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

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