CA1142750A - Devices for displaying or performing operations in a "two-valued" system - Google Patents

Abstract

ABSTRACT OF THE DISCLOSURE
Disclosed are various apparatuses for displaying and perform-ing operatings upon a complete set of the sixteen binary connectives in a two-valued notational system, one that can be readily adapted to teach, to learn about, and to apply logic. The apparatus may com-prise one device having means displaying a plurality of symbols, where-in each symbol represents one of the binary connectives, the means hav-ing the plurality of symbols in a predetermined configuration. The apparatus may comprise are than one device, in which case the devices are movable and belong to a set, and when used in a selected combina-tion, the combination of movable devices is capable of being assembled in a predetermined comformation. Whether the apparatus comprises one or more devices, each of the symbols has a shape selected to indi-cate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein the shape has iconicity, frame consistency, and eusymmetry with re-spect to the quadrants of Cartesian coordinates, and wherein the alignment of the shape is symmetry positional with respect to the x-y axes of the Cartesian coordinates. Further the shape is taken from at least six letter-shapes capable of generating 16 symbols that can be readily assigned a phonetic value, the symbols having four levels of symmetry such that two of the symbols are two-way self-flippable and self-rotatable, two of the symbols are not self-flippable but are self-rotatable, four of the symbols are one-way self-flippable but are not self-rotatable, and eight of the symbols are neither self-flippable nor self-rotatable. The symbols, the configuration of the symbols, the set of movable devices, and the conformation of the devices are selec-ted so that the one device is, or the devices are, adaptable by trans-formations taken from the group consisting of reflections, rotations, translations, counterchanges, and combinations thereof to display and perform the operations.

Description

V

DEVICES FOR DISPLAYING OR PERFORMING
OPERATIONS IN A TWO-VALUED SYSTEM

ABSTRACT OF THE DISCLOSURE
Disclosed are various apparatuses for displaying and perform-ing operations upon a complete set of the sixteen binary connectives in a two-valued notational system, one that can be readily adapted to teach, to learn about, and to apply logic. The apparatus may com-prise one device having means displaying a plurality of symbols, where-in each symbol represents one of the binary connectives, the means hav-ing the plurality of symbols in a predetermined configuration. The apparatus may comprise more than one device, in which case the devices are movable and belong to a set, and when used in a selected combina-tion, the combination of movable devices is capable of being assembled in a predetermined conformation. Whether the apparatus comprises one or more devices, each of the symbols has a shape selected to indi-cate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein the shape has iconicity, frame consistency, and eusymmetry with re-spect to the quadrants of Cartesian coordinates, and wherein the alignment of the shape is symmetry positional with respect to the x-y axes of the Cartesian coordinates. Further the shape is taken from at least six letter-shapes capable of generating 16 symbols that can be readily assigned a phonetic value, the symbols having four levels of symmetry such that two of the symbols are two-way self-flippable and self-rotatable, two of the symbols are not self-flippable but are self-rotatable, four of the symbols are one-way self-flippable but are not self-rotatable, and eight of the symbols are neither self-flippable nor self-rotatable. The symbols, the configuration of the symbols, the set of movable devices, and the conformation of the devices are selec-ted so that the one device is, or the devices are, adaptable by trans-formations taken from the group consisting of reflections, rotations, 11~275~

translations, counterchanges, and combinations thereof to display and perform the operations.
BACKGROUND AND GENERAL DESCRIPTION
This invention relates generally to useful and aesthetic de-vices for computing, teaching, demonstrating, and displaying the inner relationships in the algebra of logic, so that ordinary operations in symbolic logic can be performed with extreme calculational ease.
As is known to those skilled in the discipline of symbolic logic, there are sixteen binary connectives that exhaust all possible second-order true-false combinations for the two-valued logic of two sentences. As is also known to those skilled in this disciplineg the performance of logical operations is limited by a number of inadequa-cies and complexities in the current notational systems. For instance, the most common notational system presently being used is the one historically evolved in the first decade of the 20th century by Peano-Whitehead-Russell, hereinafter called the PWR system.
In practice, the PWR system has several limitations. This system, for example, does not have a unique and easily recognizable symbol for each of the sixteen binary connectives (see TABLE I). The few symbols in use are a mixture of analogical and alphabetical shapes;
they exist as separate entities---each an island unto itself. The in~
terrelationships among these symbols are not immediately transparent, nor are they easily discernible by inspection; consequently, these in-terrelationships are not readily available to facilitate logical cal-culations. As a result, the symbols are cumbersome and abstract, and the interrelationships must be memorized before one is able to perform logical operations. Also, when these operations are performed, the PWR
system lacks the ability to use a few simple rules that apply uniformly wnen the symbols are made to act upon themselves. Furthermore, this prior system cannot be embodied or used in physical models that clearly display the interrelationships among these symbols. The absence of physical models is a decided disadvantage when it comes to teaching, demonstrating, and using the underlying structures of symbolic logic.

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Another current notational system in use is the Polish system which employs a different upper-case letter of the English alphabet to represent each of the sixteen binary connectives (see TABLE I). The alphabetical symbols of the Polish system, as in the PWR system, are also separate and isolated, which again puts a burden on memory and ab-stract thought, particularly when one wishes to consider the interrela-tionships among these symbols. Not designed to be in keeping with a few simple uniform rules of manipulation, the alphabetical symbols of the Polish system also cannot be acted upon directly to perform logical operations. The cumbersomeness with which interrelationships are han-dled in the Polish system also precludes this notation from being easily embodied in physical models for displaying, teaching, and performing logical operations.
A third notational system, one less prevalent than the previ-ously discussed PWR and Polish systems, is the McCulloch system wherein each of the sixteen binary connectives is represented by an all-common X-frame (Greek letter "chi") to which is added the appropriate combina-tion of dots in the four regions that surround the intersection at the center of the X-frame (see TABLE I). Although the logical meaning of each McCulloch symbol is geometrically and visually displayed, these symbols do not have distinct mnemonic values by which they can be easi-ly identified. Also, this notation cannot be used unless the writer constantly repeats the X-frame, once for each symbol. In addition, the McCulloch system has not heretofore been used with a few simple uniform rules by which to flip and rotate the symbols themselves. Moreover, rules of this kind are available for these symbols only if the user is willing to endure a very awkward situation, perceptually.
As is also well-known to those skilled in the discipline of symbolic logic, there are over a dozen additional notational systems that are variations, for the most part modifications, extensions, or ad-mixtures, of the three basic systems discussed above. In general, these variations in the notational systems include the several disadvantages listed above with respect to the three basic systems of PWR, Polish, and McCulloch.
In view of the above background and the existing notational 271:~0 systems used in symbolic logic, it is an aspect of this invention to present a notational system for which carefully combined features yield advantages that overcome the above noted disadvantages of the cur-rent notational systems including, in particular, embodiments of the system in physical devices.
The notational system presented herein is a direct continu-ation of, in some ways a completion of, the notational efforts of Charles Sanders Peirce (1839 - 1914). Expression of his efforts can be found in his Manuscripts 429, 431, and especially 530, as numbered in R. S. Robin's annotated catalogue of Peirce's papers. Perspective in regard to the works of Peirce is best obtained by way of the scholarly efforts of Max H. Fisch (1900 - ), who is presently the General Ed-itor of the Peirce Edition Project. This project entails the prepara-tion of a new edition of Peirce's writings, expected to run to twenty volumes. For an example of Professor Fisch's work, see the article en-titled "Peirce's Arisbe: The Greek influence in his later life,"
which is found in the journal called "Transactions of the Charles S.
Peirce Society" (1971, Volume 7, Pages 187-210).
First and foremost, my notational system, specifically the notational system herein, is a lesson in man-sign engineering. That is to say, the act of notation building is conducted in such a way that the new symbolism meets two requirements. First, it possesses mind-brain economies that fit the psychological characteristics of the person.
Second, it possesses the same interrelatedness among its symbols that exists among the logical meanings being expressed. In effect, each sym-bol of the system, like a small organism, is well adapted both to the society of people who will use it and to the society of symbols that will be used with it.
Fullfilling the requirements of man-sign engineering leads to a new approach. Unlike the decimal system used for numbers, which is a notation that is base consistent and value positional, the notational system herein is frame consistent and symmetry positional. That is to say, logical operations are performed by means of non-numerical motions, namely, by changing the positions of the symbols when symmetrically placing them in different orientations.

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The notational system herein, unlike the PWR system, is com-plete in the sense that a special set of shapes is assigned to all of the sixteen binary connectives, namely, certain shapes that are taken from the lower-case letters of the English alphabet (see TABLE I). Un-like the Polish system, the notational system also gives a central role to some geometric properties that are, by careful selection, an inherent part of the letter-shapes themselves. In addition, unlike the McCulloch system, the notational system is phonetic. Each symbol is identified with a distinct associational and mnemonic sound value, in most cases the same one that is assigned to the corresponding letter of the alpha-bet in normal use for reading. Consequently, the combined advantages of the notational system not only go beyond tne fragmentary advantages of the PWR system, but they also give the geoemtric advantages of the Mc-Culloch system to a Polish-like system, and conversely, they give the phonetic advantages of the Polish system to a McCulloch-like system.
A key consideration is how best to think about the fact that the notational system presented herein is constructed from symbols that are letter-shapes. Heavily weighted toward opposite extremes, the Po-lish and McCulloch systems are split-brain notations. That is to say, the Polish system favors the alphabetic and algebraic side, the letter side of letter-shapes; and the McCulloch system favors the iconic and geometric side, the shape side of letter-shapes. The PWR notation is so much in the fragmentary and analogical direction that it does not, in any systematic way, really participate in this distinction. In these terms, and by a contrast that grows out of a proper synthesis, the notational system is a combined brain, better yet, a unified brain notation. That is to say, this notational system simultaneously and structurally incor-porates both sides of the algebraic and geometric extremes, both the letter and the shape aspects of letter-shapes, thereby making it possi-ble to design a symmetry positional notation that is phonetic-iconic.
Unlike the PWR and Polish systems, the notational system here-in has iconicity. That is to say, the visual meaning assigned to each letter-shape is a matching image of its logical meaning. Furthermore, by carefully adapting iconicity to another feature called frame consis-tency, the notational system has eusymmetry. That is to say, also unlike , 0 the PWR and Polish systems, and unlike the McCulloch system as hereto-fore practiced, each letter-shape in the notational system has been carefully selected so that, as a matter of good symmetry, it uniformly participates in one all-common set of geometric orientations, namely, the same system of flips and rotations.
Another important and unique characteristic of the notational system presented herein is that each of the symbols has transformational facility. Logical operations are performed automatically because, unlike the currently used systems of PWR, Polish, and McCulloch, the physical operation of flipping and rotating acts upon the letter-shapes themselves. As a result, also unlike the PWR, Polish, and McCulloch systems, the transformational facility of this invention not only re-duces significantly the need for abstract rules but also, when perform-ing logical operations, greatly simplifies the nature of these rules.
In accordance with this invention and in the same act that es-tablishes frame consistency, each letter-shape in the notational system herein is assigned its logical meaning by carefully relating it to an all-common basic square, one that usually remains unwritten, but even more important, one that is always retained at the mental level. In contrast, the PWR and Polish systems employ no such frame of meaning, and the X-frame in the McCulloch system is not retained at the mental level but, instead, must be written each time a symbol is used. As de-scribed below, as other features will show, the unwritten, mental square constitutes a fundamental part of the notational system.
In addition, to facilitate understanding of the deep commonal-ities between logic and mathematics, it is a feature of the notational system herein to adapt the unwritten, mental square so that it contains the traditional order assigned to the x-y coordinates of analytic geom-etry. To accomplish this, the TT, TF, FT, and FF compartments of an or-dinary Venn diagram are contracted to the smallest possible size, namely, set-regions reduced to the size of points, which thereby establishes the limit case of state space reduction. Next, treated as elements having point set size, the corners of the frame-consistent basic square are coded to represent the entries in the ordinary truth table for two sen-tences (TT, TF, FT, FF), that is, coded so that they are placed in ~l~L~ 5~

Cartesian order. Thus, in the same act that establishes not only frame consistency but also iconicity and eusymmetry, the patterns of true and false at the four coded corners are arranged to match the patterns of plus and minus in the four quadrants of the x-y axes, as typically em-ployed in analytic geometry.
The notational system presented herein has a good think-write ratio. This condition follows from the way in which the above described features are combined with the delicate balance contributed by the men-tal role and the unwritten aspect of the all-common basic square. On the one hand, unlike prior notations that expect the user to think too much, such as the PWR and Polish systems, an operator employing the notational system can avoid many unnecessary, abstract mental manipula-tions and the memory work that goes with them. On the other hand, un-like prior notations that expect the user to write too much, such as the McCulloch system, the operator does not engage in unnecessary, repeti-tious work-writing of the all-common reference frame when employing the new symbols. Consequently, for each symbol of the notational system, as it participates in the good think-write ratio, the basic square in thought on the one hand and the minimal letter-shape in writing on the other are evenly weighted and greatly reduced man-sign components of the total symbolic act-process.
A consequence of the several features of this invention is that, unlike the PWR and McCulloch systems, the notational system herein has typographical potential. This statement is conservative: twelve of the sixteen lower-case letter-shapes are already included on the ordinary keyboard of a standard typewriter. Adding only four symbols to the key-board makes the logic alphabet, like the Polish system, completely typo-graphical. These symbols ( ~ ), old shapes in new posi-tions, are obtained when the c-letter is flipped from left to right and when the h-letter is flipped both ways and rotated through a half-turn.
Another consequence of the several features of this invention relates to the use of parentheses. As a matter of prior practice, the PWR system is parenthesis-bound and the Polish system is parenthesis-free. In contrast, in keeping with the mental economy of the moment, the notational system herein can be used with or without parentheses, 5~3 even in a mixed way, if the operator so chooses.
Furthermore, a unique and important advantage of the notation-al system herein is that it facilitates the use of a large family of physical embodiments or models that can be employed in computing, teach-ing, and demonstrating standard logical operations. Finally, exception-al far beyond the prior notational systems discussed above, the physical embodiments of this invention can be displayed with great clarity, both visually and tactually, thereby not only fostering learning at the sensorimotor level but also making explicit in an elegant and aesthetic manner the underlying structures that inhabit elementary symbolic logic.
The invention in one aspect as claimed pertains to an appara-tus for displaying and performing operations upon a complete set of the sixteen binary connectives in a two-valued notational system. The appa-ratus includes at least one device, with the device having means display-ing a plurality of symbols, wherein each symbol represents one of the binary connectives. The display means has the plurality of symbols in a predetermined configuration. When the apparatus includes more than one device, the devices are movable and belong to a set, and when used in a selected combination, the combination of movable devices is capable of being assembled in a predetermined conformation. Each of the symbols has a shape selected to indicate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein the shape has iconicity, frame consistency, and eusymmetry with respect to the quadrants of Cartesian coordinates, and wherein the alignment of the shape is symmetry positional with respect to the x-y axes of the Cartesian coordinates. Further, the shape is taken from at least six letter-shapes capable of generating 16 symbols that can be readily assigned a phonetic value, the symbols having four levels of symmetry such that two of the symbols are two-way self-flippable and self-rotatable, two of the symbols are not self-flippable but are self rotatable, four of the symbols are one-way self-flippable but are not self-rotatable, and eight of the symbols are neither self-flippable nor self-rotatable. The symbols, the configuration of the symbols, the set of movable devices, and the conformation of the devices are selected so that the at least one device is, or the devices are, adaptable by transformations taken from the group consisting of reflec-tions, rotations, translations, counterchanges, and combinations thereof to display and perform the operations. The invention may also be ex-tended to apparatus for displaying and performing a selected set of the z(2 ) n-ary connectives obtained from a finite number of elements (A, B, C, D, ... n) in a two-valued notational system.
In summary, in reference to the sixteen binary connectives, my notational system is unique in that it has a systematically pursued and a carefully combined set of special properties. It is complete, geomet-ric, and phonetic; it has iconicity, frame consistency, and eusymmetry;its unwritten, mental basic square is placed in Cartesian orientation;
it is based on lower-case letter-shapes that are symmetry positional.
It has transformational facility, typographical potential, and a good think-write ratio; if preferred, it is parenthesis-free. It consists of a society of symbols for which manipulatory structure has been designed to reflect logical structure.
Especially, as a primary consequence of the foregoing, it easily lends itself to the construction of a large family of physical models which in turn reflect both the matching manipulatory structure and the underlying logical structure.
The following TABLE I is a comparison of the major ways of ex-pressing the sixteen binary connectives, including the three above noted prior systems and the notational system presented herein.

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TABLE
Truth Notation No. Table In Words P W R Polish McCulloch Herein 1 FFFF Contradiction Contradiction O X

2 FFFT Not-A and Not-B ~ A ~J B X ~ p

3 FFTF Not-A and B rJ A B M X b

4 FTFF A and Not-B A rJ B L ~ q

5 TFFF A and B A B K ~ d

6 TTFF A A I ~ c

7 TFTF B B H ~ u

8 TFFT A equivalent B A - B E ~ s

9 FTTF A or else B A ~ B J * z

10 FTFT Not-B ~ B G ~ n

11 FFTT Not-A ~ A F

12 FTTT Not-A or Not-B A / B D ~ h

13 TFTT if A, then B A :~ B C ~.

14 TTFT if B, then A B :~ A B ~

15 TTTF A or B A ~ B A ~

16 TTTT TautologyTautology V ~ x EXEMPLARY EMBODIMENTS
Further objects and features of this invention will become apparent from a review of several embodiments thereof taken in conjunc-tion with the accompanying drawings in which:
FIGURE la is an illustration of the all-common basic square that contains the external code for the (T,F)-valued logic of (A,B) sentences;
FIGURE lb is an illustration of a binary letter, the d-letter, when it is placed on the basic square and when it is related to the ex-ternal code, FIGURE 2 shows two examples of how the internal code of the notational system herein is used to represent the common binary connec-tives "and" and "or";
FIGURE 3 illustrates the sixteen binary letters of the nota-tional system herein when they are placed in a clock-compass arrangement, FIGURE 4 is an illustration of binary letters placed on movable chips, showing how combinations of A-negation and B-negation are perform-ed by the manipulation of the chips;
FIGURE 5a is an illustration of two examples that show how the logical operation of complementation, also called counterchange, is applied to a binary letter;
FIGURE 5b is another illustration of binary letters placed on movable chips, showing how complementation along with A-negation and B-negation is performed by the manipulation of the chips;
FIGURE 6 is an illustration in which four-cell composite chips, one for each binary letter, have been placed in a clock-compass arrange-ment, and for which the chip cells are used to perform the logical oper-ations of Identity, A-negation, B-negation, and (A,B)-negation;
FIGURE 7 is an illustration of an eight-cell composite chip (shown for the o-letter), which is generated for any binary letter by pairing those two four-cell chips in FIGURE 6 that are symmetrically located either across the center of or across the polar axis of the clock-compass, and which is used to perform the additional operation of complementation, FIGURE 8a is an illustration of a four-cell composite chip, ~L 1 ~2'~j0 wherein the same ce11s appear such as is shown in FIGURE 6 and wherein the binary letters are arranged side by side in a rectangle;
FIGURE 8b is an illustration of a four-cell cube-chip wherein the rectangle in FIGURE 8a is wrapped around the four sides of a movable cube;
FIGURE 8c illustrates how the binary letters are stacked into five subsets, called binomial sub-towers, each sub-tower containing those cube-ch;ps, separate or joined, for which the binary letters have an equal number of stems;
FIGURES 9a and 9b illustrate a vertical arrangement of the notational system in a logic tower, which may be constructed, among sev-eral possibilities, from separate cube-chips such as is shown in FIGURE
8b, from cube-joined sub-towers such as is shown in FIGURE 8c, or from a single piece of material;
FIGURE lOa illustrates a horizontal arrangement of the nota-tional system in a flip stick, which is constructed from material that is movable and preferably transparent;
FIGURE lOb is an illustration of a portion of a horizontal ar-rangement of the notational system in a logic alphabet board, which is provided with movable and replaceable blocks in the shape of binary letters;
FIGURE 11 is an illustration of a symmetrical arrangement of the notational system in a logic bug, which is constructed as a single and movable composite chip of all sixteen binary letters;
FIGURE 12 is an illustration of another symmetrical arrange-ment of the notational system in a polar network, which is also con-structed as a single and movable composite chip of all sixteen binary letters;
FIGURE 13 is an illustration of a four-block arrangement of the notational system in a stem-count finder, in which the blocks are moved to give immediate visual access to the binomial pattern of stem changes (1 4 6 4 1) by which each binary letter differs from the other binary letters;
FIGURE 14a is an illustration of a unique arrangement of the binary connectives, called a logical garnet, for which the binary letters 'iJ50 are placed at the vertices of a shadow rhombic dodecahedron, otherwise conceived of as a 4-dimensional cube that has been compressed into a 3-dimensional space;
FIGURE 14b is an illustration of the binomial (1 4 6 4 1) and symmetrical arrangement of the binary letters that is obtained when, starting with the o-vertex placed in front and viewed in polar perspec-tive, the logical garnet in FIGURE 14a is subjected to five vertical cross-sections that are successively placed one behind the other;
FIGURE 14c is an illustration of the same sequence of cross-sections obtained in FIGURE 14b, but in which a set of chess pieces has been substituted for the binary letters, and for which the chess-set correspondence in kind and in frequency matches exactly the symbols of the notational system;
FIGURE 15 is an illustration of one member in a set of special chess pieces, called a knight cube-chip, with its base portion such as is shown in FIGURE 8b, with its chess-piece portion such as is specified in FIGURE 14c, and with its placement in the rhombic dodecahedron such that it is stationed at the c-letter vertex located at the far right in FIGURE 14a and at Position 6 in FIGURE 14b;
FIGURE 16 is an illustration of a 3-dimensional model of a 4-dimensional cube, also called a tesseract, with vertices that are occu-pied by a set of cube-chips, one for each binary letter, such as are shown in FIGURES 8b or 8c;
FIGURE 17 is an illustration of an eight-cell arrangement of logical garnets, wherein the logical garnet in FIGURE 14a is first placed in the near lower-left octant (000) and then it is kaleidoscopi-cally acted upon by three mutually perpendicular mirrors, thereby show-ing in one view the manner in which the eight combinations of negation operate on all sixteen binary letters of the notational system;
FIGURE 18 is an illustration of ternary connectives placed on movable chips, showing how the scope of the notational system is extend-ed so that side-by-side pairs of binary chips are used to obtain ternary chips for three sentences (A,B,C), and for which the xh-pair is the counterchange mate of the od-pair;
FIGURE 19 is an illustration of quaternary connectives placed on movable chips, showing how two pairs of side-by-side binary chips, or how one pair of one-above-the-other ternary chips, are used to obtain quaternary chips for four sentences (A,B,C,D), and for which the quadru-plet on the right (xxxh) is the counterchange mate of the one on the left (oood).
Sixteen Binary Letters The notational system presented herein is a special set of symbols that are used to represent the sixteen binary connectives. Now an ordinary part of modern symbolic logic, these sixteen connectives are a fundamental part of the logic of sentences, also called the proposi-tional calculus. The symbols of the notational system are listed in the last column of TABLE I, as is shown above. These symbols are called binary letters, one each for the sixteen binary connectives. The shapes of all the binary letters have been carefully selected from the lower-case letters of the English alphabet. Twelve of these letter-shapes have phonetic values that are the same as those used in ordinary read-ing. The other four of these letter-shapes ( ~ ) have positions that are different from those used in ordinary print and, therefore, have been assigned their own phonetic values.
The logical meaning assigned to each binary letter is deter-mined by two codes, one external and the other internal. The external code is assigned to the four corners of a basic square, as is shown in FIGURE la. More specifically, the truth table for two sentences (TT, TF, FT, FF) is assigned to the four corners of the basic square, thereby serving as the meaningful parts of a mental frame of reference.
The basic square is always retained at the mental level and, used in this way except for purposes of illustration, it remains un-written when logical operations are expressed. Furthermore, the two mirror planes (mA and mB) in FIGURE la have been included to function as the symmetry elements that will activate a system of flips and rota-tions. These symmetry elements are represented by the lines that cut across the basic square. The position of these lines shows that the vertical and horizontal axes of the basic square coincide with the x-y coordinates commonly used in analytic geometry. By this arrangement, ~4Z~

the fcur corners of the basic square reside in the four quadrants of the Cartesian coordinate system; more specifically, they reside on the diag-onal axes that cut through and that bisect the four quadrants. This arrangement is the basis from which the patterns of true and false are externally coded to match the patterns of plus and minus.
Each binary letter consists of an arrangement of stems that number from zero at the least to four at the most. These stems, in ref-erence to their positions, are always located in close proximity to the coded corners of the basic square. Assigned to these stems is the in-ternal code that determines the logical meaning given to each symbol.
Two simple images will show how a unified code is obtained from the external and the internal codes of the new symbolism. First, as is given in FIGURE la, the truth table for two sentences (TT, TF, FT, FF) is externally assigned to the four corners of the basic square.
Second, the four stems (TTTT) of the lower-case x-letter of the nota-tional system herein, as is given in TABLE I, are internally extended, like arms and legs, until they reach into the corners of the basic square. According to this unified code, when the x-letter is framed against the basic square (TT, TF, FT, FF), it is a stem-determined ab-breviation for its truth table (TTTT) and is, therefore, the symbol for tautology (A x B). Consequently, no matter how the x-letter is flipped and half-turned within the basic square, and no matter in what sequence it performs these movements, it will always come to rest with its four stems located in close proximity to the four corners of the basic square.
At the other extreme, as is also given in TABLE I, the o-letter of the notational system has no stems at all (FFFF); it is the symbol for con-tradiction (A o B).
Another example of what happens when a binary letter is placed into the space occupied by the unwritten, mental square is shown in FIGURE lb. As is seen therein, the d-letter of the notational system herein has one stem (TFFF) in the upper-right corner (TT) of the basic square; it is the symbol for the and-connective, also called conjunction (A d B). In an effort to attain extreme simplicity of expression, the first line of FIGURE 2 also shows how the d-letter, when given its logi-cal meaning, is related both externally to the all-common basic square and internally to the corresponding truth table. Following the same format, the second line of FIGURE 2 shows how the three stems (TTTF) of the ~ -letter of the notational system are used to stand for the or-connective, also called disjunction (A ~ B).
All of the possible combinations of stems, exactly sixteen, have been built into the binary letters. That is to say, these stem combinations are an inherent part of the letter-shapes. Therefore, these combinations are also shown in the last column of TABLE I, and in FIGURE 8c, where by stem frequency (0 1 2 3 4) and by symbol incidence (1 4 6 4 1) they have been placed in binomial order. Eight (- 4 - 4 -) of the binary letters (p b q d h ~ ~ ~ ) are called tall letters; like a flagpole, each of these odd-stemmed symbols is pulled long in the up-down direction. The other eight (1 - 6 - 1) of the binary letters (o c u s z n ~ x) are called squat letters; like a box, each of these even-stemmed symbols is pushed both ways into a compact space. The iconicity of each binary letter is determined by the number of stems, one each for the incidence of T's in its truth table. The eusymmetry of each binary letter is determined by the positions of the stems, al-ways located in close proximity to the corners of the basic square. It is the all-common basic square that provides each binary letter of the logic alphabet with frame consistency.
Figure 3 shows a special arrangement of the sixteen symbols of the notational system presented herein. The same unified code, as already described for FIGURE 2, has been used to place each of the six-teen binary letters in its corresponding basic square. As a convenience for display purposes, the bold dots in FIGURE 3 stand for the incidence of T's in the corresponding truth table for each binary letter (see TABLE I). A further feature of the arrangement shown in FIGURE 3 is that the sixteen stations for the binary letters are associated with the positions of a clock-compass, the encircled numbers 1 through 12 indicating the clock positions, and the encircled letters N, E, W, and S representing the compass positions.
Another look at the bold dots shown in FIGURE 3 gives further emphasis to the special way in which the notational system herein makes use of the ordinary letter-shapes that are already a part of the il4Z~SO

- 17 -English alphabet. The unified code prescribes that each stem-tip on the internal side must be close to a bold dot on the external side.
It follows that each binary letter is a "topological cursive"; in other words, topological because proximity relations of the stem-tips are the means by which the symbols are given their logical meaning, and cursive because all of the symbols, except the x-letter, are written by means of one short motion of the pen. From this point of view, again in reference to the bold dots and the stem-tips close to them, the notational system is a special set of stem-tip icons.
Sixteen Binary Chips The notational system presented herein is embodied in a spe-cial set of manipulatory chips, one each for the sixteen binary letters.
Each chip along with the binary letter on it is called a binary chip.
Binary chips can be made from any suitable material, such as plastic, wood, ceramic, or the like. They may be transparentj if opaque, they may be printed upon, painted, engraved, or otherwise mark-ed so that the letter-shapes will be clearly noticeable from both sides. They may be solid, grooved to match from both sides, cut all the way through, or whatever favors the ease of displaying the letter-shapes. A suitable transparent chip would, for instance, be construct-ed by placing a printed binary letter between two sheets of laminated plastic material. By using the same example shown in the first line of FIGURE 2, the letter-shape for the and-connective is repeated in chip form in the lower-left rectangle of FIGURE 4. As is seen therein, a separate and movable d-chip is placed between an A-chip and a B-chip, also separate and movable.
Negation and the Binary Chips A chief advantage of the notational system presented herein is the ease with which the logical operation of negation acts upon the system of manipulatory chips. Behind all of this is total adherence to one key idea, when it is systematically applied to designing a notation that is phonetic-iconic. The key idea is to treat negation as a trans-formational mirror, one for which a binary letter is flipped or re-flected 180 degrees, every time one of the two sentences (A,B) is acted 2'~

- 18 -upon by negation. Giving rise to transformational facility at its best, in fact, almost an invitation to lazy logic, this key idea be-comes one of the primary sources from which to construct a notation that is symmetry positional.
The main emphasis in FIGURE 4 is to give an example of how the system of manipulatory chips can be used to display and to perform the logical operation of negation. In what follows, the letter N
stands for (N)egation, and the asterisk is an algebraic symbol that stands for any one of the sixteen binary letters. Fundamentally, there are four combinations in which negation can be applied to two senten-ces (A,B), as follows: (1) negate neither, (A * B); (2) negate A, (NA * B)j (3) negate B, (A * NB), and (4) negate both, (NA * NB). The last three of these combinations, likewise labelled with N-letters, are shown in FIGURE 4 by the arrows that pass from one rectangle to anoth-er. These arrows activate specific rules for which a manipulatory chip, along with the letter-shape on it, is flipped and rotated.
The manipulatory rules that go with the four combinations of negation will be demonstrated for the d-chip. This example (A d B) con-tinues with what is given in the first line of FIGURE 2 and with what is repeated in the lower-left rectangle of FIGURE 4. Rule One: when neither the A-chip nor the B-chip is negated (A d B), the d-chip is sub-jected to the Identity operation, that is, it remains in place, sta-tionary and unchanged. Rule Two: when the A-chip alone is negated (NA d B), the d-chip is flipped from left to right, thereby changing the d-chip into a b-chip, as is shown in the lower-right rectangle of FIGURE
4. Rule Three: when the B-chip alone is negated (A d NB), the d-chip is flipped from top to bottom, thereby changing the d-chip into a q-chip, as is shown in the upper-left rectangle of FIGURE 4. Rule Four:
when both the A-chip and the B-chip are negated (NA d NB), the d-chip is flipped both ways in either order, or what amounts to the same thing, it is rotated a half turn of 180 degrees in the horizontal plane, there-by changing the d-chip into a p-chip, as is shown in the upper-right rectangle of FIGURE 4. In keeping with these manipulatory rules, the d-chip of this example has been subjected successively to no change, a left-right flip, an up-down flip, and a half-turn rotation.

75~

,9 In like manner, the same manipulatory rules can be applied uniformly to all of the binary letters. One requirement is that the corresponding binary chips must remain in the same general orientation, never tilted sideways but always horizontal like a box resting on the ground, in every case such as is shown in FIGURE 3. When this condi-tion is maintained, every one of the binary chips can be meaningfully subjected to no change, a left-right flip, an up-down flip, and a half-turn rotation. In this way the above-identified combinations of negation act upon the notational system presented herein.
Complementation and the Binary Chips Complementation is a special case of negation. In this case the scope of negation is extended so that, instead of acting on either of the sentences (A,B), it acts on the binary connective itself, as represented by its binary letter. Negating any binary letter, that is, the directive to find its complement, is shown by placing the letter N
next to the asterisk itself, as in the expression (A N* B). When the (N*)-operation is applied to the system of manipulatory chips, addition-al rules are generated for the notational system herein.
Figure 5a illustrates the manner in which the notational sys-tem is used to perform the (N*)-operation of complementation. The com-plement of any binary letter (A N* B) is another binary letter (A * B), one that is obtained by counterchanging the former to yield the latter.
The rule for counterchanging any binary letter is to reverse the pres-ence and absence of its stems and, therefore, to reverse the T's and F's in its truth table (see TABLE I). For instance, the looped arrow on the left side of FIGURE 5a indicates that the`h-letter in (A h B) is the complement of the d-letter in (A Nd B); the four positions at (TFFF) have been counterchanged to obtain the stems for (FTTT). In like man-ner, the x-letter (TTTT) is the counterchange of the o-letter (FFFF).
In addition, the direction of the looped arrow is reversible because the letter-shapes of the notational system subdivide into complementary pairs, which in keeping with the corresponding patterns of stem rever-sals appear as counterchange mates.
The main emphasis in FIGURE 5b is to show what happens to the o same example (A d B) when the operation of counterchange is added to the four combinations of negation illustrated in FIGURE 4. When counter-change is applied to the d-chip, four more combinations of negation are introduced and so are the manipulatory rules that go with them. Rule Five: when counterchange (Nd) is added to (A * B), the d-chip is ex-changed for an h-chip, as is shown in the first line of FIGURE 5b.
Rule Six: when counterchange (Nd) is added to (NA * B), the d-chip is flipped from left to right and then it is exchanged for a ~ -chip, as is shown in the second line of FIGURE 5b. Rule Seven: when counter-change (Nd) is added to (A * NB), the d-chip is flipped from top to bottom and then it is exchanged for a ~ -chip, as is shown in the third line of FIGURE 5b. Rule Eight: when counterchange (Nd) is added to (NA * NB), the d-chip is flipped both ways or it is rotated, and then it is exchanged for a ~ -chip, as is shown in the fourth line of FIGURE 5b. In keeping with the four additional combinations of nega-tion, the d-chip of this example is first subjected to Rules One, Two, Three, and Four, and then these outcomes are exchanged for the corre-sponding counterchange mates.
As is noted above, the four letter-shapes of the notational system presented herein that are placed in new positions can be as-signed their own meaningful phonetic values. Appropriate phonetic val-ues, for instance, would be as follows. The ~ -letter is called "yor", for "why" and "or". The ~ -letter is called "mif", for "mu" and "if".
The ~ -letter is called "rif", for (r~otated "if". The ~ -letter is called "rAy", for (r)eflected "A". By analogy, the c-letter (see TABLE
I) could be called "Acke", for "A" and the k-sound of a hard c. As is evident, these phonetic values have been taken from the pool of assoc-ciations contained in their ties to the history of the alphabet, to the logical meanings that need to be assigned, and to the logical operations that are being performed.
Complementary pairs of binary chips can also be specified by making use of the orderly arrangements of the notational system herein that have already been given in TABLE I and FIGURE 3. When sets of bi-nary chips are placed in these patterns, the counterchange mate of any binary chip is located with respect to a center of symmetry. Such as in ~427S(~
.

TABLE I, this center is the midpoint of the vertical column (between the sz-chips), for example, the mate of the o-chip at the top of the column is the x-chip at the bottom of the column, and vice versa. Such as in FIGURE 3, this center is the midpoint of the clock-compass (shown as an eight-pointed cross); for example, the mate of a d-chip at 11 o'clock is the h-chip at 5 o'clock, and vice versa. In conclusion, for arrangements of the notational system such as are shown in TABLE I and FIGURE 3, the (N*)-operation is reduced to the visual search for two chip mates, each symmetrically located with respect to the other.
Sub-Societies of Binary Chips Sub-societies are manipulatory structures that match exactly the underlying algebraic structures that are an inherent part of the logic of two sentences (A,B). Consequently, these sub-societies not only govern the interrelationships among the sixteen binary chips but also offer a clear exhibit of the transformational facility of the nota-tional system presented herein. Each sub-society contains just enough binary chips so that it is a factor of sixteen, especially the factors two, four, and eight. Examples of these sub-societies will be described.
Each binary chip is a one-chip sub-society, an island unto itself, under the operation of Identity. Under the operation of com-plementation, there are eight pairs of two-chip sub-societies. These are the same pairs of counterchange mates, described above, that are symmetrically situated with respect to the midpoint of the last column of TABLE I and the midpoint of the clock-compass in FIGURE 3.
As is well-known to anyone skilled in the fundamentals of ab-stract algebra, particularly group theory, the pattern of changes un-derlying a four-chip sub-society is called a Klein 4-group. The nature of a Klein 4-group is also described in F. J. Budden's book entitled "The Fascination Gf Groups" (Cambridge University Press, London, 1972, Pages 139 and 149).
Two examples, among others, will demonstrate the transforma-tional existence of four-chip sub-societies. The first example is mostly a repeat of FIGURE 4 when it is looked at in a broader context.

'7i~C~

The added consideration shows that it is not necessary to start with the d-chip in the lower-left rectangle. Instead, it is possible to start from any rectangle and then, by applying the first four manipulatory rules of negation, to follow the arrows to the other three rectangles.
That is to say, precisely these four manipulatory rules acting on the binary chips p, b, q, and d will generate a sub-society of transforma-tions so that any one of the four chips is converted into itself (Iden-tity) and the other three. In effect, the test for a sub-society is the convertibility of all of its chips into each other---under the same set of operations. The second example of a four-chip sub-society is ob-tained when the binary chips (h ~ ~ ~ ) at the right side of FIGURE 5b are also subjected to the first four manipulatory rules of negation.
The transformational facility of the notational system herein also makes it possible to identify an eight-cell sub-society. This larger sub-society consists of the binary chips contained in both FIGURES
4 and 5b. These binary chips (p b q d h ~ ~ ~ ) are the tall-letter chips that constitute the odd-stemmed half of the notational system.
When the eight manipulatory rules of negation act upon the eight tall-letter chips, they generate an 8-group of logical transformations. In other words, by way of negation any tall-letter chip can be defined in terms of any other tall-letter chip. That is to say, uslng the eight manipulatory rules makes it possible to convert any tall-letter chip in-to any other tall-letter chip, such being the system of interconnec-tions, or arrows that connect, all of the symbols with each other in this special and larger sub-society of binary chips. It is worth noting that the same 8-group of transformations, still in reference to the tall-letter chips, is also activated as a subset that resides at the vertices of the cube-cores that occupy the eight logical garnets that are shown in FIGURE 17.
Binary Chips for the 128 Basic Transformations A summary of the system of manipulatory chips to the extent that it has been described above is contained in the series of numbers 3, 8, 16, 64, and 128. Indicating how this series of numbers is ob-tained will further reveal the compact simplicity and at the same time the transformational comprehensiveness of the notational system pre-sented herein.
Three reduces to a minimum the number of basic changes, that is, the primitive operations, that are strong enough to generate the system of manipulatory chips. These primitive operations, the same as Rules Two, Five, and Three, described above, are as follows: left-right flip for (NA * B); counterchange for (A N* B); and up-down flip for (A * NB). These primitive operations, always given in the same order as a triplet, may be expressed in several ways: in symbols, as (NA, N*, NB); with words added, as NA-flip, (N*)-counterchange, NB-flip; or simply, as flip, counterchange, flip. Hereinafter this triplet will be referred to as flip-mate-flip.
Eight is the number of manipulatory rules that are obtained from all combinations of flip-mate-flip. For purposes of convenience, a compact code for these rules will be established. An upper-case O-symbol will stand for the ABSENCE of negation (not to be confused with the lower-case binary o-letter); as before, the letter N will stand for the PRESENCE of negation. Rule One was given for (A * B), which is now expressed as (OA O* OB), or more compactly, as the triplet 000. Rule Eight was given for (NA N* NB), which as another example is now re-duced to NNN. In keeping with the flip-mate-flip operations described above, the left N is for A-negation, the middle N is for counterchange, and the right N is for B-negation. When this code is applied to all combinations of flip-mate-flip, the eight rules of negation reappear in terms of the following triplets: (l) 000, (2) NOO, (3) OON, (4) NON;
(5) ONO, (6) NNO, (7) ONN, and (8) NNN. This code will be used below;
for example, as is shown in FIGURE 17. It should be noted that the four triplets before the semicolon are without complementation (-O-) and the other four triplets include it (-N-).
The numbers 16, 64, and 128 are obtained in the following manner. A complete table of transformations is constructed from two half-tables. The first half-table is generated when the 16 binary chips are acted upon by the first four triplets of flip-mate-flip (without complementation). This half-table contains 64 basic transformations that are shown in the four columns of FIGURE 9b. The second half-table is generated when the 16 binary chips are acted upon by the other four manipulatory triplets (with complementation). This half-table of 64 basic transformations is also shown in the same four columns of FIGURE
9b. but this time these columns are taken in reverse order. The com-plete table of transformations, arranged in 16 rows and 8 columns that contain 128 binary chips, is likewise compressed into the four columns of FIGURE 9bi the four columns going down omit complementation and the same four going up include it. The complete table of 128 basic trans-formations shows that the system of manipulatory chips is exact and analytic.
In summary, the notational system presented herein has been designed so that the same manipulatory rules for flip-mate-flip can be uniformly applied to each of the binary chips---one at a time, in any combination, or all at once. It is this condition that makes it possi-ble to generate a large family of physical models, only some of which are illustrated in the following drawings. These embodiments, as ex-pected, will repeatedly call attention to the 128 basic transformations that are generated by flip-mate-flip. Of course, it will be appreciated that the notational system is not limited to this particular set of transformations.
Composite Chips The binary chips of this invention may also become the parts and pieces of several arrangements that are compact, meaningful, and very helpful, both to use and to learn about the two-valued logic of two sentences. No matter how joined or separate are the components therein, these arrangements are called composite chips. All of the em-bodiments that follow belong to composite chips. The simple forms are given directly below. After that, the more elaborate forms will be described in separate subsections.
Composite chips are built to favor three extremes. First, such as is shown in FIGURE 8b, they embody the patterns of logical op-erations in which the binary chips participate. Second, such as is shown in FIGURE 6, they embody the symmetry properties that are an in-herent part of the letter-shapes themselves. Also, such as are shown ~5 0 in FIGURES lOa and lOb, a few mixtures of these two extremes will be included. Third, both the logical patterns and the symmetry proper-ties are embodied in the same model so that, as a two-way unity, they are isomorphically incorporated into a single structure. This third kind, particularly the logical garnets such as are shown in FIGURES
14a and 17, will be given special attention.
Two-cell composite chips are not included in the drawings.
They are constructed from pairs of binary chips that contain counter-change mates. Eight pairs of them are obtained when the bottom half of the last column of TABLE I is bent around to match the top half.
The binary chips in these pairs may be separate or joined, as when an o-chip and an x-chip are pushed together so that they have a common edge across the middle or down the middle.
Four-cell composite chips are shown in FIGURE 6. These chips are used to illustrate, to display, or to allow performance of the op-eration of negation, when the first four combinations of flip-mate-flip (without complementation) act upon all of the binary chips of this in-vention. The routine is as follows. Rule One for Identity (000) leaves each binary letter unchanged, as is shown in the lower-left quadrant of each four-cell chip. Rule Two for A-negation (NOO) flips each binary letter into the lower-right quadrant of each four-cell chip. Rule Three for B-negation (OON) flips each binary letter into the upper-left quadrant of each four-cell chip. Rule Four for (A,B)-negation (NON) rotates each binary letter into the upper-right quadrant of each four-cell chip. When these four combinations of flip-mate-flip are applied simultaneously to all of the binary letters, a clock-compass arrange-ment of one-cell chips, such as is shown in FIGURE 3, is converted into a clock-compass arrangement of four-cell composite chips, such as is shown in FIGURE 6.
For example, again calling attention to the Klein-group sub-society of binary chips such as is shown in FIGURE 4, the p-letter at 7 o'clock in FIGURE 6 yields a four-cell composite chip by respectively remaining a p-chip, flipping into a q-chip, flipping into a b-chip, and rotating into a d-chip. In like manner, the first four combinations of flip-mate-flip acting upon the sixteen binary chips embody the half-table of 64 basic transformations that are shown in the four columns of FIGURE 9b.
Another model of a four-cell composite chip, one with compo-nents in a different arrangement, is illustrated in FIGURE 8a. In this case, after applying the first four rules of negation to the b-letter, the resulting binary chips are placed in a rectangle. In like manner, such a rectangle of four-cell chips can be obtained for all of the bi-nary letters. When this is done, the same four columns in the half-table of 64 basic transformations, such as is shown in FIGURE 9b, are arranged from left to right so that the last column of FIGURE 9b is shifted to the position of the first chip on the left hand side of FIGURE 8a.
Eight-cell composite chips are obtained by making extended use of two of the four-cell panels that are shown in FIGURE 6, in such a way that the eight combinations of negation are applied to the same binary letter. Not included among the drawings, these pairs of panels are also placed side by side or one above the other, again separate or joined at the common edge. Other arrangements of these pairs of panels are described in the next subsectîon where, such as is shown in FIGURE 7, they take the shape of eight-cell cube-chips. When using a full set of sixteen, one for each binary letter, any arrangement of eight-cell com-posite chips is capable of embodying the complete table of 128 basic transformations.
Combinations of Cube-Chips A cube-chip is a composite chip that is obtained when some combination of binary chips is placed on some combination of the four sides, or faces, of an ordinary cube. One-cell cube-chips, such as is shown in FIGURE 8c, are marked on the front faces. Two-cell cube-chips are not included among the drawings. They are marked on the front and back faces, usually showing complementary pairs of binary letters. Ro-tating such a set of two-cell cube-chips, which applies to the mate part of flip-mate-flip (ONO), carries the corresponding back faces and the counterchange mates to the front. Another rotation of this kind reinstates the original condition.

Z ~

A four-cell cube-chip, such as is shown in FIGURE 8b, is ob-tained by wrapping the rectangle of binary chips in FIGURE 8a around the four sides of a cube. A set of cube chips of this kind can be a-dapted to many situations, only a few of which will be mentioned. Such cube-chips, for instance, could appear as blocks hanging in a crib, as beads on a bracelet, as parts of a necklace, or the like, for decora-tive or display purposes. Other examples, such as are shown in FIGURES
8c, 9a, 14a, 15, 16, and 17, emphasize the computing and teaching features of these cube-chips.
An eight-cell cube-chip is an arrangement that follows from having all combinations of flip-mate-fl`;p act upon the same binary let-ter. For example, the eight-cell cube-chip that is shown in FIGURE 7 is obtained when the eight rules of negation are applied to the o-letter.
In this case, the first four rules (without complementation) yield four o-letters and the next four rules (with complementation) subjoin four x-letters; both sets of four are oriented such as is shown in FIGURE 7.
In like manner, a full set of eight-cell cube-chips, one for each binary letter, can be placed at the stations of a clock-compass arrangement such as are shown in FIGURES 3 and 6. When this is done, as mentioned above, any arrangement of eight-cell composite chips, in this case cube-chips, is capable of embodying the complete table of 128 basic transformations.
Eight-cell cube-chips, one for each binary letter, can also be obtained directly from larger components by carefully selecting two-panel combinations from a full set of four-cell composite chips (see FIGURE 6). By end-on contraposition, such as is shown in FIGURE 7, the two panels become the front and back faces of a cube-chip, thereby making it unnecessary to apply the eight rules of negation one by one. Each pair of panels is selected in two ways, depending on the logical opera-tion that is embodied in the front-back direction of the cube-chip such as is shown in FIGURE 7.
The two ways of constructing eight-cell cube-chips are de-scribed in reference to the clock-compass arrangement given in FIGURE 6.
First, if the front-back direction is represented by complementation (ONO), which sends duality (NNN) through the center of the cube, the Z ~

~ 28 ~
four-cell panels that are paired must be located symmetrically across the center of the clock-compass arrangement. Second, in reverse, if the front-back direction is represented by duality (NNN), which sends comple-mentation (ONO) through the center, the panels that are paired must be located symmetrically across the polar axis. The first arrangement of the eight rules of negation is the more readily apparent from FIGURE 6;
the second is useful in the development of further embodiments of this invention. An example of the latter is shown in FIGURE 17.
Binomial Sub-Towers FIGURE 8c illustrates how binomial sub-towers incorporate the binomial subsets (1 4 6 4 1) that constitute the notational system pre-sented herein. Among the several embodiments of this model, the cube-chips in each sub-tower may be one-cell chips, that is, displaying only one binary letter on the front face, such as is shown in FIGURE 8c. The cube-chips may also have two cells (not included among the drawings), four cells such as is shown in FIGURE 8b, or eight cells such as is shown in FIGURE 7. The cube-chips may be separate or joined, but in either case, preferably rotatable. The cube-chips may be detachable and capable of being scrambled, before they are reassembled. In particular, the one-cell cube-chips shown in FIGURE 8c have been ordered from left to right so that each sub-tower contains those binary letters (1 4 6 4 1) that have an equal number of stems (O 1 2 3 4).
Binomial sub-towers, such as is shown in FIGURE 8c, are used to display and to perform logical operations. The example that follows will consider a specific model for which a set of four-cell cube-chips, such as is shown in FIGURE 8b, has been stacked into the arrangement given in FIGURE 8c, thereby making it possible to apply the eight combi-nations of negation. The physical movements that go with the first four combinations of negation are applied to all of the sub-towers as follows:
(000), in place and unchanged; (NOO), quarter turn from the left; (OON), quarter turn from the right; and (NON), half turn in either direction.
The other combinations of negation introduce complementation (ONO, NNO, ONN, NNN); counterchange mates are located symmetrically across the center of the middle sub-tower, and are therefore positioned across the 2~ 5(~

center of the front of the total display.
For example, when using the model in FIGURE 8c after it has been adapted to four-cell cube-chips, and when (NNN) acts on (A p B), the p-letter found at the bottom of the second sub-tower (as counted from left to right) is converted into an h-letter found on the half-turned cube-chip located at the top of the fourth sub-tower. In like manner, the complete table of 128 basic transformations is embodied in, and can be displayed and performed by, the binomial sub-towers con-structed in accordance with this invention.
Logic Tower A logic tower of this invention is illustrated in FIGURES 9a and 9b. This tower is preferably constructed from wrap-around cube-chips that are separate and rotatable, but it can also be made from cube-joined sub-towers, or from a single piece of material.
In regard to logical operations, the physical movements are the same as those described above (see FIGURE 8c) for the binomial sub-towers: (000), in place and unchanged; (NOO), quarter turn from the left; (OON), quarter turn from the right; and (NON), half turn in either direction. The other combinations of negation introduce complementation (ONO, NNO, ONN, NNN); counterchange mates are located symmetrically across the up-down midpoint of the logic tower. As a convenient guide, as can be determined from the numbered positions in FIGURE 9b, seventeen is always the sum of the positions of counterchange mates.
For example, when (ONN) acts on (A h B), the h-letter that is found on the front face of FIGURE 9b, shown at Position 12, is converted into a q-letter that is found by rotating the logic tower a quarter turn from the right and, shown at position 5, by jumping symmetrically across the midpoint of the second column of FIGURE 9b. In like manner, the com-plete table of 128 basic transformations is embodied in, and can be dis-played and performed by, the logic tower of this invention.
Flipstick A flipstick modification of this invention is illustrated in FIGURE lOa. The flipstick is constructed from material that is movable and preferably transparent, thereby allowing all of the binary letters o to be manipulated easily at the same time. The order of the binary letters in FIGURE lOa is the same as the order already given both in the last column of TABLE I and on the front face of the logic tower in FIGURE 9b.
In regard to logical operations, the physical movements that act upon the flipstick, used as a single object, are as follows: (000), in place and unchanged; (NOO), a left-right flip; (OON), an up-down flip;
and (NON), a half turn in either direction. The other combinations in-troduce complementation (ONO, NNO, ONN, NNN); counterchange mates are located symmetrically with respect to the midpoint of the flipstick (be-tween the sz-letters). As a convenience, using a flipstick for which the notational system herein has been spaced according to the binomial sub-sets (1 4 6 4 1), such as is shown in FIGURE lOa, makes it easy for the eye to find counterchange mates.
For example, when (NNO) acts on (A b B), the b-letter found on the front face is converted first into a d-letter, following a left-right flip of the flipstick, and then into an h-letter, located symmetrically across the middle from the post-flip d-letter. In like manner, the com-plete table of 128 basic transformations is embodied in, and can be dis-played and performed by, the flipstick of this invention.
Logic Alphabet Board A further modification of this invention consists of a logic alphabet board, in part illustrated in FIGURE lOb. This board is a more elementary version of a flipstick, one that has been built to favor the use of the hand. As an educational device, especially at the sensori-motor level, the outside frame of this model is placed in the orientation shown in FIGURE lOa, and then a set of movable and replaceable blocks in the shape of binary letters is put into the cutout openings.
In regard to logical operations, the outside frame is flipped or rotated (000, NOO, OON, NON) and, with mates that are located symmet-rically across the middle of the board, any letter-block is counter-changed (ONO, NNO, ONN, NNN). In general, whatever manipulatory rule carries the outside frame of the logic alphabet board to a new location, the same rule carries all of the letter-blocks back to their places in the outside frame. Some binary letters, when this is done, remain un-changed in both orientations of the outside frame. These binary let-ters always belong to the squat half of the notational system (o c u s z n ~ x).
For example, when (OON) acts on the whole row of letter-blocks (OA O* OB), the outside frame is relocated while flipping it from top to bottom, whereupon executing the same flip with each letter-block not only assures that the logical operation has been carried out but it also guarantees that each letter-block, by that manipulatory motion, can be reinserted into the outside frame. Four binary letters (o ~ c x) in this example remain unchanged in both orientations of the outside framej a perfect analogue of the required logical operation (OON), they are in-ternally symmetrical with respect to the up-down flip. In like manner, the complete table of 128 basic transformations is embodied in, and can be displayed and performed by, the logic alphabet board of this inven-tion.
Logic Buy A composite chip called a logic bug is illustrated in FIGURE
11. The logic bug is single, movable, and preferably transparent. It is obtained from a front-view, single-plane presentation of the special rhombic dodecahedron shown in FIGURE 14a. The symmetrical placement of all of the binary letters with respect to the enlarged o-letter is what gives this composite chip the appearance of having "arms", "legs", and "inside parts", like a bug.
In regard to logical operations, the logic bug is subjected to the same flips and half turns (000, NO~, OON, NON); counterchange mates are located across the center of the enlarged o-letter (ONO, NNO, ONN, NON). As seen in FIGURE 11, for even-stemmed binary letters, the coun-terchange pairs are located symmetrically with respect to the enlarged o-letter. For odd-stemmed binary letters, the counterchange pairs are located alternatively inside and outside of the enlarged o-letter.
Placing the ( ~ c)-letters at the wings and the (u n)-letters at the head-tail puts these binary letters in special categories; they are lo-cated at special symmetry positions. That is to say, the ( ~ c)-letters ~14Z~'5~) -are up-down self-flips, the (u n)-letters are left-right self-flips, and the members of each pair are counterchange mates to each other.
As an example, when (NN0) acts on (A n B), the n-letter is converted into a u-letter in a special way; the logic bug, as expected, is flipped from left to right, thereby leaving the n-letter unchanged as a result of the self-flip, and then the unchanged n-letter is replaced by its counterchange mate, located across the middle and at the head end of the logic bug. In like manner, the complete table of 128 basic transformations is embodied in, and can be displayed and performed by, the logic bug of this invention.
Polar Network A composite chip in the form of a polar network is illustrated in FIGURE 12. This chip is also single, movable, and preferably trans-parent. It is obtained from a front-view, peek-in presentation of the special rhombic dodecahedron shown in FIGURE 14a. From this view, the binary letters, located at the vertices in FIGURE 14a, are symmetrically placed with respect to the network formed by the connecting edges. Such a polar network, as well as the logic bug, could appear as a toy, as a paper weight, as a mantel piece, or the like, for decorative purposes.
The computing and teaching features are mentioned below.
In regard to logical operations, flips, half turns, and coun-terchanges are the same as those described above (see FIGURE 11) for the logic bug. Placing the sz- and ox-letters across the center of the polar network also puts these binary letters in special categories; they are also located at special symmetry positions. The sz-letters are not self-flips but they are self-rotates and counterchange mates. The ox-letters are self-flips for both A-negation and B-negation, along with being self-rotates and counterchange mates.
As an example, when (NNN) acts on (A o B), the o-letter is con-verted into an x-letter in a special way: the polar network is given either the two flips or a half turn, thereby leaving the self-flipping and self-rotating o-letter unchanged, and then the unchanged o-letter is replaced by its counterchange mate. In like manner, the complete table of 128 basic transformations is embodied in, and can be displayed and s~5 O

performed by, the polar network of this invention.
Stem-Count Finder A modification of this invention, called a stem-count finder, is illustrated in FIGURE 13. This modification is constructed from four blocks that are placed in a clock-compass arrangement, with the corners of these blocks marked with binary letters, such as are shown in FIGURES
3 and 13. The stem-count finder is used to obtain immediate visual ac-cess to the binomial pattern (1 4 6 4 1) that describes the way in which each binary letter, including itself, is so many stem changes (0 1 2 3 4) different from all of the binary letters. This same binomial pattern of stem differences always goes from any one outside corner of the four-block arrangement to the opposite outside corner.
For example, when the four blocks are placed as is shown in FIGURE 13, the o-letter is zero stem changes away from itself (o), one away from (p b q d), two away from (c u s z n ~ ), three away from (h ~ ~ ~ ), and four away from (x). This pattern for the o-letter is the same pattern already used for the flipstick, such as is shown in FIG-URE lOa. For the x-letter, which is the full complement of the o-letter, the blocks remain in the same positions and the same pattern is read in the opposite direction from the opposite corner. For the c-letter, with the blocks still in the same positions, the same pattern starts at 12 o'clock and ends at 6 o'clock. In general, this basic pattern always starts at one outside corner, goes to the nearest outside four, cuts across through the middle six, comes to the outside four on the other side, and finally arrives at the opposite outside (counterchange) corner.
The same basic pattern, from one outside corner to the opposite outside corner, can be obtained for all of the binary letters, but not until the four blocks of the stem-count finder are moved into four dif-ferent arrangements, each one for a specific 4-set of binary letters.
Each arrangement, resulting from special crossovers, is obtained by sliding ADJACENT PAIRS of blocks past each other, sliding them across the (J,K)-diagonals shown in FIGURE 13. Each crossover brings another 4-set of binary letters into the (9, 12, 6, 3) o'clock corners of the clock-compass. In reference to FIGURE 13, the four arrangements of the f~ 1 ~0 blocks, along with the accompanying 4-sets of outside corners, are as follows: no crossovers (o c ~ x); J-diagonal crossover (d q ~ h);
K-diagonal crossover (p ~ b ~ ); and (J,K)-diagonal crossovers (s n u z).
Consequently, no more than two diagonal crossovers of the stem-count finder is needed to place any binary letter in an outside corner, from which in one sweep of the eye is read the (1 4 6 4 1) pattern of stem changes, thereby immediately comparing that binary letter with all of the binary letters. In effect, for the notational system presented herein, the stem-count finder is an adaptation whereby, with extreme convenience, one is able to look from any vertex, through the middle, to the opposite vertex of a 4-dimensional cube, another model of which is shown in FIGURE 16.
The Logi_al Garnet The logical garnet of this invention, as is illustrated in FIGURE 14a, is recognized as a geometrical form that has the shape of a special rhombic dodecahedron. This rhombic dodecahedron is not the same as an ordinary crystal of garnet, which has 14 vertices. Instead, it is a shadow rhombic dodecahedron, one that is obtained when a regular 4-dimensional cube is compressed into a 3-dimensional space and, therefore, one that has 16 vertices. The nature of a shadow rhombic dodecahedron is also described in H. S. M. Coxeter's book entitled "Regular Polytopes"
(Third Edition, Dover, 1973, Pages 255-258). It follows that the logi-cal garnet is a "solid shadow"; in other words, a solid because it occu-pies ordinary space in three dimensions and a shadow because it is com-pressed from a higher dimensional form. When the compression takes place and the solid shadow is constructed, two of tne 16 vertices are fused at the center; consequently, the center of the logical garnet is called a "co-center". Special attention will need to be given to the presence and the function of this co-center.
The primary reason for constructing the logical garnet is that it possesses the optimal amount of symmetry. That is to say, the symme-try properties that go with performins the logical operation of negation are isomorphic to, in other words, they match exactly, the symmetry 5~

properties of the binary letters themselves. In effect, the logical garnet is a fundamental embodiment of the notational system presented herein.
In the next step, on a cut-away stick-figure of the solid shadow, such as is shown in FIGURE 14a, all of the binary letters are assigned to the vertices of the logical garnet. It further follows that two of the binary letters, in particular two squat letters, the sz-letters, are placed at the co-center and that the remaining fourteen are placed in two surrounding subsets, the tall-eight in the first orbit at the vertices of an internal cube-core, and the other six squat letters in the second orbit at the vertices of an over-all encasing octahedron.
Another consideration is that several varieties of cube-chips may be placed at the vertices of the logical garnet. These cube-chips may have one binary letter on each front face, such as are shown in FIG-URES 8c and 14a. They may have the same binary letter on all four faces, for which no drawing is given. They may have rectangular chips wrapped around the four faces, such as is shown in FIGURE 8b. They may have four cells on the front face such as is shown in FIGURE 6. They may have eight cells constructed from the end-on contraposition of two four-cell panels, such as is shown in FIGURE 7. They may be stationary or, the same as the logical garnet as a whole, preferably rotatable. A further consideration, described below, is that any of these cube-chips may be used as the base portion of a special set of chess pieces, such as the example that is shown in FIGURE 15. Although not all of the possibili-ties have been mentioned, each variety of cube-chips is another way of embodying the complete table of 128 basic transformations that is gener-ated by all combinations of flip-mate-flip.
Another view of the logical garnet is shown in FIGURE 14b.
This view is obtained in two steps. First, the logical garnet, such as is shown in FIGURE 14a, is oriented so that the o-letter vertex is placed in front and then it is viewed in polar perspective. Second, the logical garnet in this position is subjected to five vertical cross-sections, each one successively behind the other. These cross-sections, such as is shown in FIGURE 14b, slice the notational system herein into its bi-nomial subsets (1 4 6 4 1). The binary letters in each cross-section are SO

symmetrically arranged, and the sz-letters occupy the co-center, as is shown in the center of the middle cross-section in FIGURE 14b.
The binomial subsets of the notational system herein (1 4 6 4 1), as is now shown in FIGURE 14b, also appeared in the last column of TABLE I, on the binomial sub-towers in FIGURE 8c, on the front face of the logic tower in FIGURES 9a and 9b, on the flipstick in FIGURE lOa, on the logic alphabet board described for FIGURE lOb, and from 9 to 3 o'clock on the stem~count finder in FIGURE 13.
Use of the logical garnet to display and to perform logical operations is explained in the next two subsections, specifically, under what are called Chess-Set Cube-Chips and Eight Logical Garnets.
Chess-Set Cube-Chips A unique feature of the notational system presented herein is the perfect correspondence that exists between the binary letters on the one hand, in regard to levels of symmetry and the incidence at each level, and chess pieces on the other, in regard to the power of these pieces and the incidence of each kind. Another unique feature of this invention is the likewise correspondence between the chess pieces and the vertices of the logical garnet. To illustrate this double corre-spondence, a set of chess pieces has been substituted in FIGURE 14c for the binary letters in FIGURE 14b.
The correspondence between binary letters and chess pieces is as follows. On the highest level of symmetry (Queen and King), the ox-letters are self-flippable and self-rotatable. On the second level (Rooks), the sz-letters are not self-flippable either way but they are self-rotatable. On the third level (Knights and Bishops), the ( ~ c)-and (u n) letters are self-flippable only one way but they are not self-rotatable. On the lowest level of symmetry (Pawns), which constitutes the eight-chip sub-society described above, the tall-eight, odd-stemmed letters (p b q d h ~ ~ ~ ) are neither self-flippable nor self-rotatable. In this manner, unique and unexpected, the chess-set corre-spondence is very helpful in displaying the different patterns that ex-ist not only among the levels of symmetry but also among the sub-societies within the binary letters, as they participate in the complete ~ 2 ~7 table of 128 basic transformations.
FIGURE 15 illustrates how the chess-set correspondence is used to construct a knight cube-chip, in this particular case, one that has four cells. A full set of these special cube-chips is obtained accord-ing to the following routine. Each base portion is adapted from a four-cell cube-chip such as is shown in FIGURE 8b, each chess-piece portion is situated in the over-all pattern such as is shown in FIGURE 14c, and each chess-set cube-chip is placed at a vertex of the logical garnet such as is shown in FIGURE 14a. It should be recalled that these cube-chips may be rigidly attached to the logical garnet, or the base portions, at least, may be independently rotatable.
In regard to logical operations for chess-set cube-chips such as is shown in FIGURE 15, quarter turns (NOO, OON) and half turns (NON) are the same as those described for the logic tower (FIGURE 9a). Coun-terchange mates (ONO) are located symmetrically with respect to the co-center of the logical garnet, with the sz-Rooks occupying the co-mate co-center.
For example, when four-cell chess-chips such as is shown in FIGURE 15 are attached to the logical garnet such as is shown in FIGURE
14a, and when (NNO) acts on (A b B), the b-letter is converted into an h-letter. More specifically, the logical garnet as a whole is rotated a quarter turn from the left, thereby showing the left-side faces of all of the chess chips. This movement carries the b-pawn cube-chip at the near upper-left to the near upper-right, thereby and thereon showing a d-letter, which in turn has its counterchange h-mate showing on the pawn cube-chip located across the co-center at the far lower-left. In like manner, the complete table of 128 basic transformations is embodied in, and can be displayed and performed by, the chess-set logical garnet of this invention.
Eight Logical Garnets FIGURE 17 is an illustration of eight logical garnets that are presented as a schematic model of what happens when one logical garnet is acted upon by three external mirrors. A complete model, in all of its detail, is constructed from eight components, wherein the binary l~Z~ SV

letters (128 of them) are assigned to eight sets of cube-vertices, all of them properly situated and labelled. More specifically, the fundamental eight-cell arrangement of this invention is generated, in all of its de-tail, when the logical garnet in FIGURE 14a is placed in the near lower-left octant (000) of FIGURE 17 and then, after being externally framed by three mutually perpendicular planes, it is kaleidoscopically acted upon by three primitive mirrors that are located in these planes. These mir-ror planes are schematically illustrated in the center of FIGURE 17.
In general, any 4-set of logical garnets that are located on the same side of any primitive mirror is reflected, simultaneously, to the other side of that mirror. As expected, the left-right mirror plane (NOO) is A-negation and the up-down mirror plane (OON) is B-negation.
The surprise is that duality (NNN) is being treated as an independent mirror plane, such as is shown in FIGURE 17, wherein it is placed in the front-back direction. It should be recalled that in FIGURE 6 the dual (NNN) of a four-cell chip is another four-cell chip that is located sym-metrically across the polar axis of the clock compass, these pairs by end-on contraposition giving rise to eight-cell cube-chips, such as is shown in FIGURE 7. As an additional source of information, the eight-cell structure of mirror reflections is also described in F. J. Budden's book entitled "The Fascination of Groups" (Cambridge University Press, London, 1972, Pages 277-278).
In regard to logical operations, the eight-cell arrangement of logical garnets, such as is shown in FIGURE 17, is a basic lesson in transformational facility. This condition follows from the fact that all of the 128 binary letters in FIGURE 17, in eight subsets of sixteen, are oriented in their respective logical garnets so that, in a single presen-tation, all of the possible flips, half turns, and counterchanges are cast, by mirror reflections, into a large scale repeat of the same small scale movements of the individual symbols. It further follows that FIG-URE 17 is easily partitioned into subsets of logical transformations.
The 16 identity transformations, such as is shown on the front face of FIGURE 9b, are situated at the (000) identity garnet of FIGURE 17, which is a repeat of FIGURE 14a. The 64 transformations, such as is shown in the four columns of FIGURE 9b, are situated in the front four garnets of ~l~Z7S~

FIGURE 17 (000, NOO, OON, NON). The other 64 transformations, such as is shown in the same four columns of FIGURE 9b, when they are taken in re-verse order for the additional operation of counterchange, are situated in the back four garnets of FIGURE 17 (ONO, NNO, ONN, NNN). Consequent-ly, in terms of the symmetry properties of the binary letters, the eight-cell arrangement of logical garnets is an exact embodiment of the complete table of 128 basic transformations. From this approach, the physical model illustrated in FIGURE 17 may also serve as an introduc-tion to the crystallography of logic.
In regard to a specific example for FIGURE 17, when the four triplets (000), (NOO), (OON), and (NON) act on (A o B), the o-letter is repeatedly converted into itself, always another o-letter. These o-letters are located at the head-on, closest cube-vertices of the front four garnets (FIGURE 17). This outcome emphasizes that the o-letter (Queen) is self-flippable and self-rotatable. The same outcome also ap-pears on the four-cell chip of o-letters at 9 o'clock (FIGURE 6). When the example at hand is extended so that the four triplets (ONO), (NNO), (ONN), and (NNN) act on (A o B), the o-letter is repeatedly converted into x-letters. These x-letters are located at the tail-away, hidden cube-vertices on the back of the back four garnets (FIGURE 17). In each case not only serving as a counterchange mate with respect to the center point of the total model where the mirror planes intersect, the x-letter (King) is also self-flippable and self-rotatable. The same outcome also appears on the four-cell chip of x-letters at 3 o'clock (FIGURE 6). Com-bining the four head-on o-letters with the four tail-away x-letters is another way of constructing the corresponding eight-cell cube-chip (FIG-URE 7).
In like manner, each binary letter in the (000) identity garnet generates its own set of eight outcomes, one for each garnet. The cube-vertices for each set of eight outcomes take the shape of a distinct rectangular box that is symmetrically situated with respect to the center point that is equally distant from all eight garnets. The binary letters in the 8-sets of vertices from all sixteen oF these between-garnet boxes constitute the complete table of 128 basic transformations, such as is embodied in FIGURE 17.

75~

Three Internal Mirrors The dynamic use of three internal mirrors does not generate an eight-cell arrangement of logical garnets, such as is shown in FIGURE 17.
Instead, using three mirrors in this way leads to internal shifting that takes place within only one logical garnet, such as is shown in FIGURE
14a. For example, by way of a cross-section through the logical garnet that is shown in FIGURE 14a, the primitive, internal mirror (NOO) cuts down through the rniddle from front to back, for which the (o u s z n x) letters lie in this mirror plane. When this (NOO) mirror is activated, the two halves of the logical garnet, one on the left and one on the right, are by reflection carried both ways into each other. As an in-herent part of the same transformation, the binary letters that lie in the (NOO) mirror plane, except for the sz co-center, are self-flippable from left to right.
In like manner, the same logical garnet can be internally sub-jected to the eight combinations of mirror reflections, namely~ 000, NOO, OON, NON, along with ONO, NNO, ONN, and NNN. As a result, this pro-cess does not generate a static eight-cell arrangement of logical gar-nets, such as is shown in FIGURE 17. Instead, it leads to eight arrange-ments of internal shifting that dynamically takes place within only one logical garnet. These arrangements are another way of specifying the (8 x 16) table for the complete set of 128 basic transformations. In this case, at their best, the symmetry properties inherent in the binary letters give direct evidence that they match the symmetry properties of the logical operations being performed, and vice versa.
Tesseract of Cube-Chips A tesseract of cube-chips, such as is illustrated in FIGURE 16, is constructed to serve as a 3-dimensional model of a 4-dimensional cube.
This model consists of a stick-figure of eight cube-vertices that is placed inside of a larger stick-figure of eight cube-vertices, and for which the eight pairs of corresponding cube-vertices are connected by radiating spokes. Again, in reference to the variety of cube-chips de-scribed for the logical garnet (FIGURE 14a), there are several kinds of cube-chips that can be placed at the sixteen vertices of a tesseract. It 1~2~0 should be recalled that these cube-chips may be rigidly attached to the tesseract, or they may be independently rotatable.
In reference to logical operations for cube-chips such as are shown in FIGURES 8b and 15, quarter turns (NOO, OON) and half turns (NON) are the same as those described for the logic tower (FIGURE 9a). Coun-terchange mates (ONO) are located on a diagonal that passes through the center of the tesseract, such that the mate for any vertex in one stick-figure cube is at the opposite vertex in the other stick-figure cube.
The mate of the o-letter, for instance, is across the center at the x-letter, which is located in the far upper-right of the smaller stick-figure cube.
For example, when using four-cell cube-chips such as is shown in FIGURE 8b, and when (NNN) acts on (A u B), the u-letter is converted into itself, another u-letter. More specifically, the tesseract is given a half turn in either direction, thereby showing the back faces of all of the cube-chips. This movement carries the u-chip at the far upper-left to the near upper-right, thereby and thereon showing an n-letter, which in turn has its counterchange u-mate showing on the cube-chip located inside of the tesseract across the center at the far lower-left. That is to say, the half turn of the u-letter is neutralized by the counterchange of the n-letter, thereby yielding itself, another u-letter. In like man-ner, the complete table of 128 basic transformations is embodied in, and can be displayed and performed by, the tesseract adapted to the notation-al system presented herein.
More Elaborate Embodiments The embodiments considered above have repeatedly called atten-tion to the complete table of 128 basic transformations. In other words, these embodiments have emphasized that eight combinations of negation acting on the sixteen binary letters constitute a fundamental part of the two-valued logic of two sentences. It will, of course, be appreciated that there are many additional varieties of these embodiments.
Several examples will show this point by placing previously described components into previously described arrangements. An eight-cell of flipsticks places components such as is shown in FIGURE lOa into the arrangement such as is shown in FIGURE 17. A logical garnet of eight-cell cube-chips places components such as is shown in FIGURE 7 in-to the arrangement such as is shown in FIGURE 14a. For a more complex logical structure, a clock compass of logical garnets places components such as is shown in FIGURE 14a into the arrangement such as is shown in FIGURE 3. Even more complex, but still in reference to the two-valued logic of two sentences, it is meaningful to construct a third order logical garnet, for which the components such as is shown in FIGURE 14a are placed at the vertices of FIGURE 14a, which in turn is repeated 16 times and then stacked into a super-chip having the same arrangement such as is shown in FIGURE 14a.
All of these examples have one thing in common. Binary chips are the basic building blocks, on which the binary letters are the atoms that appear in the proper number and orientation.
Extension of the Notation and Combinations of Composite Chips Use of the binary chips, one for each of the sixteen binary connectives and all of them as a fundamental part of two-valued logic, need not be limited to two sentences (A,B). The same approach, such as has been described for the above embodiments, can also be extended to a universe of more than two sentences. For example, 256 arrangements of binary chips are needed to represent the 256 ternary connectives for three sentences (A,B,C). Likewise, 65,536 arrangements of binary chips are needed to represent the 65,536 quaternary connectives for four sen-tences (A,B,C,D). Even though arrangements of binary chips can be ex-tended to any number of sentences, the ordinary use of hand-operated manipulatory chips is especially convenient when the notational system presented herein is applied to any combination of four sentences. It would be possible to use wooden trays and mirrors, also machines, to cope with the rapidly growing number of connectives that go with a large number of sentences.
Ternary connectives are represented by ternary chips, for which pairs of binary chips are placed side by side. By this arrangement, 16 times 16 yields the 256 two-cell chips needed to represent all of the ternary connectives. For example, two o-chips placed side by side, that s~

is, a two-cell oo-chip, is the ternary chip for contradiction. The primitive rules for A-negation, B-negation, and counterchange are re-tained. For example, the xx-chip, also called tautology, is the coun-terchange mate of the oo-chip. C-negation is the added operation, for which the two binary chips are exchanged from left to right. For exam-ple, C-negation of the xd-chip is the dx-chip. Figure 18 further illus-trates how ternary connectives are embodied in the extended system of manipulatory chips. In this case, when negation (N) acts on conjunc-tion (od), it yields the corresponding counterchange mate (xh), such as is also shown as single operations on both sides of FIGURE 5a.
In like manner, when all combinations of (A,B,C)-negation and counterchange act upon the 256 ternary chips, the number obtained from 16 times 256 yields a complete table of 4096 basic transformations that exist in the two-valued logic of three sentences.
Quaternary connectives are represented by quaternary chips, for which pairs of ternary chips are placed one above the other. By this ar-rangement, 256 times 256 yields the 65,536 four-cell chips needed to rep-resent all of the quaternary connectives. For example, placing two oo-chips one above the other yields the quaternary chip for contradic-tion. This chip is a four-cell chip of o-letters, similar to what is shown at 9 o'clock in FIGURE 6, but which, as separable components, is now being used in another context for another purpose. Likewise, the separable four-cell of x-letters, similar to what is shown at 3 o'clock, is the corresponding counterchange mate. D-negation is the added opera-tion, for which the two ternary chips are exchanged from top to bottom.
For example, D-negation of a quaternary chip that had od-above and oo-below, which is the one for conjunction, is another quaternary chip that has oo-above and od-below. FIGURE 19 further illustrates how quaternary connectives are embodied in the extended system of manipulatory chips.
In this case, when negation (N) acts on conjunction (oood), it yields the corresponding counterchange mate (xxxh).
In like manner, when all combinations of (A,B,C,D)-negation and counterchange act upon the 65,536 quaternary chips, the number obtained from 32 times 65,536 yields a complete table of 2,097,152 basic trans-formations that exist in t5~e two-valued logic of four sentences.

7~3 It is thus apparent that the use of manipulatory chips can be extended to any number of sentences. Composite chips for n-ary connec-tives are arranged according to a simple rule. For an odd number of sentences, two (n-l)-ary composite chips are placed side by side, such as is shown in FIGURE 18; for an even number of sentences, two (n-l)-ary composite chips are placed one above the other, such as is shown in FIGURE 19. Some ingenuity is required to handle the large number of components in these patterns of composite chips.
Also in like manner, several physical models can be built for the logical structures that go with additional sentences. Although the above presentation has favored those models that embody the 128 basic transformations for two sentences, it will be appreciated that even more elaborate models can be custom-designed, in a similar fashion, to reflect the systems of composite chips for more than two sentences.

Claims (89)

The embodiments of the invention in which an exclusive property or priviledge is claimed are defined as follows:

1. An apparatus for displaying and performing operations upon a complete set of the sixteen binary connectives in a two-valued notational system, said apparatus comprising:
at least one device;
said at least one device having means displaying a plu-rality of symbols, wherein each symbol represents one of said binary connectives, said means having said plurality of symbols in a predeter-mined configuration, and when more than one device, said devices being movable and belonging to a set, and when used in a selected combination, said combination of movable devices capable of being assembled in a predetermined conformation;
each of said symbols having a shape selected to indicate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein said shape has iconicity, frame consistency, and eusymmetry with respect to said quadrants of Cartesian coordinates, and wherein the alignment of said shape is symmetry positional with respect to the x-y axes of said Cartesian coordinates;
and further wherein said shape is taken from at least six letter-shapes capable of generating 16 symbols that can be readily assigned a phonetic value, said symbols having four levels of symmetry such that two of said symbols are two-way self-flippable and self-rotatable, two of said symbols are not self-flippable but are self-rotatable, four of said symbols are one-way self-flippable but are not self-rotatable, and eight of said symbols are neither self-flippable nor self-rotatable; and said symbols, said configuration of said symbols, said set of movable devices, and said conformation of said devices selected so that said at least one device is, or said devices are, adaptable by transformations taken from the group consisting of reflections, rota-tions, translations, counterchanges, and combinations thereof to display and perform said operations.

2. The apparatus of claim 1 wherein each said at least six letter-shapes is a different shape selected from the font consisting of o, b, s, ? , ? , and x.

3. The apparatus of claim 1 wherein there are eight letter-shapes selected from the font consisting of o, b, s, z, n, ? , ? , and x.

4. The apparatus of claim 1 wherein there are sixteen letter-shapes selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ? , ? , ? , and x.

5. An apparatus for displaying and performing operations upon a complete set of the sixteen binary connectives in a two-valued notational system, said apparatus comprising:
one device having means displaying a plurality of sym-bols, each symbol representing one of said binary connectives, said means having said plurality of symbols in a predetermined configuration;
each of said symbols having a shape selected to indi-cate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein said shape has iconicity, frame consistency, and eusymmetry with respect to said quadrants of Cartesian coordinates, and wherein the alignment of said shape is symmetry positional with respect to the x-y axes of said Cartesian coordinates;
and further wherein said shape is taken from at least six letter-shapes capable of generating 16 symbols that can be readily assigned a phonetic value, said symbols having four levels of symmetry such that two of said symbols are two-way self-flippable and self-rotatable, two of said symbols are not self-flippable but are self-rotatable, four of said symbols are one-way self-flippable but are not self-rotatable, and eight of said symbols-are neither self-flippable nor self-rotatable; and said symbols and said configuration of said symbols selected so that said device is adaptable by transformations taken from the group consisting of reflections, rotations, translations, counter-changes, and combinations thereof to display and perform said operations.

6. The apparatus of claim 5 wherein said means includes pairs of areas such that each second area is opposed to each first area, said configuration selected such that first and second symbols are placed in said first and second areas and such that each second-area symbol is a component-matching repeat in see-through orientation of the corresponding first-area symbol.

7. The apparatus of claim 6 wherein each second-area symbol is a same-shape repeat in see-through orientation of said corresponding first-area symbol.

8. The apparatus of claim 5, 6, or 7 wherein each said at least six letter-shapes is a different shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ?, h, ?, ? , ?, and x.

9. The apparatus of claim 6 wherein said first-area symbols are arranged in a side-by-side relationship in a selected direction.

10. The apparatus of claim 6 wherein said first-area symbols are arranged from left to right in a selected permutation of the 16 combinations of the said four components of said letter-shapes.

11. The apparatus of claim 6 wherein said means comprise symbol-shaped openings that extend from each selected first area to its corresponding second area.

12. The apparatus of claims 9, 10, or 11 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ?, ?, ?, and x.

13. The apparatus of claim 5 wherein said means includes sets of four areas, said configuration in each set selected such that each of said symbols is placed in a first area and such that one of the x-axis, y-axis, and xy-axes component-matching flips of said each sym-bol is placed in each of the remaining areas of said set.

14. The apparatus of claim 13 wherein said four areas are visible from four selected orientations of said device and wherein the components of said first-area symbols are arranged in a selected permu-tation of the sixteen combinations of the four components of said letter-shapes.

15. The apparatus of claim 14 wherein said device comprises a tower and said first-area symbols are arranged in a vertical direc-tion and said four areas are visible each from a different compass direction.

16. The apparatus of claims 13, 14, and 15 wherein each of said symbols is a different letter-shape selected from the font consist-ing of o, p, b, q, d, c, u, s, z, n, ? , h, ?, ?, ?, and x.

17. An apparatus of claim 5 wherein said predetermined con-figuration includes four groups of symbols that have been placed in four groups of locations, said symbols and said locations coordinated such that, with respect to the symmetry introduced into a selected area by a specified set of x-y axes, the level of symmetry contained in each group of said locations matches the corresponding level of symmetry contained in the shapes of the corresponding group of said symbols, and further wherein said levels of symmetry contained in said locations are such that (1) two of said symbols in said first group are two-way self-flippable and self-rotatable, (2) two of said symbols in said second group are not self-flippable but are self-rotatable, (3) four of said symbols in said third group are one-way self-flippable but are not self-rotatable, and (4) eight of said symbols in said fourth group are neither self-flippable nor self-rotatable.

18. The apparatus of claim 17 wherein said first, second, third, and fourth groups of locations are selected such that the combi-nations of said components of said symbols are the same respectively as exemplified by the stem combinations of the symbols in the following four groups of letter-shapes: (1) o, x; (2) s, z; (3) c, u, n, ? ;
and (4) p, b, q, d, h, ?, ?, ?.

19. The apparatus of claim 17 wherein said four groups of symbols in said four groups of locations take on an appearance shaped like a bug, thereby showing (1) the o-thorax on an x-frame, (2) the sz-middle of said o-thorax, (3) the head, tail, and wings (u n ? c), and (4) the four legs (p b q d) along with the inside joints of these legs (h ? ? ? ).

20. The apparatus of claim 17 wherein said four groups of symbols in said four groups of locations are connected by lines to give the appearance of a network, thereby showing (1) the four arms of an x-grid radiating from an enlarged o-circle, (2) the sz-middle in said o-circle, (3) the one-way self-flippable symbols in proximity to the vertical and horizontal axes of said o-circle (u n ? c), and (4) the odd-component symbols in proximity to left and right diagonal axes selected with respect to the center of said o-circle (p b q d h ? ? ?).

21. The apparatus of claim 17 or 18 wherein said each area is the first of a pair of areas such that each second area is opposed to said each area, said configuration selected such that first and second symbols are placed in said first and second areas and such that each second-area symbol is a component-matching repeat in see-through orien-tation of the corresponding first-area symbol.

22. The apparatus of claim 17 or 18 wherein said each area is the first of a pair of areas such that each second area is opposed to said each area, said configuration selected such that first and second symbols are placed in said first and second areas and such that each second-area symbol is a same-shape repeat in see-through orientation of the corresponding first-area symbol.

23. The apparatus of claim 17 or 18 wherein said device is constructed such that each symbol has a non-planar shape, said shape occupying three dimensions, and said device including means for inter-connecting said three-dimensional shapes in said predetermined config-uration.

24. The apparatus of claim 5 wherein said device is at least one rhombic dodecahedron and wherein said means displaying said symbols in said predetermined configuration include means in proximity to the co-center and each of the vertices of said rhombic dodecahedron.

25. The apparatus of claim 24 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ?, ?, ?, and x.

26. The apparatus of claim 24 or 25 wherein said device com-prises said at least one rhombic dodecahedron and a plurality of like rhombic dodecahedra arranged in reference thereto, the arrangement and orientation of said plurality of like rhombic dodecahedra being deter-mined by the reflections obtained when at least one set of intersecting mirrors acts upon said at least one rhombic dodecahedron.

27. The apparatus of claim 24 or 25 wherein said device com-prises said at least one rhombic dodecahedron and seven like rhombic dodecahedra arranged in reference thereto, the arrangement and orienta-tion of said seven like rhombic dodecahedra being determined by the re-flections obtained when three intersecting mirrors act upon said at least one rhombic dodecahedron.

28. The apparatus of claim 24 or 25 wherein said means in proximity to said co-center and each of said vertices include at least one symbol and at least one piece of a set of sixteen chess pieces.

29. The apparatus of claim 24 or 25 wherein each said means in proximity to said co-center and each of said vertices includes pairs of areas such that each second area is opposed to each first area, said configuration selected such that first and second symbols are placed in said first and second areas and such that each second-area symbol is a component-matching repeat in see-through orientation of the correspond-ing first-area symbol.

30. The apparatus of claim 24 or 25 wherein each said means in proximity to said co-center and each of said vertices includes pairs of areas such that each second area is opposed to each first area, said configuration selected such that first and second symbols are placed in said first and second areas and such that each second-area symbol is a same-shape repeat in see-through orientation of the corresponding first-area symbol.

31. The apparatus of claim 24 wherein each said means in proximity to said co-center and each of said vertices includes a set of four areas, said configuration in each said set selected such that each of said symbols is placed in a first area and such that one of the x-axis, y-axis, and xy-axes component-matching flips of said each sym-bol is placed in each oF the remaining areas of said set.

32. The apparatus of claim 31 wherein each said set of four areas is arranged in quadrants.

33. The apparatus of claim 31 wherein each said means in proximity to said co-center and each of said vertices includes first and second sets of four areas, each symbol visible from said second set of four areas being a counterchange mate respectively of a corresponding symbol visible from said first set of four areas.

34. The apparatus of claim 31, 32, or 33 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ?, ?, ?, and x.

35. The apparatus of claim 5 wherein said device is at least one tesseract and wherein said means displaying said symbols in said predetermined configuration include means in proximity to each of the sixteen vertices of said tesseract.

36. The apparatus of claim 35 wherein said device comprises said at least one tesseract and a plurality of like tesseracts arranged in reference thereto, the arrangement and orientation of said plurality of like tesseracts being determined by the reflections obtained when at least one set of intersecting mirrors acts upon said at least one tesseract.

37. The apparatus of claim 35 or 36 wherein each of said sym-bols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ?, ?, ?, and x.

38. An apparatus for displaying and performing operations upon a complete set of the sixteen binary connectives in a two-valued notational system, said apparatus comprising:
a plurality of devices assembled in a predetermined conformation;
each of said devices having means displaying a plurality of symbols, wherein each said symbol represents one of said binary con-nectives, said means having said plurality of symbols in a predetermined configuration;
each of said symbols having a shape selected to indicate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein said shape has iconicity, frame consistency, and eusymmetry with respect to said quadrants of Cartesian coordinates, and wherein the alignment of said shape is symmetry positional with respect to the x-y axes of said Cartesian coordinates;

and further wherein said shape is taken from at least six letter-shapes capable of generating 16 symbols that can be readily assigned a phonetic value, said symbols having four levels of symmetry such that two of said symbols are two-way self-flippable and self-rotatable, two of said symbols are not self-flippable but are self-rotatable, four of said symbols are one-way self-flippable but are not self-rotatable, and eight of said symbols are neither self-flippable nor self-rotatable; and said devices, said configuration of said symbols, and said conformation of said devices selected so that said devices are adaptable by transformations taken from the group consisting of re-flections, rotations, translations, counterchanges, and combinations thereof to display and perform said operations.

39. The apparatus of claim 38 wherein each said means of each said device displaying said symbols comprises four areas, said config-uration selected such that one of said symbols is placed in a first area and such that one of the x-axis, y-axis, and xy-axes component-matching flips of said one symbol is placed in each of the remaining said four areas.

40. The apparatus of claim 39 wherein said predetermined con-formation comprises said devices being juxtaposed in an end-to-end re-lationship in a selected direction such that said four areas are visi-ble from four selected orientations of each of said devices, and where-in the letter-shape components of said first-area symbols are arranged in a selected permutation of the sixteen combinations of the four com-ponents of said letter-shapes.

41. The apparatus of claim 38, 39, or 40 wherein each said at least six letter-shapes is a different shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ? , ?, ?, and x.

42. The apparatus of claim 40 wherein said devices comprise a plurality of cubes and said end-to-end relationship is in the vertical direction, said four areas visible from the four sides of each of said cubes.

43. The apparatus of claim 38 wherein each said means of each said device includes one of five subsets of first-area symbols, and wherein each of said subsets of said symbols has an equal number of from zero to four letter-shape components.

44. The apparatus of claim 43 wherein said configuration in each of said five subsets includes a second-area symbol opposed to each of said first-area symbols such that each second-area symbol is a component-matching repeat in see-through orientation of the correspond-ing first-area symbol.

45. The apparatus of claim 43 wherein said configuration in each of said five subsets includes second-area, third-area, and fourth-area symbols obtained respectively from the x-axis, y-axis, and xy-axes component-matching flips of each corresponding said first-area symbols.

46. The apparatus of claim 43, 44, or 45 wherein said first-area symbols are different letter-shapes selected from the font consist-ing of the following subsets: (1) o; (2) p, b, q, d; (3) c, u, s, z, n, ? (4) h, ?, ?, ? ; and (5) x.

47. The apparatus of claim 44 or 45 wherein said conformation comprises said five subset devices assembled in a side-by-side relation-ship in a selected direction, wherein said five subsets are arranged in an increasing or decreasing sequence of said four components, and fur-ther wherein said area symbols are visible from selected orientations of each of said devices.

48. The apparatus of claim 44 or 45 wherein said conformation comprises 16 devices, each device displaying one of said first-area sym-bols, said devices assembled in an end-to-end relationship in a selected direction, said devices further assembled in said five subsets in an increasing or decreasing sequence of said four components, and wherein said area symbols are visible each from a different compass direction.

49. The apparatus of claim 38 wherein said devices comprise a recessed device having a plurality of symbol-shaped recesses therein and a plurality of movable symbol-shaped devices having corresponding sizes and shapes of said symbol-shaped recesses.

50. The apparatus of claim 49 wherein said recesses are open-ings extending completely through said recessed device.

51. The apparatus of claim 49 or 50 wherein said symbol-shaped recesses in said recessed device are arranged in a selected per-mutation of the sixteen combinations of said four components of said letter-shapes.

52. The apparatus of claim 42, 49, or 50 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ? , ? , ?, and x.

53. The apparatus of claim 38 wherein said conformation of said devices comprises a device assembled in proximity to the co-center and each of the vertices of at least one rhombic dodecahedron, said de-vices located in said rhombic dodecahedron by interconnecting means in-cluding means permitting individual rotational movement of said devices.

54. The apparatus of claim 53 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ? , ? , ?, and x.

55. The apparatus of claim 53 or 54 wherein each said device in proximity to said co-center and each of said vertices includes pairs of areas such that each second area is opposed to each first area, said configuration selected such that first and second symbols are placed in first and second areas and such that each second-area symbol is a component-matching repeat in see-through orientation of the corresponding first-area symbol.

56. The apparatus in claim 53 or 54 wherein each said device in proximity to said co-center and each of said vertices includes a set of four areas, said configuration in each said set selected such that each of said symbols is placed in a first area and such that one of the x-axis, y-axis, xy-axes component-matching flips of said each symbol is placed in each of the remaining areas of said set.

57. The apparatus of claim 53 or 54 wherein each said device in proximity to said co-center and each of said vertices includes first and second sets of four areas, each symbol visible from each area in said first set being no change, x-axis, y-axis, and xy-axes component-matching flips of a first-area symbol, and each symbol visible from each area in said second set being the component-matching counterchange mate of a corresponding symbol in said first set.

58. The apparatus of claim 53 or 54 wherein said conformation comprises said at least one rhombic dodecahedron and a plurality of like rhombic dodecahedra arranged in reference thereto, the arrangement and orientation of said plurality of like rhombic dodecahedra being deter-mined by the reflections obtained when at least one set of intersecting mirrors acts upon at least one rhombic dodecahedron.

59. The apparatus of claim 53 or 54 wherein said conformation comprises said at least one rhombic dodecahedron and seven like rhombic dodecahedra arranged in reference thereto, the arrangement and orienta-tion of said seven like rhombic dodecahedra being determined by the re-flections obtained when three intersecting mirrors act upon said at least one rhombic dodecahedron.

60. The apparatus of claim 38 wherein said plurality of de-vices comprises at least one rhombic dodecahedron displaying said con-figuration of said symbols and at least one set of intersecting mirrors, said rhombic dodecahedron oriented with respect to said mirrors such that images of a plurality of said rhombic dodecahedron is visibly gen-erated by the reflections obtained when said mirrors act on said rhombic dodecahedron.

61. The apparatus of claim 60 wherein said plurality of de-vices comprise said at least one rhombic dodecahedron and three inter-secting mirrors, said rhombic dodecahedron being oriented with respect to said three mirrors such that images of seven rhombic dodecahedra are visibly generated by the reflections obtained when said three mirrors act on said at least one rhombic dodecahedron.

62. The apparatus of claim 38 wherein said conformation of said devices comprises a device assembled in proximity to each of the sixteen vertices of at least one tesseract, said devices located in said tesseract by interconnecting means including means permitting individual rotational movement of said devices.

63. The apparatus of claim 62 wherein said conformation com-prises at least one tesseract and a plurality of like tesseracts ar-ranged in reference thereto, the arrangement and orientation of said plurality of tesseracts being determined by the reflections obtained when at least one set of intersecting mirrors acts upon said at least one tesseract.

64. The apparatus of claim 38 wherein said plurality of de-vices comprise at least one device displaying said configuration of said symbols and at least one set of intersecting mirrors, said device oft-ented with respect to said mirrors such that images of a plurality of said device is visibly generated by the reflections obtained when said mirrors act on said device.

65. The apparatus of claim 62, 63, or 64 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ? , ? , ?, and x.

66. An apparatus for displaying and performing operations upon a complete set of the sixteen binary connectives in a two-valued system, said apparatus comprising:
a set of selectively movable devices, each of said devices having means displaying at least two symbols, wherein each said symbol represents one of said binary connec-tives, said means having said at least two symbols in a predetermined configuration;
each of said symbols having a shape selected to indicate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein said shape has iconicity, frame consistency, and eusymmetry with respect to said quadrants of Cartesian coordiantes, and wherein the alignment of said shape is symmetry positional with respect to the x-y axes of said Cartesian coordinates, and further wherein said shape is taken from at least six letter-shapes capable of generating 16 symbols that can be readily assigned a phonetic value, said symbols having four levels of symmetry such that two of said symbols are two-way self-flippable and self-rotatable, two of said symbols are not self-flippable but are self-rotatable, four of said symbols are one-way self-flippable but are not self-rotatable, and eight of said symbols are neither self-flippable nor self-rotatable; and said at least two symbols and said configuration of said at least two symbols selected so that said selectively movable devices are adaptable by selected assembly and by transformations taken from the group consisting of reflections, rotations, translations, counterchanges, and combinations thereof to display and perform said operations.

67. The apparatus of claim 66 wherein each selected device in-cludes two symbols in said predetermined configuration such that said two symbols are placed in first and second areas and such that the second-area symbol is a y-axis component-matching flip of the corresponding first-area symbol.

68. The apparatus of claim 66 wherein each selected device includes two symbols in said predetermined configuration such that said two symbols are placed in first and second areas and such that the second-area symbol is an x-axis component-matching flip of the corre-sponding first-area symbol.

69. The apparatus of claim 66 wherein each selected device includes two symbols in said predetermined configuration such that said two symbols are placed in first and second areas and such that the second-area symbol is an xy-plane component-matching half-turn rotation of the corresponding first-area symbol.

70. The apparatus of claim 67, 68, or 69 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ?, ?. ?, and x.

71. The apparatus of claim 66 wherein each selected device includes two symbols in said predetermined configuration such that said two symbols are placed in first and second areas and such that the second-area symbol is a counterchange mate of the symbol visible from the corresponding first area.

72. The apparatus of claim 66 wherein each selected device includes four symbols in said predetermined configuration such that said four symbols are placed in a set of four areas, such that a first of said symbols is placed in a first area, and such that one of the x-axis, y-axis, and xy-axes component-matching flips of said first-area symbol is placed in each of the remaining areas of said set.

73. The apparatus of claim 66 wherein each selected device includes eight symbols in said predetermined configuration such that said eight symbols are placed in first and second sets of four areas, each symbol visible from each area in said first set being no change, x-axis, y-axis, and xy-axes component-matching flips of a first area symbol, and each symbol visible from each area in said second set being a coun-terchange mate of a corresponding symbol in said first set.

74. The apparatus of claim 72 or 73 wherein said areas of each said device are visible each from a different direction.

75. The apparatus of claim 72 or 73 wherein there are at least sixteen devices and each device includes a shape representing one of the shapes in a set of sixteen chess pieces.

76. The apparatus of claim 71, 72, or 73 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ?, ?, ?, and x.

77. The apparatus of claim 72 wherein said four areas are further arranged in a side-by-side relationship in a selected direction.

78. The apparatus of claim 72 wherein said four areas are fur-ther arranged in quadrants.

79. The apparatus of claim 66 comprising four devices assem-bled in quadrants, one device in each quadrant, each said device includ-ing a set of four areas also arranged in quadrants and each of the six-teen areas having one of said symbols, wherein said four devices are selectively movable past each other in side-by-side pairs to display four basic arrangements of said four devices, wherein each of the four symbols at the outside quadrant areas in each of said four basic ar-rangements becomes the first member in an orderly progression of changes in the letter-shape components of said symbols (0, 1, 2, 3, 4), and fur-ther wherein the pattern across each said progression always occurs in the same binomial frequency of said symbols (1, 4, 6, 4, 1), whereby said apparatus in said four basic arrangements displays said pattern of component changes by which a selected symbol differs from all of said symbols, including said selected symbol, which always appears as said first member in said progression of component changes.

80. The apparatus of claim 79 wherein said four devices com-prise four blocks, said four blocks selectively placed in said four basic arrangements to display said sixteen progressions, such that each pro-gression (a) begins at one of the four outside corner positions of a clock-compass configuration (at 3, 6, 9, or 12 o'clock), (b) passes across said binomial frequency of said symbols (1, 4, 6, 4, 1), and (c) goes to the opposite corner position of said clock-compass configuration.

81. The apparatus of claim 80 wherein said quadrant area sets of symbols, one said area set visible from each of said four blocks, are sorted into the same arrangement as the four columns of quadrant area sets in the following clock-compass configuration of letter-shapes:
(1) (2) (3) (4) 2 - ? , 5 - h, 8 - b, 11 - d, 3 - x, 6 - ? , 9 - o, 12 - c, 4 - ? , 7 - p, 10 - q, 1 - ? , E - s; S - n; W - z; N - u.

82. The apparatus of claim 66, 79, or 80 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ?, ?, ?, and x.

83. An apparatus for displaying and performing operations upon a complete set of the sixteen binary connectives in a two-valued notational system, said apparatus comprising:
a set of at least six movable devices, said devices capa-ble of being used in a selected combination, said combination also capa-ble of being assembled in a predetermined conformation;
each of said devices having means displaying a plurality of symbols wherein each said symbol represents one of said binary con-nectives, said means having said plurality of symbols in a predetermined configuration;

each said means for displaying including a pair of areas such that a second area is opposed to a first area, said configuration selected such that first and second symbols are placed in said first and second areas and such that second-area symbol is a component-matching repeat in see-through orientation of first-area symbol;
each of said symbols having a shape selected to indicate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein said shape has iconicity, frame consistency, and eusymmetry with respect to said quadrants of Cartesian coordinates, and wherein the alignment of said shape is symmetry positional with respect to the x-y axes of said Cartesian coordinates;
and further wherein said shape is taken from at least six letter-shapes capable of generating 15 symbols that can be readily assigned a phonetic value, said symbols having four levels of symmetry such that two of said symbols are two-way self-flippable and self-rotatable, two of said symbols are not self-flippable but are self-rotatable, four of said symbols are one-way self-flippable but are not self-rotatable, and eight of said symbols are neither self-flippable nor self-rotatable; and said symbols, said configuration of said symbols, and said conformation of said devices selected so that said combined movable devices are adaptable by assembly and by transformations taken from the group consisting of reflections, rotations, translations, counterchanges, and combinations thereof to display and perform said operations.

84. The apparatus of claim 83 wherein there are at least 16 devices and each device includes a shape representing one of the shapes in a set of sixteen chess pieces.

85. The apparatus of claim 83 wherein each one of said devices is an object having the shape of one of said letter-shape symbols.

86. The apparatus of claim 83, 84, or 85 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ? , ? , ?, and x.

87. The apparatus of claim 83 wherein said devices form a set of sixteen chess pieces such that the shapes of (1) the Queen and King, (2) the Rooks, (3) the Knights and Bishops, and (4) the Pawns are re-spectively selected from the font of letter-shapes displayed by the following four groups of letter-shapes: (1) o, x; (2) s, z; (3) c, u, n, ? ; and (4) p, b, q, d, h, ? , ? , ? .

88. An apparatus for displaying and performing a selected set of the 2(2n) n-ary connectives obtained from a finite number of elements (A, B, C, ... n) in a two-valued notational system, said apparatus com-prising:
a set of movable devices, said devices capable of being used in a selected combination, said combination also capable of being assembled in a predetermined conformation;
each of said devices having means displaying a plurality of symbols, wherein each said symbol represents one of said binary con-nectives, said means having said plurality of symbols in a predetermined configuration;
each said means for displaying including a pair of areas such that second area is opposed to first area, said configuration se-lected such that first and second symbols are placed in said first and second areas and such that second-area symbol is a component-matching repeat in see-through orientation of first-area symbol;
each of said symbols having a shape selected to indicate a selected number of from zero to four components arranged with respect to the quadrants of a set of Cartesian coordinates, wherein said shape has iconicity, frame consistency, and eusymmetry with respect to said quadrants of Cartesian coordinates, and wherein the alignment of said shape is symmetry positional with respect to the x-y axes of said Cartesian coordinates;
and further wherein said shape is taken from at least six letter-shapes capable of generating 16 symbols that can be readily assigned a phonetic value, said symbols having four levels of symmetry such that two of said symbols are two-way self-flippable and self-rotatable, two of said symbols are not self-flippable but are self-rotatable, four of said symbols are one-way self-flippable but are not self-rotatable, and eight of said symbols are neither self-flippable nor self-rotatable; and said symbols, said configuration of said symbols, and said conformation of said devices selected so that said devices are adaptable by rearrangement and by transformations taken from the group consisting of reflections, rotations, translations, counterchanges, and combinations thereof to display and perform said operations.

89. An apparatus of claim 88 wherein each of said symbols is a different letter-shape selected from the font consisting of o, p, b, q, d, c, u, s, z, n, ? , h, ? , ? , ?, and x.