Duality

Abstract

The first people to take the view that incompleteness and inconsistency are somehow equally reasonable, seem to have been Da Costa and the Brazilian school of logicians, and independently the Routleys. The idea expressed itself in Brazilian logic in the paraconsistent C-logics, dualising by abandoning the law of Noncontradiction ~(A & ~ A), rather than Excluded Middle A V ~ A as in intuitionism; and additionally adopting the opposite Double Negation axiom to intuitionism. The Routleys proposed their *-operation on theories (see Definition 3.4 or below), which had the property that for any background logic satisfying minimal conditions, the * of an incomplete theory is an inconsistent theory and vice versa. The capacity to admit both inconsistent and incomplete theories was seen as essential to, and explicative of, relevance, at least at the propositional level. Neither the Brazilians nor the Routleys appealed to topological duality, which has only become clear more recently, but which would seem to be an expression of Brazilian intuitions.