Universal duality

  By universal duality we mean a duality which permeates prima facie different regions. For instance mathematical logic, algebra,  language, ontology, topology and physics.  Despite the core ternary structure in Hegel, it is easy to argue that duality and opposition also play a similarly fundamental (if subordinate) role.  So we might speak additional of a universal structure which is a kind of 'combination' of the two sides of this universal duality - that is, a move from a more implicit or less concrete kind of mutual dependence between the two poles into a more intimate fusion. In itself, of course, all this is not new. What is remarkable is just how formal, precise and analytically this perspective can verified, developed, and argued for.  Duality is generative. That is each pole of a given concrete duality can itself split into its own duality which may reflect on its own the plane the initial duality, which in turn embodies the universal duality.  Fundamental dualities are for instance the continuous vs. the discrete, open set vs. closed set, mass nouns vs. count nouns,  imperfect vs. perfect tense, intensional vs. extensional, possibility vs. necessity,  disjunction vs. conjunction,  existential quantifier vs. universal quantifier,  colimit vs. limit,  left adjoint vs. right adjoint,  addition vs. multiplication,  introduction vs. elimination rules in natural deduction,  classical vs. intuitionistic, etc. An interesting task would be to somehow unified a great part of these dualities from a single perspective, for instance, the category theoretic one could be seen as a least a partial candidate for a universal duality, the others being special cases. In real life many of the above dualities are found to a certain extent unified and harmonized into an interconnected whole: disjunction and conjunction are both part of logic and satisfy precise interaction laws such as distributivity.

  It seems that few have delved as deeply into the symmetries and dualities of logic than Jean-Yves Girard.   For he split the members of the initial dualities between true and false, conjunction and disjunction, into a further additive vs. multiplicative pairs,  and also splitting $\rightarrow$ as well as introducing 'new particles', the pair of exponentials.  A further task would be to split the quantifiers as well and to give detailed interpretations of all these Girard pairs with special focus on Hegel's logic. One place we could start would be in the category of quantity with special focus on the mass noun vs. count noun distinction and then move through essence to finally try to attempt an interpretation of notion and its three components universality, particularity and individuality with a special focus on intensionality, induction and non-standard quantification. The duality of quantifiers  analogous to $\otimes, \oplus, \sqcup, \sqcap$ is strongly related to our previous considerations and holism and the relationship between local and global.  For instance the two extremes: global information can be contained locally or else locally the can be no trace of the non-trivial global information present.  The following are key: weakening, contraction, context dependence and context sharing.