A Note concerning the Place of Contradictions in the Ontologies of Constitution

In this first section we start with defining the notions of inconsistency and para-consistency, we give an example of an inconsistency and clarify what according to us is the basic problem with respect to the occurrence of inconsistencies. We are then in a position to state the aim of this paper. We have a contradiction when both A and ∼ A are true.1 We have negation-incompleteness when neither A nor ∼ A are true. We could refer to the first as “knots”, and the second as “cracks”, because these words convey the fact that a contradiction or an incompleteness encountered in a theory or in everyday life is never apprehended in a detached manner as just another neutral fact.2 We will give all our attention to the concept of inconsistency. (Apparently knots pose more of a problem for understanding than do cracks, although of course there is the famous Gödel theorem.) In Classical Logic the presence of a contradiction leads to triviality: everything becomes true (logical “explosion” as some logicians call it, in formal language written as A,∼ A ⊢ B). A paraconsistent logic avoids the principle of explosion: there are contradictions that can be true, without everything becoming true. Being non-logicians we try to understand what a paraconsistent logic really does. With the help of an example we might get an intuitive grasp. From