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The doctrine of coincidentia oppositorum (the coincidence of opposites), the interpenetration, interdependence and unification of opposites has long been one of the defining characteristics of mystical (as opposed to philosophical) thought. Mystics of various persuasions have generally held that such paradoxes are the best means of expressing within language, truths about a whole that is sundered by the very operation of language itself. Any effort, it is said, to analyze these paradoxes and provide them with logical sense is doomed from the start because logic itself rests upon assumptions, such as the principles of non-contradiction and excluded middle, that are violated by the mystical ideas.

Mathematical development of a dialectical logic with truth-values in a complex field.

The coincidentia oppositorum is a common trope in many religious traditions, particularly those with a mystical or initiatory aspect, and fields of knowledge such as Psychology and Quantum Physics, with wave-particle duality. The aim of this paper is to present a mathematical theory of the coincidence of opposites, and where truth-values are expressed in a complex field. A propositional coincidentia oppositorum algebra is developed.

Although the literature of paraconsistent logics is abundant, the authors think that this is the first time that a dialectical logic has been developed with truth-values belonging to a complex field. The impossibility of reaching an absolute truth from opposite propositions is discovered, both being true, because of the existence of irrational numbers in the truth-values.