The Principle of Duality in Integral Relational Logic has evolved from the Principle of Duality in projective geometry. For instance, Blaise Pascal and Charles Julien Brianchon proved a pair of dual theorems in 1639 and 1810, respectively. Pascal first showed that if six points ABCDEF are placed on a conic section and joined as in the left-hand-side diagram in what Pascal called his ‘Magic Hexagram’, then their points of intersection, LMN, are collinear. Briachon then showed that if six lines are drawn tangentially to a conic section to form a hexagon, as abcedf, then the lines joining opposite vertices, lmn, intersect at a single point.
The relationship between these two theorems can best be seen from an observation made by Florimond de Beaune, a friend and student of René Descartes in the seventeenth century: a curve may be regarded both as the path of a moving point and as the envelope of a moving line. In general, whatever theorem can be proved about points and lines in projective geometry has a dual or reciprocal theorem about lines and points, where lines and points are interchanged.
This notion of duality between points and lines in projective geometry is generalized in IRL by noticing that whenever we form a concept we always also form its opposite in some sense, such as black and white, female and male, tall and short, and so on and so forth. We can thus state the proposition D—the Principle of Duality: A complete conceptual model of the Universe consists entirely of dual sets.
D is a rather unusual proposition in that it is a universal truth applicable in all situations. It is quite easy to show that D is false. For instance, a collection of entities without a common attribute do not form a set, which we usually call miscellaneous, called the axiom of choice in mathematics. However, all we are doing by proving that D is false is proving its validity in all circumstances. In the terms of Hegel’s dialectical logic, if ‘D is true’ is the thesis and ‘D is false’ is the antithesis, then ‘D is true’ is the synthesis. There is thus a primary-secondary relationship between the truth and falsity of the Principle of Duality, illustrated in this diagram.
Three other constructs arise from the Principle of Duality. The Circle of Duality shows how a spectrum of values between opposite extremes can be accommodated in IRL, thus showing that Aristotle’s Law of Excluded Middle is not true in all circumstances. The Triangle of Duality shows how all three possible relationships between opposites relate to each other, putting Aristotle’s either-or Law of Contradiction in its proper perspective. And the Cross of Duality shows how two or more pairs of opposites can be arranged together, such as Ken Wilber’s four-quadrants model, called AQAL, short for “all quadrants, all levels”, which is short for “all quadrants, all levels, all lines, all states, all types”.
The Principle of Duality also helps to distinguish between dualism, duality, and Nonduality, which we can associate with the three tiers of consciousness in Ken Wilber’s model. For people in the first tier often put a great barrier between opposites, especially when they need to defend and hold on to the belief systems that give them a precarious sense of security and identity in life. “If you’re not with us, you’re against us,” is a well-known example of dualism at work.
In contrast, when people broaden the horizon of their vision, looking at both sides of situations, this is a clear sign of our innate, natural intelligence at work. There is no longer a barrier between the opposites, which we can call duality. This principle can be further expanded in the third tier of the spectrum of consciousness by recognizing that Ineffable, Nondual Consciousness embraces all opposites in the relativistic world of form, enabling us to transcend the categories with the Principle of Unity, the fundamental design principle of the Universe.