Was Lewis Carroll an Amazing Oppositional Geometer?
Alessio Moretti · History and Philosophy of Logic · 2014
Some Carrollian posthumous manuscripts reveal, in addition to his famous ‘logical diagrams’, two mysterious ‘logical charts’. The first chart, a strange network making out of fourteen logical sentences a large 2D ‘triangle’ containing three smaller ones, has been shown (by Richards, in 1986) equivalent—modulo the rediscovery of a fourth smaller triangle implicit in Carroll's global picture—to a 3D tetrahedron, the four triangular faces of which are the 3+1 Carrollian complex triangles. As it happens, such an until now very mysterious 3D logical shape—slightly deformed—has been rediscovered, independently from Carroll and much later, by a logician (Sauriol), a mathematician (Angot-Pellissier) and a linguist (Smessaert) studying the geometry of the ‘opposition relations’, that is, the mathematical generalisations of the ‘logical square’. We show that inside what is called equivalently ‘n-opposition theory’, ‘oppositional geometry’ or ‘logical geometry’, Carroll's first chart corresponds exactly, duly reshaped by a logic-preserving geometrical transformation, to the ‘oppositional tetrahexahedron’, which is a very powerful ‘oppositional closure’, the elegant 3D entanglement of a ‘logical cube’ and six ‘logical hexagons’ (and therefore eighteen ‘logical squares’). The temptation is therefore high to consider the author of Alice's Adventures in Wonderland as a forerunner of oppositional geometry.