Multiple-Conclusion Lp and Default Classicality

Abstract

Philosophical applications of familiar paracomplete and paraconsistent logics often rely on an idea of ‘default classicality’. With respect to the paraconsistent logic LP (the dual of Strong Kleene or K3), such ‘default classicality’ is standardly cashed out via an LP-based nonmonotonic logic due to Priest (1991, 2006a). In this paper, I offer an alternative approach via a monotonic multiple-conclusion version of LP.

Paradox and Logical Revision. A Short Introduction

“…Begin with the assumption that truth naïvely plays capture and release (Beall 2007a(Beall , 2009, in the following minimal sense:…”

“…'paradoxical' sentences are glutty, and ECQ must be given up. 4 But, unless a new conditional is added to the language (about which more in a moment), :-E, and indeed !-E (modus ponens) must be given up too: these rules fail to preserve truth for any / that is both true and false (Priest 2006b;Beall 2009). Kripke (1975) famously showed how to construct models for languages in which the truth-predicate is fully transparent, provided LEM and :-I (among other rules) are suitably restricted.…”

“…Partly for this reason, Beall (2011Beall ( , 2014b has recently advocated, following Goodship (1996), a detachment-free glut-theoretic approach to paradox-one that gives up the project of defining a 'suitable' conditional, and decidedly embraces the failure of !-E displayed by basic paraconsistent logics. Horsten (2009) essentially advocates a dual strategy in a paraconsistent setting.…”

“…4 See e.g. Asenjo (1966), Asenjo and Tamburino (1975), Priest (1979), Goodship (1996) Beall (2009), Beall (2011. 5 The standard conditional-involving Curry Paradox involves a sentence c identical to TrðpcqÞ !…”

“…To be sure, such a revision is not to be taken lightly (see Terzian, THIS VOLUME), and there is no shortage of classical treatments, either hierarchical (Tarski 1936;Parsons 1974b;Burge 1979;Williamson 1998;Glanzberg 2001Glanzberg , 2004aSchurz 2011), or non-hierarchical (Kripke 1975;McGee 1991;Gupta and Belnap 1993;Maudlin 2004;Leitgeb 2005;Simmons 1993, 2000. But, it has been argued, the alternatives are dire (Kripke 1975;Field 2008), the naïve semantic principles are non-negotiable (Field 2008;Beall 2009; for a criticism of their arguments, see Zardini, THIS VOLUME), and there might be independent reasons for putting the blame on the logic in the first place (Ripley, THIS VOLUME; Zardini, THIS VOLUME). So how can logic be revised on the face of semantic paradox?…”