Reply to Bjørdal

Abstract

§1. In Bjørdal (2010), Bjørdal presents a paraconsistent set theory in which ∀x(x = x) is a theorem. The author rightly claims that, while not trivializing (in the sense of proving everything), results like this are to be avoided. The set theory presented in Bjørdal (2010) is based on that of Weber (2010b), but with an introduced definition of identity-which is used, in effect, as a new axiom. With this added notion of identity, the non-self-identity of every object does in fact obtain; and so the set theory presented by Bjørdal is inadequate. 1 Notably, the form of identity so used is not a part of Weber (2010b), which is a paraconsistent set theory in the relevant logic T L Q. Contrary to impressions, 2 what Bjørdal has shown is that a particular form of identity is not supported in the suggested framework of Weber (2010b). Bjørdal has given evidence for the practical maximality of paraconsistent set theory in T L Q. As it was intended to be, this system would be the strongest possible that both endorses a full naive comprehension axiom, while at the same time avoiding triviality. Adding anything more, however apparently innocuous, however wafer thin, will end in disaster. In this note I show three things. First, I emphasize that the trivializing identity principle is not a part of the system in Weber (2010b). Then I explain why it should not be, either, and not only for ad hoc reasons. One can, of course, introduce a connective = in the manner of Bjørdal (2010), and then derive untoward results. (The same can be said of 'tonk'.) The substantial point is whether =, so defined, is identity. We will see that, in the context of a relevant, paraconsistent solution to the naive set theory problem, it is not. Finally, I recall what principles of identity are in the system, and argue that these are more than adequate. §2. The set theory of Weber (2010b) is couched in an intensional logic, with relevant implication connective →, and is intended to solve the naive set theory problem, characterized by two criteria: