Paradoxes and Failures of Cut

Abstract

This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaning-one which takes meaning to be constituted by assertibility and deniability conditions-and shows how the usual multiple-conclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical system-ST-that conservatively extends classical logic with a fully transparent truth predicate. This system is shown to allow for classical reasoning over the full (truth-involving) vocabulary, but to be nontransitive. Some special cases where transitivity does hold are outlined. ST is also shown to give rise to a familiar sort of model for nonclassical logics: Kripke fixed points on the Strong Kleene valuation scheme. Finally, to give a theory of paradoxical sentences, a distinction is drawn between two varieties of assertion and two varieties of denial. On one variety, paradoxical sentences cannot be either asserted or denied; on the other, they must be both asserted and denied. The target theory is compared favorably to more familiar related systems, and some objections are considered and responded to.

Contractions of Noncontractive Consequence Relations

“…It is only in the presence of applications of these truth rules, then, that mastt becomes nontransitive. 4 mastt is very closely related to the STTT of [3], the G1cT of [29], and the ST of [28], but it is not identical to any of these, as mastt includes both additive and multiplicative vocabulary, to make for easier comparisons with maalt.…”

“…For more on noncontractive approaches to paradoxes, see [25,11,20,36,2,33,15]. For more on nontransitive approaches, see [34,35,28,3,32]. Both approaches are susceptible to a cut-elimination style non-triviality proof (see [20,36] for the noncontractive approaches, and [28] for the nontransitive approaches).…”

“…Γ, Γ : ∆, ∆ Moreover, the particular counterexamples to contraction that occur in noncontractive solutions to the paradoxes of truth and validity become, in those solutions' contractions, exactly the counterexamples to cut put forward in the nontransitive approaches to those same paradoxes recommended in [28,3]. So the machinery of contractions not only provides noncontractivists with a new response to familiar objections to their view, it also provides all of us with a new window on connections between contraction and transitivity, connections that may underly some of the similarities between noncontractive and nontransitive approaches to paradoxes of truth and validity.…”

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.…”

Naive Validity, Internalization, and Substructural Approaches to Paradox

“…Here it might also be helpful to use the distinction between strict and tolerant assertions from Ripley (2013). That π is true (i.e., Val( π , ⊥ )) can only be asserted tolerantly, which means that it does not rule out the strict denial of ⊥.…”