QUANTUM GRAVITATIONAL VS. QUANTUM LOGIC:
VIRTUALLY LEFT-HANDED REAL TRINARY LOGIC

Where 0 = 1, 10 = , and 1 = 0, and 00 = = 11 = 0 = 1, the cornerstone,

which has a left-handed 1-diagonal, cannot be derived by the method of external multiplication. As so underivable, it actually transcends handedness. As such it is the foundational template of quantum gravitational logic. It is irreducibly the ‘square essence’ of the infinite lattice of 0’s, ’s, and 1’s, as here depicted:1

Precisely as infinite, the infinite lattice is not able to be mirrored, as it is not able to be gotten outside of. In fact, where 0 = 1, 10 = , and 1 = 0, and 00 = = 11 = 0 = 1, the attempt at external multiplication, in attempting to reproduce the virtually left-handed diagonal of 1’s, either produces a distorted square (a 2, 5, 2 distribution of 0’s, 1’s, and ’s), as here illustrated:

or, in order to maintain the fundamental pattern of the original distribution, must produce a mirror reversal of the ‘square essence’, which, however, can be achieved only by substituting U (= ‘intermediate’) -- here in place of -- for 1, producing the right-handed U-diagonal, where the following relations obtain: 0U = 1, 10 = U, 1U = 0, and 11 = 1, but 00 = 0, UU = U, and UU ≠ 00 ≠11, that is, U ≠ 0 ≠ 1:2

Here U is truly the ‘intermediate’ between 0 and 1, while the latter retain their binary exclusiveness from each other apart from U, and each, therefore, from U.3

That such a right-handed trinary logic -- the virtual reverse and reduction of real trinary logic to the logic of superpositional intermediation -- is radically binary, literally can be seen in the form of the infinitely expanding triangle produced by the infinite iterative sequential summing of the logical sequence, 1, 0, U, 1, 0, U, 1, 0, U, . . ., where the logical operator is Hellerstein's "pivot,"4 as here illustrated:

Note the doubling (binary) pattern of horizontally-based triangles of U's, and the 2, 5, 2, 14, 2, 5, 2, 41, . . ., pattern of the edges of the edge-sharing triangles of U's. This 2, 5, 2, . . ., repetition fairly reflects the distortion of the trinary logic ‘square essence’ pattern (the 2 0’s, 5 1’s, and 2 ’s in the externally derived square illustrated above) which is introduced by the attempt to define the trinary relations with reference to an exterior multiplication.  The radically binary 2-based nature of right-handed trinary logic is further manifest in the fact that the sequence of the number of U's constituting the edges of the edge-sharing triangles of U's whose hypotenuses extend to the edges of the major infinitely expanding triangle is 2, 5, 14, 41, 122, . . ., that is, where x = 3, this sequence is 2, 2 + x, 2 + x + x2, 2 + x + x2 + x3, 2 + x + x2 + x3 + x4, . . ..

Contrast with this the tripling (trinary) pattern of horizontally-based triangles of 1's in the infinitely expanding triangle produced by the real trinary logic geometric series infinite iterative sequential summing, where the sequence of such triangles is 1, 3, 9, 27, 81, . . ., that is, immediately the geometric series where x = 3, and where the latter's edge-sharing triangles of 1's -- all of whose hypotenuses extend to the edges of the major infinitely expanding triangle -- unlike the 1 : 2 ratio in which the edge-sharing triangles of U's do so in the case of the infinitely expanding triangle illustrated above -- immediately expand according to the geometric series 1, 4, 16, 64, . . ., as here illustrated:5

The last mentioned geometric proportion is equivalently 1, 1+(1
× 3), 1+(5 × 3), 1+(21 × 3), 1+(85 × 3), . . ., where the multipliers of 3 are the increasing powers of the constant denominator (.125) of the odd-numbered rows of the infinite lattice composed of the infinitely alternating series of natural and odd numbers which constitutes the infinite series which is the differences between the squares of the natural numbers, the differences between those differences, the differences between those . . ., ad infinitum.6 To say that real trinary logic is virtually left-handed is to say that prior to the attempt to derive the relations of the terms from external multiplication, that is, to derive them from something outside the square essence of the infinite lattice -- therefore, from something essentially outside the infinite array -- it is itself neither left- nor right-handed. Real trinary quantum gravitational logic is the logic of infinite inclusive difference.7

Notes

1 Cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), Section III.1. In general, what follows below should be compared to the critique of Peirce’s real triadic duality, ibid., "Afterword 2," pp. 287ff.  Likewise see also D.G. Leahy, Faith and Philosophy: The Historical Impact (Aldershot, 2003), Chapter 7.
2 The writer is indebted to N. Hellerstein who in correspondence prompted by this web called attention to his newly defined "pivot" operator for a ternary logic in which the third term is intermediate between 0 and 1. This logical operator relates three values -- here, 0, U, and 1 -- in the ways indicated. It is the exemplary instance of the right-handed ternary logic illustrated in the text. N. Hellerstein is the author of Diamond: A Paradox Logic (Knots and Everything, Vol. 14; River Edge, 1997).
3 The fundamental distinction between real trinary logic and ternary logic which is essentially binary is that the 3rd value in the latter, U, is superpositional, that is, can only be the intermediate it is by not being the intermediated terms in distinction from it and from each other, whereas, in the former, the 3rd value, , is ‘transpositional’, that is, is the mean proportional of the terms either and both of which it is through the intermediation of universal unity. The former is the logic of quantum computation. The latter is the logic of quantum gravitational computation. Cf., on this web, Logical Unity & the Four Physical Forces.  For the correspondence of the real trinary logic equation 0 = 00 = = 1 to XY = X2 = Y2 = Z, when uniquely in the case of the absolute dead center cube (D.G. Leahy, Foundation: Matter the Body Itself, pp. 533ff.) X = (x/6)4/x3 and Y = 8x3/x4, see, on this web, The Deep Epidermal Surface: The Cornerstone Construction Order, Minimum Order Tetrahedron Hypercube, & Absolute Dead Center Hypercube.  In this connection, see also, on this web, The Magnitude of Being.
4 See, above, n. 2.
5 For detailed analysis of the properties of this infinitely expanding triangle, see, on this web, Real Trinary Logic Geometric Series Infinite Iterative Sequential Summing. Note that the infinitely expanding triangle produced by the real trinary logic geometric series infinite iterative sequential summing also produces immediately the biological geometric series governing cell multiplication and, therefore, all organic growth (ibid., note 5).
6 Ibid.
7 Cf., however, on this web, Thinking Creation Ex Nihilo, where it is shown that the cornerstone of this virtually left-handed logic contains within itself the (equally virtually) right-turning line of temporality of the created universe, so that the virtually left-handed 1-1-1 diagonal of the cornerstone has as its logical equivalent the 0 (= 1) - 1 - 0 (= 1) right-handed turn of this line within the cornerstone:

For the fact that gravitational interactions are virtually the asymmetric weak force interactions governing neutrinos whose electric charge is zero and whose mass is zero or near zero ("nothing" or "almost nothing"), see, on this web, Logical Unity & the Four Physical Forces.

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