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Mirror Logic and How to Construct Master Equivalences | |
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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 flip-mate-flip and flip operate on (A * B). The three rules R1-R2-R3 are active when (N)egations enter and leave the over-dots. R4 is active when the under-arcs 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 well-formed atomic equivalences that can be written under this logical form. All combinations honor the equivalence sign, when all combinations of the four symmetry operations (f-m-f 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 over-dots (NNN -) and the under-arcs (- - - C). The organizing and the unifying power of the logic alphabet lies in the open-faced 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 box-X 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 1st-order sunburst of 48 logical garnets. Doing the same thing on both sides generates a 2nd-order sunburst. Repeating the same thing when the same Master Equivalence takes on n-copies of (A * B) gives rise to an nth-order sunburst. The self-similar 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 over-dots and three under-arcs are added to each repetition of (A * B). | |
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