12. SYNTHESIS IN THE GEOMETRY OF LOGIC In the last lecture, we saw the orderly way in which 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 the patterns we were dealing with originate in the mind. Recognizing these orderly patterns, Kant suggested that reason itself contains a fixed, "architectonic" structure. And his promotion of what he called reason's "architectonic unity" is an inseparable aspect of his a priori approach. For his assertion that there are certain necessary conditions for the possibility of any human experience (see above, Lecture 7) assumes that human reason operates in accordance with a fixed order. Because reason fixes this order--this architectonic--for us, philosophers ought to do their best to understand and follow it whenever they adopt an a priori perspective in their philosophizing (i.e., whenever they ask what the mind imposes upon experience, rather than what 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. Logic is one kind of a priori perspective, so we should not be surprised to find such patterns playing an important role in this branch of philosophy. However, logical patterns do not relate only to our a priori ways of thinking. They also relate very closely to the way we actually live our lives. That is one reason why I ended the previous lecture with an example from Chinese philosophy. In ancient China, the I Ching was never regarded merely as a logical table of a priori thought-forms. Most (or perhaps even all) of the people who used it were not even aware of its neat, logical structure, as a perfect 6LAR. Rather, they regarded it intuitively, as a reflection of the ever-present changes in the real situations of human life. In the real world, of course, 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 might wish to view the line in Figure 11.2b 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 from the Chinese tradition which performs this same, synthetic function, even though it can also serve as a map of an analytic relation. I am thinking here of the famous "Tai Chi" symbol, which depicts the opposition between the forces of yin (dark) and yang (light). As shown in Figure 12.1, this symbol can be regarded as simply another way of mapping a 2LAR. However, in the Chinese tradition its primary symbolic value was quite different, for it was regarded as a pictorial expression of the fact that in real life opposite concepts, experiences, forces, etc., not only depend on each other for their own existence, but actually merge into each other through the passage of time. This is why the two halves are shaped in the form of teardrops, connoting movement. Moreover, at the very center of the large part of each "teardrop" we find the opposite force. This, like the arrows on the axes of the 2LAR cross, represents the way in which opposites converge upon each other. We saw in Lecture 10 that this tendency of opposites to be "the same", as Heraclitus put it, is actually the proper subject of synthetic logic, not analytic logic. So today I would like to explore how the Geometry of Logic can be used to construct accurate pictures of logically synthetic relations. Like analytic logic, synthetic logic also starts from a point, but the point is now regarded as already containing within itself a pair of opposites. Why? Because synthetic logic is based not on the laws of identity and noncontradiction, 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, (from A to -A), but in two (from x to A and -A simultaneously). Thus the geometrical figure best representing this "simple", or "first-level", synthetic relation (which I will abbreviate as "1LSR") is a triangle. This threefold process can refer either to the original synthetic division of a nonidentical point into two opposites, as shown in Figure 12.2a, or to the subsequent synthetic integration of two opposites into a new whole, as shown in Figure 12.2b. [Figure 9.3: Analytic and Synthetic Propositions] (a) The Original Synthesis (b) The Final Synthesis [Figure 12.2: The Triangle as a Map of a 1LSR] Ordinarily, whenever we are working with 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" which arises out of the two "known" terms, "+" and "-", preserving what is essential in each, yet going beyond them both. However, when these two types of synthetic triangle are pictured together (see e.g., Figure 6.2), it is better to label the synthetic term in Figure 12.2a with a "0", to represent its function as the common origin of the two opposites, while labelling the synthetic term in Figure 12.2b with a "1", to represent its function as the final reunification of the two estranged opposites. Another way of mapping a 1LSR would be to use the circle given in Figure 11.2a, labelling the circumference with an "x". This would be 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 labelling the circumference ought therefore to fulfill a synthetic function 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 will treat the former as the standard 1LSR map. The second level of synthetic relation (2SLR) would be constructed by regarding each of the three terms, "+", "-", and "x", as generating its own synthetic relation. This would give rise to the following nine components of a 2SLR: ++ -+ x+ +- -- x- +x -x xx A good map for a 2SLR would be 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 you will experiment with some of these higher levels on your own. The reason I don't want to spend time examining the higher-level synthetic relations is that the simple 1LSR itself has another application which is both more interesting and more significant. For, just as we saw in Lecture 9 that analysis and synthesis are best regarded as complementary functions, so also analytic and synthetic logic have their most profound application in the Geometry of Logic when they are combined together in a single map. The simplest way of doing this is to combine a 1LAR with a 1LSR, by putting together two intersecting triangles to form 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 the manner shown in Figure 12.3, with the first term in each component representing the analytic opposition between the two triangles. This map could be used to explore the logical relationships between any two sets of three concepts which 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 onto the diagram in Figure 12.3, such that the concepts placed in opposition to each other really do have characteristics which make them complementary opposites. We could begin this task by associating the "-" triangle with the philosophical concepts and the "+" triangle with the religious concepts, thus defining the basic 1LAR. But once again, I will let you experiment for yourself with the other details, or with other examples of your own making. Another way of integrating analytic and synthetic relations is to combine the simple 1LSR with a 2LAR. The twelve components (3x4 = 12) of the resulting "twelvefold compound relation" (or "12CR") could, of course, be mapped onto a twelve-pointed star; but I think a better way is simply to map them onto a circle, especially since Figure 12.3: The Star of David as a 6CR the map would then resemble the familiar figure of the face of a clock. In addition, by using a circle, we can leave the center open, to be filled in with whatever figure represents the specific set of logical relations we wish to highlight among the many that exist between the twelve components. For example, in Figure 12.4, I have placed a cross inside of the circle, thus dividing it into its four main (2LAR) quadrants. However, we could also use a line, a triangle, a square, or combinations of these, to highlight other logical relations implicit within this map. [Figure 12.4: The Circle as a Map for a 12CR] What use is a complex map like this? One obvious point is that Figure 12.4 coincides exactly with the traditional signs of the zodiac, which are divided into four groups of three in exactly the same way. But even apart from the light it might shed on the rational origin of such ancient "wisdom", which is generally scoffed at by modern philosophers, we can find 12CRs operating in many diverse areas of human life and thought. Why, for instance, do we divide the year into twelve months (four seasons, each with three months)? Or the day into twelve 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 7, his set of twelve categories fits the same pattern of three sets of four [see Figure 7.6]. Furthermore, as I have argued in Kant's System of Perspectives, Kant also used the same twelvefold pattern in constructing the arguments which compose his Critical systems--indeed, this pattern is the basic form of his own "architectonic plan". Moreover, other academic disciplines 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. Other examples could be cited. However, a detailed explanation of how such applications of the 12CR actually operate is beyond the scope of this introductory course. Let us instead turn our attention back to synthetic logic itself, in order to get a better idea of how it operates. The main function of synthetic logic is to shock us into seeing new perspectives. Once we realize this, it becomes easier to understand how it is possible for a proposition to be meaningful even though it breaks the law of noncontradiction. The explanation is that such propositions do not actually break Aristotle's law in its fullest form. Aristotle himself recognized that "A" could be identical with "-A" if the "A" in question is being viewed from two different perspectives. That is why in defining this law he added "at the same time, in the same respect" 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 his law does not hold. But most of us find it quite difficult to look at a familiar subject in a new way. What synthetic logic does is to bring us face to face with some exceptional way of thinking about or looking at a familiar subject; and in so doing, it fires our imagination with insight. Synthetic logic has, in fact, been used by some philosophers 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 10), can be regarded as a way of prodding the reader on to discover new insights about "the Way" or about "God", respectively. Likewise, this is, I believe, the most fruitful way of interpreting Hegel's famous "dialectical" logic (see Figure 10.3). His idea that in human history changes occur whenever two opposite forces clash and give rise to a new reality (called the "synthesis") is best regarded as a description of the process according to which human perspectives change. And whenever 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 confusion 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 very practical ways of applying synthetic logic. Edward de Bono (1933-) is not so much a professional philosopher as an educator. 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 synthetic logic is not just an abstract principle which is difficult or impossible to apply, but also a practical tool, which can be used to help us solve many different sorts of real- life problems. In his book, The Use of Lateral Thinking, for example, de Bono distinguishes between our ordinary, "horizontal" way of thinking and "lateral" thinking, which always seeks to look at a given situation from a new perspective. (Obviously, the former corresponds to what I call "analytic logic", and the latter to what I call "synthetic logic".) 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 our perspective is too narrow. That is why it often helps in such situations to take a short break from our efforts: when we return, we are more likely to feel free to change the way we are looking at the problem; and often we discover that the solution was right under our nose all along! For example, 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 chicken with a fork and knife, I asked him how he performed this difficult task with such ease. His answer was simple: "You are trying to remove the chicken from the bone; what you need to do is to remove the bone from the chicken." This is lateral thinking! And it worked: 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 beyond what seemed before like an absolute opposition between "It is easy to get the chicken off the bone when I use my fingers" and "It is difficult to get the chicken off the bone when I use a fork and knife". The new perspective, "remove the bone from 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 No, de Bono suggests a tool for making new discoveries which, as the very title reveals, is even more obviously rooted in synthetic logic. In this book de Bono coins a new word, "po", as a way 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, a short form of the phrase "Presuppose the Opposite". In order to show how this new word 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 that we experiment with various "po situations". To perform such an experiment, we must use "po" as an adjective, modifying a word about which we wish to think creatively; but our description of the characteristics of that word must then presuppose the opposite of whatever we normally think about the object, activity, or situation to which the word refers. If we think about how things would be different if this po situation were really the case, de Bono says we will find it much easier to gain new insights. Let's give this experiment a try. Imagine I am dissatisfied with my teaching method, and I want to think of some new, creative way of teaching 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 which, in real life is usually not true about teachers. What shall we 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, which intentionally contradicts this common assumption, can serve as an example of a typical po situation. What would 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 task with humility (if not with fear and trembling!), knowing that I would probably be learning much more than my students. As a result I would certainly need to respect my students, just as teachers normally hope their students will respect them. Moreover, I would try my best to get the students themselves to talk, 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 are the ones who know best what the subject is all about, a po teacher would be very foolish not to give them a chance to share what they know. If I now step back from this po situation, and re-enter the "real world", I find that I have stumbled upon several new ideas about how I can improve my teaching: I should be humble enough to learn from my students, respect them as equals in the adventure of learning, allow my students to ask and answer questions, and give them opportunities to discuss issues among themselves. The first time I gave this lecture, I had not prepared these insights beforehand: they just came to me as I was experimenting with de Bono's method in front of the class. Yet I think these are really very good insights, don't you? If so, it is important to understand that they did not come to me because I am especially clever; they came because I was willing to use lateral thinking, to adopt a new perspective in thinking about a familiar subject. You can prove this for yourself simply by using the same method to think creatively about something you wish to improve. Just remember: we learned these lessons by intentionally adopting a perspective we knew was 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 have confidence that they have, because in my experience, 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 trained by their teachers 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 would be regarded as speaking nonsense! Nevertheless, as we have seen, synthetic logic exhibits patterns just as much as analytic logic; so if we define logic as "patterns of words", then synthetic and analytic logic clearly ought to have an equal right to be called "logic". (Philosophers trained in the eastern way of thinking, incidentally, sometimes develop a prejudice in favor of synthetic logic, which is in the end no better than the western prejudice. A "good" philosopher will be able to appreciate the value of using both.) Perhaps another reason why beginners find it so easy to accept synthetic logic is that it actually requires less formal training to use synthetic logic than analytic logic: whereas analytic logic is the logic of knowledge (especially thinking), synthetic logic is the logic of experience (especially intuiting). Accordingly, we could call synthetic logic the logic of life. QUESTIONS FOR FURTHER THOUGHT 1. Could triangles be used to map an analytic relation? 2. What would higher levels of compound relations (i.e., beyond 12CR) look like? 3. Would it be possible to form a synthesis between "x" and "+" (or "x" and "- ")? 4. What would po lateral thinking be like? RECOMMENDED READINGS 1. Edward de Bono, The Use of Lateral Thinking (Harmondsworth, Middlesex: Penguin Books, 1967). 2. Edward de Bono, Po: Beyond yes and no (Harmondsworth, Middlesex: Penguin Books, 1972). 3. Lao Tzu, Tao Te Ching: The book of meaning and life, tr. (Chinese-German) Richard Wilhelm and (German-English) H.G. Ostwald (London: Routledge & Kegan Paul, 1985). 4. F. Boerwinkel, Inclusive Thinking, tr. Hubert Hoskins (London: Lutterworth Press, 1971).