Naïve validity

Abstract

Beall and Murzi (J Philos 110(3):143-165, 2013) introduce an objectlinguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules (over sufficiently expressive base theories). As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field (Notre Dame J Form Log 58(1):1-19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a non-classical solution to the semantic paradoxes need not be substructural. The aim of this paper is to respond to Field's objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi's principles: grounded validity. The notion, first introduced by Nicolai and Rossi (J Philos Log. doi:10.1007/s10992-017-9438-x, 2017), is a generalisation of Kripke's notion of grounded truth (J Philos 72:690-716, 1975), and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic (nor, more generally, of a revisionary approach to semantic paradox), we take the notion of naïve

Logical Nihilism and the Logic of ‘prem’

“…While this framework is certainly a good starting point since it permits a change in context from premise to conclusion, it will deliver us logical nihilism in the sense discussed by Russell (2018). This is as expected since the trivalent valuations obtained by dropping c, the result of which are valuations known as strong Kleene, is the typical tool for obtaining theories of transparent truth and naive validity based on a non-reflexive logic, see for example (Fjellstad, 2017), but also (French, 2016), (Nicolai and Rossi, 2018) and (Murzi and Rossi, 2017). After all, one virtue with that approach is that one can use standard rules from a bilateral sequent calculus for classical logic to define → and ¬.…”

“…While this is perhaps not the immediate candidate for a logical law to question when things get tough, there is some recent research on the use of a non-reflexive logic to accommodate self-referential definitions in order to block the set-theoretic and truth-theoretic paradoxes. Examples include (Gilmore, 1986), (Greenough, 2001), (Schroeder-Heister, 2016), (French, 2016), (Fjellstad, 2017), (Nicolai and Rossi, 2018) and (Murzi and Rossi, 2017). The non-reflexive logics presented in the literature to that purpose share the feature that they do not have any valid inferences that support uniform substitution of any formulas for propositional variables.…”

Non-reflexivity and Revenge

“…What is more, they can also be motivated by the need to model reasonings involving expressions of agreement and disagreement, just like the principles for naïve truth. For instance, the following reasoning can be taken to motivate VD (see [22] for more details):…”

“…As we're about to see, the theory validates versions of Validity Proof and Validity Detachment. Since the new construction is but a natural generalisation of Kripke's original one, [22] argue, in response to Field, that grounded validity is just as legitimate as a naïve conception of truth based on Kripke's 7 Strictly speaking, Cut is not employed in the above derivation. This is because the formulation of VDm essentially already incorporates a form of Cut (or meta-inferential modus ponens), and is therefore unavailable to non-transitive theories.…”

“…Val and as a meta-inference, as in our VDm (the 'm' stands for 'meta-inference'). A derivation of via a v-Curry reasoning involving both Cut and VD can be easily obtained via a simple modification of the above derivation (see [22], §1.3). We here focus on the meta-inferential VDm, because our target are non-reflexive theories such as the one developed in Section 2, which can be consistently closed under VDm but not VD.…”