follows we will let the logic alphabet focus on four rules. By design, these ... their numerals, did use an abacus;аthey did not, however, have a slide rule. Modern ...
Let The Mirrors Do The Thinking and Let The People Do The Reflecting
By Shea Zellweger 2008
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Introduction
Our story begins with a simple example. Suppose that someone asked you to keep a record of your thoughts, exactly, and in terms of the symbols given, when you are making an effort to multiply XVI times LXIV. Also suppose that, refusing to give up, you finally arrive at the right answer, which happens to be MXXIV. We are sure that you would have had a much easier time of it, to solve this problem, if you would have found that 16 times 64 equals 1024.
This example not only looks at what we think and what we write. It also looks at the mental tools, the signs and symbols, that we are using when that thinking and that writing is taking place. How we got these mental tools is a long story, one that now includes the presence of some new developments. Our main idea comes from calling attention to a deep commonality that cuts across the parallel streams of development that in recent millennia have unfolded in the ways and byways of evolutionary notation. It took many centuries of collective search to devise a placevalue notation for counting. Likewise to devise a soundvalue notation for reading. Likewise to devise a notevalue notation for singing. And so forth, for each neurologically specialized ability; in effect, a different specialized notation for each specialized ability. These observations, easily recognized in the history of evolutionary notation, strongly suggest that every kind of intelligence needs its own kind of notation. In what follows, with emphasis on a fastforward recapitulation, we will run a replay of what happened when Europe took several centuries to go from MXXIV to 1024. This replay in not for numbers. It is for another specialized ability. It is for logic, when it is recast in terms of a shape value notation. Modern logic starts in the middle of the 1800s and, as is well recognized, with the work of George Boole. This means that we have had only about 150 years to think up and to grow into the symbols we now use for symbolic logic. These symbols, and they are only symbols, leave a lot to be desired. We hope that we can draw you into taking a look at a lesson in lazy logic. If you follow us all the way, we hope to leave you with a new set of signs, signs that are much better than any you have seen yet. Not only will it be much easier for you to use them. Even mirrors will be able to use them.
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From Roman Numerals to Arabic Numerals Roman numerals start with some of the number values, such as I, V, X, and then the right combinations are used to express any of the inbetween values, such as IV, VII, IX. Modern logic does the same thing when it starts with a few heavyduty connectives, such as "and" (TFFF), "or" (TTTF), "if" (TFTT), and then "not" (Negation) and "equivalence" (Equivalence) are added to this mix to express the other relations in the special 16set of (A, B) relations (1 4 6 4 1). Roman numerals are loaded with difficulties because they do not lay bare, in any transparent way, the interrelations among the number values. Notice, instead, that we use Arabic numerals when we build a multiplication table. When modern logic uses (dot, vee, horseshoe) to express (and, or, if), it does not lay bare the networks of interrelations that are a fundamental part of the 16 binary connectives taken as a total system. Unfortunately, in contrast to what we will see later, the state of the art is such that symbolic logic continues to remain many, many miles away from coming up with its own multiplication table. Now continue with the comparison. Begin by looking upon Arabic numberals so that they become a primary set of ten distinct and separate abacus settings that have been carried to the mental level (09). Likewise, now look upon the Xstem Logic Alphabet (XLA) so that it becomes a primary set of 16 coded, iconic Venn truth tables that have been carried to the mental level (ox). What a paradox! The disparity in the extremes could not be more disturbing. It is obvious that we have such high standards for number symbols; in effect, Arabic numerals. But, squirm as we will, wiggle as we may, it is very odd indeed that the standards we are now using for logic signs continue to remain so much lower. The next move is obvious. Take what we already know about Arabic numberals and do no more than repeat it for a new notation for logic. By now the challenge should be clear. It continues to emphasize the working analogy. Roman numerals are to Arabic numerals as the symbols in common use for the binary connectives, such as (dot, vee, horseshoe) for (and, or, if), are to what? What follows will introduce you to the "Xstem Logic Alphabet." Then comes the part about the mirrors.
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Peirce as the Primary Precursor of the Logic Alphabet In passing realize that in 1902 Charles Sanders Peirce (18391914), American philosopher and logician, devised a notation that is like the logic alphabet. His manuscripts, the Simplest Mathematics (MSS 430, 431, 429), were published (1933) only in selected parts. This was done in such a way that his notation was desymbolized and recast so that its shapevalue properties were lost. Today these manuscripts with few exceptions are totally ignored. Even though the second author devised the Logic Alphabet a full decade before seeing what Peirce had done, the development of the logic alphabet is best thought of as a direct continuation of Peirce's 1902 boxX notation for all of the 16binary connectives. Logic for Peirce was strong in the logic of relations and, likewise, what follows will be relationstheory generated. Logic for Peirce was also centered in what he established as the study of signs and sign action, a study that has been given modern form, now called semiotics (also spelled semeiotics). Looking again at our working analogy, it is obvious that, when we go from Roman numerals to Arabic numerals, the key step lies in becoming acquainted with the code that goes with the new set of numerals. Such a key step repeats in what follows. But first, the 4by4 pattern shown above is one representation of the logic alphabet.
The 16 signs in this shapevalue notation, which could also be called a positionvalue notation, are placed in a standard configuration that is called a ClockCompass. Twelve positions are placed around the outside and the four directions are in the middle. This means that the same sign will always be placed in the same position. For example, the dletter is always at 11 o'clock and the hletter is always at 5 o'clock, both of which are located symmetrically across the center of the clockcompass. Note that Peirce used just the compass image when he described his own 4by4 configuration. As already mentioned, the construction, development, and use of the logic alphabet as a total system is best thought of as a direct continuation of what Peirce accomplished when he devised his boxX notation in 1902. previous page | next page
ThreeWay Signs that Cofunction as Index, Icon, and Symbol Let LS stand for Letter Shape and then go directly to (A and B), also written as a pattern of four finger pairs (A TFFF B). Now use this connective as a model for what we have in mind for all of the 16 connectives. This takes a big step. It treats all of the connectives as a total system, with all parts present and no parts inactive. In (1), begin by seeing that (A and B) is equivalent to itself, (A and B). In (2), the pair of fingers in which the Touching takes place puts a T in the upperright corner. In (3), a
basic square puts an enlarged doT in the same corner. In (4), the LS in (A d B) has a sTem in the same corner. By design, also centered in Peirce and his threeway emphasis on sign making (index, icon, symbol), note especially that (2), (3), and (4) have become the main parts in a triple isomorphism, Sowa (1998) and Shin (2002) come the closest to this analysis. In brief, when the logic alphabet was devised in 196162, it became one more logic in an infinite variety of logics, and now it continues to function as a fully developed multimodal representation system. The "and" in (1) is repeated three times: in (2) as a square truth table, in (3) as a matching coded dot square, and in (4) as a corresponding squareoutlined stem shape. The dletter is a simple cursive that has enough teamwork built into its carefully selected subareas so that, as needed, it is geometrically capable of being inserted to meet any one of the signobject conditions in (2), (3), and (4). Consequently, all in one stroke and also in terms of Peirce's three classes of signobject relations, (i) the dletter in reference to the square truth table is an index; (ii) in reference to the dotsquare, it is an icon; and (iii) in reference the concept called conjunction, it is a symbol. When all three are conjoined and carried in the same multimodal channel, this indexical, iconic, symbolic dletter is not only serving as a threeway compound for conjunction (A TFFF B). It also runs headon into a nmemonic accident. The last letter of the word "and" happens to be the same dletter that appears in (A d B). The external code (below) puts the finger pairs (TT, TF, FT, FF) at the four corners of the allcommon basic square. This groundform repeats the order in common use for the plusminus quadrants in Cartesian coordinates. The internal code puts a sTem so that it comes close to Touching a corner, when the finger pair in the same corner is counted as True. See that the hletter (FTTT) has stem positions that are opposite exactly to where the dletter (TFFF) does and does not have stems. These opposites (h, d) are at both ends of the word "hand." In (5), by way of a special act of abstraction, like an eagle off to the sky, we come to the asterisk in general. It has become an algebraic symbol. Like an x when it stands for any number, the asterisk stands for any of the 16 binary connectives.
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SelfSame Transformations for Symmetry Operations, Mental Operations and Logical Operations The logic alphabet is a shapevalue notation. It has 16 signs, all cursives, arranged from left to right, as shown on a Flipstick. Again, to repeat the same pattern (1 4 6 4 1), the oletter has no stems (all zeros) and the xletter has four stems (all bentin ones). These LSs have been selected with great care so that, as a total set, all of it is now serving as a new notation. It has been shapespecified to encode some fundamental properties in algebra, in geometry, in symmetry, and most of all, in logic. As a matter of high fidelity and in keeping with the requirements of good fit, the logic alphabet is very sensitive to the networks of interrelations that exist in layers throughout the 16 binary connectives. The logic alphabet introduces a shapevalue alphabet that does its best to favor what happens when we activate logical operations. This is what makes the LSs like Arabic numerals. When we multiply Arabic numerals, we use symbols in such a way that the rules fit the calculations. Likewise, in what follows we will let the logic alphabet focus on four rules. By design, these rules will fit the logical operations. These rules introduce the game called flipmateflip and flip. This game, also called fmf and f, is a shorthand way of specifying how the four rules will operate on any LS in particular, such as (A d B), or on all of the LSs in general (A * B). R is for Rule, N is for Negation, and C is for Conversion. The C, also called Commuting, reverses the two sides, from (A, B) to (B, A), and vice versa.
The four symmetry rules are as follows. R1 negates A. The NA flip is from left to right; (NA d B) changes to (A b B). R2 negates the LS. N* is the mate of any LS because all stem places are reversed; (A Nd B) changes to (A h B). This tells us that the end letters of "hand" are mates; also, the hletter is the Nand gate, sometimes expressed as the S(h)effer stroke (A / B). R3 negates B. The NB flip is from top to bottom; (A d NB) changes to (A q B). R4 converts (A, B). This flip is along the dotsquare diagonal that goes from upperright to lowerleft; (A d B) remains (B d A) because the dot in the dletter dotsquare in (3) stays in place. This example is testing the system. Obviously, under conversion, (A d B) must remain (B d A) because conjunction (A TFFF B) is commutative. One example will hint at what usually remains unsaid. It is inserted here to indicate how much sign engineering and architectural design went into the selection of the LSs. Those who use this notation may never notice that a super truth table (2 2 4 8) has been built into the SelfFlippable (SF) Self Rotatable (SR) construction of the LSs. In other words, there are four levels of symmetryasymmetry in the LSs. Extremely clarifying and visually uncanny, the four levels happen to line up exactly with the four levels of power in a chess set, when one chess piece is assigned to each LS. Not only introduced as stemcoded truth tables and not only serving as relational icons, the LSs are also topological cursives, at least to the extent that they must bend and twist, and stem, in just the right places or the system will not work. Two of the LSs are twoway SF and SR. Two of them are not SF but they are SR. Four of them are oneway SF but not SR. And eight of them are neither SF nor SR. Which LS belongs in which subset will remain an exercise. Likewise for specifying which chess pieces go in which subsets (2 2 4 8).
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Mirror Logic and the Algebra of Abstract Groups All of this runs head on into the algebra of abstract groups. All of this forces a certain kind of behavior on the logic alphabet. The flipmateflip and flip rules will become four mirrors and, under the four rulegoverned mirrors, the 16 LSs will be forced to act all alike. This will lead to consequences that are deep and very rich. What follows gives only a small part of the story. Now we have it! Table I is the new multiplication table. Akin to Arabic numerals and the use of zero, the 4tuple (OOOO) is acting both as the rulefree condition and as a combinatory identity element. The 16 combinations of the four rules along the side (N, N, N, C), which line up with the 16 combinations of the four mirrors, fmf and f in that order, are acting on all of the 16 LSs across the top (A * B). This (16 x 16) table is loaded with transformational symmetry. It contains layers of abstract structure. Rarely in a lifetime do we stand in enough light to become aware of the underlying networks of interrelations, always present, usually ignored, and now laid bare. Again, without exception, all and every one of the 16 LSs must obey the same symmetry rules. An easy way to learn about Table I is to notice that Arabic numerals encourage the use of models, such as an abacus and a slide rule. The Romans, in spite of their numerals, did use an abacus; they did not, however, have a slide rule. Modern logic, even more limited than the Romans, does not use handheld models with which to make logical calculations. In contrast, the logic alphabet is a notation that stumbles all over itself, so many are the models.
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Flipstick and ClockCompass
A standard row of LSs (lower left above) occupies the homerow of a model that is called a Flipstick. It repeats the first row of Table I, and it has another set of LSs on its backside, in seethrough placement. R1 flips it to the right, R3 flips it upward, and (R1, R3) rotates it around to the upper right. Note especially that a flipstick, taken as a single unit, is acting on all of the LSs, on all of them at the same time and on all of them in the same way. Again, all of this releases the transformational power contained in a small number of symmetry rules (fmf and f).
The diamondshaped 4by4 pattern on the left is called a ClockCompass. This view of the logic alphabet has 12 LSs around the outside, at the positions of a clock, and 4 LSs in the middle, at the directions of a compass. The eight oddstemmed LSs ( 4 4 ) are tall and black. The eight even stemmed LSs (1 6 1) are squat and white. The four LSs that have been added at the corners repeat the onestemmed cursives. The changes in these positions are isomorphic to what happens when combinations of (R1, R3) act on the subset (p, b, q, d). In other words, the symmetry operations for the combinations of (R1, R3), also called the Klein 4group for (N – N ), are acting on this 4set of LSs. previous page | next page
Logic Bug and Logical Garnet The composite display on the right is called a Logic Bug. It is a 2D arrangement of the 16 LSs. It comes from pushing nonstop against a headon view of the Logical Garnet (below) until it becomes as flat as a pancake. Like the 1D arrangement on the Flipstick, it can also be subjected to the identityflipfliprotate symmetry changes. Right here the logic bug is telling us that a symmetry operation that performs a logical calculation has become something very special and very powerful. Not only is it mirror driven. It is also making a symmetry calculation. "And that," said the Logic Bug, fully in the tradition of Froedel, Montessori and Piaget, "is precisely what I have been designed to do."
Now we can make a statement for R2. Look symmetrically across the center in these models to find the mate of any LS. The 16 LSs in the Logical Garnet (below) have been placed at the vertices of a special polyhedron, one that takes the shape of a rhombic dodecahedron. In a league all by itself, this model is called a shadow rhombic dodecahedron. In other words, the 16 vertices of a 4cube have been shadowed, or projected, into 3D. Consequently, this model absorbs the flipmateflip and flip symmetry properties that have been built into the LSs and that now appear in Table I. These are the same symmetry properties that isomorphically line up with the networks of interrelations that inhabit the 16 binary connectives. In other words, and this is a significant point, the same symmetry calculations are activated when the symmetrydriven LSs isomorphically circulate throughout the networks of interrelations.
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Mirror Logic and Networks of Abstract Graphs The figures on this page go with the top half of Table I. In another context, the network on the right also appears in Ramon Lull (1235? 1315). The network on the right is for the odd stemmed LSs ( 4 4 ); the network below is for the evenstemmed LSs (1 6 1). The eight LSs on the right occupy the vertices of a cube and the eight below are at the vertices of an octahedron (two at the cocenter). These two geometric solids are seen as 3D objects that interpenetrate in the A symmetry approach to the 16 binary Logical Garnet. It follows that the connectals should leave no doubt that we Logical Garnet becomes a key player have a very unusual tiger by the tail. because it is fully able to let 3D Another insight comes from realizing that mirrors act on a 3D arrangement of a Arabic numerals happen to be an abacus 4D count of the 2D signs, all of them that has been lifted to the mental level. serving as a special 16set of Likewise, the LSs are small VennCarroll connectals (not numerals). The same Cartesian diagrams raised to the mental set of stem shapes appears in the dot level. Ordinary Venn diagrams are cement squares on the Clock Compass. These heavy stuck to the page. So are the lines configurations happen to display the and squares that frame an ordinary logic alphabet in a particular way, but, crossword puzzle. In contrast, using the once we know what to do, there are logic alphabet is using a shape alphabet in a many ways to embody the same spatial logic. It is even more fluid and primary structure in a large family of flexible than playing a game of ordinary handheld models. scrabble. When the LSs operate at the mental level, the logical form under consideration becomes the scrabble frame. Then the LSs become a primitive set of VennCarrollCartesian fragments. Then the LSs are pushed around in a thought world open to variations in the logical structure. This repeats another way that Arabic numerals are superior to Roman numerals. The act of writing the LSs has become an integral part of making symmetry calculations, thereby introducing
another way to engage in computational writing.
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An EightCell of Logical Garnets
What happens when a Logical Garnet becomes a basic unit that is used to display more complex structure? The three mirrors (for flipmateflip) could be put at right angles to each other. Another way to set up a vector space is to let R2 go through the center, a center of inversion, which forces duality (NNN) to become the 3rd dimension (front to back). This has been done for the 8cell of Logical Garnets shown below. This model is for the 128 cells in the top half of Table I. Taken from the first row of Table I, the identity garnet (OOO) has been placed in the dashlined rectangle. The other seven garnets are now in perfect alignment with the other seven rows in the top half of Table I. Also seen as the product of (N, N, N, ) times (A * B), the eight 16sets of LSs in this model come from taking the product of two Boolean lattices, in this case from the product of the operations (1 3 3 1) times the connectals (1 4 6 4 1). We want no part of misplaced axiomatics. Along with emphasizing the importance of notational primitives, we do not try to isolate a small number of axiomatic primitives (such as "and", "nand", "nor") that will best cut into the abstract structure that contains the product of the Boolean lattices. Note that R4 adds a diagonal mirror to the symmetry operations. This introduces the bottom half of Table I, which will then include exactly all of the 256 cells (16 by 16) in the new multiplication table.
The wellordered complexity in Table I deserves to be made more explicit. This table is obtained from (1 4 6 4 1) times (1 4 6 4 1). The product of these two Boolean lattices can also be seen as what happens when a symmetrybased 16set obtained from the four combinatory logical operations along the left side (N, N, N, C) is acting on a customselected symmetrydesigned 16 set of notational primitives, namely, the special set of binary connectals across the top (A * B), also called the logic alphabet. In effect, this line of system building, starting with a 16set of notational primitives, is a readymade introduction to Clifford algebras.
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A Sunburst of FortyEight Logical Garnets The eight rows in the top half of Table I generate the 8cell of Logical Garnets. A model for the 16 rows in all of Table I would have 16 garnets. This covers ordinary infix (A * B), so that the 16 garnets would appear (8, 8) on the front and back faces of a large 3cube. A model that allows the asterisk to be placed in three positions generates a table that has 48 rows. Along with (A * B), the three positions include (* AB) and (AB *). In other words, the sunburst absorbs the product of all combinations and all permutations of the three elements in (A * B). This product is called a wreath product. Specifically, (8 x 6) activates 48 symmetry regions located eight each on the six faces of the large 3cube, the same large 3cube that is serving as a containing frame for the sunburst of 48 Logical Garnets. The eight garnets on each of the six faces are easily collected into three pairs of opposite faces (3 x 16), so that the sunburst gives us a symmetryloaded onemodel view of ordinary infix (A * B), regular Polish prefix (* A B), and reverse Polish postfix (A B *). Loaded with both the beauty and the elegance of deep symmetry, it just so happens that the three pairs of opposite faces also belong to the one and only one perfect 3coloring of a 3 cube. This carries us headon into what we find in crystallography, where the same emphasis is placed not on the three elements in (A * B) but on the three elements in the Miller Index (hkl). Note, for example, that crystallographers call on the same 48group to describe and classify a special piece of carbon, namely, the gemstone diamond. Consequently, this approach opens the door to the crystallography of logic. The logic alphabet at work can also be generalized to obtain a sunburst fractal. First, start with the unending series: (A * B), (A * B), (A * B) . . . , when all of them are connected by equivalence signs. Next, add three overdots and three underarcs to each (A * B), thereby constructing an unending wreath product that is acting on each term in an unending Master Equivalence. The sunburst at the first (A * B) becomes the motif and then, taking the next step in the change of scale, each next higherorder selfsimilar sunburst is obtained from each previous (A * B). This generates an infinite fractal nesting of sunbursts. Simple, complex, transparent, now the unending Master Equivalence is standing in another rare moment. This moment displays the sunburst fractal as an elegent, hyperdimensional, analytic polyform, one that is correct, exact, and beautiful. previous page | next page
Hindsight on Some Conveniently Ignored History Becoming familiar with the logic alphabet will give us some new eyes when we look through the history of logic. Two examples will show that a new set of predecessors come into view when we shift to a shapevalue notation. First, in 1903 Davis gave an 8group interpretation to De Morgan's eight forms of a proposition. For us, this is the same subgroup (N, N, N, ) that covers both the top half of Table I and the corresponding 8cell of logical garnets. Note especially that Davis was one of Peirce's students. By then, as noted by Ladd Franklin (1880), Peirce was already using his own symbol for "if, then" (the claw), a symbol that could be subjected to a symmetry operation. Also by then Peirce had already devised his boxX notation (1902). Second, fifteen years before the 1902 in Peirce, Renton (1887) was very precise about what for us covers the eight symmetries on the front face of the large 3 cube. He published a short 16page pamphlet that left out the mate operation (N, , N, C). Believe it or not, as early as 1887 Renton also published the 8group table for (D4), the one that lets the eight combinations of (negation, negation, conversion) act not on (A, B) but on (subject a, predicate b). There it was, nicely made explicit. Renton even included some special vocabulary for the eight operations. The four for (NN) were called nonverse, adverse, reverse, converse, and the four that added conversion ( C) were called inverse, severse, deverse, and proverse. A onepage review of Renton (1888) gave considerable space to a standard interpretation of the syllogism. But the review did not even acknowledge that Renton gave us an 8group description of the "Versation of the Proposition." It is worth noting that Renton did not use a notation that had the symmetry properties built into the symbols themselves. For us the same 8group is another subgroup, (D4) for (N, , N, C), the one that covers the first four and the third four rows of Table I. In fact, with praise for Renton, (D4) is at its best for the logic alphabet when the eight symmetries of a rigid square act on every one of the 16 LSs on the Clock Compass. It is worth noting that, once while at the Peirce Edition Project in Indianapolis, Max Fisch (19001995) emphasized that Frege and Peirce, two of the most outstanding logicians of the 1800s, both came up with dimensionalized notations. Frege had a system of branching conditionals, but, even within the Peirce community, very few people know that Peirce had TWO dimensionalized notations. The first (1896+) is the betterknown Existential Graphs and the second (1902+) is the one that is like the logic alphabet, namely, his rarely
recognized boxX notation for the 16 binary connectives. The code for this one is very simple. All combinations of the four sides of a box were added to enclose (negate) the 16 combinations of the four, small subareas (TT, TF, FT, FF) that surround the center of an allcommon Xframe. Historically, look at all of the 20th century. Did we get pulled into some misplaced axiomatics, namely, into a strong formal push to isolate the smallest convenient set of rigorously complete connectives ("and", "or", "if") for the propositional calculus? If not, then what were the prevaling mindsets that prompted two to three generations of logicians to read past the untapped potential that stands fully present and clearly exposed in the many pages of Peirce's manuscripts? previous page | next page
Simplicity and the Generalization of De Morgan's Laws
Mirror logic, by analogy from Chomsky, can also be cast as transformational logic. The point here rests on the fact that the symmetry properties contained in square truth tables, as in (2), have been built into the shapevalue structure of the LSs, as in (4). Consequently, the signmaking act has been completed in such a way that the formal conditions in the syntax of the LSs have been designed and then specified so that they have been put into a simple and direct isomorphism with the interrelational requirements that exist in the semantics of the truth tables. The model below is made from a set of slotaccepting movable blocks that form and resolve equivalences. The setting on this Equivalence Board shows one of De Morgan's laws. In "dot, vee", (NA dot NB) is equivalent to N(A vee B). In words, it says that (NotA and NotB) is equivalent to (A Notor B). In finger pairs, the two expressions become (NotA TFFF NotB) and (A NotTTTF B). As given in the slotted and carefully framed set of moveable blocks shown below, the mirrors can be used to demonstrate that the equivalence, now fully serving as a miniature equilibrium, is "balanced" (valid). When we rotate on the left (NON) and mate on the right (ONO), both sides become (A p B): in words, (neither A nor B); in finger pairs, (A FFFT B). It so happens that the pletter in (A p B) is the Nor gate, sometimes expressed as the down dagger (A | B) and also known as the (p)eirce connective.
Resolving equivalences when they are written in the Logic Alphabet repeats what is done to resolve identities in trigonometry. Change either side into the other side, or change both sides into the same third expression. De Morgan's laws, both of them, come to no more than (i) rotate the LS on the left, (ii) mate the LS on the right, and (iii) thereby obtain the same third expression, that is, the same LS on both sides. In effect, serving as a nice example of cognitive ergonomics at its best, and again calling attention to a fundamental triple isomorphism, the selfsame transformations for the symmetry operations line up with the mental operations in such a way that both line up with the logic operations. This arrangement makes it easy to introduce a generalization of De Morgan's laws. As an exercise, find another pair of LSs, and more challenging, all possible pairs, that can be subjected to the same pattern of equivalencedetermined symmetry mirrors, namely, (NON) on the left and (ONO) on the right, as these two negation triplets now appear in the top row of the Equivalence Board above.
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Mirror Logic and How to Construct Master Equivalences Unfortunately it so happens that the algebra of logic does not have enough algebra in it. In reply, the Generalized Equivalence given below adds some algebra to the algebra of logic. As a key step in the present approach, this Generalized Equivalence comes into existence when all combinations of the four mirrors in flipmateflip and flip operate on (A * B). The three rules R1R2R3 are active when (N)egations enter and leave the overdots. R4 is active when the underarcs commute the (A, B) order. This Master Equivalence reaches across a total of 4096 atomic equivalences. This total is equal to all and exactly all of the wellformed atomic equivalences that can be written under this logical form. All combinations honor the equivalence sign, when all combinations of the four symmetry operations (fmf and f) act on both sides of this Master Equivalence. The Master Equivalence below belongs to an algebraic system that introduces a special act of abstraction, one that subordinates three kinds of variables. (i) The Master Equivalence contains propositional variables, as indicated by (A, B). (ii) It contains connective variables, as indicated by the asterisks (*, *). (iii) It contains operation variables, as indicated by the overdots (NNN ) and the underarcs ( C). The organizing and the unifying power of the logic alphabet lies in the openfaced ease with which the three classes of variables, all under the same abstract forms and all under the same system, have been built into the same dimensionalized cognitive frame. This satisfies the same challenge that Peirce, coming very close indeed, was grappling with in 1902, when he devised his boxX notation for the 16 binary connectives.
Extending this Master Equivalence so that it covers all combinations and all permutations of (A * B) on just the left side of the equivalence sign comes back to the wreath product. This generates a 1storder sunburst of 48 logical garnets. Doing the same thing on both sides generates a 2ndorder sunburst. Repeating the same thing when the same Master Equivalence takes on ncopies of (A * B) gives rise to an nthorder sunburst. The selfsimilar repetitions of the same motif,
namely, the sunburst, quickly gives rise to a cosmically immersed, infinitely nested, hyperdimensional fractal. All of this from no more than some carefully placed symmetry that is acting on a Master Equivalence that contains an unending repetition of (A * B), when three overdots and three underarcs are added to each repetition of (A * B). previous page | next page
A TopDown Approach to Sign Factors Engineering Why is the logic alphabet doing such a good job when it circulates in the networks of interrelations? Because we are going at this whole thing the other way around. Instead of selecting signs for the parts, such as "vee" for (A or B), and then trying to struggle our way back into the larger system, what we have done is turned around and going in the opposite direction. We went to the larger system first, to the interrelational symmetry properties contained in the square part of square truth tables, and then we designed the parts, namely, a special set of signs in a special notation (connectals), so that they will fit the needs of the larger system. In brief, very much in line with Peirce and 1902, our sign making has a Gestalt bias. In this case, the bias puts the emphasis on the polyhedral and on the crystallographic structures that are generated when a certain set of symmetry transformations is acting on a shapevalue notation for the 16 binary connectives. The bias is straightforward and direct. Let the systemic properties of the whole determine the formation of the parts. This bias cuts a claryifying swath across the full length of the 1900's. We can now say that, breaking with the standard format carried in the traditional mold, any notation that is limited to (and, or, if) is not enough. In effect, this bias exposes a faulty assumption, one that is still carried strong in the conventional wisdom and one that still looks to economydriven axiomatics to justify notation building for the propositional calculus. Ironically, the counter view started as early as Pierce and his 1902 BoxX notation. Like it or not, all those tables, all those diagrams, all those models introduced by the Xstem Logic Alphabet leaves us walking full stride into the 21st Century. This approach also gives rise to a run of frontline questions. How about an optical computer? What about isomorphisms that lead into crystallography and into quantum physics? What about a specialized computer language? What about another form of Arabic numerals, when the 16 LSs in another font are used as number symbols that are alphanumeric, binary stempositional, and hexadecimal? What about semiotics and cognitive ergonomics? What about mental development and the psychology of logic education, especially for children? What about the standard presentation of symbolic logic as it continues to appear in the textbooks today? It is easy enough to say, "Let the fingers do the walking." By now it is obvious, we hope, that, when we construct the logic signs with great care, it will be easy to say "Let the mirrors do the thinking and let the people do the reflecting." One more nudge. Suppose that you have lived all of your life back in the age of Roman numerals. Also suppose that one day quite by accident you read a short summary of Arabic numerals. How would you have reacted to the jolt carried in that much change? We suggest that what you now know about the logic alphabet leaves you, and whatever you do in logic, very much in the same position. previous page | beginning
