Identity taken seriously: a non-classical approach

Abstract

Identification of distinct items is a basic technique in mathematics. However, identification suffers from a certain weakness of resolve in that it is (classically) accompanied by dropping the original disidentification, which causes a loss of information about the theory which sources the identity. This article proposes an alternative, namely keeping the disidentification along with the identification. This produces an inconsistent theory which is generally an extension of the source theory. The concept of a Dunn–Meyer extension is defined to study these properties. It is seen that this technique is sensitive to the choice of background logic, particularly RM3 as opposed to closed-set logic. By employing the Routley functor, a best-choice logic is found for this construction.