11. ANALYSIS IN THE GEOMETRY OF LOGIC In the first lecture on logic we learned that logic--analytic logic, that is--abstracts from the concrete truth of a proposition, and focuses attention first and foremost on its bare mathematical form, its truth value. Today I want to explore some of the ways in which this bare form can actually be converted into a pictorial form. Philosophers since Aristotle, and even before that, have almost universally recognized that logic and mathematics are closely related disciplines. Until about 150 years ago, most philosophers would have said this relationship is confined primarily to arithmetic, where functions such as addition, subtraction, multiplication and division have clear analogies to logical operators such as "and", "not", etc. But then a scholar named George Boole (1815-1864) wrote a book defending what he called the "Algebra of Logic". He demonstrated that algebraic relations are also closely related in many ways to logical relations. Although Boole's ideas are far too complex to examine in this introductory 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 diagrams in a way that conforms to what I call the "Geometry of Logic". In this and the following lecture I want to explain in detail just how these and other diagrams actually function. In the process we will also have the opportunity to look at more examples of analytic and synthetic logic, respectively. There exists a thoroughgoing analogy between the structure of many simple geometrical figures and the most fundamental kinds of logical distinctions, which has rarely, if ever, been fully acknowledged in the past. The starting point of this analogy is the analytic law of identity (A=A), which posits that a thing "is what it is". In order to choose a diagram which can accurately represent this simplest of all logical laws, all we need to do is think of the simplest of all geometrical figures: a point. [Figure 11.1: The Point as a Map of an Identical Relation] Technically, the point merely posits itself as a single position, with no real extension in any direction, though of course, the black spot in Figure 11.1 which represents a point must have some extension in order for us to see its position. The function of the law of noncontradiction is to extend the identical "A" of the law of identity beyond itself, to its opposite, "-A". The geometrical figure which extends 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 by using the inside and outside of a circle, as shown in Figure 11.2. Note that I have labelled these figures with a mere "+" and "-". These symbols are derived directly from the law of noncontradiction, simply by dropping the "A" from 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 concerned with nothing but the bare logical form of the sets of concepts we use. Since this simple distinction arises out of the laws of analytic logic, I refer to it as a "first- level analytic relation" (or "1LAR"). As we shall see, representing this law with the simpler equation, "+ = -" (i.e., positivity is not negativity), makes it much easier to work with more complex, higher levels of logical opposition. (a) The Circle (b) The Line [Figure 11.2: Two Ways of Mapping a 1LAR] The circle and line segment can be used as maps of virtually any distinction between two opposite terms. Such distinctions, as we learned from Chuang Tzu in the last lecture, are a commonplace in our ordinary ways of thinking about the world. We naturally divide things into pairs of opposites: male and female, 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 only when there is an imbalance between the two terms in question--as, for example, only when one acts as a limitation on the other. Now if we were to stop here, the Geometry of Logic would not be a very interesting discipline. No one has any trouble seeing the logical relationship between 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 helpful only when 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 7.2 and 7.7), helped us, I believe, to 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 most useful 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 in which each side of a pair of opposed concepts is itself broken down into a further 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 whereby we divide "one day" into two more or less equal and opposite halves, which we call "daytime" and "nighttime" (i.e., "not daytime"). This is a good example of a typical 1LAR. However, as with most 1LARs, if we try to apply this strict division to every moment in a day, we find there are certain times during the day when we hesitate to say whether it is "daytime" or "nighttime"; and as a result, we make a further analytic division, between "dusk" and "dawn". In order to translate this into the form of our logical apparatus, using "+" 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 from 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 a simple and balanced way of representing such a four-fold relation. But the same 2LAR can also be represented by the four corners of a square (see Figure 3.3). I map the four components onto the cross and the square in the following ways: (a) The Cross (b) The Square [Figure 11.3: 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 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 which, though they may not be any better than some alternative set of rules, can help us avoid confusion and inconsistency in constructing our maps. These rules are, quite simply: (1) a "+" component is placed above and/or to the left of a "-" component whenever possible, with priority given to the first terms in each component; (2) an arrow between two components with the same term in the first position points away from the pure component; (3) an arrow between two components with different terms in the first position points towards 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 11.3a 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 term will be a "+" on one side and a "-" on the other. By contrast, the components mapped onto the square in Figure 11.3b are organized according to contradictory opposites. That means the component at any given corner of the square does not overlap at all with 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. This latter map is, in fact, the one geometrical figure that can be found fairly consistently in most logic textbooks. For it is the formal basis of what is commonly referred to as "the square of opposition". This square has proved to be very helpful in clarifying for logicians the formal relations between propositions which 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 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 any set of four opposing concepts. The first two can be called "primary" types. The first is represented by the first term in each component, which, as we can see in Figure 11.3a, 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 which also share something in common. This common factor is represented by the first term shared by both components on a given axis of the cross: + 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 helpful to draw a diagonal line through the center of the cross, either from the top right to the bottom left, or from the top left to the bottom right. The former diagonal line, as shown in Figures 1.1, 6.3, and 10.1, 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 "-+"). Although I have not included this type of diagonal line in the maps used so far, it would be appropriate to add it to the cross any time we want to call special attention to the two pairs of concepts which are diametrically opposed in a given 2LAR. Understanding the complex web of logical relationships which exists within any set of concepts composing a 2LAR helps us to see that the cross cannot properly be used to map the relationship between any randomly chosen set of four concepts. Or at least, if we use it in this way, we may not be using the cross to represent the logical form of a 2LAR. In that case the cross will only be, at best, a nice picture, and at worst, a misleading over- simplification. For only sets of concepts which 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 to 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 the opposite time of day? This gives rise to four possible situations, which correspond 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 (= "dawn") -- No, it is not obvious, and no, it is not lighter (= "dusk") This 2LAR can therefore be mapped onto the cross, as shown in Figure 11.4a: (a) Four Parts of a Day (b) Four Weather Conditions [Figure 11.4: Two Examples of 2LARs, Mapped onto the Cross] Perhaps I should also mention that we cannot produce a proper 2LAR by combining any randomly chosen pair of two questions. Or at least, the resulting 2LAR will in some cases be one in which one or more of the possible combinations of answers describes a self-contradictory concept, or an impossible situation. For this reason, I sometimes use the term "perfect" to refer to a 2LAR (or any other logical relation) in which all the logically possible components also represent real possibilities. For example, consider the two questions: (1) Is it raining? and (2) Is the sun shining? At first it appears as if only three of the four combinations of answers to these questions depict real possibilities. If we answer "Yes" to both questions, then it might seem that we have discovered an impossible combination, since (at least here on earth) it must be cloudy, not sunny, in order for it to rain. If this were the case, then these two questions would compose an imperfect 2LAR. However, as often happens when we use the Geometry of Logic as an aid to our reflection, if we think further about this fourth option, we will realize that it does represent a real possibility. 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 we can see in Figure 11.4b, represents a perfect 2LAR, while at the same time teaching us that such maps can help us to 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 combined with a "Yes" answer to the first question.) Remember the map of the four elements I gave in the third lecture (see Figure 3.4)? Now that we 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 "++" and water 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., fuel) in order to continue burning; water can mix with air (as in soda) and with 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 which in real life correspond 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 there will be 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 simple formula: t C = 2 where "C" refers to the total number of different components possible and "t" refers to the number of +/- terms in each component (which, 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 total number 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 will be three, and the total number of components will be eight (23 = 8). The higher the level of analytic relation, the more complex is the map that would have 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 is constructed around a set of 64 "hexagrams", each of which represents some kind of life situation. Scholars believe the book was used primarily for predicting future events: in some arbitrary way, such as throwing dice, a person would choose two of the 64 hexagrams, and the transformation from one to the other would then be used as the basis upon which a prediction would be made. For our purposes, of course, it is not the predictive power of the I Ching that interests us, but its logical form. For the 64 hexagrams actually function as the 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 can translate 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 sphere, 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: [Figure 11.5: A Map of the 6LAR in the I Ching] Don't worry if this map confuses you. It is intended to present the logical form of a highly complex system of concepts at a glance. However, if you are not familiar with the system, the map is not likely to be very meaningful. Nevertheless, I would like to end today simply by pointing out that this 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. As we will see in the next lecture, the Geometry of Logic itself is also not limited to such analytic applications, but can actually touch upon the way we live our life. QUESTIONS FOR FURTHER THOUGHT 1. What would a map of a 3LAR look like? 2. Could there be a half-level of analytic division (e.g., a 11/2LAR)? 3. Could the cross be used to map a synthetic relation? 4. Is there such a thing as a magic number? RECOMMENDED READINGS 1. Stephen Palmquist, "Analysis and Synthesis in the Geometry of Logic", Indian Philosophical Quarterly 19.1 (January 1992), pp.1-14. (See also The Geometry of Logic (unpublished manuscript).) 2. George Boole, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities (London: Dover Publications, 1854). 3. Robert Lawlor, Sacred Geometry: Philosophy and practice (London: Thames and Hudson, 1982). 4. Nigel Pennick, Sacred Geometry: Symbolism and purpose in religious structures (Wellingborough, Northamptonshire: Turnstone Press, 1980).