Sebastian G.W. Speitel

Carnap’s (Categoricity) Problem concerns the relationship between (rules of) inference and model-theoretic values. In particular, it asks whether proof-theoretic constraints are ‘strong enough’ to uniquely determine intended semantic values. Carnap [20] demonstrated that already in the classical bivalent setting this is not the case for the majority of the usual logical constants. To remedy this underdetermination of ‘semantics by syntax’ a variety of solution strategies has been explored in the literature. This article is a philosophical-logical survey of these attempts, comparing them with respect to scope, motivation, and success. Besides the mathematical interest held by Carnap’s Problem, the underdetermination it uncovers has significant consequences for a variety of philosophical projects and positions, warranting a systematic study of attempts at resolving it.

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.

The existence of non-standard models of first-order Peano-Arithmetic (PA) threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to the natural number structure is possible on the basis of our best arithmetical theories. The move to logics stronger than FOL is denied to the moderate realist on the grounds that it merely shifts the indeterminacy “one level up” into the meta-theory by, illegitimately, assuming the determinacy of the notions needed to formulate such logics. This paper argues that the challenge can be met. We show how the quantifier “there are infinitely many” can be uniquely determined in a naturalistically acceptable fashion and thus be used in the formulation of a theory of arithmetic. We compare the approach pursued here with Field’s justification of the same device and the popular strategy of invoking a second-order formalism, and argue that it is more robust than either of the alternative proposals.

This paper reflects on the limits of logical form set by a novel criterion of logicality proposed in (Bonnay and Speitel, 2021). The interest stems from the fact that the delineation of logical terms according to the criterion exceeds the boundaries of standard first-order logic. Among ‘novel’ logical terms is the quantifier “there are infinitely many”. Since the structure of the natural numbers is categorically characterisable in a language including this quantifier we ask: does this imply that arithmetical forms have been reduced to logical forms? And, in general, what other conditions need to be satisfied for a form to qualify as “fully logical”? We survey answers to these questions.