PALMQUIST.tree.13.13. Mapping Analytic Relations

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13. Mapping Analytic Relations

by Stephen Palmquist ([email protected])

Inthefirstlectureonlogicwelearnedthatlogic-analyticlogic,thatis-abstracts

f rom the concrete truth of a proposition, and focuses attention first and

foremost on its bare (essentially mathematical) form, its truth value. This

week I want to explore some ways of converting this bare form into a richer,

ictorial form.

Philosophers since Aristotle, and even before that, have almost universally

recognized that logic and mathematics are closely related disciplines. Until

the middle of the nineteenth century, most philosophers would have said

this relationship is confined primarily to arithmetic, where f unctions such

as

add

ition,su

btr

action, mu

ltiplication and d

ivis

ion have clear

analogies

tological operators such as "and", "not", etc. But then a scholar named George

Boole (1815-1864)wrote a book defendingwhat he called the " Algebra of

Logic". He demonstrated that algebraic relations are also closely related in

manyways to logical relations.

Although Boole's ideas are far too complex to examine in an

intro?/span>ductory course, I have mentioned his discovery because I

believe a similar discovery awaits us in the area of geometry. For this reason

I have already been using, throughout these lectures, several simple dia-grams in a way that conforms towhat I call the "Geometry of Logic". This

week I shall explain in detail just how these and other diagrams actually

f unction as precise "maps" of logical relations.The first two lectures will

examine ways of constructing maps that correspond to analytic and

synthetic relations, respectively. Lecture 15 will then provide numerous

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examples of howwe can use such maps to encourage and deepen our

insight.

A thoroughgoing analogy can be constructed between the structure of many

simple geometrical figures and the most f undamental kinds of logical

distinctions, though this has rarely, if ever, been f ully acknowledged in thepast. The starting point of this analogy is the analytic law of identity (A=A);

it posits that a thing "is what it is". To choose a diagram that can accurately

represent this simplest of all logical laws, all we need

todoisthinkofthesimplestofall geometrical figures: a point. Techni-


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cally, a point exists merely as a single position, with noreal extension in any direction, though of course, theblack spot representing a point in Figure V.1 must havesome extension in order for us to see its position.

Figure V.1: The Point as a Map of an Identical Relation

The f unction of the law of noncontradiction is to contrast the solitary "A" of

the law of identity with its opposite, "-A". The geometrical figure thatextends a point beyond itself in a single direction is called a line. There are,

of course, two kinds of line: straight and curved. So also, there are two good

ways of depicting the logical opposition between "A" and "-A" in the form of

a geometrical figure: by using the two ends of a line segment, or byusing the

inside and outside of a circle, as shown in Figure V.2:

(a) The Circle(b) The Line

Figure V.2: Two Ways of Mapping a 1LAR

Note that I have labeled these figures with a mere "+" and "-". These symbols

are derived directly f rom the law of noncontradiction, simply by dropping

the "A" f rom both sides of the "+A≠-A" equation. The "A" is a formal

representation for "some content", so dropping this symbol implies, quite

rightly, that in the Geometry of Logic we are concernedwithnothing b

u

t the

bare logical form of the sets of concepts we use.

Sincethissimpledistinctionarisesoutofthelawsof analytic logic, I refer to it

as a "first-level analytic relation" (or "1LAR"). As we shall see, representing

this lawwith the simpler equation, "+ ≠ -" (i.e., positivity is not negativity),

makes it much easier towork with more complex, higher levels of logical


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opposition.

The circle and line segment can be used as maps of virtually any distinction

between two opposite terms. Such distinctions, as we learned f rom Chuang

Tzu last week, are a commonplace in our ordinaryways of thinking about

the world. We naturally divide things into pairs of opposites: male andfemale,day and night, hot and cold, etc. In most cases I believe the line

segment offers the most appropriate way of representing such distinctions.

Since the circle marks out a boundary between "outside" and "inside",we

should employ this figure onlywhen there is an imbalance between the two

terms in question-as, for example,when one acts as a limitation on the

other, but not vice versa.

Now ifwe were to stop here, the Geometry of Logic would not be a very

interesting subject.No one has any trouble seeing the logical relationshipbetween a pair of opposite terms, to say nothing of a single term in its

relation to itself. Using points, line segments, or circles in such a way is

helpf ul onlywhen the terms in question do not define an obvious

opposition. This is especially true of the circle. For example, using a circle

to represent Kant's distinction between our necessary ignorance and our

possible knowledge, as we did in Lecture 7 (see Figures III.5 and III.10),

helped us fix in our minds the proper relationship between these two,with

the former limiting the extent of the latter.

In any case, one of the most interesting and usef ul tools in the Geometry of

Logic arises out of the simple application of the law of noncontradiction to

itself. By this I am referring to cases involving each side of a pair of opposed

concepts being itself broken down into a f urther pair of two opposing

concepts. As an example, let's consider the familiar concept "one day". We

all know how to perform the simple analytic process wherebywe divide

"one day" into two more or less equal and opposite halves, called "daytime"

and "nighttime" (i.e., "not daytime"). This is a good example of a typical

1LAR.However, as with most 1LARs, ifwe try to apply this strict division to

every moment in a day,we find there are certain times during the daywhen

we hesitate to saywhether it is "daytime" or "nighttime"; and as a result,we

make a f urther analytic division, between "dusk" and "dawn".

In order to translate this into the form of our logical apparatus, using "+"


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and "-" combinations to replace the actual content of our distinctions, all

we need to do is add another "+" and "-" term, in turn, to each of the original

terms f rom the simple 1LAR. This gives rise to the following four

"components" (i.e., combinations of one or more +/- terms) of a

"second-level analytic relation" (or "2LAR"):

--+--+++

I call the first and last components (i.e., "--" and "++") pure, because both

terms are the same,whereas I call the middle two components (i.e., "+-" and

"-+") mixed, because they both combine one "+" and one "-".

If one pair of opposites is represented by a single line segment, then two can

best be represented by a combination of two line segments. As we have seen

on numerous occasions already, the four end points of a cross can serve as asimple and balancedway of representing such a four-fold relation. But the

same 2LAR can also be represented by the four corners of a square (cf.

Figure II.3). I map the four components onto the cross and the square in the

following ways:

(a) The Cross(b) The Square

Figure V.3: Two Ways of Mapping a 2LAR

The position of the four components and the direction of the arrows on

each of these maps is, in a sense, arbitrary. In other words, the same


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components could be arranged in a number of different ways and still

represent a 2LAR just as accurately.However, after experimenting with all

the different ways of constructing such maps, I have come to the conclusion

that these two examples represent the most common and appropriate

patterns. Moreover, the above maps both follow a fixed set of rules that can

helpus avoid conf usion and inconsistency in constructing our maps-thoughthey may not be any better than some alternative set of rules. The rules I

have chosen are, quite simply: (1) a "+" component is placed above and /or to

the left of a "-" componentwhenever possible, giving priority to the term(s)

that come first in each component; (2) an arrow between two components

with the same term in the first position points away f rom the pure

component; (3) an arrow between two components with different terms in

the first position points toward the pure component; and (4) an arrow

between two components that each contain only one term (i.e., the simple

opposites "+" and "-") should be double-headed, to depict the tension or

balance between them.

The components are mapped onto the cross in Figure V.3a according to

their complementary opposites. That means the two components located at

opposite ends of each line segment will share one common term. For

example, the first term in both components might be a "+", while the second

termwill be a "+" on one side and a "-" on the other. By contrast, the

components mapped onto the square in Figure V.3b are organized according

to contradictory opposites. That means the component at any given corner

of the square does not overlap at allwith the component at the opposite

corner. For example, if the component at one corner has a "+" in the first

position, the component at the opposite corner must have a "-" in that

position; and likewise for the second position.

The square is, in fact, the one geometrical figure that can be found fairly

consistently in most logic textbooks. For it is the formal basis ofwhat is

commonlyr

eferr

ed

to as

"thes

qu

ar

e of oppos

ition". This s

qu

ar

e has

pr

oved

to be very helpf ul in clarifying for logicians the formal relations between

propositions that are opposed to each other in different ways (namely, as

"contradictions" or as "contraries").However, I do not wish to dwell on that

well-known application here. Instead, since I've already used the cross as a

map on numerous occasions in these lectures, let's look more closely at how


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it can represent the relationships between complementary opposites.

The cross enables us to visualize four distinct types of "first-level" logical

relationships (i.e., simple +/- oppositions) between such sets of four

opposing concepts. The first two can be called "primary" types. The first is

represented by the first term in each component; as we can see in FigureV.3a, it is the same on both ends of each axis of the cross. So the first term in

each component actually labels the axis itself: the vertical axis can

therefore be called the "+" axis, and the horizontal axis can be called the "-"

axis. The second type is represented by the second term in each component,

and denotes the opposition between the two ends of any given axis. So the

second term in each component mapped onto the 2LAR cross represents a

"polar" (i.e., complementary) opposition-an opposition between two

concepts that also share something in common. The common factor is

represented by the first term of both components on a given axis of thecross: + for the vertical and - for the horizontal.

The third and fourth types of first-level relationships visible on the cross

can be called "subordinate" types, because they are not as evident as the two

"primary" types.Hence,when we want to call attention to them, it is helpf ul

to draw a diagonal line through the center of the cross, either f rom the top

right to the bottom left, or f rom the top left to the bottom right. The former

diagonal line, as shown in Figures I.1, III.3, and IV.5, calls our attention to

the secondary complementary relationship existing between the

components with different first terms, but the same second term (i.e.,

between "--" and "+-", and between "-+" and "++"). The latter diagonal line

highlights the fourth type of first-level relationship, between pairs of

contradictory opposites (i.e., between the two pure components, "++" and

"--", and between the two mixed components, "+-" and "-+"). I have not

included this type of diagonal line in the maps used so far, but it would be

appropriate to add it to the cross any time we want to call special attention

to the tw

o pairs

of concepts

that ar

ed

iametr

ically oppos

ed

in a given2

LAR.

Understanding the complex web of logical relationships that exists within

any set of concepts composing a 2LAR helpsussee that the cross cannot

properly be used to map the relationship between any randomly chosen set

of four concepts. Or at least, ifwe use it in this way,we may not be using the


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cross to represent the logical form of a 2LAR. In that case the cross will only

be, at best, a nice picture, and atworst, a misleading over-simplification.

For only sets of concepts that can be shown to exhibit the set of

interrelationships defined above, and representable by the four +/-

components of a 2LAR, ought to be mapped onto the cross.

Having given this warning, I can now add that there is actually quite a

simple method of testing any set of four concepts that we think might be

related according to the form of a 2LAR. All we need to do is find two

yes-or-no questions whose answers, when put together, give rise to simple

descriptions of the four concepts we have before us. Thus, for example, in

order to prove that the four concepts mentioned above, "daytime",

"nighttime", "dusk", and "dawn", compose a 2LAR, all we need to do is posit

the two questions: (1) Is it obviously either daytime or nighttime (as

opposed to being a transition period)? and (2) Is it lighter now than at theopposite time of day? This gives rise to four possible situations,

corresponding to the four components of a 2LAR as follows:

++Yes, it is obvious, and yes, it is lighter (= "daytime")

+- Yes, it is obvious, but no, it is not lighter (= "nighttime")

-+No, it is not obvious, but yes, it is lighter (= "da wn")

--No, it is not obvious, and no, it is not lighter (= "dusk")

This demonstrates that the four terms in question can be mapped properly

onto the 2LAR cross, as shown in Figure V.4a.

Perhaps I should also mention that we cannot produce a proper 2LAR by

combining any randomly chosen pair of two questions. Or at least, we must

be prepared for the possibility that in attempting to construct a 2LAR, one

or more of the possible combinations of answers might end up describing aself-contradictory concept, or an impossible situation. For this reason, I use

the term "perfect" to refer to a 2LAR(or any other logical relation) inwhich

all the logically possible components also represent real possibilities. For

example, consider the two questions:

(1)Isitraining?and(2)Isthesunshining?Atfirst,onlythreeofthe


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(a) Four Parts of a Day(b) Four Weather Conditions

Figure V.4: TwoExamples Mapped onto the 2LAR Cross

four combinations of answers to these questions appear to depict real

possibilities. If we answer "Yes" to both questions, then it might seem thatwe have discovered an impossible combination, since (at least here on

earth) it is cloudy, not sunny,whenever it rains. If this were the case, then

these two questions would compose an imper fect2LAR.However, ifwe

think f urther about this fourth option, we will realize that it does represent

a real possibility. (As we shall see throughout this week, such surprises

often pop up when we use the Geometry of Logic as an aid to our reflection.)

For the sun does sometimes shine while it is raining: this is what is

happening whenever we see a rainbow! Hence even this example, as shown

in Figure V.4b, represents a perfect 2LAR,while at the same timeillustrating how such maps can help us gain new insights. (Incidentally, if

the second question were "Is it cloudy?", then this would be an imperfect

2LAR, since a "No" answer could not be combinedwith a "Yes" answer to

the first question.)

Remember the map of the four elements I gave in Lecture 4 (see Figure

II.4)?Now thatwe have analyzed the formal structure of distinctions

mapped onto the cross, we can actually test that traditional set of concepts

to see if it represents a perfect 2LAR. If fire is "++" andwater is "--", then we

would expect these to be contradictory opposites. And they are. Water puts

out fire, and fire changes water into vapor. Likewise, if earth is "-+" and air

is "+-",we would expect earth and air to be similarly resistant. And they are.

Earth and air do not mix! What about the complementary opposites? Here

we find equally appropriate results: fire needs air and earth (i.e., f uel) in


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order to continue burning;water can mixwith air (as in soda) andwith

earth (as in mud). So even though the ancient Greeks had not developed the

Geometry of Logic, they were intuitively able to choose, as their four basic

elements, materials that correspond in real life to the form of a perfect

2LAR.

Of course, there are actually more than four physical "elements" in the

universe; likewise, a day can be divided into more than just four parts, and

the weather has far more than just four variations! In the same way, the

process of analytic division can and does go on and on, forming increasingly

complex patterns of relations between groups of concepts. In this course we

have no time to examine the complex relations created by these "higher

levels" of analytic division.However, I would like to mention one final

example. But first I should point out that, no matter how far we go in

making analytic divisions, the patterns will always follow this very simpleformula:

C = 2t

where "C" refers to the total number of different components possible and

"t" refers to the number of +/- terms in each component. The latter,

incidentally, is always identical to the number of the level. Thus, as we have

seen, the number of divisions required to construct a 2LAR is two, the

number of terms in each resulting component is also two, and the totalnumber of components is four (22= 4). Likewise, the number of divisions

required to construct a 3LAR is three, the number of terms in each resulting

component is three, and the total number of components is eight (23= 8).

The higher the level of analytic relation, the more complex is the map that

has to be constructed to give an accurate picture of all the logical relations

involved. One good example of such a complex system can be found in the

ancient Chinese book of wisdom, the I Ching. This book describes a set of 64

"hexagrams" (i.e., six-part pictures), each representing some kind of life

situation. The bookwas originallyused primarily for predicting f uture

events: in some arbitraryway, such as throwing dice, a person selects two of

the 64 hexagrams, and the trans formation f rom one to the other is then

used as the basis for answering a question,usually about how some present

situation will change in the f uture. (Thus it is also calledThe Book of


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Changes.) For our purposes, of course, the predictive power of the I Ching is

not its main attraction; rather, its logical form is what interests us. For the

64 hexagrams actually f unction as six-term components of a 6LAR. The

traditional way of representing this system of logical possibilities is to use

sets of six solid or broken lines to define each hexagram. By simply

replacing the solid lines with a "+" and the broken lines with a "-", we cantranslate this system directly into the one developed above. If we arrange

the components according to their contradictory opposites (as is normally

done in using the I Ching), then the intricate relationships between these

hexagrams can be mapped onto a s phere, which,when projected onto a

plane surface so that the opposite poles of the sphere are represented as the

center and the circumference of a circle, looks like this:


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Figure V.5: A Map of the 6LAR in the I Ching

Don't worry if this map conf uses you. It is intended to present the logical

form of a highly complex system of concepts at a glance. If you are not

familiar with the system, the map is not likely to be very meaningf ul.

Nevertheless, I would like to end this lecture simply by pointing out thatthis map bears a striking resemblance to the symmetrical pictures used in

some eastern religions (called "mandalas"). Such mandalas are constructed

not in order to clarify the logical structure of a set of concepts, but rather, in

order to stimulate new insights (and eventually, "enlightenment") in those

who use them as tools for meditation (see DW 157-159). As we shall see in

the next lecture, the Geometry of Logic itself is also not limited to such

analytic applications, but can actually touch upon the waywe live our life.

14. Mapping Synthetic Relations

In the last lecture,we saw the orderlyway logical patterns are constructed

when we use analytic logic in our thinking. This kind of pattern,we found,

can be directly related to the patterns exhibited by some simple geometrical

figures. This fact should not surprise us. For in both cases such patterns

originate in the mind. Recognizing these orderly patterns, Kant suggested

that reason itself contains a fixed, architectonic structure. And his

promotion ofwhat he called reason's "architectonic unity" is an inseparable

aspect of his a priori approach. For his assertion that there are certainnecessary conditions for the possibility of any

humanexperience(seeLecture8)assumeshumanreasonoperatesaccording to

a fixed order. Because reason fixes this order-this architectonic -for us,

philosophers ought to do their best to understand and follow itwhenever

they adopt an a priori perspective in their philosophizing (i.e.,whenever

they askwhat the mind imposes upon experience, rather thanwhat it draws

out of experience). Kant believed philosophers ought to allow these

patterns to serve as an a priori "plan" for the construction of a philosophical

system, much as a building contractor uses the architect's blueprints as the

plan for constructing a building. It is nowonder, then, that Kant regarded

Pythagoras (c.569-c.475 B.C.), not Thales, as the first genuine philosopher

(see OST 392); for Pythagoras focused not on metaphysical issues, but on

mathematics and number mysticism.


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Logic is one kind of a priori perspective (see Figure IV.4), so we should not

be surprised to find such numerical patterns playing an important role in

this branch of philosophy.However, logical patterns do not relate only to

our a prioriways of thinking. As Pythagoras recognized, they also relate

very closely to the waywe live our lives. That is one reason I ended the

previous lecture with an example f rom Chinese philosophy. In ancientChina, the I Ching was never regarded merely as a logical table of a priori

thought-forms. Most-perhaps even all-who used itwere not even aware of

its neat, logical structure, as a perfect 6LAR. Rather, they used it intuitively,

as a reflection of the ever-present changes in their daily life situations. In

the real world, things do not remain eternally opposed to each other, as our

concepts might lead us to believe. Instead, opposites gradually fade into

each other by passing through an infinite series of degrees. Once we

recognize this fact, we mightwish to view the line in Figure V.2b no longer

as representing an absolute separation, requiring a choice between two

discrete kinds, but as representing a continuum, containing infinitely many

degrees.

There is, in fact, another symbol f rom the Chinese tradition that performs

this same, synthetic f unction, even though it can also serve as a map of an

analytic relation. I am thinking here of the famous "Tai Chi" symbol,

depicting the opposition between the forces of yin (dark) and yang (light).

As shown in Figure V.6, this symbol can be regarded as simply another way

of mapping a 2LAR.However, in the Chinese tradi-

tionitsprimarysymbolicvaluewasquitedifferent, foritwasregarded


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as a pictorial expression of the fact that in reallife opposite concepts, experiences, forces, etc.,not only depend on each other for their ownexistence, but actually merge into each other

through the passage of time. This is why the twohalves are shaped in the form of teardrops,connoting movement. Moreover, at the verycenter of the large part of each "teardrop"we find

the opposite force. This, like the arrow on eachaxis of the 2LAR cross, represents

FigureV.6: The2LAR

Implicit in the Tai Chi

the way opposites converge upon each other.

We saw in Lecture 12 that this tendency of opposites to be "the same", as

Heraclitus put it, is actually the proper subject ofs

ynthetic, not analytic,logic. So I would now like to explore how to use the Geometry of Logic to

construct accurate pictures of logically synthetic relations. Like analytic

logic, synthetic logic also starts f rom a point, but the point is now regarded

as already containingwithin itself a pair of opposites.

Why?Becausesyntheticlogicisbasednotonthelawsof identityandnon?/i>cont

but on the laws of nonidentity (A≠A) and contradiction (A=-A). Hence, in

order to picture its extension,we must draw a line not in one direction,

(f rom A to -A), but in two (f rom x to A and -A simultaneously). Thus the

geometrical figure best representing this "simple" or"first-level"syntheticrelation(abbreviated"1LSR")isatriangle.This threefold

process can refer either to the original synthetic division of a nonidentical

point into two opposites or to the synthetic integration of two opposites

into a newwhole (cf. Figures I.4 and I.2), as shown in Figure V.7. Ordinarily,

whenever we are workingwith only a single triangle, it is best to use an "x"

sign to represent the third term in a synthetic relation. For this third term is

in a sense an "unknown" that arises out of the two "known" terms, "+" and

"-", preservingwhat is essentialineach,yetgoingbeyondthem both.However,

when these two


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(a) The Original Synthesis(b) The Final Synthesis

Figure V.7: The Triangle as a Map of a 1LSR

types of synthetic triangle are pictured together (cf. Figures III.2 and V.7),

the best way to depict the overall logical form is to label the original

synthetic termwith a "0", to represent its f unction as the common source of

the two opposites,while labeling the concluding synthetic termwith a "1",

to represent its f unction as the final reunification of the two estranged

opposites.

Another way of mapping a 1LSR is to use the circle given in Figure V.2a,

labeling the circumferencewith an "x". This is appropriate because the

boundary participates in both the outside and the inside of the circle, just as

"x" participates in both "+" and "-". Whenever we use a circle as a logical

map, the concept labeling the circumference ought therefore to f ulfill a

synthetic f unction in relation to the two opposite concepts it separates.

However, synthetic logic, like analytic logic, also has higher levels of

relations; and the triangle has a more natural application to these higher

relations than the circle, so I'll treat the former as the standard 1LSR map.

The second level of synthetic relation (2SLR) can be constructed by

regarding each of the three terms, "+", "-", and "x", as generating its own

synthetic relation. This gives rise to the nine components of a 2SLR:

++-+x+

+---x-

+x-xxx


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A good map for a 2SLR is a nine-pointed star, composed of a set of three

intersecting triangles, though there are other possibilities as well. For our

present purposes, we need not go into the details of these higher-level

synthetic relations. Instead, it will be enough merely to point out that the

formula governing the patterns that will appear on each level is:

C = 3t

where "C" refers once again to the total number of different components

possible and "t" refers both to the number of different terms and to the

number of the level. I hope youwill experiment with some of these higher

levels on your own.

Mapping the regularity of higher-level logical relations, as illustrated in

Figure V.5, is more appropriate to analytic relations than to syntheticrelations. This is because analytic relations are produced by dividing

wholes into discreet parts, whereas synthetic relations are produced by

integrating parts to produce larger wholes. Because these newwholes

combine opposites together in typically mysteriousways, the higher-levels

tend to produce complex networks of relations that appear to be chaotic.

Instead of mapping an example of such a higher-level synthetic relation

here, I shall therefore discuss how synthetic logic can shed new light on one

of the most interesting developments in science during the last quarter of

the twentieth century. "Chaos Theory", also called "non-linear dynamics", is

a rather surprising new area of mathematical physics that has great

potential to explain some of the most mysterious aspects of human life. The

theory claims, in a nutshell, that order comes out of chaos: when we observe

whole systems, the parts seem to have haphazard relations with each other,

yet on lower levels, the same system can exhibit a high degree of order. The

typical illustration of the long-range effect chaos can have on the world is

the claim that "the flapping of a butterfly's wings inNew York may cause

the weather to change in Hong Kong".How can this be true, when there is

no observable cause-and-effect relation between the two? I believe the

answer lies in regarding chaos as a higher-level synthetic relation. In this

case, the "cause" being referred to here must not be interpreted as an

ordinary cause of the sort that can be understood through analytic logic.

Rather, it is like the mutual interaction between a huge collection of


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interwoven 1LSR triangles, whose combined syntheses are not subject to

precise analysis.

A good reason for not spending more time examining higher-level synthetic

relations is that the 1LSR has another application that is easier to map and

so also, more usef ul to philosophy. For, just as we saw in Lecture 11 thatanalysis and synthesis are best regarded as complementary f unctions, so

also analytic and synthetic logic have their most profound applications in

the Geometry of Logic when they are joined together in a single map. The

simplest way of doing this is to combine a 1LARwitha1LSR,byputting

together two intersecting triangles to form


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a "star of David". The six components (2x3 = 6)of the resulting "sixfold compound relation" (or"6CR") can be placed on such a map in themanner shown in Figure V.8,with the first termin each component representing the analyticopposition between the two triangles. Thisfigure can be constructed by sliding together

the two triangles in Figure V.7, then rotating theentire figure counter-clockwise by 30? The 0and 1 vertices become the -x and +x,respectively.

Figure V.8: The Star

of David as a 6CR

This map can be used to explore the logical relationships between any two

sets of three concepts we believe might be related in this way. For example,

one of my students once came up with the idea of comparing the famous

philosophical triad, "truth, goodness, and beauty" with the famous religious

triad, "faith, hope, and love". The way to test whether or not these six

concepts make up a legitimate 6CR is to find a way of mapping them ontothe diagram in Figure V.8, such that the concepts placed in opposition to

each other really do have characteristics that make them complementary

opposites. We could begin this task by associating the "-" trianglewith the

hilosophical concepts and the "+" trianglewith the religious concepts, thus

defining the basic 1LAR. But once again, I prefer to let you experiment for

yourselfwith the other details, or with other examples of your own making.

Another way of integrating analytic and synthetic relations is to combine

the simple 1LSRwith a 2LAR. The twelve components (3x4 = 12) of the

resulting "twelvefold compound relation" (or "12CR") can,of course,be

mapped onto a twelve-pointed star;but I think a better way is

simplytomapthemontoacircle,especiallysincethemapthenresembles the

familiar figure of the face of a clock. In addition, by using a circle,we can

leave the center open, to be filled inwith whatever figure represents the


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specific set of logical relations we wish to highlight among the many that

exist between the twelve components. For example, in Figure

V.9,Ihaveplacedacrossinsidethecircle,thusdividingit into its four


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main (2LAR) quadrants. However,we couldalso use a line, a triangle, a square, orcombinations of these, to highlight otherlogical relations implicit within this map.

What use is a complex map like this? Oneobvious point is thatFigureV.9coincidesexactlywiththe traditionalsigns of the zodiac,which are divided into fourgroups of three in exactly the sameway. Buteven apart f rom the light it might shed on therational origin of such ancient"wisdom",generallyscoffedatby

Figure V.9: The Circle

as a Map for a 12CR

philosophers nowadays, we can find 12CRs operating in many diverse areas

of human life and thought. Why, for instance,dowe divide the year into

tw

elve months

(four s

eas

ons

, eachw

ith thr

ee months

)? Or

thed

ay intotwelve hours? It's easy to pass off such facts as merely arbitrary

conventions. But perhaps they have their origin in the very structure of

rational thinking! This was Kant's conviction; for, as we saw in Lecture 8,

his list of twelve categories fits the same pattern of four sets of three (see

Figure III.9). Furthermore, as I have argued in Kant' s System of

Pers pectives, Kant also used the same twelvefold pattern in constructing

the arguments that compose his Critical systems-indeed, this pattern is the

basic form of his "architectonic plan".

Other academicdisciplines have no shortage of twelvefold distinctions with

exactly the same structure. A famous scientist named Maxwell, for example,

discovered in the nineteenth century that there are twelve distinct forms of

electromagnetic forces, and that they can be grouped into four sets of three

types. More recently, quantum physicists have discovered exactly twelve


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different types of "quarks", the basic building block of matter. Numerous

examples like these could be cited. But a detailed explanation of how such

applications of the 12CR actually operate is beyond the scope of this

introductory course. Instead,we shall turn our attention in the next lecture

back to synthetic logic itself, in order to gain a better understanding of how

it operates and of how the Geometry of Logic facilitates the process of having new insights by providing visual representations of the new

perspectives synthetic logic provides.

15. Mapping Insights ontoNew Perspectives

The main f unction of synthetic logic is to shock us into seeing new

ers pectives. Once we realize this, it becomes easier to understand how it is

possible for a proposition to be meaningf ul even though it breaks the law of

noncontradiction. The explanation is that such propositions do not actuallybreak Aristotle's law in its strictest sense. Aristotle himself recognized that

"A" could be identical with "-A" if the "A" in question is being viewed f rom

two different perspectives. That is why in defining this law he added "at the

same time, in the same res pect" to the words "A thing cannot both be and not

be". Things change in time, and they can be described differently when

looked at in different ways, so in these cases the A≠-A law does not hold. But

most of us find it quite difficult to look at a familiar subject in a newway.

What synthetic logic does is to bring us face to facewith an exceptionalway

of thinking about or looking at a familiar subject; and in so doing, it fires our

imagination with insight.

This is where synthetic logic and the Geometry of Logic share a common

f unction. For both are instrumental in providing us with the means to

develop our capacity to see old issues in newways, and in so doing, to

deepen our insight intowhatever is perplexing us. Indeed, taken together,

these two logical tools probably constitute philosophy's most usef ul

practical application. For as we shall see, a clear understanding of these

tools can assist you in thinking andwriting more clearly and more

insightf ully in virtually any area, not just when dealingwith philosophical

issues. Let us therefore look first at synthetic logic on its own, and then

move f rom there into a discussion of how geometrical maps can be used in a

similar way to promote clarity and insight.


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Syntheticlogichas,infact,alreadybeenusedbysomephilosophers to show

how new insights come about. For instance, the perplexing contradictions

of Chuang Tzu and the string of negations proposed by Pseudo-Dionysius

(see Lecture 12) can be regarded as away of prodding the reader to discover

new insights about "the Way" or about "God", respectively. Likewise, this is

the most f ruitf ulway of interpretingHegel's famous "dialectical" logic (seeFigure IV.7): his idea that changes occur in human historywhenever two

opposite forces clash and give rise to a new reality, called the "synthesis", is

best regarded as a description of the process whereby human pers pectives

change. Andwhenever our perspective changes, a new insight normally

accompanies the change. But unfortunately,Hegel's language is so complex,

and his arguments so difficult to follow, that many people end up with more

conf usion than insight after reading one of his books. So a better approach

for our purposes will be to look at a contemporary scholar who has

developed some ways of applying synthetic logic on a very down-to-earth

level.

EdwarddeBono(1933-)isnotsomuchaprofessionalphilosopherasaneducator

ar excellence.Nevertheless, some of the principles he discusses in his

many books are closely related to various philosophical concerns,

especially in the area of logic. For his main concern is to teach people how

to think creatively. In the process of doing so, he demonstrates that the laws

of synthetic logic are not just abstract principles that are difficult or

impossible to apply, but are effective tools that can be used to help us solve

many different sorts of real-life problems. In his book, The U se of Lateral

Thinking, for example, de Bono uses geometrical terms to distinguish

between our ordinary, "horizontal"way of thinking and the "lateral"

thinking that always seeks to look at old situations f rom new perspectives.

(Obviously, the former corresponds to analytic logic and the latter to

synthetic logic, though de Bono does not use these terms.) He suggests that

whenever we have the feeling we are "stuck"with a problem we cannot

solve, the reason is not that there is no solution in sight, but that ourperspective is too narrow. That is why it often helps in such situations to

take a short break f rom our efforts: when we return,we are more likely to

feel f ree to change the waywe are looking at the problem; and often we

discover that the solutionwas right under our nose all along!


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Let me illustrate lateral thinkingwith a personal story. When I was a boy I

used to have a great deal of trouble eating chicken with a fork and knife. I

always preferred to use my fingers. When I noticed one day how easily my

grandfather ate chickenwith a fork and knife, I asked him how he

performed this difficult taskwith such ease.His answer was simple: "You

are trying to remove the chicken f rom the bone; what you need to do is toremove the bone f rom the chicken." This is lateral thinking! And itworked:

all along the bones had been disturbing my enjoyment of one of my favorite

foods; but when I changed the way I thought about my task, the disturbance

virtually vanished! It is also an example of using synthetic logic, because my

grandfather's suggestion enabled me to pass beyondwhat seemed before

like an absolute opposition between "It is easy to get the chicken off the

bonewhen I use my fingers" and "It is difficult to get the chicken off the

bonewhen I use a fork and knife". The new perspective, "remove the bone

f rom the chicken" enabled me to synthesize the "It is easy" of the first

proposition with the "use a fork and knife" of the second proposition.

Lateral thinking always cuts across our former way of thinking in just this

way, much as the vertical axis of a cross cuts across the horizontal axis.

In another book, called Po: Beyond Yes and N o, de Bono suggests another

tool for making newdiscoveries. As the very title reveals, this new tool is

rooted in synthetic logic even more obviously than lateral thinking. In this

book de Bono coins a newword, "po", as away of responding to questions

whose proper answer is neither "yes" nor "no" (or both "yes" and "no"). The

letters "P-O", he POints out, are found together in many words that play an

imPOrtant role in creative thinking, such as "hyPOthesis", "POetry",

"POssibility", "POtential", "POsitive thinking", and "supPOse". "PO" can also

be regarded as an acronym, an abbreviation of the phrase "Presuppose the

Opposite". In order to show how this newword can actually help us develop

our ability to gain new insights-i.e., to see new POssibilities, new

opPOrtunities, just over the horizon of our present perspective-de Bono

suggests we experiment with various "po situations". To perform such anexperiment, we must use "po" as an ad jective, modifying a wordwe wish to

think about creatively; but our description of the characteristics of that

word must then presu ppose the opposite ofwhatever we normally think

about the objects, activities, or situations related to the word. If we think

about how things would be different if this po situation were really the case,


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de Bono assures us that gaining new insights will become much easier.

Let's give this kind of experiment a try. Imagine I am dissatisfiedwith my

teachingmethod,andIwanttothinkof somenew,creativewayofteaching my

classes. In order to treat this as a po situation, I must say to myself "Po

teachers are...", and complete the sentence with something that is usuallynot true about real life teachers. What shallwe say?How about: "Po

teachers know less than their students." This is just a random choice of one

among many of the characteristics of the student-teacher relationship. But

we do normally assume that teachers know more than their students, so the

above statement, by intentionally contradicting this common assumption,

can serve as a good example of a po situation. Whatwould happen if

students really did know more than their students? Well, for one thing, if I

were assigned the task of teaching in such a situation, I would approach my

taskwith humility (if not with fear and trembling!), knowing that I wouldprobably be learning much more than my students. As a result I would

certainly need to res pect my students, and the common expectation that

students ought to look up to me as their teacher would not be so obviously

ustified. Moreover, I would try to encourage students themselves to talk

more, either by asking them questions in class, by having them ask me

questions, or by dividing them into groups and having them talk with each

other. For, since po students know best what the subject is all about, a po

teacher would be very foolish not to give them ample opportunity to share

what they know.

If I now step back f rom this po situation, and re-enter the "real world", I

find I have stumbled upon several new ideas about how I can improve my

teaching: I should be humble enough to learn f rom my students, respect

them as equals in the adventure of learning, not be upset if they show some

disrespect toward me, encourage them to ask and answer questions, and

give them opportunities to discuss issues among themselves. The first time

I gave thi

s lect

ure,I

had

not pr

epar

ed

thes

e ins

ights

befor

ehand

: they jus

tcame to me as I was experimenting with de Bono's method in f ront of the

class. Yet I think these are really very good insights, don't you? If so, it is

important to remember that they did

notcometomebecauseIamespeciallyclever;theycamebecauseIused po to

think laterally, thus leading me to adopt a surprising new perspective on a


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familiar subject. You can prove this for yourself simply by using the same

method to reflect on any area youwish to improve or any topic

youneedtoviewwithf reshinsight.Justremember:pothinkingstimulates

insights because it causes us intentionally to adopt a perspective we know

is contradictory to the real situation-a practical application of synthetic

logic if ever there was one!

I hope the foregoing examples have helped you see the great value -indeed,

the necessity-of using synthetic logic. I'm confident that they have, because

over the years I've noticed that beginning philosophy students often find it

easier to grasp synthetic logic than do professional philosophers! This, no

doubt, is partly because western philosophers are often taught to have a

prejudice in favor of the exclusive validity of analytic logic. In some

traditions logic is defined as "analysis"; so of course, anyone who tries to

propose a nonanalytic logic is regarded as speaking nonsense!Nevertheless, as we have seen, synthetic logic exhibits patterns just as

much as analytic logic; so ifwe define logic as "patterns ofwords", then

synthetic and analytic logic clearly ought to have an equal right to be called

"logic". (Philosophers trained in easternways of thinking, incidentally,

sometimes develop a prejudice in favor of synthetic logic; in the end this is

no better than the western prejudice. A "good" philosopher will be able to

appreciate the value of using both.) Perhaps another reason beginners can

accept synthetic logic so easily is that it actually requires less formal

training to use synthetic logic than

analyticlogic:whereasanalyticlogicisthelogicof knowledge(especially

thinking), synthetic logic is the logic of experience (especially intuiting). In

this sense,we can call synthetic logic the logic of life.

Ifyouarereadingthisasastudent,yourlifeislikelytobefocused largely on

studying,writing papers, and taking tests. With this in mind, I shalldevote

the rest of this lecture to suggesting how an awareness of perspectives can

be an aid

to impr

oving your wr

itings

kills

-a topic thats

hou

ld

inter

es

t allreaders, especially thosewriting insight papers. We have already seen that

insights tend to arise when we learn to shift our perspective (as in lateral

and po thinking) and that synthetic logic is the logic that governs such

changes; we shall now proceed to examine how an ability to map our

perspectives according to the principles of the Geometry of Logic can


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improve our receptivity to insight still f urther.

First, let me warn you that before you actually use a logical map in a paper

or essay, you should caref ully assess whether the reader(s) will be receptive

to thinking in pictures. Some people have a natural preference for this type

of thinking, while others seem to be virtually incapable of understanding it.My own doctoral dissertation at Oxfordwas initially rejected because one

of my examiners had an allergic reaction to my use of diagrams. He claimed

my thesis contained "publishable material", but not as long as itwas filled

with diagrams based on the Geometry of Logic. Ironically, the chapter

where I defended myuse of diagrams (Chapter III of KSP ) had at that time

already been accepted for publication in a very reputable professional

ournal! Nevertheless, I had to rewrite my dissertation, removing the

diagrams, before it was deemed acceptable by that examiner. This

illustrates that a person's response to a diagram may have more to dowithhis or her bias (e.g., an unquestioned myth about what an academic thesis

should look like) than with any rationally justifiable objection to pictorial

thinking. If you think your reader(s) might have such a bias, you can still

use diagrams to help organize your thinking and stimulate insights; but it

would bewise not to include your diagrams in the final version of your

essay. But if your teacher likes using diagrams or at least has an open mind

about such things, including the actual diagram can be an impressive way of

making a good essay even better.

The most basic use for a logical map is in outlining the overall flow of your

essay, just as I did for this book in Figure I.1. What you may not have noticed

is that the 2LAR map provides a pattern that can serve as a universal guide

for constructing a clear and complete argument. In its simplest application,

as shown in Figure V.10, the pure components (--and++)standforthe

Introductory and Concluding parts


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Figure V.10: The Four Parts of an Organized Essay

of your essay. In awell-organized essay these will not be mere "before and

after" summaries ofwhat is contained in the other parts. Rather, a good

introduction sketches out the basic limiting conditions on the topic, just as

"recognition of ignorance" did for this course. Likewise, a good conclusion

leaves the reader with clear and interesting practical applications and /or

ideas for f uture research. As we shall see in Part Four, the "wonder of

silence"will do just that for our study of philosophy. The impure

components (+- and -+), by contrast,will present two opposite perspectives

on the topic at hand. In our case, thinking (logic) and doing (wisdom) are

the two opposites that occupy our attention in Parts Two and Three.

Viewing these opposites as two pers pectives can lead to an especially

insightf ul essay in cases where the views examined in these two parts tend

to be regarded as competing theories or approaches. If you can effectively

demonstrate how the two are actually compatible and /or give a clear

account ofwhy certain incompatibilities are unavoidable, youwill be well

on your way towriting an impressive piece.

Having a predetermined plan for the format ofwhat you intend towrite may

seem at first like an illegitimate procedure: since the realworld is not so

neatly divided, how canwe know in advance whether the topicwill actually

it into such a neat, logical pattern? Kantwould reply that such a question

ignores the fact that reason itself has an essentially architectonic nature.

That is, our thinking is (or ought to be!) orderly and patterned, so in any

essay that involves rational thinking, that order ought not to be left to

chance. Of course, the content of any essay cannot be predetermined in thisway. But if the essay is one that can benefit by being written in a clear and

orderlyway, then selecting a pattern as common as the 2LAR will virtually

guarantee an increase in its level of clarity and persuasiveness. Some essays

may be so detailed that theywill require a more complex pattern, such as

the 12CR used in organizing


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thiscourse(anditssequel, DW ,aswellas KSP and KCR).Thealternative

approach, adopted by most writers even of highly abstract philosophical

essays, is simply to divide the essay into a haphazard number of sections

without following any rule. Yet this leaves the reader totally clueless as to

why the essay is divided up in just this way, and not some other.

By far the greatest benefit that comes f rom using the Geometry of Logic to

re-plan a piece of writing is that doing so calls attention to gaps and

previously undetected connections between the various themes being

considered. In the first two lectures this week, I gave several examples of

how geometrical maps can be used to help promote insights. (Remember

the rainbow?) The potential for giving other such examples is so great that I

could easily fill a whole book with them! But for our purposes itwill be

enough to provide one more example to illustrate how a map can assist us in

deepening our insight by discovering a new perspective on an old, familiartopic.

When I was preparing the present edition of this book, I had taught

Introduction to Philosophy more than thirty times, always using something

like Figures I.1 and I.3 on the first day as a preview ofwhat students could

expect. Then one day after a Philosophy Cafe meeting here in Hong Kong, I

was discussing the nature of silencewith one of the participants. Suddenly

as I spoke I realized that Parts Two and Four of this course can be described

as the two ways human beings experience meaning. The word "meaning"

can therefore label the vertical pole on Figure I.3. As I went home that night,

this image of the labeled vertical pole caused me towonder: How, then,

should the horizontal pole be labeled?Had I not organized this course using

the Geometry of Logic, this questionwould surely never have arisen. But it

now became so obvious that I was amazed I had never in 13 years thought of

this issue! For severalweeks I reflected on this matter without coming up

with an answer. Then, in a conversation with a former student, I finally sat

dow

n and dr

ew

thed

iagr

am. Seeing the hor

iz

ontal polew

ith "ignor

ance" onone end and "knowledge" on the other stimulated me all in a flash to think of

two good answers: Parts One and Three both dealwith reality, but f rom two

different perspectives (ultimate and non-ultimate); but a more naturalway

of contrasting this with "meaning" is to refer to it as "existence". My new

insight was now complete: the overall aim of this course is to share a vision


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ofwhat it means to exist.

QUESTIONS FOR FURTHER THOUGHT/DIALOGUE

1.A.Howwould you map a compound relation higher than a 12CR?

B. Could there be a half -level of analytic division (e.g., a 11 /2LAR)?

2.A. Could a cross be used to map a synthetic relation?

B. Could a triangle be used to map an analytic relation?

3.A. Could "x" and "+" (or "x" and "-") be synthesi z ed?

B. Are there really any magic numbers?

4.A. Whatwould po lateral thinking be like?

B. Is it possible to have an insight that could never be mapped?

R ECOMMENDED R EADINGS

1. George Boole, An I nvestigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and P robabilities (London:

Dover Publications, 1854).

2. Stephen Palmquist, Kant' s System of Pers pectives, Ch. III, "The

Architectonic Form of Kant's Copernican System" ( KSP 67-103).

3. Stephen Palmquist, The Geometry of Logic (unpublished; workingdraft

available at http://www.hkbu.edu.hk/~ppp/gl/toc.html).

4. Underwood Dudley, Numerology: Or , what Pythagoras wrought

(Washington D.C.: The Mathematical Association of America, 1997), Ch.2,

"Pythagoras", pp.5-16.

5. Robert Lawlor, Sacred Geometry: Philosophy and practice (London:


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Thames and Hudson, 1982).

6. Edward de Bono, The U se of Lateral Thinking (Harmondsworth,

Middlesex: Penguin Books, 1967).

7.Edw

ard d

e Bono, Po: Beyond

yes

and

no (H

ar

mondsw

or

th, Midd

les

ex:Penguin Books, 1972).

8. Jonathan W. Schooler, Marte Fallshore, and Stephen M. Fiore, "Putting

Insight into Perspective", Epilogue in R.J. Sternberg and J.E. Davidson (ed.),

The N ature of I nsight (Cambridge, Mass.: MIT Press, 1995), pp.559-587.

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