NOT.PragmaticHypercube4The ideas of OG are new and often confirmed scholars in logic are puzzled (and even irritated) not to be able to understand by themselves at once what is going on here. As it happens this is normal and is mainly due to the fact that OG is not stricto sensu reducible to logic alone (although it has many relations with it), but pertains to a more general mathematical view. For that aim, in this section we provide 15 short and quick tutorials (T1-T15) for getting acquainted with the main ideas of OG (with even a glimpse on LG). Feedback welcome

[ T1: square | T2: hexagon | T3: cube | T4: n-oppositions | T5: tetrahexahedron | T6: examples of β3 | T7: γ-structures | T8: (γ-β-α)-translation | T9: β-equivalence | T10: p^q-semantics | T11: poly-simplexes | T12: sheaf-translation | T13: logical geometry | T14: bitstrings | T15: oppositional dynamics ]

An overview of some of the points to be grasped through these tutorials (T1-T15):

  • the difference between static, dynamic and intensive oppositions (philosophically, pre-logically speaking)
  • the Aristotelian notion of opposition (i.e. the logical square, with its 4 “colours”: red contradiction, blue contrariety, green subcontrariety and grey subalternation)
  • the notion of n-opposition (i.e. the difference between the logical square, hexagon, cube, …)
  • the notion of oppositional bi-simplex of dimension m (i.e. how it is generated with geometrical simplexes)
  • the difference between the α-structures, the β-structures and the γ-structures (i.e. why it is very important not to confuse them)
  • the translation rule allowing to go from the γ- to the β- and from the β- to the α-structures
  • the notion of Aristotelian p^q-semantics (with its Aristotelian p^q-lattice), which allows oppositions to be p-valued and finer-grained
  • corresponding to the previous, the geometrical notion of oppositional p-simplex (of dimension m)
  • the sheaf-theoretical technique which extends to poly-simplexes what the translation rule could do for the bi-simplexes
  • the generative combinatorics and the main ideas of logical geometry (LG) and its difference with respect to OG
  • the “bitstrings” tool of LG
  • the notion of opposition dynamics (a tool for the dynamic oppositions)
  • the notion of opposition field (a tool for the intensive oppositions)