Conservatively Extending Classical Logic With Transparent Truth

Abstract

This paper shows how to conservatively extend a classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truth-involving) vocabulary. However, not all classical metainferences are preserved; in particular, the resulting logical system is nontransitive. Some limits on this nontransitivity are adumbrated, and two proof systems are presented and shown to be sound and complete. (One proof system features admissible Cut, but the other does not.)

Vagueness, Truth and Permissive Consequence

“…The main difference with Kripke's approach is that we define logical consequence in terms of strict-to-tolerant consequence, that is, a sentence A follows from a set Γ of formulae provided there is no model where all the formulae of Γ take value 1 and where A takes value 0. Like the Strong Kleene definition of consequence (or indeed the LP one, or the tolerant-to-strict), this notion of consequence supports transparency (see [Ripley, 2011] for a proof of this result). One of the essential benefits of this choice, however, which sets it apart from the other schemes, is that if an inference involving a Tfree sentence is classically valid, then it remains STTT-valid for any uniform substitution over the full vocabulary (see [Ripley, 2011]).…”

“…Like the Strong Kleene definition of consequence (or indeed the LP one, or the tolerant-to-strict), this notion of consequence supports transparency (see [Ripley, 2011] for a proof of this result). One of the essential benefits of this choice, however, which sets it apart from the other schemes, is that if an inference involving a Tfree sentence is classically valid, then it remains STTT-valid for any uniform substitution over the full vocabulary (see [Ripley, 2011]). One particular consequence of this is the fact that all T -equivalences are STTT-valid.…”

“…By inferentialism, we mean the view on which linguistic meanings are to be explained by which inferences are valid, and more specifically the bilateralist view on which the validity of arguments itself is to be explained by general constraints on the speech acts of assertion and de-nial, or acceptance and rejection more generally (see [Rumfitt, 2000], [Ripley,201x], and [Malinowski, 1990]). 3 This interpretation is arguably the most adequate when it comes to incorporating a transparent truth predicate in the language (see [Ripley, 2011] and [Cobreros et al,201xb] and below for more ample discussion). This interpretation is not mandated, however.…”

“…The notion of permissive consequence we are concerned with was originally introduced with an aim to solving the sorites paradox (see [Zardini, 2008]), and in order to account for the semantics and pragmatics of vague predicates more generally (see , 1 [Cobreros et al, 2012b], [Cobreros et al, 2012a]). It soon became apparent, however, that it could be applied in a natural way to the treatment of the semantic paradoxes and in particular to the Liar Paradox (see [Ripley, 2011], [Cobreros et al,201xb]). …”

Paradox and Logical Revision. A Short Introduction

“…7 Ripley (2012Ripley ( , 2013a has further investigated the logic and developed an inferentialist argument against Cut, a version of which is presented in Ripley (THIS VOLUME). An anti-realist nontransitive approach is presented in Tennant (2014), on the basis of ideas first introduced in Tennant (1982Tennant ( , 1987Tennant ( , 1995Tennant ( , 1997.…”