressions in which the
same term occurs in the same position at least once and a different term
occurs in the same position at least once, such as between "++" and "+-".
The alternative rule would be to regard dualities arising out of the first-level
expressions as distinctions between contradictory opposites--i.e. between
two expressions in which the same term never occurs in the same position,
such as between "++" and "--". Since at least two terms are required to
define either type of opposition, first-level analytic relations, taken on their
own, are ambiguous. But on higher levels real relations can often be
expressed in either form. Indeed, it is the interplay between these two types
of opposition which is at the root of much of the perspectival
transformation which will be discussed in this chapter.
Expanding Figure 2.1 according to the rule of polar opposition
entails adding a second term (either a "+" or a "-") to the one-term
expressions on each side of the original relation. Thus "+" gives rise to
"++" and "+-", while "-" gives rise to "-+" and "--" [see Figure 2.2a].
Following the rule of contradictory opposition, by contrast, entails viewing
one side of the original relation as "pure" (+) and the other side as
"impure", or "mixed" (-). (A pure expression is one whose
(a) Using Polar Oppositions (b) Using Contradictory
Oppositions
Figure 2.2:
The Expanded Map of Second-Level Analytic Division
terms are all the same; a mixed expression is one whose terms differ. These
terms can also be applied to the relation between corresponding terms in a
pair of expressions.) Thus "+" gives rise to the two pure expressions "++"
and "--", while "-" gives rise to the two mixed expressions "+-" and "-+"
[see Figure 2.2b].
The two vertical lines in both versions of the "expanded" map
represent the logical relations unique to the second level. These relations are
denoted by the second term in each of the four expressions, the pattern for
which is the same in both maps. (The first terms are identical in each pair of
second-level expressions in 2.2a, but opposites in 2.2b.) Dropping the first
term in each of the four second-level expressions in each map makes it
evident that the second terms are related in pairs identical to the pair of
terms which describe first-level division. The only difference is in the
location of the arrowheads on these new lines with respect to the logical
expressions, for they sometimes point from "+" to "-", and at other times
from "-" to "+". The rule followed in determining the direction for these
arrows is that on the "+" (right) side of the first-level division the second-
level arrow is repelled by the "+" in the second term of both expressions,
so it points towards the "-", while on the "-" side the arrow is repelled by
the "-" in the second term, so it points towards the "+". As a result of this
rule, the pure expressions have priority over the mixed expressions, as they
should, in both maps:in Figure 2.2a they label the starting-points of the two
second-level lines, and in 2.2b they label the line which is expanded from
the starting-point of the first-level line.
This new rule for mapping the direction of second-level arrows can
be correlated with the active-passive distinction (used to determine the
direction of the first-level arrow in Figure 2.1) by noting the two ways of
interpreting the operation symbolized by an arrow. On the first level "+"
side of both maps in Figure 2.2 the arrow symbolizes the active push of the
second-level "+" out towards its corresponding "-"; on the "-" side, by
contrast, the arrow symbolizes the active pull of the second-level "+" away
from its source in "-". So in both twofold relations the positive can be
regarded as the active term. This difference in the interpretation of the
arrow is required by the rule which determines its placement, and is
intuitively consistent with numerous empirical facts, such as the repelling
characteristic of electrical charges which, for example, causes paired
electrons to oscillate in opposite directions.
The same kind of perspectival transformation crops up in much
religious symbolism, such as in the story of Creation in Genesis, where
man (+) is given dominion over the world (-):on the "+" side of the first-
level relation we are told that Eve (+-) is formed out of the rib of Adam
(++), so that the logical development proceeds from "+" (male) to "-"
(female); but on the "-" side we are told that light (-+) is made to shine in an
otherwise dark (--) world, so that on this side of the first-level relation the
logical development proceeds from "-" (dark) to "+" (light). The use of
male-female symbolism in various religious traditions itself differs
according to the same pattern. As mentioned in 1.4, a good example of this
is the contrast between Hindu and Buddhist cosmologies:the Hindu's
attention is directed towards the earth (-), so he tends to regard the female
as the source (--) from which the male springs (-+); the Buddhist's
attention, by contrast, is focussed more on the heavenly (+), so he tends to
regard male symbolism as a way of representing the source (++) from
which the female springs (+-).These symbolic traditions should not be
regarded as incompatible or contradictory, but as complementary or polar
opposites. For in many respects they are based on the same logical map,
which is merely being viewed from a different perspective and therefore
interpreted with different rules. Bharti points out, for example, that in both
traditions "the redeeming function is assigned to the dynamic [i.e. the pure]
principle" [B528:210]. Likewise, the Protestant tends to emphasize
individual salvation through an active profession of faith (+) in the saving
role of the True Man, Jesus (+), while the Catholic puts more weight on
corporate salvation through the (relatively) passive acceptance of and
obedience to a tradition (-), which usually emphasizes the Mothering role of
the True Woman, Mary (-).Numerous unnecessary clashes between such
rival religious traditions could be avoided if their adherents would
recognize the perspectival relationship between their respective
interpretations of religious symbolism.
Given the legitimacy of interpreting the arrow differently in one and
the same map, we must now seek to alleviate the disconnected character of
the map used in Figure 2.2 by slightly altering its shape. Instead of
analytically separating the lines representing the two second-level dualities,
they can be depicted as originating at the two end-points of the first-level
line and bisecting each other at right angles as they proceed from their
primary second-level term ("+" on the "+" side and "-" on the "-" side) to
the corresponding secondary term. The resulting map integrates the two
second-level lines which were separated in the expanded form:
(a) Assuming Polar Opposition (b) Assuming Contradictory
Opposition
Figure 2.3:
The Integrated Version of the Expanded Maps
The main problem with both the integrated and the expanded form of the
map of second-level analytic division is that the same map is used to
represent operations which are actually quite different. Our next task,
therefore, is to alter these integrated maps in ways which correspond to the
differences in the logical forms they represent.
When the two expanded maps of second-level analytic division are
integrated, the first-level lines actually turn out to be redundant in both
cases, since the relation between the two second-level lines is now the same
as that between the end-points of the first-level line. This is reflected in the
notation by the fact that the single term at either end of the first-level line is
identical to the second term in both of the second-level expressions which
are directly underneath it. The dashed lines on the other three sides of the
"x" in Figures 2.3a and b enable all possible first-level relations to be
explicated. The two pairs of parallel lines can then be labelled with "+" and
"-" signs of their own, with the top and right, as usual, corresponding to
the "+". However, this notation is fully accurate only for the map of
contradictory opposition (hence the brackets in Figure 2.3a), where the
terms labelling the two horizontal lines are identical to the first term in the
two expressions connected by these lines, and those labelling the vertical
lines are identical to the second term in the two corresponding expressions.
Because the positions of "--" and "+-" are interchanged in Figure 2.3a, the
horizontal lines now connect contradictory expressions, though the vertical
lines represent the same second-level polarity as they do in Figure 2.3b. In
fact, in both maps, the pair of horizontal lines connect the same expressions
as the pair of integrated second-level lines in the other map. Thus in Figure
2.3a the "pure-mixed" distinction is between the two horizontal lines, or
between the starting-points and end-points of the two second-level arrows,
while in Figure 2.3b the same distinction is between the starting-points and
end-points of the two first-level lines, or between the two second-level
arrows as such. This not only shows that the first-level distinctions are
contained within the second-level distinctions, but also hints at the
reciprocal relationship between these two methods of mapping. This
reciprocity can be made clear by connecting the midpoints of the horizontal
lines in Figure 2.3a with a line, and reflecting the whole figure across the
"++/--" diagonal; these lines then turn out to form the same map as that
given for contradictory opposites in Figure 2.2b! The same holds for the
horizontal lines in Figure 2.3b, which turn out to be the reflection of the
map given for polar opposites in Figure 2.2a.
The maps in Figure 2.3 require only slight alteration to produce the
two standard maps for second-level analytic division. If we begin by
omitting the first-level lines in Figure 2.3a, which were in any case
unsuitable, the remaining "x" can be rotated counterclockwise until the
"++" expression is at the top and the "--" is at the left. The resulting cross
[see Figure 2.4a] is an accurate map of the polar oppositions produced by
this type of second-level analytic division, since it connects pairs of
expressions which are partially similar. The "x" in Figure 2.3b, by
contrast, is not so appropriate as it stands, since the pairs of expressions it
connects are totally dissimilar (i.e. contradictory) opposites. The map can
be improved by omitting the two diagonal arrows and retaining the four
first-level lines in Figure 2.3b, extending them until they intersect, and
plotting the four second-level expressions onto their points of intersection.
The resulting square [see Figure 2.4b], the diagonals of which form the
original "x", is an accurate map of the contradictory oppositions produced
by this type of second-level analytic division, since it separates the
expressions which stand in contradictory opposition (i.e. the "++" and "--"
on the one hand, and the "+-" and "-+" on the other) by placing them
diagonally opposite each other. The difference between the two types of
opposition can therefore be depicted geometrically in terms of the difference
between opposite poles of a cross (polarity) and opposite vertices of a
square (contradiction).
(a) Polar Opposition (b) Contradictory
Opposition
Mapped onto the Cross Mapped onto the Square
Figure 2.4:
The Standard Maps of Second-Level Analytic Division
Both of these standard maps actually contain both types of
opposition, but in each case one type is clearly more prominent than the
other. We will see shortly, for example, that the quadrants of the cross in
Figure 2.4a can be interpreted as contradictory opposites. Likewise, the
two horizontal lines of the square in Figure 2.4b depict the first-level
relation between top (+) and bottom (-), while their end-points both depict
the second-level relation between right (+) and left (-); and conversely, the
two vertical lines depict the second-level relation between right (+) and left
(-), while their end-points depict the first-level relation between top (+) and
bottom (-). Nevertheless, using only one map for both types of opposition
would fail to bring out their distinct, yet reciprocal, relation. Moreover, as
we have seen, the cross as a map of contradictory opposition would
wrongly imply that the expressions are related as polar opposites of a
common line segment, when in fact they have no common factor
whatsoever. The cross should be preserved, therefore, as a map for polar
opposition alone, and the square for contradictory opposition, as accurately
depicted in Figure 2.4.
These two standard maps are the most important rectilinear figures
in the entire Geometry of Logic. For they reflect the perspectival
relationship between polar and contradictory oppositions. That is, polar
opposition can be transformed into contradictory opposition (and vice
versa) by changing the perspective. A "perspectival transformation" (i.e. a
change in perspective) is, for the Geometry of Logic, a change in the rules
for mapping, combined with a corresponding change in the geometrical
map. The most fundamental perspectival transformation is precisely the one
between polarity and contradiction, and symbolized by the cross and the
square, namely inversion. "Inversion" is an appropriate name for this
operation, not only because the change in the rules which transforms polar
opposition into contradictory opposition entails an interchange (or
inversion) of the positions onto which the "--" and "+-" expressions are
mapped in Figures 2.2-4, but also because the cross is actually the
geometrical inversion of the square!
Throughout this book the geometrical operation of inversion is
assumed to refer to the interplay between the center and the circumference
of a figure [see note 1.12]. This is different from the traditional geometrical
uses of the term, as when it refers to operations such as relating the outside
of a figure to the inside, or passing all the parts of a figure through a point
outside the figure, thus projecting it on the other side. In such cases the
inversion of a circle causes the center to be projected out to infinity,
whereas for our purposes it is projected only as far as the circumference.
The inverse geometrical relation between the cross and the square
can be verified by a simple empirical exercise. On a square piece of paper
draw a cross whose vertical and horizontal axes extend the length and
width of the paper. Draw over the lines to make them thick. Then cut the
paper along the lines so that each line shows on both sides of the cut.
Rotate each of the four smaller squares 180\ (so that each quadrant is
inverted, its inner corner becoming its outer corner), and arrange them so
that the corners which were formerly on the outside now touch in the
middle. The result is a square. If the original cross is labelled with the
standard pattern of arrowheads and expressions given in Figure 2.4a, then
each arrowhead is split into two and is paired with its opposite at the
midpoints of the four sides of the square, with the vertical lines pointing up
(+) and the horizontal lines pointing left (-); the expressions are also paired
off at each midpoint, with "++" and "+-" labelling the vertical (+) lines and
"--" and "-+" labelling the horizontal (-) lines [see Figure 2.5]. The
expressions at opposite corners of each quadrant (as defined by the dashed
cross) always differ in their first terms; hence, as we shall see below, the
second term can be used to define each quadrant. With a few minor
adjustments--viz. by placing the defining expression for each quadrant at its
outer vertex and suitably rearranging the arrowheads to depict a circuitous
counterclockwise development--this map turns out to be identical to the
standard square given in Figure 2.4b.
Figure 2.5: The Geometrical Inversion of the
Standard Cross
The cross in Figure 2.4a divides its surface into four equal
quadrants, each of which is connected to all the other quadrants, either by a
common line, or by the point at the center. Labelling these quadrants is
another good way of demonstrating the inverse logical relation between
polar and contradictory opposition. The expressions used to label the
quadrants can be defined by mapping either the first or the second terms in
the two expressions labelling the poles on either side of it. In the former
case the quadrant's label consists of the two first terms in two successive
expressions [as in Figure 2.6a], and in the latter it consists of the two
second terms [see Figure 2.6b].
(a) By First-Term Relations (b) By Second-Term Relations
Figure 2.6: Second-Level Inversion,
Mapped onto the Quadrants of the Cross
The identity between the expressions labelling adjacent quadrants in
Figure 2.6a is a direct result of the similarity in the logical status of the
pairs of expressions which gave rise to them:the two "+-" quadrants arise
out of the contradictory opposition between two expressions which are
both pure (viz. "++" and "--") or both impure (viz. "+-" and "-+"),
whereas the two "-+" quadrants arise out of the polar opposition between
one pure and one mixed expression (viz. "--" and "+-", or "-+" and "++").
Thus the two "+-" quadrants in Figure 2.6a express the same relation as the
two horizontal lines in Figure 2.3a, and so also the same relation as the two
vertical lines in Figure 2.2b.
The results of relating the quadrants of the cross according to their
second-term relations, as mapped in Figure 2.6b, are even more
interesting. For the four expressions are related according to exactly the
same pattern as those mapped onto the square in Figure 2.4b. This
suggests that the two standard maps for second-level analytic division can
be combined in a single map of polar and contradictory opposition
Figure 2.7:
The Standard Second-Level Maps, Combined
[see Figure 2.7]. This combined form makes it easy to see how the former
can be transformed into the latter through the operation of inversion, which
is here accomplished by defining the vertices of the square by taking the
second terms from the expressions labelling two successive
(counterclockwise) poles of the cross.@The arrows of the square are
arranged, as in Figure 2.4b, in a counterclockwise circuit [see note 2.10].
Its two horizontal arrows connect the same expressions as the arrows
which serve as the axes of the cross. The square's two vertical arrows
depict secondary polar relations which are covertly contained within the
cross, as indicated by the dashed arrows in the "++" and "--" quadrants.
The standard cross keeps these secondary polar opposites separated in
order to stress the primary polar relations between expressions whose first
terms are identical. (The former will henceforth be called "first-term
polarity", and the latter "second-term polarity", where "polarity" refers to
the term which differs between the two expressions. As such, contradictory
second-level expressions are examples of "first-and-second-term polarity".)
When the first-term polarities are connected, however, the cross too forms
exactly the same complete circuit. This transformation of the cross gives
rise to a map which will be used to represent synthetic integration in
Chapter Four.
Three additional points should be mentioned briefly in connection
with this description of the operation of second-level inversion. The first is
that applying the same operation to the expressions labelling the square
does not (as we might predict) transform it back into the cross. Rather it
always yields the same version of the contradictory pattern of oppositions,
only shifted 45\ either clockwise or counterclockwise, whether the first or
second-level relations are used and no matter which direction the operation
proceeds. This operation can therefore be used, as we shall see, to define
the sides of the square in terms of contradictory opposites.
The second point is that changing the direction in which the
expressions are considered in the operation of inversion (whether of the
cross or of the square) always yields the same pattern, except that the
positions of the "+-" and "-+" expressions are interchanged. That this is so,
and that Figure 2.7 depicts the most appropriate analytic form of inverse
relation between the cross and the square, can be made clear by listing the
eight types of transformation which arise out of the combination of the
three distinctions (i.e. the third-level analytic relation [see 2.3]) between
cross (+) and square (-), between first-level (+) and second-level (-)
relations, and between clockwise (+) and counterclockwise (-) development
[see Figure 2.8; cf. Appendix 2A]. The switch between clockwise and
counterclockwise entails reversing all the arrows in the case of the square;
but for the cross a reversal of the horizontal axis (and so also a change in
the interpretation of its arrowhead) is sufficient. In the latter case reversing
the vertical arrow would entail changing the starting-point of the map as
well, which complicates matters unnecessarily. Incidentally, the
interchange between "+-" and "-+" could also be used to transform all the
crosses in Figures 2.2-4,6-7 from maps of second-term polarity into maps
of first-term polarity. But we need not discuss this variation, since the latter
follows exactly the same pattern of transformations as the second-term
polarity which we have taken and will take as standard.
(a) Cross; First Term; (b) Cross; First Term;
Clockwise (+++) Counterclockwise (++-)
(c) Cross; Second Term;* (d) Cross; Second Term;
Clockwise (+-+) Counterclockwise (+--)
(e) Square; First Term; (f) Square; First Term;
Clockwise (-++) Counterclockwise (-+-)
(g) Square; Second Term;* (h) Square; Second Term;
Clockwise (--+) Counterclockwise (---)
Figure 2.8: Eight Versions of Second-Level Inversion
* Standard Maps
Finally, we should note the obvious similarity between our use of
the cross as a logical map and Descartes' use of the same figure when he
invented co-ordinate geometry [see W516:81-3]: "the interest of co-ordinate
geometry lies in the fact that it relates together geometry, which started as
the science of space, and algebra, which has its origin in the science of
number" [84]. By assigning gradual numerical values to each axis, the
whole range of possible mathematical numbers can be depicted. For the
"real" numbers are mapped onto the horizontal axis, the "imaginary"
numbers are mapped onto the vertical axis, and the "complex imaginary"
numbers are found in the spaces of the four quadrants [cf. 74-5]. By using
this same figure as a key symbol in the Geometry of Logic we add to its
pedigree the potential for symbolizing the essential characteristics of the
"science of thought". It should be noted, however, that our standard use of
the cross in analytic logic assumes counterclockwise development
beginning with the "+-" quadrant in Figure 2.8d, whereas co-ordinate
geometry assumes clockwise development, beginning with the "++"
quadrant, according to the pattern given in Figure 2.8c.
Having now established the standard maps for the second-level
analytic form of relation, both in its polar form and as transformed into
contradictory opposition through the operation of inversion, we are
prepared to discuss what can be called its "range of application"--i.e. the
extent to which this abstract form of relation can be applied to various sorts
of concrete content [cf. 1.3]. Virtually any time two sets of two elements
are compared or interrelated, the process involved is an instance of second-
level analytic division of one sort or another.
An example can be chosen almost at random from B501:759, where
Baum is discussing the pros and cons of various editorial policies with
(a) Mapped in Expanded Form
(b) Mapped onto the Cross
(c)Mapped onto the Cross and Square
Figure 2.9:
The 2LAR Alternatives in Editorial Policy
There are four logically possible atemporal acquaintance-relation
ships between any two parties. For example, the author and editor could
each know the identity of the other [++]; the author could know the identity
of the editor but not vice versa [+-]; the editor could know the identity of
the author but not vice versa [-+]; or neither could know the identity of the
other [--]. Baum rightly refers to this fourfold distinction as a logical one,
though he does not elaborate on its logical character. His distinction can be
readily mapped in expanded form [see Figure 2.9a] as a distinction on the
first level between "author knows editor" (+) and "author doesn't know
editor" (-), and on the second level between "editor knows author" (+) and
"editor doesn't know author" (-). When mapped onto the cross [see Figure
2.9b], using arrows in the expressions to denote knowledge, the former
distinction is represented by the relation between the axes and the latter as
the relations between their endpoints.@But Figures 2.9a and b do not
reduce this distinction to its most fundamental components. This is done by
distinguishing on the first level between the editor, who actively judges (+)
the material submitted, and the author, who passively awaits (-) the
decision, and on the second level between one's knowledge of the identity
of the other (+) and the lack thereof (-).The resulting second-level analytic
division can be mapped onto the cross, as in Figure 2.9c. When the
quadrants are defined [as in Figure 2.6b] and mapped onto a square, the
logical basis of Baum's fourfold distinction is fully explicated. Since all
four alternatives describe empirically possible situations, and since they are
organized according to a definite "2+2" pattern, this example can be called a
"perfect" application of the second-level analytic form of relation to a
concrete situation.
Many conceptual distinctions which are made in ordinary language
are "imperfect" in one way or another. For example, the four combinations
of concepts which result from a second-level analytic division could
produce one or more expressions which are self-contradictory, either
logically or empirically. Thus "raining"/"not-raining" and "cloudy"/"not-
cloudy" yield four logically possible weather situations; but one of these
(viz. "raining"/ "not-cloudy") does not correspond with the way the real
world is structured. Another imperfect instantiation would be one in which
all four expressions describe possible situations, but in which three of the
expressions are more closely interrelated than the fourth, thus resulting in a
"3 + 1" pattern rather than a "2 + 2" pattern. The logical significance of the
former type of pattern, an example of which would be the conception of
"space-time" (three dimensions of space and one dimension of time), will
be brought out in 4.4. In this chapter we will limit our attention to examples
of perfect analytic divisions as applications of their corresponding forms of
relation.
In ancient Greek philosophy one of the best examples of a perfect
second-level analytic division is the division of all natural "elements" into
fire, air, water and earth. Plato describes these elements as the four
essential "qualities" of all things, which are always found intermingled in
various ways in the empirical world [P494:49A-50A]. The image which is
often used in ancient cosmologies is that these elements "form four
concentric spherical layers, with fire on the outside (in the stars) and earth
at the centre" [C495:246]. As Plato puts it, "God ['o qeoz] set water and air
between fire and earth, and made them, so far as possible, proportional to
one another, so that as fire is to air, so is air to water, and as air is to water,
so is water to earth" [P494:32A-B]. Thus the "extremes" which do not
touch are fire (++) and earth (--), while the "means" which relate each
element proportionally to the others, are air (+-) and water (-+). Plato's
threefold proportion can therefore be expressed conceptually as:
fire air water
---- = ---- = ----
air water earth
This proportion can be made more complete by adding a fourth
relation between the center (earth) and the circumference (fire). (This
accords well with the cosmological fact that the earth's core is made of
molten (i.e. fiery) rock. Thus the logical form of this proportion is most
accurately expressed as:
++ +- -+ --
-- : -- : : -- : --
+- -+ -- ++
The relation between the oppositions in this full form of Plato's
proportion can be expressed in the following rule:positive polarity is to
mixed contradiction as negative polarity is to pure contradiction. The circuit
which is described here can best be mapped onto a square [see Figure
2.10a], with its capacity to represent the relation between polarity and
contradiction.
The suitability of mapping Plato's four elements in this way can be
confirmed by explicating the qualities which correspond to each. The
(a) Mapped onto the Square (b) Their Qualities, Mapped onto the Cross
Figure 2.10:
The Logical Relations between the "Four Elements"
primary quality of fire is that it is hot; likewise water is wet, air is cold, and
earth is dry. The oppositions between hot and cold and between wet and
dry are both polar, since the former pair both measure high and low
temperature, and the latter high and low humidity. Hence they can be
mapped most accurately onto the cross, as in Figure 2.10b, with their first-
level distinction depicted as the relation between the two poles (cf. the
vertical lines in Figure 2.10a). The inverse relation between these two maps
can be verified by noting how any two successive (counterclockwise)
second-level relations, as mapped onto the cross, can be combined to
define the vertex of the corresponding quadrant of the square [cf. Figure
2.6b]. Thus "high/high" defines the position of fire in Plato's cosmology,
"low/low", that of earth; likewise "high/low" corresponds to the position of
air and "low/high" to that of water. This fourfold division of elements is
often regarded as an attempt to set down the first principles of natural
science [cf. 1.1], and as such is disregarded with a chuckle by modern
scientists and philosophers of science. But to regard it in this way is to
miss the point altogether. For in this as in any a priori cosmology, the point
of the theory is to make logical, not empirical, distinctions.
Numerous general philosophical concepts can be categorized and
interrelated in a helpful manner by means of the formal tools provided by
the Geometry of Logic. One of the many examples is the way in which
Kant combines the analytic-synthetic distinction with the a priori-a
posteriori distinction in his theory of knowledge. A detailed account of his
use of these terms would be out of place here [see P1:4.2], but we can
mention a few interesting implications of their interrelationship, which
become apparent when the two first-level analytic relations are mapped
together as a perfect second-level analytic relation. If the analytic-synthetic
distinction is regarded as primary (i.e. first-level), and so mapped onto the
axes of the cross, then the a priori-a posteriori distinction will describe the
(second-level) relation between the end-points of each axis. Kant says that
analysis presupposes synthesis [K105:133], since intuition is more
fundamental epistemologically than conceptualization; hence it is
appropriate to map the former onto the vertical (+) axis, and the latter onto
the horizontal (-) axis. Elsewhere he stresses that "all our knowledge
begins with experience", yet "it does not follow that it all arises out of
experience" [K105:1]; hence the a posteriori is best correlated with a "+"
value, though it is fundamental only when connected with synthesis, a
priori (-) being fundamental when the distinction is put in the context of
analysis [see Figure 2.11].
Figure 2.11:
Kant's Primary Epistemological Distinctions
Many critics of Kant would regard this as an imperfect second-level
analytic division:only the two pure expressions ('synthetic a posteriori" and
"analytic a priori") are universally accepted as legitimate pairs of concepts;
the coherence of the two mixed expressions ('synthetic a priori" and
"analytic a posteriori") is often questioned, or (in the latter case) denied
without argument. I have argued in P1:4.2-4, however, that the division is
a perfect one, since all four combinations have a legitimate field of
application. The argument is supported by the correspondence which can
be seen to hold between these expressions and Kant's four main
"perspectives" (which guide the development of his epistemological system
[see 6.4; s.a. P1:4.3]), when we perform the operation of inversion on the
second-level terms mapped onto the cross in Figure 2.11. Proceeding in the
standard counterclockwise direction, we can see that the "transcendental" is
concerned with the a posteriori basis of the a priori; the "logical" is
concerned with the a priori basis of the a priori; the "empirical" is
concerned with the a priori basis of the a posteriori; and the "practical" is
concerned with the a posteriori basis of the a posteriori. (Note that for each
perspective there is also a first-term polarity which always includes both the
analytic and the synthetic [cf. Figure 2.6a]. This hints at the reciprocity
between analysis and synthesis, which will be treated in detail in Chapter
Four.) Without a clear understanding of the interplay between these
perspectives, Kant's first Critique is bound to be misinterpreted.
A similar example of a set of philosophical concepts which can be
organized according to the pattern of second-level analytic division is the
first-level relation between epistemology (+) and ontology (-), as combined
with the second-level relation between subjectivity (+) and objectivity (-).
These relations can be mapped onto the standard cross as shown in Figure
2.12. The inversion of this map reveals a square describing four "modes of
knowledge", or ways of approaching the empirical world: "detached
reasoning" and "passionate faith" require purely objective or subjective
approaches, respectively, while "artistic imagination" and "scientific
observation" require a mixture of both. In itself, mapping the modes of
knowledge in this way obviously adds nothing concrete to the content of
philosophical knowledge. But it is
Figure 2.12: Four Modes of Knowledge
not intended to do so, any more than the "four elements" of Classical Greek
philosophy are intended to describe a set of fundamental categories for the
"periodic table" of atomic substances. Instead, such distinctions are
intended to provide a formal framework according to which real
philosophical knowledge can be organized and clarified. Such organization
and clarification of concepts, both general and particular ones, is the most
important function of philosophy.
Gazing over the horizon of Western philosophy brings into view a
rather different geometrical figure, one which perfectly symbolizes the
second-level analytic form of relation. The "t'ai chi" symbol has already
been mentioned in 1.4, where it was used as an illustration of how unity is
divided into duality. What was not revealed is that at the heart of each of the
principles depicted in Figure 1.6 is traditionally placed a small portion of
the other. Using our notation to label this more complete version of the
famous Chinese symbol yields the map shown in Figure 2.13a.@ By
simply rotating this map counterclockwise by 225\
(a) In its Traditional Position (b) Rotated to Correspond, to the Square
(c) Rotated and Altered to Correspond to the Cross
Figure 2.13: The T'ai Chi Symbol as a Map of
Second-Level Analytic Division
(c) Rotated and Altered to Correspond to the Cross (or clockwise by 135\),
the position of the four expressions is transformed so that it coincides
exactly with that of the square [see Figure 2.13b; cf. Figure 2.4b], with the
contradictory oppositions at diagonals and the polar oppositions where the
sides of the square would be. Altering the rules for mapping so that the
small portions at the heart of the two main sections represent contradictory,
rather than polar, relations, and rotating the original map counterclockwise
by 315\ (or clockwise by 45\), transforms the position of the four
expressions so that it coincides exactly with that of the cross [see Figure
2.13c; cf. Figure 2.4a]. Note that the direction in which the two "fish"
traditionally "swim" is counterclockwise. This is one reason why the t'ai
chi correlates so easily with the counterclockwise versions of the cross and
square, which we have taken as the standard direction for analytic maps.
The t'ai chi can be altered, so that it coincides with the variations given in
Figure 2.8a,c,e and g, simply by flipping it over (i.e. "reflecting" it, as in a
mirror). This transforms the implicit flow of the two bulges from
counterclockwise into clockwise development.
However, it is not necessary to look beyond the horizon of Western
logic in order to find examples of geometrical figures which have already
been used as maps of second-level analytic relations. Boole lists the four
"constituents" produced by any two-variable function as "xy, x(1-y), (1-
x)y###(1-x)(1-y)": "these classes together make up the universe"
[B506:6.2]. Langer describes the same four classes in L483:166 with the
equations "W x B = 0", "W x -B = 0", "-W x B = 0", 1$/// and "-W x -B =
0". She then maps these four classes (assuming the left / circle to represent
the class of W's and the right circle the class of 2B's) onto the map given in
Figure 2.14a, which follows the method of diagramming initially
introduced by Euler in the eighteenth century [see S515:73]:
(a) Using Traditional Notation (b) Using "+"
and "-"
Figure 2.14:
The Four Possible Relations between Classes W and B
Of this map Langer exclaims [L483:166]: "Here is the whole
system in a form so simple, so visible, that all its relations may be
intuitively grasped--the height and ideal of logical explication." While I
agree with Langer's judgment as it stands, I would nevertheless wish to
add that this "height and ideal" can be reached in its purest form by
abstracting from all actual class relations and using the same logical map to
represent not specific class relations, but the form of relation which serves
as their basis. The resulting map [see Figure 2.14b] is even simpler and
more intuitively valuable. Moreover, it enables us to make obvious
correlations between the traditional map and the maps we have been using,
so that key distinctions, such as that between polar and contradictory
opposition, can be applied directly and unambiguously to the study of class
relations in general.
Another diagram which finds its way in one form or another into
many beginning logic textbooks is the famous "square of opposition" [see
e.g. K502:55-6,551]. The square of opposition is a map of the logical
relationship between the four primary two-term categorical propositions, or
indeed between any four propositions or concepts which display a similar
form. This common logical device is a clear example of the inverted form
of second-level analytic division, for its primary purpose is to explicate the
contradictory relations between the propositions placed at opposite corners
of the square. (In order to emphasize these contradictory relations, the
square is sometimes replaced by a sideways cross [as in Figure 2.3b],
connecting the contradictory propositions; but the misleading implications
of this alteration have already been discussed.) That the square should be
so widely accepted as an appropriate geometrical representation for this
logical device is evidence of the power of intuitive symbols to clarify
formal-logical relations, and confirms the legitimacy of our choice of maps.
Moreover, our use of the cross and the square as logically distinct models
for mapping formal-logical relations provides an easy way of
distinguishing between a "proper" square of opposition and an "improper"
one.
Before considering the traditional square of opposition, it will be
helpful to examine the formal basis of the propositional logic which it is
intended to describe. The four fundamental logical propositions can be
listed using both Aristotelian terminology and the more precise modern
terminology of universal and particular quantification [see e.g.R517:161-3;
S515:44,46-8; L483:106-7] as follows:
All S is P =(s): s is P= universal affirmation;
No S is P =(s): -(s is P) = universal negation;
Some S is P = (Es): s is P= particular affirmation;
Some S is not P = (Es): -(s is P) = particular negation.
The superiority of the modern terminology is revealed by the ease
with which it can be mapped onto the standard maps for the second-level
analytic form of relation. (The traditional terminology is misleading
primarily because "No S is P" really means "All S is not P".) The first-level
relation is between the "quantifier" (+) and the "quantified" (-), which can
be mapped in expanded form [cf. Figure 2.2a] by dividing the former into
"universal" ("(S):") and "particular" ("(Es):") quantifiers, and the latter into
quantified "negation" ("-(s is P)") and "affirmation" ("+(s is P)") [see
Figure 2.15a]. The same relations can be mapped directly onto the cross, as
is Figure 2.15b, so that the full set of four propositions can be derived by
performing the operation of inversion on the original second-level relation.
Thus a universal quantifier (++) applied to a quantified negation (--) results
in universal negation (+-), which is expressed symbolically as "(s): -(s is
P)"; etc.@
(a) Mapped in Expanded Form
(b) Mapped onto the Cross and Square
Figure 2.15: Universal and Particular Quantification
Consider now two of the most commonly cited examples of the square of
opposition [see e.g. C500:156,160; K502:55,86], both drawn from
Aristotle:
(a) A Proper Square (b) An Improper Square
Figure 2.16:The Traditional Square of Opposition 1
Figure 2.16a is a proper square of opposition because both terms in each
pair of diagonals are opposite to the corresponding term in the expression
diagonal to it. But the "opposing" terms in 2.16b stand not in a
contradictory (as these terms are often carelessly taken to be), but in a
polar, relation to their opposites, for they each have one common term
(either "necessary" or "possible"), and only one that differs (viz. the
assertion/negation of "s"). Hence the latter is an improper example of the
square of opposition.
The validity of this judgment can be tested by making suitable
alterations in the distinctions given in Figure 2.16 so that both sets can be
mapped together onto the standard maps for analytic division. In the former
set [as in Figures 2.15b and 2.16a] the first-level relation can be assumed
to be between "s is..." (+) and "s is not..." (-), and the second-level
relation (unlike Figure 2.16b), between "impossible" (+) and "necessary"
(-)--the former being a more accurate description of the contradictory
opposite of the latter than "possible". We can now construct a map of the
proper relationship between the distinctions given in Figures 2.16a and b--
i.e. the relationship between the type of operation which is performed by a
proposition (a) and the modal status of the result (b). By following exactly
the same procedure as was used to construct the map in Figure 2.15b, we
can map the first set of distinctions directly onto the vertices of the square
[see Figure 2.17]. (Note that the "+-" and "-+" expressions are
interchanged from the traditional positions given in Figure 2.16a. This is
because the latter figure presupposes a clockwise development, whereas we
have chosen to use counterclockwise development as the standard direction
for analytic division.) The second set of distinctions now turns out to
define the sides of the square--i.e. to describe the result of inverting the
square's vertices:
Figure 2.17:
The Proper Relations between Figures 2.16a and b
The logical outcome of applying the types of quantification given at
two successive vertices of the square in Figure 2.17 to a common content is
specified by the proposition labelling the side of the square which mediates
between them. The suitability of this way of mapping the distinctions is
verified by the fact that each threefold relation both properly describes the
actual logical relation and perfectly corresponds to the pattern set by the
second-level analytic form of relation. Thus, it is impossible for universal
affirmation to entail universal negation; it is necessary for universal
negation to entail particular negation; it is not necessary for particular
negation and particular affirmation to apply to the same concepts in
different situations; and it is not impossible for particular affirmation and
universal affirmation to apply to the same concepts in different situations.
The clarity and order of the map in Figure 2.17, especially as compared to
the inconsistencies which result from a loose employment of the traditional
square of opposition [cf. Figure 2.16], is a good example of how the new
tools developed by the Geometry of Logic can be used both to clarify and
to correct traditional logical distinctions--even those as old as the square of
opposition.
Of the myriad other examples which could be cited at this point, I
shall dwell on just one more. Not only can propositions be formally
classified according to the second-level analytic form of relation, but the
grammatical relation between words themselves can also be described along
these lines. Grammarians disagree on just how to define the main parts of
speech, but they generally agree that the primary constituents are nouns,
adjectives, verbs and adverbs. The second-level analytic form of the
grammatical relations between these four parts of speech can be
demonstrated by distinguishing on the first level between the functions of
modification (+) and predication (-), and on the second level between the
active (+) and passive (-) role which a word can play in either of these. The
four resulting combinations are mapped onto the cross in Figure 2.18.
Applying the operation of inversion to the successive poles of this cross
then enables us to map the logical relations between the four grammatical
parts of speech onto the vertices of the square:
Figure 2.18: The Four Grammatical Parts of Speech
That the main grammatical components of a sentence should fit so
perfectly into this formal-logical structure is no coincidence. For grammar
is a rational analysis of language, so it is only natural for it to follow a
pattern set by an analytic form of relation. There are, of course, modern
grammarians and linguists who believe such classifications "are an outdated
concept that can never give us more than a rough approximation to the truth
about English grammar. Nevertheless, they do give us a rough
approximation at least" [H485:63]. Here, as elsewhere in our discussion,
we see that the process of distilling the unorganized variety found in the
empirical world into the flask of order and clarity must be recognized for
what it is:an imposition of the logical perspective upon the empirical.
The same rebuttal can be made against those who argue: "The
traditional view that there are a limited number of propositional forms
departure from which constitutes irregularity is absurd. Every proposition
has a determinate form which is its logical form" [S515:79]. The second
point is undoubtedly correct, but only insofar as logicians assume that their
role is to observe the empirical structure of language, like a scientist. The
first point is mistaken because it fails to recognize that the "traditional view"
can be legitimately upheld by logicians who see their role not as extracting
the logical from the empirical, but as imposing the logical upon the
empirical. Modern logic, with its emphasis on the importance of the
"universe of discourse", is always primarily interested in the truth (i.e. the
content) or the applicability (i.e. the existence or non-existence) of its
propositional distinctions, whereas traditional logic abstracts from the real
world or any other concrete context. But the fact that traditional logic is
more interested in first establishing the validity (i.e. the form) of clear-cut
logical distinctions does not render it "absurd", so long as the logician is
aware of the limitations of the perspective he is assuming. Once this crucial
point is recognized, we can understand and applaud the scientific analysis
of language, which seeks to look beyond the logical structure of the
empirical, and into the empirical structure of the empirical; yet at the same
time we can continue to appreciate the value of categorizing propositional or
grammatical relations in the way we have done in this section.
Our discussion of second-level analytic division and sampling of its
various applications has been necessarily lengthy, for of all operations in
the Geometry of Logic, the inversion of polarity into contradiction, which
first arises on this level, is the most important to understand. All the
essential distinctions in analytic logic operate at this second level, so our
discussion of all subsequent levels will look back to the principles of
division, mapping, transformation and inversion established in this section.
Only when the details of second-level analytic division are grasped, so that
performing this operation and inverting its results become second nature,
will the more complex (and subtle) operations dealt with in the remainder of
this book be comprehensible.
2.3 Polar and Contradictory Opposition
in Third-Level Analytic Division
Just as second-level analytic division is the operation which
interrelates two pairs of single-level analytic relations, so also third-level
analytic division is the operation which interrelates three pairs of single-
level analytic relations. The former deals with two variables, and is
therefore represented by two-term expressions, whereas the latter deals
with three variables, and is therefore represented by three-term
expressions. The addition of a third term transforms the number of possible
expressions from four to eight. The third-level analytic form of relation can
be mapped onto a number of geometrical figures, each of which brings out
slightly different aspects of this level of analytic division. After discussing
these various maps and mentioning a few of their applications, this chapter
will conclude with a discussion of analytic division on the fourth level.
The logical relation between third-level analytic division and the two
lower levels of division can be readily seen by expanding the map given in
Figure 2.2 still further, so that each of the four main end-points is
converted into a two-fold division. By following the rules for mapping
outlined in 2.1, we are then able to map all three levels onto a single figure,
with the eight expressions organized according to third-term polarity [see
Figure 2.19]. (Using lines to connect other types of polarity, including the
thoroughgoing polarity entailed in contradictory opposition, was judged in
2.2 to be misleading.) Four lines must be added to the expanded map of
second-level analytic division to produce the expanded map of third-level
division. Hence there is a total of seven different single-level relations
inherent within the third-level form of relation. (The "6 + 1 = 7" pattern
which is evident in this map--i.e. the six higher-level lines plus the one line
representing first-level analytic division as such--will play an important role
in Part Two.) Similarly, there are three different second-
(a) Assuming Polar Oppositions
(b) Assuming Contradictory Oppositions
Figure 2.19:
The Expanded Map of Third-Level Analytic Division
level expanded maps in Figure 2.19: one on the right, one on the left, and
the original one in the middle.
The direction of the four arrows denoting the third-level relations is
determined by applying the same rule as was used to determine the
direction of the two second-level arrows:on any given level the arrow
points towards the expression with a "-" in the final term if the penultimate
term is a "+", and towards the expression with a "+" in the final term if the
penultimate term is a "-". As a result the even levels display a clockwise
development while the odd levels display a counterclockwise (or in the case
of the first level, backwards) development.
On the third and higher levels the expanded form of map is too
awkward to be of much use as an intuitive tool for mapping real relations,
though it does provide an invaluable method of analyzing the
interconnections between all the single-level relations which are implicitly
contained within each higher level. As in 2.2, this problem can be avoided
by mapping the relevant expressions onto the standard cross and square.
The third-level polar oppositions will first be mapped onto an expanded
cross (rather than an expanded line, as in Figures 2.2 and 19). The
resulting map will then be used to construct several versions of the "double
cross". After reviewing the rules which determine the form chosen for
mapping the expressions and arrowheads onto the double cross, this new
standard map will be inverted so that it can be used as a map for
contradictory oppositions--an operation which, as it turns out, produces the
same result as "expanding" the square itself. A brief discussion of several
three-dimensional maps and their relation to their two-dimensional
equivalents will then prepare us to apply the third-level form of relation to
real relations.
Expanding each endpoint of the standard cross into a line segment
perpendicular to the corresponding line gives rise to the map of third-level
analytic division in Figure 2.20. The first two terms of each expression are
derived directly from the expression labelling the corresponding end point
of the cross. Each new endpoint is labelled with a third "+" or "-" using the
rules set out in 2.1. And the arrows follow the same pattern as that
specified for Figure 2.19.
Figure 2.20:
The Expanded Standard Second-Level Cross
It is interesting to note that the combination of either the second or the third
terms in any two successive expressions which are on different lines, and
thus are closest to the point where the lines would meet to form the corner
of a square, yields exactly the same pattern of expressions as that mapped
around the standard square [see Figure 2.4b]. When the second terms are
used this operation is, of course, the operation described as "inversion" in
2.2, since the third terms have to be ignored. When the third terms are used
the direction has to be reversed in order to produce the same result. That is,
clockwise succession of third terms in Figure 2.20 yields the same result as
the counterclockwise succession in Figure 2.7b. This implies that
expanding the second-level cross to a third level is in some sense the
reversal of the operation of second-level inversion.
The expanded form of the standard second-level cross can be
transformed into an integrated map of third-level analytic division in several
ways. One way is by connecting the end-points on each opposite pair of
perimeter lines in Figure 2.20, and then omitting all other lines. This
procedure results in the construction of the double cross shown in Figure
2.21a. The "+" and "-" in the second term of each expression now refers to
the status of the end-point on which the expression is mapped. The third
term of each expression refers to the status of the lines in its opposition to
the line which is parallel to it. And the first term refers to the status of the
pair of parallel lines, as opposed to the pair which is perpendicular to it.
But this pattern should not be accepted as standard, because the logical
order in which the terms of each expression are used does not correspond
directly to the geometrical order of the parts of the map. That is, the
oppositions between (1) the two pairs of lines###(2) the two lines in a pair,
and (3) the two end points of a line do not correspond to the first, second
and third terms in each expression. Instead, the three
(a) Assuming 1-2-3 Polarity (b) Assuming 1-3-2 Polarity
Figure 2.21: The Double Cross as an Integrated Map
of Third-Level Polar Oppositions
Figure 2.22: The Second-Level Relations in the
Standard Double Cross
(a) Assuming Second-Level Polar (b) Assuming Third-Term
and Contradictory Opposition Polar Opposition
Figure 2.23: The Fully Integrated Double Cross
terms refer to the first, third and second of these geometrical oppositions,
respectively. So this form of the double cross can be referred to as an
example of "1-3-2 polarity".
Another way of constructing the double cross from the expanded
single cross yields a more adequate result. Each of the four perimeter lines
can be rotated on its starting-point by 90\ and extended to the opposite side
of the map. Omitting the original single cross in the middle results in the
double cross in Figure 2.21b, wherein the third term in each opposition
refers to the status of the end-points, the second term refers to the status of
the line, and the first term refers to the status of the pair of lines. The exact
correspondence between geometrical order and logical order which
characterizes this version of the double cross makes this example of "1-2-3
polarity" a good candidate for a standard map of third-level analytic
division. The suitability of this pattern of expressions and arrowheads is
brought out in the Appendix to this chapter, where it is shown to be the
only pattern out of 645,120 possible variations which fits the requirements
of the rules we have set down for the analytic mapping of polar opposition
in the Geometry of Logic.
The obvious affinity between the third- and second-level crosses
gives rise to a number of interesting correlations. For instance, if we ignore
the second term in each three-term expression, then the two lines in each
pair of parallel axes in the double cross are labelled with identical
expressions. The parallel axes could then be merged into one, thus
transforming the double cross back into the single cross. Alternatively, this
version of the double cross can be analyzed in terms of two single crosses,
related by a diagonal reflection. The shading in Figure 2.22 makes this
clear:the dark cross can be transformed into the light cross by reflecting it
across either of the dashed diagonal lines.@Omitting the first term in each
three-term expression on this map leaves two crosses which are labelled
with exactly the same pattern of expressions as the standard second-level
cross. These sets of two-term expressions overlap when the double cross is
folded (i.e. reflected) across the "pure" diagonal (i.e. the one connecting
the pure expressions); only the first term in each pair of three-term
expressions differs. But when the map is folded across the "mixed"
diagonal (i.e. the one connecting the top left to the bottom right corner), all
three terms in each pair of overlapping expressions differ. Reflection across
the pure diagonal therefore gives rise to first-term polar opposition, while
reflection across the mixed diagonal gives rise to contradictory opposition
in third-level analytic division.
Significantly, these two diagonals themselves form a second-level
cross which can be labelled with the inverted pattern of two-term
expressions, either by using the third term of the two successive
expressions on either side of each corner or by simply using the second and
third terms in one of these adjacent expressions. For the two end-points of
the pure diagonal, both operations yield the unambiguous results "++" and
"--". But for the two end-points of the mixed diagonal both operations yield
a choice between "+-" and "-+". The reason for this uncertainty is that one
result will be obtained if clockwise development is assumed, and the other
if counterclockwise development is assumed [cf. Figures 2.8e and f].
A final variation of the double cross is suggested by the fact that the
standard double cross can be transformed into either the standard second-
level cross or the sideways cross. For these two types of "single" cross can
be combined to form a "fully integrated" double cross, in which the four
axes bisect each other at a common center [see Figure 2.23]. The resulting
figure can be used to depict the perspectival relationship between second-
level polar and contradictory opposition by combining them onto a single
map:a first-level distinction can be made between the standard (lighter)
cross (+) and the sideways (darker) cross (-), and the second and third
terms of each expression can then be taken directly from the standard
patterns for second-level polar and contradictory oppositions, respectively
[see Figure 2.23a].@ This exemplifies the way in which variations on a
lower level can converge onto a single, higher-level map. Nevertheless, we
cannot use it as the standard double cross, since it requires a mixture of
third-term polarity (on the horizontal and vertical axes) with second-and-
third-term polarity (on the diagonal axes).
A version of the fully integrated cross which consistently assumes
the standard final-term polarity is given in Figure 2.23b. Here the two
darker axes (-) are opposed to the two lighter axes (+) on the first level, as
in 2.23a; but the "-++" and "--+" expressions are interchanged, so that in
both crosses the second and third terms of each expression (as well as the
direction of the arrows) duplicate the standard pattern for the second-level
cross. (The dashed line indicates the place where this map can now be
folded so that all contradictory expressions overlap.) A 45\ rotation of the
"+" cross counterclockwise, or of the "-" cross clockwise, reveals the first-
term polarity between the two crosses (i.e. the second and third terms of all
overlapping expressions would then be identical). This rotational symmetry
symbolizes the reciprocal relationship between analysis (-) and synthesis
(+), which will be examined in greater detail in Chapter Four. The highly
integrated and consistent representation of the third-level form of relation,
which this version of the double cross gives us, makes it as suitable a
candidate for a standard map of third-level polar opposition as the version
given in Figure 2.22. But, as we shall see shortly, the inversion of the
latter is slightly more appropriate than that of the former, so we shall
continue to regard the version given in Figure 2.22 as the standard map.
Before examining the way in which the double cross can be
transformed into a map of contradictory opposition through the operation of
third-level inversion, it will be helpful to give a clear and concise
restatement of the rules we have been following. This will insure against
possible misunderstanding as to just how these rules have enabled us to
narrow down the patterns of mapping so exclusively. Table 2.3 describes
in general terms the distinctions which determine the pattern for mapping
the expressions and arrowheads appropriate to any level of analytic division
onto the corresponding version of the cross, assuming that higher levels
will use the version of the cross given in Figure 2.22. (Some of the
distinctions to which the terms can apply when mapped onto other figures,
such as the fully integrated double cross, are given in brackets.)
Table 2.3: Rules for Mapping
Analytic Expressions onto the Cross
For the double cross in Figure 2.22, then, this means (1) that the
first term in each of the eight expressions is "+" for both vertical lines and
"-" for both horizontal lines; (2) that the second term is "+" for the right or
top line in each pair and "-" for the left or bottom line in each pair; (3) that
the third term is "+" when the expression is at the right or top end of the
line and "-" when it is at the left or bottom end; and (4) that the arrow
points away from any expression whose second and third terms are alike
and towards any expression whose second and third terms differ. Each of
these defining rules narrows down the number of alternative maps which fit
the required description [see Appendix A2]; and altogether they reduce the
number from 645,120 all the way to 1.
The key to inverting the standard double cross in such a way as to
transform it into a suitable map of third-level contradictory opposition is to
notice the unlabeled square which is formed by the intersection of the four
axes. For the dashed diagonal lines in Figure 2.22 not only define the
vertices of a large square around the perimeter of the double cross; they
also intersect the vertices of this smaller square. These eight vertices can be
used to map the contradictory oppositions in a far more adequate form than
merely folding the double cross across the mixed diagonal. This can be
done by drawing the outer square and omitting the lines of the cross which
then connect the two squares, so that the main (first-level) opposition is
now not between the vertical and horizontal pairs of lines, but between the
outer (+) and inner (-) squares. In Figure 2.24 the lines with arrowheads
specified connect the same expressions and point in the same direction as
those on the four axes of the double cross (both in Figure 2.22 and in
Figure 2.23); similarly, the lines without arrowheads connect the
expressions which are at the same ends of parallel lines. This results in a
clockwise (+) development around the outer square and a counter-
Figure 2.24: The Double Square as a Standard Map
of Third-Level Contradictory Opposition
clockwise (-) development around the inner square.@The second and third
terms of each expression are identical to those in the corresponding
positions on the second-level squares for clockwise and counterclockwise
development, respectively [see Figures 2.8e-f], so they could be derived by
the same process of comparing quadrants as was used in Figure 2.6b.
Contradictory pairs of expressions are now located on completely different
squares, thus accurately representing their incompatibility. Their opposition
is further represented by the fact that one of the squares must be reflected
across its mixed axis (so that the position and direction of the arrowheads
on the two squares agree with, rather than oppose, each other), in order for
the pairs of opposite expressions to be relocated at the same corner of the
opposite square.
This standard "double square" can be regarded as the third-level
inversion of the standard double cross, according to the same geometrical
analogy used in 2.2 to name the operation of transforming the second-level
cross into the square. But the process of inverting the double cross to
produce the double square is rather more complicated than second-level
inversion. For the result of merely bisecting the standard double cross
vertically and horizontally then rotating each of the resulting sections by
180\ simply confuses matters by enlarging the central square by a factor of
four (assuming that the nine sections in
(a) Using Second-Level Inversion (b) Using Diagonal Inversion
Figure 2.25:The Inversion of the Double Cross1
the double cross start out as equal in size), and merging at the center of
each line two expressions (and an arrowhead) whose third-level terms
alone are opposites, as in Figure 2.25a. Another option is to cut the double
cross through the center along the diagonal lines given in Figure 2.22, then
rotate each section by 180\. This operation produces the "squared cross"
given in Figure 2.25b, which connects opposite expressions at two corners
and expressions with only first term opposition at the other two corners.
The position of the arrowheads and expressions could be easily transferred
to the corresponding end-points of the lines, as we did in 2.2 for Figure
2.5, without changing the logical relations being represented. However,
neither of these maps has the potential to be used as a standard map because
they do not consistently represent contradictory expressions in
contradictory positions, as does the double square.
Third-level inversion separates two procedures which are subtly
combined in second-level inversion:the cutting of the map into four
quadrants, and the cutting of the map along the lines. Because the lines of
the double cross do not coincide with the division of the figure into four
quadrants, these two procedures must be performed separately, thus
dividing the figure into sixteen segments [see Figure 2.26b]. Rotating each
segment by 180\ gives rise to the map of third-level contradictory
opposition shown in Figure 2.26a.@ This "simple" version of third-level
inversion thus transforms the double cross into a cross within a square, the
very figure used in 2.2 as the standard map for representing the
combination of the second-level analytic form of relation (the cross) with its
inverted form (the square). This correspondence is further highlighted by
the fact that the second and third terms in each expression correspond
exactly to the second-level values which can be assigned to the vertices of
the square (as shown in brackets [cf. Figure 2.26a and 2.22]). Another
significant point about this map is that the arrowheads have all been cut in
half and are now technically located only inside the mixed quadrants of
Figure 2.26a. (The other side of each half-arrowhead is included as a
dotted line simply to make it look more like an arrowhead.) This indicates
that the set of mixed expressions in any form of relation is dynamic,
whereas the set of pure expressions is static--a correlation which is further
supported by the fact that it is the mixed expression which vary when the
direction of logical development is altered, whereas the pure expressions
remain unchanged. (This distinction will be developed further in 4.4.)
In order to transform Figure 2.26a into the standard map already
posited for third-level contradictory opposition (viz. the double square in
Figure 2.24), it is necessary to perform a second-level inversion on the
interior cross. This produces another square of the same size [see Figure
2.26b], whose expressions and arrowheads are arranged in a pattern
strikingly similar to that of the square in Figure 2.26a. Figure 2.26c
combines both maps, together with dotted lines showing the position of the
original cut and dashed lines showing the position of the original double
cross (which was also cut). The overall three-step operation which
transforms the double cross into this duplicated square can be referred to as
"radical" third-level inversion.
(a) Simple Third-Level Inversion (b) Second-Level
Inversion, of
the Interior Cross
(c) Radical Third-Level Inversion
Figure 2.26: The Inversion of the Standard Double
Cross
(a) Simple Third-Level Inversion (b) Double Second-Level Inversion
Figure 2.27:
The Inversion of the Fully integrated Double Cross
The arrows and expressions of the two overlapping squares in
Figure 2.26c are differentiated by putting the ones referring to the square in
Figure 2.26a on the outside and those referring to the square in Figure
2.26b on the inside. Note that the expressions are all 1/6 of the length of a
side away from the identical expression on the other square, and that the
corresponding arrows all point in opposite directions. Two implications of
the latter point are that the geometrical inversion explicates alternatives
without determining a single consistent direction of development, and that
there needs to be some way of showing such an opposition of direction on
the standard map. Both of these requirements are satisfied by reducing the
size of one of the squares, omitting all duplicated expressions, and
mapping the expressions with a first-term "+" onto the vertices of the
corresponding quadrant of the outer square, and those with a first-term "-"
onto those of the inner square. A comparison of Figure 2.26c with the
standard map of the double square given in Figure 2.24 reveals that these
alterations do not change the pattern of relations between the expressions,
but only simplify the form in which it is represented. Radical third-level
inversion can therefore be regarded as the operation which transforms the
double cross into the double square.
The inversion of the fully integrated double cross [see Figure 2.23]
can also be done in several ways. If we simply cut along the lines and
rotate the eight resulting sections by 180\, so that the midpoint between the
two end-points of each section is placed in the center of the map, then the
map is transformed into two double crosses [see Figure 2.27a]. (One of
these corresponds to the light (+) and one to the dark (-) cross in the
original figure, which is represented by the dashed lines in Figure 2.27a.)
This indicates that the fully integrated double cross is more fundamental
than the standard double cross, since the latter is the inversion of the
former. But it also explains why the latter makes a better standard map:the
inversion of the standard map of polar opposition should result in a
standard map for contradictory opposition.
Another way of inverting the fully integrated double cross is to
invert each single cross separately [as in Figure 2.5], so that we end up
with two squares whose sides trisect the two adjacent sides of the opposite
square [see Figure 2.27b]. Each end-point of the original cross (see dashed
line) bisects one of the sides of the square created by its inversion. If we
label the eight vertices in Figure 2.27b according to the same pattern as
used in Figure 2.24, then the dark diagonal square in the former map will
correspond directly to the inner square in the latter map. The only
difference is that the first-level operation which distinguishes the two
squares is 45\ rotation in the former map and 50% reduction (or "scaling")
in the latter. These are both equally accurate geometrical representations of
the first-term distinction in third-level contradictory opposition; but the
latter is marginally preferable, because it is less complex (the sides of the
two squares do not intersect), and because it will turn out to be very similar
to an important map which will be examined shortly.
Up until now we have limited our attention almost exclusively to
two-dimensional geometrical figures. Such figures are easier to use as
logical maps than those of higher dimensions, because the latter do not lend
themselves readily to unambiguous representation in a two-dimensional
drawing. Nevertheless, if, as we suggested in 2.1, the levels of analytic
division can be compared to dimensions in space, then we should also
investigate the ways in which third-level forms of relation can be
represented by three-dimensional geometrical figures.
Perhaps the most obvious way of representing third-level analytic
relations with a three-dimensional figure is to expand the second-level cross
into a three-dimensional cross by adding a third axis, as in Figure 2.28a.@
To use this figure as a map for third-level analytic division requires using
each of the three lines to represent just one of the three levels (i.e. one of
the three distinctions between "+" and "-"). As such, the arrowheads on
each line point towards the "+", like all single-level relations. But since the
map has six axial end-points and twelve 90\ angles (or planar quadrants),
neither of these sets of parts can be used to symbolize the eight expressions
which arise at this level. What can be used is the eight spatial quadrants
which are defined by the intersection of the three lines. Unambiguously
labelling these spaces on the two-dimensional representation of the three-
dimensional cross would be difficult, not to mention the fact that the
arrowheads cannot be mapped clearly onto the spaces. This suggests that it
may be helpful to alter the diagram in some way to make the relations more
clear.
These spaces can be labelled rather less ambiguously by expanding
each axis of the cross into a square plane. The resulting "solid cross" [see
Figure 2.28b] has a number of interesting characteristics, which we will
consider in 7.2; but for our present purposes it is sufficient merely to cite it
as an adequate three-dimensional map of polar oppositions. The
expressions labelling the quadrants in Figure 2.28b draw their three terms
from the three linear axes which define the quadrant--the new third term in
each expression being derived from the "perpendicular" (i.e. front-to-back)
line [see Figure 2.28a]. Although the arrowheads still cannot be mapped
onto the spaces, they can now at least be mapped onto each second-level
square, whose vertices can be labelled with the pattern for second-level
contradictory opposition. This map is a perfect representation of third-level
polar opposition, since the expressions labelling each quadrant have exactly
the same number of terms in common with any other expression as the
number of lines which the two quadrants have in common. Thus, for
example, the
Figure 2.28: The Three-Dimensional Cross
Figure 2.29: The Third-Level Form of Relation,
Mapped onto the Cube
two-term equivalence of "+++" and "++-" is reflected by the fact that they
share the "+" segments of the first and second-level axes; the one-term
equivalence of "+++" and "-+-" is reflected by their meeting only on the
"+" segment of the second-level axis; and the contradictory opposition (i.e.
no-term equivalence) between "+++" and "---" is reflected by their sharing
no common lines whatsoever.
An alternative way of using square planes to construct a three-
dimensional map of the same form of relation is to position each square in
such a way that the other two squares bisect it perpendicularly on its
diagonals, thus forming a regular octahedron [see Figure 2.28c]. The
advantage of this version of the three-dimensional map of third-level polar
opposition is that the three-term expressions can now be mapped onto the
eight triangular faces, which are rather more clearly defined than the eight
spaces of the solid cross. The pattern according to which the three-term
expressions are mapped onto the faces of the octahedron is the same as that
used for the solid cross, so the axial line segments once again connect polar
opposites, just as they did in Figure 2.28b. But in addition, the edges of
each face now symbolize the one term which differs between it and the
expression labelling the adjacent face. Moreover, the three squares must
now be labelled according to their sides, since the six vertices now express
the same single-level relations as the end points of the three axes. A
possible disadvantage of using the octahedron as a map of the third-level
analytic form of relation is that the triangular shape of each face will be
used for a rather different purpose as a map for synthetic forms of relation
in Chapter Four.
The inversion of the three-dimensional cross produces an equally
accurate map which highlights contradictory opposition on the third level.
For if we cut along the lines in Figure 2.28a, or the planar squares in
Figures 2.28b-c, and rotate each of the eight sections by 180\ in two
different directions, then the innermost vertices of the cross will be
transformed into the outermost vertices of a cube. The rules for mapping
summarized in Table 2.3 require us to map the eight expressions in the
third-level analytic form of relation according to the same pattern as was
used to map the spaces or faces of the three-dimensional maps in Figure
2.28 [see Figure 2.29].@ Each line now connects a pair of expressions
with two equivalent terms; the opposite corners of each square face are
labelled with pairs of expressions having one term in common; and the
expressions at diagonal vertices of the cube are contradictory opposites.
The arrows follow exactly the same pattern as those of the double square in
Figure 2.24, with the vertical square on the right side of the cube
corresponding to the outer square of the double square and that on the left
side corresponding to the inner square. The horizontal lines which connect
these two squares in Figure 2.29 to form a cube all point from left to right,
because they represent the first-level relation. Since the cube so
appropriately explicates all the relations (i.e. both polar and contradictory)
in the third-level analytic form of relation, its relation to the second-level
square can be easily apprehended. For if we take away the first term in each
expression along with all the horizontal lines, we are left with two parallel,
vertical squares, which are "perpendicular" to this piece of paper, and are
labelled with the appropriate two-term expressions and arrowheads.
Likewise, omitting all the second terms and vertical lines leaves two parallel
horizontal squares, and omitting all the third terms and perpendicular lines
leaves two parallel squares which are parallel to this two-dimensional
surface--all of which are labelled with the pattern of expressions
appropriate to contradictory opposition (though the pattern of arrowheads
differs in each case).
The disadvantage of using the cube, or indeed any geometrical
figure with more than two dimensions, as a logical map is that drawing it
on a two-dimensional surface inevitably distorts the actual figure in various
ways in order to give it the appearance of a higher-dimensional figure. The
usual method is to represent any edges of a solid figure which are in the
"background" with dotted lines, so that they can be easily distinguished
from lines with which they would otherwise appear to intersect in the two-
dimensional version of the map. In a completely faithful representation of
the geometrical relations between the parts of a solid figure, however, "the
lines representing the edges intersect only at the points representing the
vertices, so that no extra crossings occur" [B520:80]. This difficulty can
sometimes be avoided by "projecting" the solid figure from a given point
(which I shall call the "point of perspective") onto a planar surface on the
other side of the figure. Simple figures such as the cube are in some
respects more faithfully represented in their projected form than in the usual
three-dimensional form; for, although the size and shape of the faces
becomes distorted (e.g. the face closest to the point of perspective now
corresponds to the unbounded surface on the outside of the planar figure),
the relations between the vertices, edges and faces are in perfect accord
with the actual relations in a real cube. What is even more significant for
our purposes is that the projection of the cube reveals the close affinity
between the cube and the standard double square as maps of the third-level
analytic form of relation.
The cube is projected by placing a point a short distance from the
center of one of its eight faces and drawing lines from this point through
each of the eight vertices to a plane parallel to the opposite face. The
resulting two-dimensional figure is, quite simply, a double square with
lines connecting the corresponding vertices of the two squares [see Figure
2.30a]. Which face is chosen as the one closest to the point of perspective
is of utmost significance, for the relation
(a) Of the Cube
(b) Of the Solid Cross
Figure 2.30:
Three-Dimensional Projection onto Two Dimensions
represented by the lines connecting this face with the opposite face will be
highlighted as the relation between the two squares in the projected figure.
Moreover, the pattern of arrowheads will differ significantly depending on
which perspective is chosen for the projection. In Figure 2.30a the
perspective is taken to be that of the first-level "+", so the point is placed
just to the right of the right, vertically-perpendicular face.@ The arrows on
the projected squares thus form a clockwise circuit, whereas, had the
perspective point been placed on the first-level "-" side of the cube, the
arrows would have formed a counterclockwise circuit. This is one good
reason for associating the clockwise (and the synthetic, as we shall see)
with a "+" perspective in relation to the counterclockwise (and analytic) as
based on a "-" perspective.
On the projected map in Figure 2.30a the type of relation which
holds between any two expressions corresponds to the number of lines
which must be traversed in order to get from one to the other by the
shortest possible path. The four possibilities are:
1. No lines traversed = identity.
2. One line traversed = polarity with two-term equivalence
3. Two lines traversed = polarity with one-term equivalence; and
4. Three lines traversed = contradiction.
(Note that these rules would not hold for the double square unless one of
the squares is reflected across its mixed diagonal so that both squares
develop in the same direction, as in the projected cube.) Applying the same
operation to the solid cross yields a large cross divided into eight sections
(four inner and four outer) by a square [see Figure 2.30b; cf. Figure
2.26a]. The eight expressions can be mapped much more clearly onto these
eight plane surfaces than onto the three-dimensional spaces to which they
correspond. The projections of the solid cross and the cube not only
confirm their inverse relation, but also make it easier to see the similarity in
the pattern of expressions used for each map. Indeed, these two three-
dimensional figures are the most highly refined maps of the third-level
analytic form of relation which have been developed in the Geometry of
Logic. For both maps are equally good representations for both polar and
contradictory opposition.
We are now well prepared to discuss some applications for analytic
division on this third level. The further we travel along the road of analytic
division the more complex becomes the formal structure with which we are
dealing, and so also the more difficult it is to find perfect examples of how
it is applied to real situations. Nevertheless, the level is still low enough to
provide numerous applications. We could use this form of relation as a way
of explaining why metaphysicians sometimes make seemingly arbitrary
eight-fold divisions, such as we find in Plato's theory of the eight "circles
in the World-Soul" [P494:36B-D], and perhaps even to explicate the logical
relations between the elements of such theories. A perfect example of such
eight-fold metaphysical division is found in the "primal arrangement" of the
famous "I Ching trigrams", which serve as symbols for the eight "houses"
of ancient Chinese astrology. Less speculative applications could be drawn
from logic, as for instance, when the basic propositional forms are
expanded from four to eight by adding the new condition that any given
combination may (+) or may not (-) exist [see e.g. S515:77]. Indeed, most
of the applications given in 2.2 could be easily extended to the third level
merely by introducing some new +/- parameter. So it will suffice to limit
our applications in this section to two examples, one taken from number
theory and one from geometry.
The first example has to do with the logical basis of our base-10
number system. We have already discussed the significance of the
numerals "0" and "1" as representations of the logical concepts of
"nothing" and "everything" [see 1.2]. These two symbols (or their
equivalents) are generally regarded as marking out the lower and upper
limits of our (or any) number system; for on their own they represent the
absence of number and the beginning of number, respectively, but when
together ("10") they represent the point at which the number system begins
to repeat itself. Most number systems which develop naturally employ eight
whole numbers between "1" and "10" (i.e. between the points at which
number begins and at which it begins to repeat itself)--in the West these are
symbolized by the Arabic numerals "2", "3", "4", "5", "6", "7", "8" and
"9". Obviously, there are many reasons why the resulting base-10 number
system has developed into the standard one in most cultures (such as the
fact that man has eight fingers and two thumbs on which to count). But
without a doubt, part of the explanation for the depth with which this
system is ingrained into our thought patterns is that it corresponds perfectly
to the logical structure of third-level analytic division.
Within the defining context of a center and its boundaries, i.e. of
"0" and "1", the eight primary numerals can be mapped onto any of the
standard third-level maps [see e.g. Figure 2.31].@ In these maps the first-
level relation is between "less than 6" (-) and "greater than 5" (+)--i.e.
between the four numerals which "ascend from two to five" (-) and the four
which "descend from nine to six" (+). The second-level relation is between
the first two (-) and the last two (+) numbers in each set of four--i.e.
between the extremes (2, 3, 8 and 9) and the means (4, 5, 6 and 7) of the
overall set of eight primary numbers. Likewise, the third-level relation is
between the means and the extremes of each set of four numbers--the
"means" being correlated with "+" when the first-level term is "-" and with
"-" when the first-level term is "+". One result of ordering the eight primary
numerals in this
Figure 2.31: The Eight Primary Numerals
(a) Mapped in Expanded Form
(b) Mapped onto the Double Cross
(c) Mapped onto the Projected Cube
way is that each pair of numbers which correspond to contradictory logical
expressions adds up to the number "11" (or "12" if the "1" in the middle is
taken into account). The logical implications of these numerical relations
will be discussed in Chapter Five.
The second example is based on a more straightforward set of
logical relations and so can serve as the archetype for any attempt to analyze
a concept into eight categories according to the third-level analytic form of
relation. The traditional geometrical division of the class of quadrilaterals,
which for the most part dates back to Euclid [E487:1. (154=?)], can be
readily depicted in terms of the following schematic:
Figure 2.30: The Schematic Third-Level
Categorization of Quadrilaterals
QUADRILATERALS ##
non-parallelograms (-)parallelograms (+) { no sides paralleltwo sides
parallelnon-rectangularrectangular (trapezoid)(--)(trapezium)(-+)(+-)(++)
scaleneisocelesscaleneisosceles rhomboid rhombus oblong square trapezoid
trapezoidtrapezium trapezium(+--)(+-+)(++-)(+++) (---) (--+)(-+-) (-++)
These three levels of analytic division are obtained, respectively, by giving
a "yes" (+) or "no" (-) answer to the following questions:
(1) Does the quadrilateral have two sets of parallel sides?
(2) Is the quadrilateral more regular than the others in its first level
group?(That is, on the first-level "+" side: Are the angles right angles? and
on the "-" side: Are two sides parallel?)
(3) Are there more sides of equal length in the figure than in its
complement?71
Using these questions to determine the logical expression which fits
each type of quadrilateral enables us to describe, for example, a rhomboid
as "+--", and in so doing to know immediately how it is related to each of
the other seven types of quadrilateral. For the logical symbol tells us that it
is a parallelogram (+) which is non-rectangular (-) and which does not have
four sides of equal length (-).When the higher-level forms of relation are
applied to concrete content so perfectly, they lend a degree of completeness
and unity of structure which is indeed elegant. In such instances one's
formal labours begin to show fruition in increasingly intricate symmetrical
patterns:logic gives birth to beauty.
2.4 Polar and Contradictory Opposition
in Fourth-Level Analytic Division
The patterns and rules which guide all transformation, both within
and between the various levels of analytic division, have been established
in the process of working through the transformations on and between the
first three levels of analytic division. Now that the procedure has been
defined and demonstrated, the same terminology and notation can be used
to investigate all subsequent inter- and intra-level transformation. The main
difficulty in dealing with higher-level forms of relation is designing
adequate higher-dimensional figures to represent the more complicated
logical structures. As a guide to how this can be done, I will conclude this
chapter with a comparatively brief discussion of fourth-level analytic
division, leaving the interested reader to pursue its full implications, and
those of the higher levels, independently.
Once again there are several helpful ways of mapping the sixteen
expressions which, as we saw in 2.1, result when the operation of analytic
division is carried to the fourth level. The fully "expanded" form, given in
Figure 2.33, is now even more awkward than its third-level counterpart
[cf. Figure 2.19a], though it still has the great advantage of clearly
displaying the cumulative effect of analytic division.@ For within this
complex map we can single out fifteen single-level relations (i.e. the fifteen
lines), seven two-level relations (i.e. sets of three lines [cf. Figure 2.2a]),
and three three-level relations (i.e. sets of seven lines [cf. Figure 2.19a]),
all of which can be labelled with the appropriate expressions simply by
omitting the first three, the first two, or the first, term in each expression,
respectively. From this the formula for determining how many distinct n-
level relations (r) are contained within any higher-level form of relation can
be seen to be: "r = 2-1".
The awkwardness of the expanded map can be reduced partially by
"doubling" the double cross. This can be done in several ways:by
expanding the double cross, thus emphasizing the eight new single-level,
polar oppositions, as in Figure 2.34a, or by constructing a "double double
cross", consisting of two sets of four parallel lines, which intersect each
other perpendicularly, and whose sixteen end-points are labelled with the
appropriate fourth-level expressions, as in Figure 2.34b; or again, by
constructing a "fully integrated fourth-level cross", in which the eight axes
all bisect each other at a common center, as in Figure 2.34c.@ The
derivation of the first figure should cause no problem, since it proceeds
along exactly the same lines as the derivation of the expanded second-level
cross in 2.3 [see Figure 2.19a]. Likewise, the second figure is derived
from the expanded double cross by omitting the double cross itself and
rotating each remaining fourth-level line on its starting-point towards the
center by 90\ and extending each arrow to the opposite side of the map [cf.
Figure 2.20]. (The same map can be derived more straightforwardly simply
by splitting each line of the double cross into two parallel lines pointing in
Figure 2.31: The Expanded Map of Fourth-Level
Analytic Division, Assuming Polar Oppositions
Figure 2.32: Doubling the Double Cross
(a) The Expanded Double Cross (b) The Fourth-Level Cross
(c) The Fully Integrated Fourth-Level Cross
opposite directions, and appending a fourth "+" or "-" on each three-term
expression in order to differentiate between them.) And the third figure
clearly follows the same pattern as the fully integrated double cross in
Figure 2.23, with the new first-level relation being between dashed lines (-)
and solid lines (+). Each of these maps is helpful, because each depicts the
same form of relation in a slightly different way.
On the fourth level, a way of mapping the same polar oppositions
arises, which is even simpler and more intuitively obvious than any of the
above-mentioned methods. If Figure 2.33 is the geometrical expression of
the equation "2 = 16" and Figure 2.34a, that of "8.2 = 16", then we should
expect there to be an appropriate geometrical representation of the equation
"4.4 = 16". Indeed there is. For each of the four poles of the second-level
cross can itself be divided according to the same fourfold pattern--an
operation which can be called "crossing the cross".
Figure 2.35: The Crossing of the Cross
Each of the four secondary crosses in Figure 2.35 is constructed out of a
pair of adjacent parallel lines taken from the fourth-level cross in Figure
2.34b. Thus the first two terms in the four expressions on each secondary
cross are identical, while the third and fourth terms conform to the same
pattern as the standard second-level cross on its own. This map is superior
to those in Figures 2.33-4 in the sense that it facilitates a quick comparison
between various sub-systems within a system. For it clearly depicts the
way in which sub-systems can be self-contained wholes (based on second-
level analytic divisions), yet can be closely related to other sub-systems
within a common System (i.e. a second-level analytic relation between sub-
systems).
Incidentally, the "grid" pattern made up of nine squares in the center
of Figure 2.34b, which is undefined when used as part of the double
double cross, could also be used to make helpful comparisons between
different levels. For example, the fourth-level expressions could be mapped
onto the sixteen points of intersection between the axes. The sixteen-fold
grid could then be categorized into four interlocking second-level squares
(4.4 = 16) or into a combined double square and double cross, with the
center of the latter overlapping the center square of the former (8+8 = 16).
In this way the second- and third-level relations within a fourth-level
system could be explicitly mapped. But this map is too similar to some of
the inverted fourth-level maps (which will be introduced shortly) to merit
any further consideration here [but see 4.2].
The operation of inverting two-dimensional fourth-level maps of
polar opposition is straightforward and requires only brief explanation.
Expanded forms such as in Figures 2.33,34a can be ignored, since this
type of map is never sufficiently integrated to be meaningfully inverted.
The best way of inverting Figure 2.34c is simply to invert each of the four
crosses separately, as in Figure 2.27b, so that the
(a) Quadruple Second-Level Inversion(b) Replacing Rotation with, Scaling
Figure 2.37: The Inversion of the Fully Integrated
Fourth-Level Cross
(a) Using Simple Second-Level(b) Using Diagonal Inversion
Inversion(Enlarged)
(c) Combining Simple Second- and(d) Radical Fourth-Level Simple Third-
Level InversionInversion
Figure 2.38: Fourth-Level Inversion of the Double
Double Cross
resulting map consists of four equal squares, each at a 22.5\ (= 90\/4) angle
from the two adjacent squares [see Figure 2.37a]. By replacing the
operation of rotation with the operation of 50% scaling, Figure 2.37a can
be transformed into Figure 2.37b.@ In the latter map the systematic
relations are easier to apprehend, since the squares do not intersect each
other.
The inversion of the double cross in Figure 2.34b poses a rather
more difficult problem. Simple second-level inversion produces a similar
figure, but with a bigger square in the middle [see Figure 2.38a; cf. Figure
2.24a], while diagonal inversion creates a squared cross with sixteen points
[see Figure 2.38b; cf. Figure 2.24b]. The dashed lines in both of these
maps represent the position of the cuts, or of the edges of the paper. Note
that the four sections of paper in diagonal inversion must overlap each
other, so the dark (-) squares are assumed to be underneath the light (+)
ones. (This overlapping explains why the inversion turns out to be smaller
in size than the original.) In both of these types of inversion, the sixteen
fourth-level expressions end up being grouped into pairs of fourth-term
polarities, each pair labelling a common point from two different directions.
Moreover, the third and fourth terms in the four expressions on each set of
parallel lines always form a perfect second-level analytic division, while the
first and second terms are identical for every expression in each set.
Nevertheless, each of these figures is too irregular to be used as the
standard map for fourth-level inversion.
A more regular figure can be constructed by dividing Figure 2.34b
into four equal quadrants and rotating each, as in second-level inversion,
then applying simple third-level inversion [as in Figure 2.26a] to the double
cross in each these sections. This "fourth-level inversion" yields the grid
shown in Figure 2.38c, which has sixteen spaces as well as sixteen end-
points around the perimeter, either of which can be labelled with the sixteen
fourth-level expressions. By choosing to label the spaces, we are able to
divide the overall map into four main squares, each of which duplicates the
pattern of the whole. This is reflected, as in Figure 2.35, by arranging the
expressions in such a way that the first two remain the same within the
larger quadrant, while the last two follow the standard pattern for second-
level contradictory opposition. Applying a further second-level inversion to
each main quadrant, and scaling down the overlapping squares, transforms
the fourth-level cross into the map given in Figure 2.29a, which clearly
depicts the presence of the four second-level sub-systems within the fourth-
level analytic form of relation. This map expresses exactly the same
relations as the map in Figure 2.37b, except that 50% scaling does for the
latter what is done for the former by rotating the small square through the
four quadrants.
By now the similarities between fourth-level inversion and its
second- and third-level counterparts should be obvious. Nevertheless, one
more inversion should be mentioned, that of the systematically refined
"crossed cross" in Figure 2.35. Each cross in this map can be simply
transformed into a square through second-level inversion, with the
expressions and arrowheads rearranged according to the pattern specified in
2.2 for contradictory opposition. Just as the four crosses in Figure 2.35 are
placed at the poles of a second-level cross, so also the four squares which
result from its fourth-level inversion should be placed at the vertices of a
second-level square [see Figure 2.36]. This map is slightly better than that
given in Figure 2.38d, since it replaces the cross in the latter with a second-
level square, as is more appropriate for symbolizing contradictory
opposition in the Geometry of Logic.
Representing fourth-level analytic division with dimensionally maps
presents some difficulties not hitherto encountered, for Euclidean
Figure 2.36: Fourth-Level Inversion of the Crossed
Cross
geometry can cope with only three dimensions. Higher-dimensional figures
are usually represented on paper by adding some suitably defined distortion
to a figure which would ordinarily be regarded as two- or three-
dimensional. Thus, for example, a four-dimensional cube could be
constructed by defining the eight vertices of a cube's front and back faces
in terms of the expressions appropriate for two sets of second-level analytic
squares [see Figure 2.39a]. Arrowheads are plotted onto these squares in a
clockwise (+) direction for the front (+) square and a counterclockwise (-)
direction for the back (-) square, as well as on all depth lines, pointing from
front to back. The diagonals of this cube can be defined as its fourth
dimension. Each of the twelve lines which mark the sides of the cube can
then be taken, together with each of the four diagonals (each of which
points from front to back), as a suitable map on which to plot the sixteen
fourth-level expressions [see Figure 2.39aii].@ The label for each line can
be determined by following the arrows around Figure 2.39ai, and adding
the expression labelling the end-point of each line to that labelling its
starting-point. Two other ways of distorting a cube in order to construct a
"four-dimensional" map for the fourth-level analytic form of relation are
given in Figure 2.39b [adapted from C521:160]. The projection of these
figures onto a two-dimensional plane would, interestingly enough, give
(a) Using the Diagonal and Sides of the Cube
(i) Its Two Second-Level Squares(ii) Its Sixteen Fourth-Level
Lines
(b) Using the Vertices of the Double Cube
Figure 2.39: The 4LAR, Mapped onto the Four-
Dimensional Cube
(i) As a Cube in a Cube (ii) As Two Cubes Slightly Offset # rise to a map
virtually the same as Figure 2.37b, with the adjacent vertices of the separate
squares connected with lines, as in Figure 2.30a. For precisely this reason,
it is usually unnecessary to appeal to such sophisticated maps as those
given in Figure 2.39 when applying higher-level forms of relation to real
relations.
The range of application for our formal maps becomes more and
more narrow as we progress to higher and higher levels of analytic
division. When an appropriate application is made, however, it enables us
to view a group of relations as a well-organized, systematic whole, whose
parts are all distinct, yet interconnected. As such this level would be an
appropriate one from which to discuss the dimensional structure of the
universe:the thoroughgoing interconnection between space and time could
be represented by mapping three-dimensional space and one-dimensional
time, respectively, onto the perimeter lines and the diagonals in Figure
2.39aii. Another, less speculative application would be to use this system
of sixteen logical expressions as a way of structuring a piece of systematic
writing in any field.22 In hopes of providing a good example of how this
can be done, and of how it can help to clarify a writer's train of thought, I
have structured Part One of the present book according to this pattern, by
organizing the material into four chapters, each with four sections. (The
form of relation on which the book as a whole is based will be discussed in
Chapter Seven.) The extent to which I have succeeded will be the extent to
which my analysis of logic fits perfectly into the form provided by the logic
of analysis. In such cases the imposed character of analytic division is more
obvious, because it is done intentionally.
The logical structure of the modern computer is one of the best
examples of how analytic forms of relation operate on various levels, but
particularly on the fourth. In 1.4 we mentioned the basic "on/off" switch as
an example of a first-level analytic relation. The equivalent terms for the
two positions of a computer's switch are "set" and "unset". Computer
"hardware" consists of an elaborate network of such simple switches,
which the "software" can use to define virtually any logical operation, by
expressing it in terms of the two simple operations "and" and "not". (For
instance, "or" can be defined as "not both 'not x' and 'not y'".) Computer
"firmware" is the logical structure which defines how the whole system
works--i.e. how the software goes about using the hardware to produce the
information we see on the screen. The simple "set/unset" (+/-) switch
forms the basic unit, called the "binary digit", or "bit". Eight bits, taken
together, form a "byte", which, as an eighth-level analytic structure, gives
rise to 256 (2) different bytes (combinations of eight bits). (In ASCII, 32
bytes are used to control the hardware, and 96 others to define the character
set, which leaves 128 unused bytes. This is equivalent to using only seven
of the eight available bits, since "96+32 = 128" and "2 = 128".) A "nibble"
is four bits, i.e. half a byte, and has sixteen variations (2). Using the nibble
as the basic unit in place of the bit enables the (fourth-level) hexadecimal
number system (1-10 and A-F) to be substituted for the more cumbersome
binary (0 and 1) number system. A byte can then be expressed by merely
two digits, and the standard "word" (16 bits) by four hexadecimal digits.
All these fundamental aspects of the structure of the computer
obviously follow the same patterns of analytic division which we have been
discussing in this chapter. Even if this were the only useful application of
the various levels of analytic division, the rapid growth in the importance of
the computer in today's society is enough to render the labours of this
chapter worthwhile as a framework in which the logic of the computer can
be readily grasped.
An investigation into higher levels of analytic division at this point
is unnecessary [but see 3.4], not only because they become more and more
difficult to apply, but also because they all follow the
Figure 2.40: A "Fractal" Map of the Tenth-Level
Analytic Form of Relation
patterns established in this chapter for lower levels. Sufficient groundwork
has been done to enable anyone who wishes to pursue this avenue further
to do so independently. For example, it should now be enough merely to
look at a map of tenth-level division to appreciate the beauty of its
symmetry:Figure 2.40 is a "fractal curve" (discussed further in 4.3)
reproduced from M536:65, which could be carried to indefinitely high
levels, if the resolution of printing and our range of vision were more
precise. But even in the form given here, its 1024 parts are related
according to a clearly organized pattern, which could be completely defined
in terms of our "+" and "-" notation.
Our attention must now turn towards synthetic forms of relation and
their relationship with the analytic forms of relation discussed in this
chapter. For it is with synthetic and compound forms of relation that our
discussion broadens out beyond the narrow, analytical boundaries of
logical thought and touches more closely on the synthetical reality of actual
experience.
NOTES TO CHAPTER TWO
1. Russell is an exception, for he mentions this rule in his discussion of
classes: "if a class has "n" members, ...there are 2n ways of selecting some
of its members (including the extreme cases where we select all or none)"
[R517:84-5]. If we assign a "+" value to each member of a class when it is
selected, and a "-" value when it is not selected, then we can easily
demonstrate the one-to-one correspondence between a particular set of class
relations and the general analytic form of relation upon which it is based.
For example, in the class "abc" the following eight sub-classes are
possible, each of which corresponds to one of the third-level expressions
given in Table 2.1d:
abc = +++ab = ++-ac = +-+a = +-- 2bc = -++b = -+-c = --+0 = --- 1
Class membership is therefore one of the many applications which are
made of the analytic forms of relation in logic.
2. Note, for example, that on every level the powers in each group of
variables add up to the same number as that of the level itself, and that all
the numerical prefixes added together come to the same total as the number
of expressions in the form of relation to which it corresponds. The latter
result can also be obtained by assigning both "A" and "B" with a numerical
value of "1". Thus on the sixth level, where "(A + B) = A + 6AB + 15AB
+ 20AB + 15AB + 6AB + B", each of the seven distinct groups of
variables has powers which add up to "6", and the total of all the numerical
prefixes is "64". The intricacies of this form of relation will be dealt with in
7.2.
3. See e.g. S515:84,104-6,170,187. In L483:162-3 Langer lists several
groups of "elementary general propositions" arising out of the
quantification of elements in a class, each of which corresponds directly to
the expressions produced by one of the first four levels of analytic division.
Similarly, in A489:1.4-7 second, third- and fourth-level forms of relation
are used to construct a "Karnaugh map", used as a tool for the
simplification of the logic expressions employed by computer design
engineers. Or again, by substituting "+" with "1" and "-" with "0", the
analytic forms of relation, as specified in Table 2.1, can be used to simplify
various calculations in a binary (base-2) number system [cf. P489:2.6-7
and A489:1.9-11]. Numerous other examples will be given in more detail
throughout this chapter.
4. Similar tables have also been constructed to "chart" the probability
(rather than the truth-value) of propositions [see e.g. H485:240].
Unfortunately, logicians typically discuss such tables without giving any
explanation of their formal-logical status or the mathematical basis of their
tautological character. The most Hodges does is to state without
explanation that tables with one, two, three and four variables will contain,
respectively, two, four, eight and sixteen "structures" (i.e. expressions)
[H485:125; s.a. 88-9].
5. See H485:132-3 for a list of formal equations which satisfy this
requirement.
6. Euclid gives the following definitions in Book I of E487 [cf. note 1.3
above]:
5. A surface is that which has length and breadth only.
6. The extremities of a surface are lines.
In his discussion of these definitions Heath refers to a number of
other definitions which clarify the concept in various ways. For Aristotle,
"a surface is a magnitude extended or continuous two ways..., or divisible
in two ways" [H488:1.170]. "Proclus remarks that...we further get a sort
of perception of [a surface] by looking at shadows, since these have no
depth (for they do not penetrate the earth) but only have 16 length and
breadth" [170]. Heath also quotes al-Nairizi,who defines "the plane surface
as that in which it is possible to draw a straight line from any point to any
other" [172]. "Leibniz...defined a plane as that surface which divides space
into two congruent parts", to which Heath adds that the complete definition
must specify "that the two congruent spaces could be slid along each other
without the surfaces ceasing to coincide" [176]. What Heath neglects to
mention is that the spaces must be able to be slid along their surfaces in any
direction.
7. This implies the following corollary to the rules for mapping specified in
2.1: "vertical" is to "horizontal" as "+" is to "-".
8. Ending up with a square has nothing to do with the fact that the shape of
the original piece of paper is square. The latter is specified only for
convenience, so that the surface on which the inverted figure is drawn will
not have holes in it, or parts that overlap. Had the cuts been made
diagonally, four triangular bits of paper would have been formed, which
would partially overlap when rotated--an operation which would end up
transforming the lines of the original cross into another cross, with the two
arrowheads pointing in at the center.
9. Incidentally, another definition of a "point" [cf. note 1.3 above] is "the
intersection of two lines" [H488:1.165]. This might be called a "second-
level analytic" definition of a point. On this basis second-level analytic
division can be regarded as including the formal relations of quaternity
(four expressions), trinity (three types of twofold division [see Figure
2.2]), duality (two levels represented by each expression), and unity (the
original, undifferentiated whole). These are symbolized geometrically by
the four end points (or quadrants), the three lines in Figure 2.2, the two
lines in Figure 2.4a, and the one surface (or the one point at the center).
10. By "successive" I mean "following upon one another when proceeding
counterclockwise around the figure". As we shall see shortly [see Figure
2.8], it would make just as much logical sense to proceed clockwise,
though it would yield a rather different result [cf. Figure 6.8].
11. As a result of their thoroughgoing identity, the map can be reduced to a
single line segment, dividing all the "+-" expressions from all the "-+"
expressions, by folding the surface along the diagonals bisecting first the
"+-" and then the "-+" quadrants, so that the four "arms" of the cross
overlap.
12. This phrase is analogous to the "range of significance" of a variable
within a proposition, which denotes "The entire class of possible values for
a variable, i.e. of individual elements it may signify" [L483:87].
13. My use of the term "perfect" is similar to Boole's use of the term
"pure": "Solutions in which there cannot, by logical divisions, be produced
any superfluous or redundant terms, may be termed pure solutions"
[B506:7.14]. However, the term "pure" has already been defined for the
Geometry of Logic as a reference to an expression whose terms are all
identical.
14. The prevalence of such imperfect analytic divisions in the empirical
world could be used to demonstrate the limited applicability of Kant's
"Copernican" perspective. If (as is traditionally, but wrongly, assumed [see
P1:5.2,6.1]) Kant had believed that we impose order (etc.) onto the object
not only in the transcendental sense, but also in the empirical sense, then he
would seem bound to force all empirical distinctions into a "perfect" mould,
which is obviously absurd.
15. See C495:343. Numerous other fourfold divisions were also made by
the ancients. Cornford quotes [in C495:69-70] from a list given by Theon,
who "enumerates ten tetractyes (sets of four things)", each of which is
based on a second-level analytic division of some sort:
Numbers: 1, 2, 3, 4. Magnitudes: point, line, surface..., solid... Simple
Bodies: fire, air, water, earth. Figures of Simple Bodies: pyramid,
octahedron, icosahedron, cube. Living Things: seed, growth in length, in
breadth, in thickness. Societies: man, village, city, nation. Faculties:
reason, knowledge, opinion, sensation. Parts of the Living Creature: body,
and the 3 parts of the soul. Seasons of the Year: spring, summer, autumn,
winter. Ages: infancy, youth, manhood, old age.
16. Instead of using a square, Stebbing [S515:59] represents "the
traditional oppositions by an unsymmetrical figure, since the symmetry of a
square is ill-adapted to represent unsymmetrical relations." Unfortunately,
she never bothers to explain the sense in which it can be said that the
relations symbolized by the square of opposition are "unsymmetrical". To
be sure, several different types of relation are mapped onto the square; but
the pattern according to which these relations are related is entirely
symmetrical.
17. The fact that any set containing three variables, each with two possible
values, can always be arranged in eight different ways is often mentioned;
but almost never is its formal-logical basis adequately explained. In
H488:1.245-6, for example, Heath simply states that three ambiguities in
the instructions for constructing a geometrical figure will give rise to eight
different ways of constructing it [cf. Figures 2.8a-h].
18. A pair of "mixed" quadrants are those through which the mixed
diagonal of the square passes--i.e. the top left and bottom right quadrants
of the standard second-level analytic square. Likewise, the other two
quadrants--the top right and bottom left--are called "pure" quadrants,
because the diagonal which bisects them connects the two pure
expressions.
19. These trigrams, like the "yin yang" distinction on which they are based
[see Figure 2.13a], are traditionally arranged according to contradictory,
rather than polar, opposites [see D491:129]. Unfortunately, those who
make use of this formal pattern tend to ignore completely (or to be
completely ignorant of) its thoroughgoing logical basis. I will elaborate on
the logical basis of the I Ching trigrams and hexagrams in 7.2.
20. For this reason Euclid and others regarded "2" as the first "number",
"1" being the essential building-block of all numbers [E487:7.1-2d;
H488:2.284].
21. This schematic is adapted from H488:1.189, where all the terms
labelled here with a "+" value are placed on the left and those with a "-"
value on the right. As well as reversing the order, I have added the logical
notation, the sample figures, and the third-level division of the trapezoid.
(An "isosceles trapezoid" has two sides of equal length, whereas a "scalene
trapezoid" has no sides of equal length.) An awareness of the logical basis
of such schematic categorization can be very useful in insuring that one
presents a complete categorization, or in filling the gaps left by others, as I
have done here.
22. Following a pattern in this way is what Kant had in mind when he
referred to reason's "architectonic unity" [see e.g. K105:861; cf. 502-3].
He is often criticized sharply for following his architectonic plan [see e.g.
W302:204-8], yet the ambiguity which results from his use of this logical
structure is due not to his inordinate adherence to it, but rather to his failure
to clarify explicitly just what the structure is, and to his inconsistency in
following it. As I have argued in P1:1.2, following such a plan is simply
following an intentional pattern in one's thinking, so that others can
understand it more readily. If the plan has its basis in formal logic, its
legitimacy is unassailable.