Geometry of Oppositions (Logic!)

Continuing with the subject of logic, here is a diagram I did after reading the work of A. Moretti [2003, 2004], H. Smessaert [2004, 2009] and R. Pellisier [2004, 2009]. These researchers have found –following previous progress achieved mainly by Blanché [1953] and Sesmat [1951] — that logical oppositions can be represented geometrically in a very consistent manner. There are many more opposition configurations than the one shown in the diagram, but this particular form, called the “logical tetraicosahedron” in the literature, is particularly interesting for my own research covering Security Management “perspectives”. The cube depicted in red at the centre of the diagram is the so-called “strong logical cube”. According to Pellisier: “…the logical cube is a three dimensional generalization of the square of oppositions…” [Pellisier, 2009]. (Note: I have “flattened” the spikes of the icosahedron –see for example the node labelled as “~a” — to better show the cube at the centre of the diagram.)

You will find many additional materials and references in Alessio Moretti’s web page:

<> means “possible”
[] means “necessary”
~ means “not”

By the way, I created this diagram mainly following R. Pellisier’s description in “Setting n-opposition” [2008]. I used the yWorks editor (yEd) which can be downloaded and used freely. It does some things better than Visio!
This is the yWorks website:

Geometry of Oppositions

Logical Tetraicosahedron according to R. Pellisier and H. Smessaert