(Global) Quantification & (Local) Definability – Some Comments on del Valle-Inclán’s Criticism of Bonnay and Westerståhl’s Solution to Carnap’s problem

Abstract In a recent paper of this journal, del Valle-Inclán ( Journal of Philosophical Logic , 53 (5), 1321–1346, 2024) criticized the proposed solution of Bonnay and Westerståhl ( Erkenntnis , 81 (4), 721–739, 2016) to Carnap’s (categoricity) problem . They there claim that Bonnay and Westerståhl avail themselves of second-order notions in order to resolve the underdetermination discovered by Carnap ( Formalization of Logic , 1943) at the first-order level, and that the proposed solution therefore does not really address the first-order case appropriately. The purpose of this brief note is to point out the highly local conception of (logical) quantification underlying del Valle-Inclán’s argument against Bonnay and Westerståhl, which severely limits the force of the counterexamples proposed and undermines the criticism. del Valle-Inclán’s paper has two main critical parts, one addressing the sufficiency of the constraint of permutation-invariance on quantifiers and the other the nature of the constraint of compositionality as used by Bonnay and Westerståhl. I here only address the former aspect of their paper, though the points made have consequences for their subsequent argument as well.