Beall and Murzi :143–165, 2013) introduce an object-linguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules. As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field :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, is a generalisation of Kripke’s notion of grounded truth, and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic, we take the notion of naïve validity to be a legitimate semantic notion that points to genuine expressive limitations of fully structural revisionary approaches. (shrink)

S. SHAPIRO (ed.), Intensional Mathematics (Studies in Logic and the Foundations of Mathematics, vol. 11 3). Amsterdam: North-Holland, 1985. v + 230 pp. $38.50/100Df.

A common objection to hierarchical approaches to truth is that they fragment the concept of truth. This paper defends hierarchical approaches in general against the objection of fragmentation. It argues that the fragmentation required is familiar and unprob-lematic, via a comparison with mathematical proof. Furthermore, it offers an explanation of the source and nature of the fragmentation of truth. Fragmentation arises because the concept exhibits a kind of failure of closure under reflection. This paper offers a more precise characterization of (...) the reflection involved, first in the setting of formal theories of truth, and then in a more general setting. (shrink)

The concept of the (full) unfolding of a schematic system is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted ? The program to determine for various systems of foundational significance was previously carried out for a system of nonfinitist arithmetic, ; it was shown that is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system (...) of finitist arithmetic, , and for an extension of that by a form of the so-called Bar Rule. It is shown that and are proof-theoretically equivalent, respectively, to Primitive Recursive Arithmetic, , and to Peano Arithmetic,. (shrink)

This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak K\"onig's (...) Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others. (shrink)