On the Ubiquity of Symmetry in Logical Geometry

The research framework of Logical Geometry investigates two major sets of logical rela- tions holding between formulas in a logical fragment or between concepts in a lexical field. On the one hand, there is the classical set of Aristotelian relations of contradiction, (sub)contrariety and subalternation/implication. On the other hand, there is the set of Duality relations of internal, ex- ternal and dual negation. Networks of logical relations are then given a visual representation by means of logical diagrams, the most well-known among which is no doubt the so-called Square of Oppositions. In this paper we investigate the omnipresent role of symmetry in Logical Geometry. This role can, first of all, be considered both from a logical/semantic perspective and from a geo- metrical/visual perspective. Secondly, symmetry turns up both on the first-order level of individual logical relations/ diagrams and on the second-order level of sets of logical relations/diagrams. This yields the four steps in our analysis: (i) symmetry within logical relations: logical & first order, (ii) symmetry between logical relations: logical & second-order, (iii) symmetry within logical dia- grams: geometrical & first-order, and (iv) symmetry between logical diagrams: geometrical & sec- ond-order.