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2. The logic of the included middle
Knowledge of the coexistence of the quantum world and the macrophysical world and the development of quantum physics has led, on the level of theory and scientific experiment, to the upheaval of what were formerly considered to be pairs of mutually exclusive contradictories (A and non-A): wave and corpuscle, continuity and discontinuity, separability and nonseparability, local causality and global causality, symmetry and breaking of symmetry, reversibility and irreversibility of time, etc.
For example, equations of quantum physics are submitted to a group of symmetries, but their solutions break these symmetries. Similarly, a group of symmetry is supposed to describe the unification of all known physical interactions but the symmetry must be broken in order to describe the difference between strong, weak, electromagnetic and gravitational interactions.
The intellectual scandal provoked by quantum mechanics consists in the fact that the pairs of contradictories that it generates are actually mutually contradictory when they are analyzed through the interpretative filter of classical logic. This logic is founded on three axioms:
1. The axiom of identity : A is A.
2. The axiom of non-contradiction : A is not non-A.
3. The axiom of the excluded middle : There exists no third term T which is at the same time A and non-A.
According to the hypothesis of the existence of a single level of Reality, the second and third axioms are obviously equivalent. The dogma of a single level of Reality, arbitrary like all dogma, is so embedded in our consciousness that even professional logicians forget to say that these two axioms are in fact distinct and independent from each other.
If one nevertheless accepts this logic which, after all, has ruled for two millennia and continues to dominate thought today (particularly in the political, social, and economic spheres) one immediately arrives at the conclusion that the pairs of contradictories advanced by quantum physics are mutually exclusive, because one cannot affirm the validity of a thing and its opposite at the same time: A and non-A.
Since the definitive formulation of quantum mechanics around 1930 the founders of the new science have been acutely aware of the problem of formulating a new "quantum logic." Subsequent to the work of Birkhoff and van Neumann a veritable flourishing of quantum logics was not long in coming [4]. The aim of these new logics was to resolve the paradoxes which quantum mechanics had created and to attempt, to the extent possible, to arrive at a predictive power stronger than that afforded by classical logic.
Most quantum logics have modified the second axiom of classical logic -- the axiom of non-contradiction -- by introducing non-contradiction with several truth values in place of the binary pair (A, non-A). These multivalent logics, whose status with respect to their predictive power remains controversial, have not taken into account one other possibility: the modification of the third axiom -- the axiom of the excluded middle.
History will credit Stéphane Lupasco with having shown that the logic of the included middle is a true logic, formalizable and formalized, multivalent (with three values: A, non-A, and T) and non-contradictory [5]. Stéphane Lupasco, like Edmund Husserl, belongs to the race of pioneers. His philosophy, which takes quantum physics as its point of departure, has been marginalized by physicists and philosophers. Curiously, on the other hand, it has had a powerful albeit underground influence among psychologists, sociologists, artists, and historians of religions. Perhaps the absence of the notion of "levels of Reality" in his philosophy obscured its substance. Many persons believed that Lupasco's logic violated the principle of non-contradiction -- whence the rather unfortunate name "logic of contradiction" -- and that it entailed the risk of endless semantic glosses. Still more, the visceral fear of introducing the idea of the included middle , with its magical resonances, only helped to increase the distrust of such a logic.
Our understanding of the axiom of the included middle -- there exists a third term T which is at the same time A and non-A -- is completely clarified once the notion of "levels of Reality" is introduced.
In order to obtain a clear image of the meaning of the included middle, we can represent the three terms of the new logic -- A, non-A, and T -- and the dynamics associated with them by a triangle in which one of the vertices is situated at one level of Reality and the two other vertices at another level of Reality. If one remains at a single level of Reality, all manifestation appears as a struggle between two contradictory elements (example: wave A and corpuscle non-A). The third dynamic, that of the T-state, is exercised at another level of Reality, where that which appears to be disunited (wave or corpuscle) is in fact united (quanton), and that which appears contradictory is perceived as non-contradictory.
It is the projection of T on one and the same level of Reality which produces the appearance of mutually exclusive, antagonistic pairs (A and non-A). A single level of Reality can only create antagonistic oppositions. It is inherently self-destructive if it is completely separated from all the other levels of Reality. A third term, let us call it T', which is situated on the same level of Reality as that of the opposites A and non-A, can accomplish their reconciliation.
The entire difference between a triad of the included middle and an Hegelian triad is clarified by consideration of the role of time . In a triad of the included middle the three terms coexist at the same moment in time . On the contrary, each of the three terms of the Hegelian triad succeeds the former in time. This is why the Hegelian triad is incapable of accomplishing the reconciliation of opposites, whereas the triad of the included middle is capable of it. In the logic of the included middle the opposites are rather contradictories : the tension between contradictories builds a unity which includes and goes beyond the sum of the two terms.
One also sees the great dangers of misunderstanding engendered by the common enough confusion made between the axiom of the excluded middle and the axiom of non-contradiction [6]. The logic of the included middle is non-contradictory in the sense that the axiom of non-contradiction is thoroughly respected, a condition which enlarges the notions of "true" and "false" in such a way that the rules of logical implication no longer concerning two terms (A and non-A) but three terms (A, non-A and T), co-existing at the same moment in time. This is a formal logic, just as any other formal logic: its rules are derived by means of a relatively simple mathematical formalism.
One can see why the logic of the included middle is not simply a metaphor like some kind of arbitrary ornament for classical logic, which would permit adventurous incursions and passages into the domain of complexity. The logic of the included middle is perhaps the privileged logic of complexity, privileged in the sense that it allows us to cross the different areas of knowledge in a coherent way, by enabling a new kind of simplicity.
The logic of the included middle does not abolish the logic of the excluded middle: it only constrains its sphere of validity. The logic of the excluded middle is certainly valid for relatively simple situations. On the contrary, the logic of the excluded middle is harmful in complex, transdisciplinary cases……………………………….
