NOT.RealBeta4Stemming from works on the geometry of logical negation (rooted in a discussion of the foundations of paraconsistent logics), “oppositional geometry” (OG, for short) is the name of what seems to be a new branch of mathematics, similar to, but different from theories like diagrammatic logic, graph theory or knot theory. Previously known as “N-opposition theory” (i.e. NOT), OG gives one of the most general frameworks explaining what are things like the “logical square (or “square of opposition”) (200 a.D.), the “logical hexagon” (1950), the “logical cube” (2004), the “logical tetrahexahedron” (1968) and so on (all seen, inside OG, as constituted of “oppositional bi-simplexes of dim. m“).

NOT.FourActorsOfNOTOne of the backbones of OG, the theory of the “oppositional poly-simplexes (of dimension m)”, which generalises the notion of bi-simplex implicit in Aristotle’s theory of opposition, is itself generated by an “ask-answer” game-theoretical device, the “Aristotelian p^q-semantics” and its “Aristotelian p^q-lattices”. By unfolding the mathematics of opposition in form of a new kind of geometry, OG gives to mathematicians, logicians, philosophers, linguists, ontologists, computer scientists and many others a general framework for modelling and handling through “oppositional structures” any “opposition phenomena” (first static, but then also dynamic). The concept of “opposition” being a very fundamental, elementary and pervasive one (recall that, for instance, “negation” – a highly important concept for logics and mathematics – is just one particular kind of opposition), OG is already finding applications in linguistics, modal logic, many-valued logic, formal ontology, artificial intelligence, philosophy and in the humanities (semiotics, anthropology, sociology, psychoanalysis, gender studies, …). OG is strongly related to some kind of twin theory, “Logical Geometry” (LG): the field of both correlated theories is growing and this website (as well as its twin LG-website) aims at giving access to anyone interested by the state of the art. In that respect, NOT-workshops (i.e. workshops joining scholars of OG, LG and similar) aim at promoting the exchange, possibly interdisciplinary, between scholars concerned in various ways with opposition phenomena and opposition structures. Any feedback, information and/or collaboration is very welcome.