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