# Carnap's categoricity problem

## Sebastian G.W. Speitel

Carnap [4] demonstrated that the usual axiomatisations of classical propositional and first-order logic fall short of ‘fully formalising’ these systems. In particular, while there is a tight correspondence between syntactic, proof-theoretic, and semantic, model-theoretic, explications of the notion of consequence for these systems, a similarly adequate correspondence between inferential and model-theoretic aspects of the meanings of their logical expressions is lacking. More precisely, Carnap showed that there is a significant mismatch between the intended model-theoretic values of the logical constants and the model-theoretic values actually determined by the usual rules of inference. The standard axiomatisations of classical propositional and first-order logic are, thus, not **categorical** for the intended model-theoretic values of the logical constants of these systems. This is **Carnap’s (categoricity) problem**.

**Carnap’s problem** has significant repercussions for a range of projects and positions in the philosophy of logic, language and mathematics. Although Carnap’s original considerations focused on classical propositional and first-order logic, its consequences have since been investigated for other classes of logical expressions, including intuitionistic connectives [1], modal operators [2], as well as generalized [3] and higher-order quantifiers [7]. Furthermore, a variety of different solution-strategies have been advanced in the literature: from modifying the language or format of the consequence relation [8,9], over re-interpretations of the notion of an inference rule [5,6], towards adopting additional constraints on the space of admissible meanings [1,3].

The goal of this talk is to survey these different solution-strategies with an eye towards their ability of solving **Carnap’s problem** in full generality and their resulting conception of (logical) meaning. An interesting upshot of this investigation concerns the, sometimes implicitly, adopted meaning-theoretic constraints by different ways of resolving Carnapian underdetermination. Moreover, **Carnap’s problem** raises interesting questions about what it takes to have **completely** grasped or characterised a logical notion, resembling a similar discussion in the philosophy of mathematics. This parallel will be explored further in the talk.

## Bibliography

- D. Bonnay, D. Westerståhl,
*Compositionality Solves Carnap’s Problem*,,vol. 81 (2016), no. 4, pp. 721–739.*Erkenntnis* - \bysame_Carnap’s Problem for Modal Logic_,
,vol. 16 (2023), no. 2, pp. 578–602.*Review of Symbolic Logic* - D. Bonnay, S.G.W. Speitel,
*The Ways of Logicality: Invariance and Categoricity*,(G. Sagi and J. Woods, editors),Cambridge University Press,2021,pp. 55–79.*The Semantic Conception of Logic. Essays on Consequence, Invariance, and Meaning* - R. Carnap,
,Harvard University Press,1943.*Formalization of Logic* - J.W. Garson,
,Cambridge University Press,2013.*What Logics Mean: From Proof Theory to Model-Theoretic Semantics* - V. McGee,
*Everything*,(G. Sher and R. Tieszen, editors),Cambridge University Press,2000,pp. 54–79.*Between Logic and Intuition: Essays in Honor of Charles Parsons* - J. Murzi, B. Topey,
*Categoricity by Convention*,,vol. 178 (2021), no. 10, pp. 3391–3420.*Philosophical Studies* - I. Rumfitt,
*‘Yes’ and ‘No’*,,vol. 109 (2000), no. 436, pp. 781–823.*Mind* - D.J. Shoesmith, T. Smiley,
,Cambridge University Press,1978.*Multiple Conclusion Logic*