The present contribution might be regarded as a kind of defense of the common sense in logic. It is demonstrated that if the classical negation is interpreted as the minimal negation with n = 2 truth values, then deviant logics can be conceived as extension of the classical bivalent frame. Such classical apprehension of negation is possible in non- classical logics as well, if truth value is internalized and bivalence is replaced by bipartition.

In order to understand negation as such, at least since Aristotle’s time, there have been many ways of conceptually modelling it. In particular, negation has been studied as inconsistency, contradictoriness, falsity, cancellation, an inversion of arrangements of truth values, etc. In this paper, making substantial use of category theory, we present three more conceptual and abstract models of negation. All of them capture negation as turning upside down the entire structure under consideration. The first proposal turns upside down the structure (...) almost literally; it is the well known construction of opposite category. The second one treats negation as a contravariant functor and the third one captures negation as adjointness. Traditionally, negation was investigated in the context of language as negation of sentences or parts of sentences, e.g. names. On the contrary we propose to negate structures globally. As a consequence of our approach we provide a solution to the ontological problem of the existence of negative states of affairs. (shrink)

The paper first introduces a cube of opposition that associates the traditional square of opposition with the dual square obtained by Piaget’s reciprocation. It is then pointed out that Blanché’s extension of the square-of-opposition structure into an conceptual hexagonal structure always relies on an abstract tripartition. Considering quadripartitions leads to organize the 16 binary connectives into a regular tetrahedron. Lastly, the cube of opposition, once interpreted in modal terms, is shown to account for a recent generalization of formal concept analysis, (...) where noticeable hexagons are also laid bare. This generalization of formal concept analysis is motivated by a parallel with bipolar possibility theory. The latter, albeit graded, is indeed based on four graded set functions that can be organized in a similar structure. (shrink)