Expressing consistency consistently

Abstract

In the paraconsistent tradition, it is fairly well‐known how difficult it is to advance a theory containing a naive truth predicate together with a (classical, consistent) consistency operator. Recently, a number of theorists have risen up to the challenge by attempting to articulate such a theory. These theorists either tinker with the idea of semantic naivety, thereby imposing restrictions on the principles governing the truth predicate, or they substantially weaken the underlying syntax theory. My aim in this paper is to offer arguments against these two proposals.

Paraconsistency in Mathematics

“…32 This push-and-pull tension between conservative and radical tendencies comes out most clearly in deciding on what language and techniques a paraconsistent mathematician might use to present their own work: using accepted classical methods for their theories, or making a revolutionary (but also untested) ascent to an inconsistent "metalanguage" beyond the classical? As several critics have pointed out, most of the attention in paraconsistency has gone to neither classical recapture nor expansion, but rather the more basic one of paraconsistent recapture: confirming that various paraconsistent systems are themselves coherent, by (classically) proving that they have classical models 30 For recent attempts at adding a consistency operator, see Barrio, Pailos, and Szmuc (2017); for problems with this and other strategies, see Rosenblatt (2021a) and cf. Omori and Weber (2019).…”