If we don’t think and speak Being, then we think and speak in oppositions and remain in oppositions. Our action then is self-defeating, banal and incomplete. Now, human thinking and speech necessarily blossoms and opens up as a series of differentiations, each one presenting us with a fragment of of the world.
The basic differentiations start with Identification and Differentiation, that is from difference bound by identity, and from identity bound by difference. The result is the complex of Particular-Universal, Abstract-Concrete, Subject-Object and External-Internal (or Other-Ego). Many authors have discussed these foundations in different ways and with different terminologies, but there is no way around the fact that we will find at the core of our experience, a hard crystalline core, a logical, irreducible structure which frames and determines our actions.
So the Gap or “Clearing of Being” (assuming Heidegger’s guidance in these matters), this “inhabiting of the Being” is itself the source of this radical ground of opposition and antinomy.
The Quadrature as a “logic” is a tool to rationalise antinomy– for example by allowing us to conceive four ontological categories (Lowe) or four causes (Aristotle); but the Quadrature can be also taken as evidence of the alienation from Being. Undoubtedly, the experience of Being occludes itself through differentiation—first into the “four causes” but then into the myriad of subjects and objects which absorb our attention.
Great effort is necessary to move from “beings” to Being (again following Heidegger’s formulation), but this movement is warranted by its own possibility: Differentiation calls for unity, and “beings” long for Being. So we are precariously sustained in the limit of this unique Human experience. Seen from this angle, the Quadratures are only a second degree observation, i.e. the expression of the result of differentiation applied to itself. The primary observation being that of negation (more precisely the two kinds of negation).
Indeed, a key to the recovery of the experience of Being (with a capital B) is the recognition of variety in negation, beyond the pale, logocentric negation we have inherited through Formal Logic. The two kinds of negation are but another way of saying that the world of oppositions arises from two dualities: Differentiation of Identity, and Identity of Differentiation.
Note: this diagram is best understood starting from the reflections published here: https://carlos-trigoso.com/2014/06/20/quadratures-person-subject-agent-and-object/ – look in particular for the reference to the works of Chris Lofting and Richard Jung.
The logical symbolism uses the standard reversed “E” to indicate “existence” not in metaphysical but in existential-experiential sense, i.e. “this exists within the experience of Being.” The overbar symbol indicates the negation as “complement”; and the standard negation symbol ( ~ ) indicates classical negation. The “F” in formulae like “F~a2 represents Anselm of Canterbury’s modal “to do” (as explained in the work of Sarah Uckelman – http://lyrawww.uvt.nl/~sluckelman/). Other symbols are more conventional or placeholders for future elaboration.
(See also: Jean –Francois Mattéi, “Heidegger et Hölderlin. Le Quadriparti” — Presses Universitaires de France, 2001 )