Nagarjuna fundamentally changed the history of Buddhism by showing that Emptiness is at the heart of logic itself, and thus showed that Emptiness is irrefutable, because it is inherent in Logic. And this was the primary reason that Buddhism was reabsorbed into Hinduism rather than remaining a Heresy. Whether this is seen as a flaw or a perfection of Logic is a matter of taste. The argument goes like this.
There are three operators in Logic AND, OR, NOT. The first two are binary operators and the last is a unary operator. These define the four logical combinatory operators AND, NAND, OR, NOR. Using these we can produce 16 truth tables which define logic as a system. The key is that there is a discontinuity between the two binary operators, and between them and the Unary operator. What are these discontinuities. Nagarjuna said that they were Emptiness itself. Emptiness is defined as A, NOT A, Both A AND Not A, Neither A Nor Not A which is the tetralemma. Emptiness is what is other than these four logical possibilities of the tetralemma. And the only thing that is clearly present that is other than these operators functioning in the Tetralemma is the actual discontinuities between these operators. In fact, it is only these discontinuities that allow the non-contradiction and excluded middle between the operators themselves, or between Unary and Binary operators.
I extend this and say that beyond the binary operators there are ternary etc operators. But going the other way there is the zeroth operator, which does nothing, as we know in computing as the null operation. But beyond this there is the neganary operator, i.e. negative one as an operator, in which the operand vanishes as well as the operator. This neganary operator when reversed gives rise ex nihilo to the first operand. Thus the neganary operator is nondually creation and annihilation at the same time. This simultaneity is the supra-rational, the other limit of reason and doxa that is the opposite of contradiction, paradox and absurdity.
See my dissertation on Emergent Design [http://about.me/emergentdesign] for more on this reduction of operator and operand toward the null point, which is nondual.
Given this analysis we see that the WorldSoul of Plato is made up of binary series and the tertiary series of powers. But the WorldSoul Series intersect at One, what Badiou calls the Ultra-One, i.e. One in its ultra-being, as singularity arising as Eventity.
The opposite of the negative operator is the sublation operator that gives us aufhebung, i.e. synthesis of contradictory opposites in conflict, but which are nihilistic opposites which through the veer, or anagogic swerve become a synthesis at another level or from another perspective. Hegel’s logic suffices to describe this process. There is the speculative or positing operator which is the opposite of the negation. If there were not the discontinuities between positing and negating, or between operators and operands, or between different kinds of operators, there would be no logic. but these differences are all the same in some sense, they are signs of existence at the heart of logic.
The very fact that Aristotle demands excluded middle and non-contradiction reveals these discontinuities. Aristotle explicitly negates the tetralemma as non-sense. But the tetralemma is merely there to exhaust the logical possibilities and to say that there is something else beyond that, i.e. Emptiness, which is not negation, but the radical difference between positing and negation. And if we go down into the Zeronary and Neganary we see that at the end there is a null nondual state which is void at the root of logic which is different from the emptiness that exists in the discontinuities between eventities.
These dual nondualities are adamantine, i.e. they cannot be changed or modified, and they are the bedrock in existence that logic itself is based on. This is represented in iconography as the Vajra in Buddhism.
As Plato says we must cut things up via their joints. But the joints are already there before the butcher arrives.
This recognition of the givenness of the discontinuities in Logic prior to the arrival of the Logician Peirce calls Precission, that differs from the Precision of analysis that occurs when we take things apart. Instead Precission is when we look at the synthesis still whole and feel out the joints within without cutting the synthesis open. When we look at Logic through the lenes of Precission then we see that it could not operate without the internal discontinuities that are essential to it, and those arise in the sublation of the logical whole itself not in any of the parts, and this extra dimension that makes logic possible is beyond the dimensionality of Logic itself because it is something we can only point to after we have exhausted the combinatorics of the logical eventities: “operator-operands”.
Responding to those commenting within the post itself where there is enough room to do so . . .
One thing we must remember is that in Sanskrit the logic is Non-count or Mass-like pervasion logic as in China. However, the nearest thing we have to a pervasion logic is the Boundary Logic of G. Spencer-Brown in Laws of Form, which has been interpreted more rigorously by N. Hellerstein Diamond and Delta Logics. See also Bricken’s Boundary Math. [http://www.boundarymath.org/]
In general there is a duality between Set-like Count and Mass Like Non-count elements in math and language, and each has its logic. Syllogistic Logic goes with Sets and Pervasion Logics (like Venn Diagrams) go with Masses (like topology and geometry). Note that concepts are almost always masses, and that all masses have counters, like a piece of furniture, or a blade of grass.
Since Buddhist Logic in both India and China are argumentatively pervasion logics I suggest looking at pervasion logics in Buddhism for examples of this kind of argumentation, taken to a high art in Tibet, and preserved by refugee priests who have brought Tibetan scholarship and scholastic traditions to the USA. Many good books on the Tibetan art of argumentation, which they pursued as intellectual jousting in their monasteries.
We also have the fascinating example of a very sophisticated operator/operand logic in the Matrix Logic of August Stern that follows out the implications of the argument given above with respect to the WorldSoul. Also I would like to mention Para-consistent Logic of Graham Priest and to that we would have to add logics of Para-completeness and para-clarity as well, for instance Fuzzy Logics, and Quantum Logics. Studies of the various species of Deviant logics would be rewarding in this respect.