3 - Metatheory and Naive Theory

Summary

This chapter looks at the methodology for dealing with paradoxes vianonclassical logic. Some standard approaches by Field, Beall, andPriest (among others) are considered and found unsatisfactory forrelying on classical logic in the “metatheory.” Theidea of paraconsistency as a universal logic is suggested instead,and the main challenge for this approach is presented: the“Feferman objection” that nonclassical logics cannotdo mathematics. This leads to a discussion of “classicalrecapture” and mathematical revisionism. The plan for using aparaconsistent “metalanguage” is outlined, the“just true” challenge is dismissed, and the goals forthe approach are laid out – namely, to develop enoughmathematics to establish the paradoxes from Chapter 1.